Merge pull request #239 from tkaaad97/fix_enable_cap
Add enums to group "EnableCap"
diff --git a/api/GL/glext.h b/api/GL/glext.h
index 39267fe..151dcf2 100644
--- a/api/GL/glext.h
+++ b/api/GL/glext.h
@@ -51,7 +51,7 @@
#define GLAPI extern
#endif
-#define GL_GLEXT_VERSION 20181130
+#define GL_GLEXT_VERSION 20190204
#include <KHR/khrplatform.h>
diff --git a/api/GL/glxext.h b/api/GL/glxext.h
index 38f6c67..bb13702 100755
--- a/api/GL/glxext.h
+++ b/api/GL/glxext.h
@@ -34,7 +34,7 @@
** https://github.com/KhronosGroup/OpenGL-Registry
*/
-#define GLX_GLXEXT_VERSION 20181130
+#define GLX_GLXEXT_VERSION 20190204
/* Generated C header for:
* API: glx
diff --git a/api/GL/wgl.h b/api/GL/wgl.h
index 3391185..afeee84 100644
--- a/api/GL/wgl.h
+++ b/api/GL/wgl.h
@@ -39,7 +39,7 @@
#include <windows.h>
#endif
-/* Generated on date 20181130 */
+/* Generated on date 20190204 */
/* Generated C header for:
* API: wgl
@@ -409,7 +409,7 @@
#define WGL_GPU_NUM_RB_AMD 0x21A7
#define WGL_GPU_NUM_SPI_AMD 0x21A8
typedef UINT (WINAPI * PFNWGLGETGPUIDSAMDPROC) (UINT maxCount, UINT *ids);
-typedef INT (WINAPI * PFNWGLGETGPUINFOAMDPROC) (UINT id, int property, GLenum dataType, UINT size, void *data);
+typedef INT (WINAPI * PFNWGLGETGPUINFOAMDPROC) (UINT id, INT property, GLenum dataType, UINT size, void *data);
typedef UINT (WINAPI * PFNWGLGETCONTEXTGPUIDAMDPROC) (HGLRC hglrc);
typedef HGLRC (WINAPI * PFNWGLCREATEASSOCIATEDCONTEXTAMDPROC) (UINT id);
typedef HGLRC (WINAPI * PFNWGLCREATEASSOCIATEDCONTEXTATTRIBSAMDPROC) (UINT id, HGLRC hShareContext, const int *attribList);
@@ -419,7 +419,7 @@
typedef VOID (WINAPI * PFNWGLBLITCONTEXTFRAMEBUFFERAMDPROC) (HGLRC dstCtx, GLint srcX0, GLint srcY0, GLint srcX1, GLint srcY1, GLint dstX0, GLint dstY0, GLint dstX1, GLint dstY1, GLbitfield mask, GLenum filter);
#ifdef WGL_WGLEXT_PROTOTYPES
UINT WINAPI wglGetGPUIDsAMD (UINT maxCount, UINT *ids);
-INT WINAPI wglGetGPUInfoAMD (UINT id, int property, GLenum dataType, UINT size, void *data);
+INT WINAPI wglGetGPUInfoAMD (UINT id, INT property, GLenum dataType, UINT size, void *data);
UINT WINAPI wglGetContextGPUIDAMD (HGLRC hglrc);
HGLRC WINAPI wglCreateAssociatedContextAMD (UINT id);
HGLRC WINAPI wglCreateAssociatedContextAttribsAMD (UINT id, HGLRC hShareContext, const int *attribList);
@@ -816,12 +816,12 @@
#define WGL_NV_present_video 1
DECLARE_HANDLE(HVIDEOOUTPUTDEVICENV);
#define WGL_NUM_VIDEO_SLOTS_NV 0x20F0
-typedef int (WINAPI * PFNWGLENUMERATEVIDEODEVICESNVPROC) (HDC hDC, HVIDEOOUTPUTDEVICENV *phDeviceList);
-typedef BOOL (WINAPI * PFNWGLBINDVIDEODEVICENVPROC) (HDC hDC, unsigned int uVideoSlot, HVIDEOOUTPUTDEVICENV hVideoDevice, const int *piAttribList);
+typedef int (WINAPI * PFNWGLENUMERATEVIDEODEVICESNVPROC) (HDC hDc, HVIDEOOUTPUTDEVICENV *phDeviceList);
+typedef BOOL (WINAPI * PFNWGLBINDVIDEODEVICENVPROC) (HDC hDc, unsigned int uVideoSlot, HVIDEOOUTPUTDEVICENV hVideoDevice, const int *piAttribList);
typedef BOOL (WINAPI * PFNWGLQUERYCURRENTCONTEXTNVPROC) (int iAttribute, int *piValue);
#ifdef WGL_WGLEXT_PROTOTYPES
-int WINAPI wglEnumerateVideoDevicesNV (HDC hDC, HVIDEOOUTPUTDEVICENV *phDeviceList);
-BOOL WINAPI wglBindVideoDeviceNV (HDC hDC, unsigned int uVideoSlot, HVIDEOOUTPUTDEVICENV hVideoDevice, const int *piAttribList);
+int WINAPI wglEnumerateVideoDevicesNV (HDC hDc, HVIDEOOUTPUTDEVICENV *phDeviceList);
+BOOL WINAPI wglBindVideoDeviceNV (HDC hDc, unsigned int uVideoSlot, HVIDEOOUTPUTDEVICENV hVideoDevice, const int *piAttribList);
BOOL WINAPI wglQueryCurrentContextNV (int iAttribute, int *piValue);
#endif
#endif /* WGL_NV_present_video */
@@ -926,14 +926,14 @@
typedef BOOL (WINAPI * PFNWGLGETSYNCVALUESOMLPROC) (HDC hdc, INT64 *ust, INT64 *msc, INT64 *sbc);
typedef BOOL (WINAPI * PFNWGLGETMSCRATEOMLPROC) (HDC hdc, INT32 *numerator, INT32 *denominator);
typedef INT64 (WINAPI * PFNWGLSWAPBUFFERSMSCOMLPROC) (HDC hdc, INT64 target_msc, INT64 divisor, INT64 remainder);
-typedef INT64 (WINAPI * PFNWGLSWAPLAYERBUFFERSMSCOMLPROC) (HDC hdc, int fuPlanes, INT64 target_msc, INT64 divisor, INT64 remainder);
+typedef INT64 (WINAPI * PFNWGLSWAPLAYERBUFFERSMSCOMLPROC) (HDC hdc, INT fuPlanes, INT64 target_msc, INT64 divisor, INT64 remainder);
typedef BOOL (WINAPI * PFNWGLWAITFORMSCOMLPROC) (HDC hdc, INT64 target_msc, INT64 divisor, INT64 remainder, INT64 *ust, INT64 *msc, INT64 *sbc);
typedef BOOL (WINAPI * PFNWGLWAITFORSBCOMLPROC) (HDC hdc, INT64 target_sbc, INT64 *ust, INT64 *msc, INT64 *sbc);
#ifdef WGL_WGLEXT_PROTOTYPES
BOOL WINAPI wglGetSyncValuesOML (HDC hdc, INT64 *ust, INT64 *msc, INT64 *sbc);
BOOL WINAPI wglGetMscRateOML (HDC hdc, INT32 *numerator, INT32 *denominator);
INT64 WINAPI wglSwapBuffersMscOML (HDC hdc, INT64 target_msc, INT64 divisor, INT64 remainder);
-INT64 WINAPI wglSwapLayerBuffersMscOML (HDC hdc, int fuPlanes, INT64 target_msc, INT64 divisor, INT64 remainder);
+INT64 WINAPI wglSwapLayerBuffersMscOML (HDC hdc, INT fuPlanes, INT64 target_msc, INT64 divisor, INT64 remainder);
BOOL WINAPI wglWaitForMscOML (HDC hdc, INT64 target_msc, INT64 divisor, INT64 remainder, INT64 *ust, INT64 *msc, INT64 *sbc);
BOOL WINAPI wglWaitForSbcOML (HDC hdc, INT64 target_sbc, INT64 *ust, INT64 *msc, INT64 *sbc);
#endif
diff --git a/api/GL/wglext.h b/api/GL/wglext.h
index a9a5a5b..e36c02a 100755
--- a/api/GL/wglext.h
+++ b/api/GL/wglext.h
@@ -39,7 +39,7 @@
#include <windows.h>
#endif
-#define WGL_WGLEXT_VERSION 20181130
+#define WGL_WGLEXT_VERSION 20190204
/* Generated C header for:
* API: wgl
@@ -318,7 +318,7 @@
#define WGL_GPU_NUM_RB_AMD 0x21A7
#define WGL_GPU_NUM_SPI_AMD 0x21A8
typedef UINT (WINAPI * PFNWGLGETGPUIDSAMDPROC) (UINT maxCount, UINT *ids);
-typedef INT (WINAPI * PFNWGLGETGPUINFOAMDPROC) (UINT id, int property, GLenum dataType, UINT size, void *data);
+typedef INT (WINAPI * PFNWGLGETGPUINFOAMDPROC) (UINT id, INT property, GLenum dataType, UINT size, void *data);
typedef UINT (WINAPI * PFNWGLGETCONTEXTGPUIDAMDPROC) (HGLRC hglrc);
typedef HGLRC (WINAPI * PFNWGLCREATEASSOCIATEDCONTEXTAMDPROC) (UINT id);
typedef HGLRC (WINAPI * PFNWGLCREATEASSOCIATEDCONTEXTATTRIBSAMDPROC) (UINT id, HGLRC hShareContext, const int *attribList);
@@ -328,7 +328,7 @@
typedef VOID (WINAPI * PFNWGLBLITCONTEXTFRAMEBUFFERAMDPROC) (HGLRC dstCtx, GLint srcX0, GLint srcY0, GLint srcX1, GLint srcY1, GLint dstX0, GLint dstY0, GLint dstX1, GLint dstY1, GLbitfield mask, GLenum filter);
#ifdef WGL_WGLEXT_PROTOTYPES
UINT WINAPI wglGetGPUIDsAMD (UINT maxCount, UINT *ids);
-INT WINAPI wglGetGPUInfoAMD (UINT id, int property, GLenum dataType, UINT size, void *data);
+INT WINAPI wglGetGPUInfoAMD (UINT id, INT property, GLenum dataType, UINT size, void *data);
UINT WINAPI wglGetContextGPUIDAMD (HGLRC hglrc);
HGLRC WINAPI wglCreateAssociatedContextAMD (UINT id);
HGLRC WINAPI wglCreateAssociatedContextAttribsAMD (UINT id, HGLRC hShareContext, const int *attribList);
@@ -725,12 +725,12 @@
#define WGL_NV_present_video 1
DECLARE_HANDLE(HVIDEOOUTPUTDEVICENV);
#define WGL_NUM_VIDEO_SLOTS_NV 0x20F0
-typedef int (WINAPI * PFNWGLENUMERATEVIDEODEVICESNVPROC) (HDC hDC, HVIDEOOUTPUTDEVICENV *phDeviceList);
-typedef BOOL (WINAPI * PFNWGLBINDVIDEODEVICENVPROC) (HDC hDC, unsigned int uVideoSlot, HVIDEOOUTPUTDEVICENV hVideoDevice, const int *piAttribList);
+typedef int (WINAPI * PFNWGLENUMERATEVIDEODEVICESNVPROC) (HDC hDc, HVIDEOOUTPUTDEVICENV *phDeviceList);
+typedef BOOL (WINAPI * PFNWGLBINDVIDEODEVICENVPROC) (HDC hDc, unsigned int uVideoSlot, HVIDEOOUTPUTDEVICENV hVideoDevice, const int *piAttribList);
typedef BOOL (WINAPI * PFNWGLQUERYCURRENTCONTEXTNVPROC) (int iAttribute, int *piValue);
#ifdef WGL_WGLEXT_PROTOTYPES
-int WINAPI wglEnumerateVideoDevicesNV (HDC hDC, HVIDEOOUTPUTDEVICENV *phDeviceList);
-BOOL WINAPI wglBindVideoDeviceNV (HDC hDC, unsigned int uVideoSlot, HVIDEOOUTPUTDEVICENV hVideoDevice, const int *piAttribList);
+int WINAPI wglEnumerateVideoDevicesNV (HDC hDc, HVIDEOOUTPUTDEVICENV *phDeviceList);
+BOOL WINAPI wglBindVideoDeviceNV (HDC hDc, unsigned int uVideoSlot, HVIDEOOUTPUTDEVICENV hVideoDevice, const int *piAttribList);
BOOL WINAPI wglQueryCurrentContextNV (int iAttribute, int *piValue);
#endif
#endif /* WGL_NV_present_video */
@@ -835,14 +835,14 @@
typedef BOOL (WINAPI * PFNWGLGETSYNCVALUESOMLPROC) (HDC hdc, INT64 *ust, INT64 *msc, INT64 *sbc);
typedef BOOL (WINAPI * PFNWGLGETMSCRATEOMLPROC) (HDC hdc, INT32 *numerator, INT32 *denominator);
typedef INT64 (WINAPI * PFNWGLSWAPBUFFERSMSCOMLPROC) (HDC hdc, INT64 target_msc, INT64 divisor, INT64 remainder);
-typedef INT64 (WINAPI * PFNWGLSWAPLAYERBUFFERSMSCOMLPROC) (HDC hdc, int fuPlanes, INT64 target_msc, INT64 divisor, INT64 remainder);
+typedef INT64 (WINAPI * PFNWGLSWAPLAYERBUFFERSMSCOMLPROC) (HDC hdc, INT fuPlanes, INT64 target_msc, INT64 divisor, INT64 remainder);
typedef BOOL (WINAPI * PFNWGLWAITFORMSCOMLPROC) (HDC hdc, INT64 target_msc, INT64 divisor, INT64 remainder, INT64 *ust, INT64 *msc, INT64 *sbc);
typedef BOOL (WINAPI * PFNWGLWAITFORSBCOMLPROC) (HDC hdc, INT64 target_sbc, INT64 *ust, INT64 *msc, INT64 *sbc);
#ifdef WGL_WGLEXT_PROTOTYPES
BOOL WINAPI wglGetSyncValuesOML (HDC hdc, INT64 *ust, INT64 *msc, INT64 *sbc);
BOOL WINAPI wglGetMscRateOML (HDC hdc, INT32 *numerator, INT32 *denominator);
INT64 WINAPI wglSwapBuffersMscOML (HDC hdc, INT64 target_msc, INT64 divisor, INT64 remainder);
-INT64 WINAPI wglSwapLayerBuffersMscOML (HDC hdc, int fuPlanes, INT64 target_msc, INT64 divisor, INT64 remainder);
+INT64 WINAPI wglSwapLayerBuffersMscOML (HDC hdc, INT fuPlanes, INT64 target_msc, INT64 divisor, INT64 remainder);
BOOL WINAPI wglWaitForMscOML (HDC hdc, INT64 target_msc, INT64 divisor, INT64 remainder, INT64 *ust, INT64 *msc, INT64 *sbc);
BOOL WINAPI wglWaitForSbcOML (HDC hdc, INT64 target_sbc, INT64 *ust, INT64 *msc, INT64 *sbc);
#endif
diff --git a/api/GLES/gl.h b/api/GLES/gl.h
index 0d49769..681ecc9 100644
--- a/api/GLES/gl.h
+++ b/api/GLES/gl.h
@@ -36,7 +36,7 @@
#include <GLES/glplatform.h>
-/* Generated on date 20181130 */
+/* Generated on date 20190204 */
/* Generated C header for:
* API: gles1
diff --git a/api/GLES/glext.h b/api/GLES/glext.h
index 41de807..f9d2852 100644
--- a/api/GLES/glext.h
+++ b/api/GLES/glext.h
@@ -38,7 +38,7 @@
#define GL_APIENTRYP GL_APIENTRY*
#endif
-/* Generated on date 20181130 */
+/* Generated on date 20190204 */
/* Generated C header for:
* API: gles1
diff --git a/api/GLES2/gl2.h b/api/GLES2/gl2.h
index cf3b7f3..d63eda1 100644
--- a/api/GLES2/gl2.h
+++ b/api/GLES2/gl2.h
@@ -44,7 +44,7 @@
#define GL_GLES_PROTOTYPES 1
#endif
-/* Generated on date 20181130 */
+/* Generated on date 20190204 */
/* Generated C header for:
* API: gles2
diff --git a/api/GLES2/gl2ext.h b/api/GLES2/gl2ext.h
index 81d712e..dccffce 100644
--- a/api/GLES2/gl2ext.h
+++ b/api/GLES2/gl2ext.h
@@ -38,7 +38,7 @@
#define GL_APIENTRYP GL_APIENTRY*
#endif
-/* Generated on date 20181130 */
+/* Generated on date 20190204 */
/* Generated C header for:
* API: gles2
diff --git a/api/GLES3/gl3.h b/api/GLES3/gl3.h
index 4570785..7c41bb4 100644
--- a/api/GLES3/gl3.h
+++ b/api/GLES3/gl3.h
@@ -44,7 +44,7 @@
#define GL_GLES_PROTOTYPES 1
#endif
-/* Generated on date 20181130 */
+/* Generated on date 20190204 */
/* Generated C header for:
* API: gles2
diff --git a/index_es.php b/index_es.php
index 0dada0f..3b2aeeb 100644
--- a/index_es.php
+++ b/index_es.php
@@ -56,11 +56,12 @@
Documentation </h2>
<ul>
-<li> OpenGL ES 3.2 Specification (May 14, 2018)
+<li> OpenGL ES 3.2 Specification (February 2, 2019)
<a href="specs/es/3.2/es_spec_3.2.pdf"> without changes marked </a>
and
<a href="specs/es/3.2/es_spec_3.2.withchanges.pdf"> with changes marked </a>. </li>
-<li> OpenGL ES Shading Language 3.20 Specification (May 11, 2018)
+<li> OpenGL ES Shading Language 3.20 Specification (December 12, 2018)
+ <a href="specs/es/3.2/GLSL_ES_Specification_3.20.html"> (HTML) </a>
<a href="specs/es/3.2/GLSL_ES_Specification_3.20.pdf"> (PDF) </a>
<li> <a href="http://www.khronos.org/opengles/sdk/docs/man32/">
OpenGL ES 3.2 Online Manual Pages.</a> </li>
diff --git a/index_gl.php b/index_gl.php
index d72b667..c96075b 100644
--- a/index_gl.php
+++ b/index_gl.php
@@ -48,7 +48,7 @@
<ul>
<li> <b> Current Specifications (OpenGL 4.6) </b>
-<li> OpenGL 4.6 API Specification (updated May 14, 2018)
+<li> OpenGL 4.6 API Specification (February 2, 2019)
<ul>
<li> <a href="specs/gl/glspec46.core.pdf"> Core Profile Specification
</a> </li>
@@ -60,7 +60,8 @@
Compatibility Profile Specification with changes marked </a>
</li>
</ul> </li>
-<li> OpenGL Shading Language 4.60 Specification (updated May 11, 2018)
+<li> OpenGL Shading Language 4.60 Specification (December 12, 2018)
+ <a href="specs/gl/GLSLangSpec.4.60.html"> (HTML) </a> </li>
<a href="specs/gl/GLSLangSpec.4.60.pdf"> (PDF) </a> </li>
<li> <a href="http://www.opengl.org/sdk/docs/man4/"> OpenGL 4.5 API and
diff --git a/specs/es/3.2/GLSL_ES_Specification_3.20.html b/specs/es/3.2/GLSL_ES_Specification_3.20.html
new file mode 100644
index 0000000..128d8d6
--- /dev/null
+++ b/specs/es/3.2/GLSL_ES_Specification_3.20.html
@@ -0,0 +1,15228 @@
+<!DOCTYPE html>
+<html lang="en">
+<head>
+<meta charset="UTF-8">
+<!--[if IE]><meta http-equiv="X-UA-Compatible" content="IE=edge"><![endif]-->
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="generator" content="Asciidoctor 1.5.7.1">
+<meta name="author" content="Robert J. Simpson, Qualcomm (Editor) ; John Kessenich, Dave Baldwin and Randi Rost (Version 1.1 Authors)">
+<title>The OpenGL ES® Shading Language, Version 3.20.5</title>
+<link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700">
+<style>
+/* Asciidoctor default stylesheet | MIT License | http://asciidoctor.org */
+/* Uncomment @import statement below to use as custom stylesheet */
+/*@import "https://fonts.googleapis.com/css?family=Open+Sans:300,300italic,400,400italic,600,600italic%7CNoto+Serif:400,400italic,700,700italic%7CDroid+Sans+Mono:400,700";*/
+article,aside,details,figcaption,figure,footer,header,hgroup,main,nav,section,summary{display:block}
+audio,canvas,video{display:inline-block}
+audio:not([controls]){display:none;height:0}
+script{display:none!important}
+html{font-family:sans-serif;-ms-text-size-adjust:100%;-webkit-text-size-adjust:100%}
+a{background:transparent}
+a:focus{outline:thin dotted}
+a:active,a:hover{outline:0}
+h1{font-size:2em;margin:.67em 0}
+abbr[title]{border-bottom:1px dotted}
+b,strong{font-weight:bold}
+dfn{font-style:italic}
+hr{-moz-box-sizing:content-box;box-sizing:content-box;height:0}
+mark{background:#ff0;color:#000}
+code,kbd,pre,samp{font-family:monospace;font-size:1em}
+pre{white-space:pre-wrap}
+q{quotes:"\201C" "\201D" "\2018" "\2019"}
+small{font-size:80%}
+sub,sup{font-size:75%;line-height:0;position:relative;vertical-align:baseline}
+sup{top:-.5em}
+sub{bottom:-.25em}
+img{border:0}
+svg:not(:root){overflow:hidden}
+figure{margin:0}
+fieldset{border:1px solid silver;margin:0 2px;padding:.35em .625em .75em}
+legend{border:0;padding:0}
+button,input,select,textarea{font-family:inherit;font-size:100%;margin:0}
+button,input{line-height:normal}
+button,select{text-transform:none}
+button,html input[type="button"],input[type="reset"],input[type="submit"]{-webkit-appearance:button;cursor:pointer}
+button[disabled],html input[disabled]{cursor:default}
+input[type="checkbox"],input[type="radio"]{box-sizing:border-box;padding:0}
+button::-moz-focus-inner,input::-moz-focus-inner{border:0;padding:0}
+textarea{overflow:auto;vertical-align:top}
+table{border-collapse:collapse;border-spacing:0}
+*,*::before,*::after{-moz-box-sizing:border-box;-webkit-box-sizing:border-box;box-sizing:border-box}
+html,body{font-size:100%}
+body{background:#fff;color:rgba(0,0,0,.8);padding:0;margin:0;font-family:"Noto Serif","DejaVu Serif",serif;font-weight:400;font-style:normal;line-height:1;position:relative;cursor:auto;tab-size:4;-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased}
+a:hover{cursor:pointer}
+img,object,embed{max-width:100%;height:auto}
+object,embed{height:100%}
+img{-ms-interpolation-mode:bicubic}
+.left{float:left!important}
+.right{float:right!important}
+.text-left{text-align:left!important}
+.text-right{text-align:right!important}
+.text-center{text-align:center!important}
+.text-justify{text-align:justify!important}
+.hide{display:none}
+img,object,svg{display:inline-block;vertical-align:middle}
+textarea{height:auto;min-height:50px}
+select{width:100%}
+.center{margin-left:auto;margin-right:auto}
+.stretch{width:100%}
+.subheader,.admonitionblock td.content>.title,.audioblock>.title,.exampleblock>.title,.imageblock>.title,.listingblock>.title,.literalblock>.title,.stemblock>.title,.openblock>.title,.paragraph>.title,.quoteblock>.title,table.tableblock>.title,.verseblock>.title,.videoblock>.title,.dlist>.title,.olist>.title,.ulist>.title,.qlist>.title,.hdlist>.title{line-height:1.45;color:#7a2518;font-weight:400;margin-top:0;margin-bottom:.25em}
+div,dl,dt,dd,ul,ol,li,h1,h2,h3,#toctitle,.sidebarblock>.content>.title,h4,h5,h6,pre,form,p,blockquote,th,td{margin:0;padding:0;direction:ltr}
+a{color:#2156a5;text-decoration:underline;line-height:inherit}
+a:hover,a:focus{color:#1d4b8f}
+a img{border:none}
+p{font-family:inherit;font-weight:400;font-size:1em;line-height:1.6;margin-bottom:1.25em;text-rendering:optimizeLegibility}
+p aside{font-size:.875em;line-height:1.35;font-style:italic}
+h1,h2,h3,#toctitle,.sidebarblock>.content>.title,h4,h5,h6{font-family:"Open Sans","DejaVu Sans",sans-serif;font-weight:300;font-style:normal;color:#ba3925;text-rendering:optimizeLegibility;margin-top:1em;margin-bottom:.5em;line-height:1.0125em}
+h1 small,h2 small,h3 small,#toctitle small,.sidebarblock>.content>.title small,h4 small,h5 small,h6 small{font-size:60%;color:#e99b8f;line-height:0}
+h1{font-size:2.125em}
+h2{font-size:1.6875em}
+h3,#toctitle,.sidebarblock>.content>.title{font-size:1.375em}
+h4,h5{font-size:1.125em}
+h6{font-size:1em}
+hr{border:solid #ddddd8;border-width:1px 0 0;clear:both;margin:1.25em 0 1.1875em;height:0}
+em,i{font-style:italic;line-height:inherit}
+strong,b{font-weight:bold;line-height:inherit}
+small{font-size:60%;line-height:inherit}
+code{font-family:"Droid Sans Mono","DejaVu Sans Mono",monospace;font-weight:400;color:rgba(0,0,0,.9)}
+ul,ol,dl{font-size:1em;line-height:1.6;margin-bottom:1.25em;list-style-position:outside;font-family:inherit}
+ul,ol{margin-left:1.5em}
+ul li ul,ul li ol{margin-left:1.25em;margin-bottom:0;font-size:1em}
+ul.square li ul,ul.circle li ul,ul.disc li ul{list-style:inherit}
+ul.square{list-style-type:square}
+ul.circle{list-style-type:circle}
+ul.disc{list-style-type:disc}
+ol li ul,ol li ol{margin-left:1.25em;margin-bottom:0}
+dl dt{margin-bottom:.3125em;font-weight:bold}
+dl dd{margin-bottom:1.25em}
+abbr,acronym{text-transform:uppercase;font-size:90%;color:rgba(0,0,0,.8);border-bottom:1px dotted #ddd;cursor:help}
+abbr{text-transform:none}
+blockquote{margin:0 0 1.25em;padding:.5625em 1.25em 0 1.1875em;border-left:1px solid #ddd}
+blockquote cite{display:block;font-size:.9375em;color:rgba(0,0,0,.6)}
+blockquote cite::before{content:"\2014 \0020"}
+blockquote cite a,blockquote cite a:visited{color:rgba(0,0,0,.6)}
+blockquote,blockquote p{line-height:1.6;color:rgba(0,0,0,.85)}
+@media screen and (min-width:768px){h1,h2,h3,#toctitle,.sidebarblock>.content>.title,h4,h5,h6{line-height:1.2}
+h1{font-size:2.75em}
+h2{font-size:2.3125em}
+h3,#toctitle,.sidebarblock>.content>.title{font-size:1.6875em}
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+<body class="book toc2 toc-left" style="max-width: 100;">
+<div id="header">
+<h1>The OpenGL ES<sup>®</sup> Shading Language, Version 3.20.5</h1>
+<div class="details">
+<span id="author" class="author">Robert J. Simpson, Qualcomm (Editor) ; John Kessenich, Dave Baldwin and Randi Rost (Version 1.1 Authors)</span><br>
+<span id="revnumber">version 3.20.5,</span>
+<span id="revdate">Wed, 12 Dec 2018 23:37:49 +0000</span>
+<br><span id="revremark">Git branch information not available</span>
+</div>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#introduction">1. Introduction</a>
+<ul class="sectlevel2">
+<li><a href="#changes">1.1. Changes</a></li>
+<li><a href="#overview">1.2. Overview</a></li>
+<li><a href="#error-handling">1.3. Error Handling</a></li>
+<li><a href="#typographical-conventions">1.4. Typographical Conventions</a></li>
+<li><a href="#compatibility">1.5. Compatibility</a></li>
+</ul>
+</li>
+<li><a href="#overview-of-opengl-shading">2. Overview of OpenGL ES Shading</a>
+<ul class="sectlevel2">
+<li><a href="#vertex-processor">2.1. Vertex Processor</a></li>
+<li><a href="#tessellation-control-processor">2.2. Tessellation Control Processor</a></li>
+<li><a href="#tessellation-evaluation-processor">2.3. Tessellation Evaluation Processor</a></li>
+<li><a href="#geometry-processor">2.4. Geometry Processor</a></li>
+<li><a href="#fragment-processor">2.5. Fragment Processor</a></li>
+<li><a href="#compute-processor">2.6. Compute Processor</a></li>
+</ul>
+</li>
+<li><a href="#basics">3. Basics</a>
+<ul class="sectlevel2">
+<li><a href="#character-set">3.1. Character Set</a></li>
+<li><a href="#source-strings">3.2. Source Strings</a></li>
+<li><a href="#version-declaration">3.3. Version Declaration</a></li>
+<li><a href="#preprocessor">3.4. Preprocessor</a></li>
+<li><a href="#comments">3.5. Comments</a></li>
+<li><a href="#tokens">3.6. Tokens</a></li>
+<li><a href="#keywords">3.7. Keywords</a></li>
+<li><a href="#identifiers">3.8. Identifiers</a></li>
+<li><a href="#definitions">3.9. Definitions</a></li>
+<li><a href="#logical-phases-of-compilation">3.10. Logical Phases of Compilation</a></li>
+</ul>
+</li>
+<li><a href="#variables-and-types">4. Variables and Types</a>
+<ul class="sectlevel2">
+<li><a href="#basic-types">4.1. Basic Types</a></li>
+<li><a href="#scoping">4.2. Scoping</a></li>
+<li><a href="#storage-qualifiers">4.3. Storage Qualifiers</a></li>
+<li><a href="#layout-qualifiers">4.4. Layout Qualifiers</a></li>
+<li><a href="#interpolation-qualifiers">4.5. Interpolation Qualifiers</a></li>
+<li><a href="#parameter-qualifiers">4.6. Parameter Qualifiers</a></li>
+<li><a href="#precision-and-precision-qualifiers">4.7. Precision and Precision Qualifiers</a></li>
+<li><a href="#variance-and-the-invariant-qualifier">4.8. Variance and the Invariant Qualifier</a></li>
+<li><a href="#the-precise-qualifier">4.9. The Precise Qualifier</a></li>
+<li><a href="#memory-qualifiers">4.10. Memory Qualifiers</a></li>
+<li><a href="#order-of-qualification">4.11. Order and Repetition of Qualification</a></li>
+<li><a href="#empty-declarations">4.12. Empty Declarations</a></li>
+</ul>
+</li>
+<li><a href="#operators-and-expressions">5. Operators and Expressions</a>
+<ul class="sectlevel2">
+<li><a href="#operators">5.1. Operators</a></li>
+<li><a href="#array-operations">5.2. Array Operations</a></li>
+<li><a href="#function-calls">5.3. Function Calls</a></li>
+<li><a href="#constructors">5.4. Constructors</a></li>
+<li><a href="#vector-components">5.5. Vector Components</a></li>
+<li><a href="#matrix-components">5.6. Matrix Components</a></li>
+<li><a href="#structure-and-array-operations">5.7. Structure and Array Operations</a></li>
+<li><a href="#assignments">5.8. Assignments</a></li>
+<li><a href="#expressions">5.9. Expressions</a></li>
+<li><a href="#vector-and-matrix-operations">5.10. Vector and Matrix Operations</a></li>
+<li><a href="#evaluation-of-expressions">5.11. Evaluation of Expressions</a></li>
+</ul>
+</li>
+<li><a href="#statements-and-structure">6. Statements and Structure</a>
+<ul class="sectlevel2">
+<li><a href="#function-definitions">6.1. Function Definitions</a></li>
+<li><a href="#selection">6.2. Selection</a></li>
+<li><a href="#iteration">6.3. Iteration</a></li>
+<li><a href="#jumps">6.4. Jumps</a></li>
+</ul>
+</li>
+<li><a href="#built-in-variables">7. Built-In Variables</a>
+<ul class="sectlevel2">
+<li><a href="#built-in-language-variables">7.1. Built-In Language Variables</a></li>
+<li><a href="#built-in-constants">7.2. Built-In Constants</a></li>
+<li><a href="#built-in-uniform-state">7.3. Built-In Uniform State</a></li>
+<li><a href="#redeclaring-built-in-blocks">7.4. Redeclaring Built-In Blocks</a></li>
+</ul>
+</li>
+<li><a href="#built-in-functions">8. Built-In Functions</a>
+<ul class="sectlevel2">
+<li><a href="#angle-and-trigonometry-functions">8.1. Angle and Trigonometry Functions</a></li>
+<li><a href="#exponential-functions">8.2. Exponential Functions</a></li>
+<li><a href="#common-functions">8.3. Common Functions</a></li>
+<li><a href="#floating-point-pack-and-unpack-functions">8.4. Floating-Point Pack and Unpack Functions</a></li>
+<li><a href="#geometric-functions">8.5. Geometric Functions</a></li>
+<li><a href="#matrix-functions">8.6. Matrix Functions</a></li>
+<li><a href="#vector-relational-functions">8.7. Vector Relational Functions</a></li>
+<li><a href="#integer-functions">8.8. Integer Functions</a></li>
+<li><a href="#texture-functions">8.9. Texture Functions</a></li>
+<li><a href="#atomic-counter-functions">8.10. Atomic Counter Functions</a></li>
+<li><a href="#atomic-memory-functions">8.11. Atomic Memory Functions</a></li>
+<li><a href="#image-functions">8.12. Image Functions</a></li>
+<li><a href="#geometry-shader-functions">8.13. Geometry Shader Functions</a></li>
+<li><a href="#fragment-processing-functions">8.14. Fragment Processing Functions</a></li>
+<li><a href="#shader-invocation-control-functions">8.15. Shader Invocation Control Functions</a></li>
+<li><a href="#shader-memory-control-functions">8.16. Shader Memory Control Functions</a></li>
+</ul>
+</li>
+<li><a href="#shader-interface-matching">9. Shader Interface Matching</a>
+<ul class="sectlevel2">
+<li><a href="#input-output-matching-by-name-in-linked-programs">9.1. Input Output Matching by Name in Linked Programs</a></li>
+<li><a href="#matching-of-qualifiers">9.2. Matching of Qualifiers</a></li>
+</ul>
+</li>
+<li><a href="#shading-language-grammar">10. Shading Language Grammar</a></li>
+<li><a href="#counting-of-inputs-and-outputs">11. Counting of Inputs and Outputs</a></li>
+<li><a href="#acknowledgments">12. Acknowledgments</a></li>
+<li><a href="#references">13. Normative References</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div id="preamble">
+<div class="sectionbody">
+<div style="page-break-after: always;"></div>
+<div class="paragraph">
+<p>Copyright © 2008-2018 The Khronos Group Inc. All Rights Reserved.</p>
+</div>
+<div class="paragraph">
+<p>This specification is protected by copyright laws and contains material
+proprietary to the Khronos Group, Inc. It or any components may not be
+reproduced, republished, distributed, transmitted, displayed, broadcast,
+or otherwise exploited in any manner without the express prior written
+permission of Khronos Group. You may use this specification for
+implementing the functionality therein, without altering or removing any
+trademark, copyright or other notice from the specification, but the
+receipt or possession of this specification does not convey any rights
+to reproduce, disclose, or distribute its contents, or to manufacture,
+use, or sell anything that it may describe, in whole or in part.</p>
+</div>
+<div class="paragraph">
+<p>Khronos Group grants express permission to any current Promoter,
+Contributor or Adopter member of Khronos to copy and redistribute
+UNMODIFIED versions of this specification in any fashion, provided that
+NO CHARGE is made for the specification and the latest available update
+of the specification for any version of the API is used whenever
+possible. Such distributed specification may be reformatted AS LONG AS
+the contents of the specification are not changed in any way. The
+specification may be incorporated into a product that is sold as long as
+such product includes significant independent work developed by the
+seller. A link to the current version of this specification on the
+Khronos Group website should be included whenever possible with
+specification distributions.</p>
+</div>
+<div class="paragraph">
+<p>Khronos Group makes no, and expressly disclaims any, representations or
+warranties, express or implied, regarding this specification, including,
+without limitation, any implied warranties of merchantability or fitness
+for a particular purpose or noninfringement of any intellectual
+property. Khronos Group makes no, and expressly disclaims any,
+warranties, express or implied, regarding the correctness, accuracy,
+completeness, timeliness, and reliability of the specification. Under no
+circumstances will the Khronos Group, or any of its Promoters,
+Contributors or Members or their respective partners, officers,
+directors, employees, agents, or representatives be liable for any
+damages, whether direct, indirect, special or consequential damages for
+lost revenues, lost profits, or otherwise, arising from or in connection
+with these materials.</p>
+</div>
+<div class="paragraph">
+<p>Khronos, Vulkan, SYCL, SPIR, WebGL, EGL, COLLADA, StreamInput, OpenVX,
+OpenKCam, glTF, OpenKODE, OpenVG, OpenWF, OpenSL ES, OpenMAX, OpenMAX
+AL, OpenMAX IL and OpenMAX DL are trademarks and WebCL is a
+certification mark of the Khronos Group Inc. OpenCL is a trademark of
+Apple Inc. and OpenGL and OpenML are registered trademarks and the
+OpenGL ES and OpenGL SC logos are trademarks of Silicon Graphics
+International used under license by Khronos. All other product names,
+trademarks, and/or company names are used solely for identification and
+belong to their respective owners.</p>
+</div>
+<div style="page-break-after: always;"></div>
+<!-- toc disabled -->
+</div>
+</div>
+<div class="sect1">
+<h2 id="introduction">1. Introduction</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>This document specifies only version 3.20 of the OpenGL ES Shading Language.
+It requires __VERSION__ to substitute 320, and requires
+<strong>#version</strong> to accept only
+<code>320 es</code>.
+If <strong>#version</strong> is declared with a smaller number, the language accepted is a
+previous version of the shading language, which will be supported depending
+on the version and type of context in the OpenGL ES API.
+See the <a href="#references">OpenGL ES Specification</a> for details on what language versions are
+supported.</p>
+</div>
+<div class="paragraph">
+<p>All references in this specification to the <a href="#references">OpenGL ES Specification</a> are to
+version 3.2.</p>
+</div>
+<div class="sect2">
+<h3 id="changes">1.1. Changes</h3>
+<div class="sect3">
+<h4 id="changes-from-glsl-es-3.2-revision-4">1.1.1. Changes from GLSL ES 3.2 revision 4</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Clarified that this specification completely defines the OpenGL ES Shading Language.
+Normatively reference C++ only for the preprocessor.</p>
+</li>
+<li>
+<p>Private GLSL issues 7, 38: Corrected the values of some builtin constants.
+The values were given correctly in the OpenGL ES Specification.</p>
+</li>
+<li>
+<p>Private GLSL issue 30: Clarify that output packing rules apply to the last
+vertex pipeline stage, not necessarily the vertex stage.</p>
+</li>
+<li>
+<p>Private GLSL issue 15: Clarify the ordering of bindings for arrays of arrays.</p>
+</li>
+<li>
+<p>Private GLSL issue 14: Uniform variables need only match at link time if they
+are statically used.</p>
+</li>
+<li>
+<p>For <strong>precise</strong> computations, the controlling expressions for
+control flow and ternary operators (<strong>?:</strong>) are not included.</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="changes-from-glsl-es-3.2-revision-3">1.1.2. Changes from GLSL ES 3.2 revision 3</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Matching of default uniforms when shaders are linked.</p>
+</li>
+<li>
+<p><em>gl_DepthRange</em> is only guaranteed to be available in the fragment
+stage.</p>
+</li>
+<li>
+<p>Clarification of definition of static use.</p>
+</li>
+<li>
+<p>Sampling behavior in the absence of <strong>sample</strong> and <strong>centroid</strong>.</p>
+</li>
+<li>
+<p>Clarified the requirements when the specification uses the terms
+<em>should</em>/<em>should not</em> and <em>undefined behavior</em>.</p>
+</li>
+<li>
+<p>Arrayed blocks cannot have layout location qualifiers on members</p>
+</li>
+<li>
+<p><strong>barrier</strong>() defines a partial order which includes tessellation control
+shader outputs.</p>
+</li>
+<li>
+<p>Vertex shader integer output qualification.</p>
+</li>
+<li>
+<p>Incorrect use of predefined pragmas.</p>
+</li>
+<li>
+<p>Clarified use of <strong>readonly</strong> and <strong>writeonly</strong> qualifiers.</p>
+</li>
+<li>
+<p>USAMPLERBUFFER added to grammar.</p>
+</li>
+<li>
+<p>Clarified precision qualifiers can be used in interface blocks.</p>
+</li>
+<li>
+<p>Clarified <strong>memoryBarrierShared</strong> only applies to the current workgroup.</p>
+</li>
+<li>
+<p>The layout qualifier <em>invocations</em> must not be zero.</p>
+</li>
+<li>
+<p>The layout qualifier <em>local_size</em> must not be zero.</p>
+</li>
+<li>
+<p>Clarified the definition of static assignment.</p>
+</li>
+<li>
+<p>Removed list of types with no default precision.</p>
+</li>
+<li>
+<p>Removed scoping rules from the grammar. Refer instead to the <a href="#scoping">scoping</a> section.</p>
+</li>
+<li>
+<p>Require a statement after the final label of a switch.</p>
+</li>
+<li>
+<p>Define <strong>gl_BoundingBox</strong>.</p>
+</li>
+<li>
+<p><strong>length</strong>() expressions returning a constant-value may not include side effects.</p>
+</li>
+<li>
+<p>Clarified that variables may be declared <strong>readonly writeonly</strong>.</p>
+</li>
+<li>
+<p>Use of constant expressions within <strong>#line</strong> directives is undefined.</p>
+</li>
+<li>
+<p><strong>gl_in</strong> can be redeclared using unsized-array syntax.</p>
+</li>
+<li>
+<p>Clarified which sampler types may be used for depth and stencil textures.</p>
+</li>
+<li>
+<p>Added order-of-operation and other explanations to the
+<a href="#the-precise-qualifier">Precise Qualifier</a> section.</p>
+</li>
+<li>
+<p>The <strong>precise</strong> qualifier applied to a block/struct applies recursively to the members.</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="changes-from-glsl-es-3.2-revision-2">1.1.3. Changes from GLSL ES 3.2 revision 2</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Updated value for <em>gl_MaxTessControlTotalOutputComponents</em></p>
+</li>
+<li>
+<p>Clarified the allowed character set for pre-processing</p>
+</li>
+<li>
+<p>Integer division wrapping behavior</p>
+</li>
+<li>
+<p>Clarified pre-processor expressions (<em>pp-constant-expression</em>)</p>
+</li>
+<li>
+<p>UBO and SSBO precisions do not need to match for linked shaders
+(consistent with GLSL ES 3.1)</p>
+</li>
+<li>
+<p><strong>modf</strong> function</p>
+</li>
+<li>
+<p>Sequence and ternary operators with <strong>void</strong> type</p>
+</li>
+<li>
+<p>Sequence and ternary operators with array types</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="changes-from-glsl-es-3.2-revision-1">1.1.4. Changes from GLSL ES 3.2 revision 1</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Signed zeros must be supported</p>
+</li>
+<li>
+<p>Layout qualifier table</p>
+</li>
+<li>
+<p>Allowed optimizations when evaluating expressions</p>
+</li>
+<li>
+<p>Updated value for <em>gl_MaxTessControlInputComponents</em></p>
+</li>
+<li>
+<p>Updated value for <em>gl_MaxTessControlOutputComponents</em></p>
+</li>
+<li>
+<p>Updated value for <em>gl_MaxTessEvaluationInputComponents</em></p>
+</li>
+<li>
+<p>Updated value for <em>gl_MaxTessEvaluationOutputComponents</em></p>
+</li>
+<li>
+<p>Updated value for <em>gl_MaxGeometryOutputComponents</em></p>
+</li>
+<li>
+<p>Require precisions in blocks to match when linking</p>
+</li>
+<li>
+<p>Updated conclusions in issues section</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="changes-from-glsl-es-3.1-revision-4">1.1.5. Changes from GLSL ES 3.1 revision 4</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Added the following extensions:</p>
+<div class="ulist">
+<ul>
+<li>
+<p><a href="https://www.opengl.org/registry/specs/KHR/blend_equation_advanced.txt">KHR_blend_equation_advanced</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_sample_variables.txt">OES_sample_variables</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_shader_image_atomic.txt">OES_shader_image_atomic</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_shader_multisample_interpolation.txt">OES_shader_multisample_interpolation</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_texture_storage_multisample_2d_array.txt">OES_texture_storage_multisample_2d_array</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_geometry_shader.txt">OES_geometry_shader</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_gpu_shader5.txt">OES_gpu_shader5</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_primitive_bounding_box.txt">OES_primitive_bounding_box</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_shader_io_blocks.txt">OES_shader_io_blocks</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_tessellation_shader.txt">OES_tessellation_shader</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_texture_buffer.txt">OES_texture_buffer</a></p>
+</li>
+<li>
+<p><a href="https://www.khronos.org/registry/gles/extensions/OES/OES_texture_cube_map_array.txt">OES_texture_cube_map_array</a></p>
+</li>
+<li>
+<p><a href="https://www.opengl.org/registry/specs/KHR/robustness.txt">KHR_robustness</a></p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="overview">1.2. Overview</h3>
+<div class="paragraph">
+<p>This document describes <em>The OpenGL ES Shading Language, version 3.20</em>.</p>
+</div>
+<div class="paragraph">
+<p>Independent compilation units written in this language are called <em>shaders</em>.
+A <em>program</em> is a set of shaders that are compiled and linked
+together.
+The aim of this document is to thoroughly specify the programming language.
+The <a href="#references">OpenGL ES Specification</a> will specify the OpenGL ES entry points used to
+manipulate and communicate with programs and shaders.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="error-handling">1.3. Error Handling</h3>
+<div class="paragraph">
+<p>Compilers, in general, accept programs that are ill-formed, due to the
+impossibility of detecting all ill-formed programs.
+Portability is only ensured for well-formed programs, which this
+specification describes.
+Compilers are encouraged to detect ill-formed programs and issue diagnostic
+messages, but are not required to do so for all cases.</p>
+</div>
+<div class="paragraph">
+<p>The compilation process is implementation-dependent but is generally split
+into a number of stages, each of which occurs at one of the following times:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A call to <em>glCompileShader</em></p>
+</li>
+<li>
+<p>A call to <em>glLinkProgram</em></p>
+</li>
+<li>
+<p>A draw call or a call to <em>glValidateProgram</em></p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>The implementation should report errors as early a possible but in any case
+must satisfy the following:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>All lexical, grammatical and semantic errors must have been detected
+following a call to <em>glLinkProgram</em></p>
+</li>
+<li>
+<p>Errors due to mismatch between the shaders (link-time errors) must have
+been detected following a call to <em>glLinkProgram</em></p>
+</li>
+<li>
+<p>Errors due to exceeding resource limits must have been detected
+following any draw call or a call to <em>glValidateProgram</em></p>
+</li>
+<li>
+<p>A call to <em>glValidateProgram</em> must report all errors associated with a
+program object given the current GL state.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Where the specification uses the terms <em>required</em>, <em>must</em>/<em>must</em> <em>not</em>,
+<em>does</em>/<em>does</em> <em>not</em>, <em>disallowed</em>, or <em>not</em> <em>supported</em>, the compiler or
+linker is required to detect and report any violations.
+Similarly when a condition or situation is an <strong>error</strong>, it must be reported.
+Use of any feature marked as <em>reserved</em> is an error.
+Where the specification uses the terms <em>should</em>/<em>should</em> <em>not</em>, <em>undefined</em>
+<em>behavior</em>, <em>undefined</em> <em>value</em> or <em>undefined</em> <em>*results*</em>, implementations
+will not produce a compile-time error but are encouraged to issue a warning
+for violations.
+The run-time behavior of the program in these cases is not constrained (and
+so may include termination or system instability).
+It is expected that systems will be designed to handle these cases
+gracefully but specification of this is outside the scope of OpenGL ES.</p>
+</div>
+<div class="paragraph">
+<p>Implementations may not in general support functionality beyond the mandated
+parts of the specification without use of the relevant extension.
+The only exceptions are:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>If a feature is marked as optional.</p>
+</li>
+<li>
+<p>Where a maximum value is stated (e.g. the maximum number of vertex
+outputs), the implementation may support a higher value than that
+specified.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Where the implementation supports more than the mandated specification,
+off-target compilers are encouraged to issue warnings if these features are
+used.</p>
+</div>
+<div class="paragraph">
+<p>The compilation process is split between the compiler and linker.
+The allocation of tasks between the compiler and linker is implementation
+dependent.
+Consequently there are many errors which may be detected either at compile
+or link time, depending on the implementation.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="typographical-conventions">1.4. Typographical Conventions</h3>
+<div class="paragraph">
+<p>Italic, bold, and font choices have been used in this specification
+primarily to improve readability.
+Code fragments use a fixed width font.
+Identifiers embedded in text are italicized.
+Keywords embedded in text are bold.
+Operators are called by their name, followed by their symbol in bold in
+parentheses.
+The clarifying grammar fragments in the text use bold for literals and
+italics for non-terminals.
+The official grammar in “<a href="#shading-language-grammar">Shading Language
+Grammar</a>” uses all capitals for terminals and lower case for
+non-terminals.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="compatibility">1.5. Compatibility</h3>
+<div class="paragraph">
+<p>The OpenGL ES 3.2 API is designed to work with GLSL ES v1.00, GLSL
+ES 3.00, GLSL ES 3.10 and GLSL ES 3.20.
+In general a shader written for versions prior to OpenGL ES 3.2
+should work without modification in OpenGL ES 3.2.</p>
+</div>
+<div class="paragraph">
+<p>When porting applications from an earlier to later version of the API, the
+following points should be noted:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Not all language constructs present in earlier versions of the language
+are available in later versions e.g. attribute and varying qualifiers
+are present in v1.00 but not v3.00.
+However, the functionality of GLSL ES 3.20 is a super-set of GLSL
+ES 3.10.</p>
+</li>
+<li>
+<p>Some features of later versions of the API require language features
+that are not present in earlier version of the language.</p>
+</li>
+<li>
+<p>It is an error to link shaders if they are written in different versions
+of the language.</p>
+</li>
+<li>
+<p>The OpenGL ES 2.0 and 3.0 APIs do not support shaders written in GLSL ES
+3.20.</p>
+</li>
+<li>
+<p>Using GLSL ES 1.00 shaders within OpenGL ES 3.x may extend the resources
+available beyond the minima specified in GLSL ES 1.0.
+Shaders which make use of this will not necessarily run on an OpenGL ES
+2.0 implementation: Similarly for GLSL ES 3.00 shaders running within
+OpenGL ES 3.2.</p>
+</li>
+<li>
+<p>Support of line continuation and support of UTF-8 characters within
+comments is optional in GLSL ES 1.00 when used with the OpenGL ES 2.0
+API.
+However, support is mandated for both of these when a GLSL ES 1.00
+shader is used with the OpenGL ES 3.x APIs.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="overview-of-opengl-shading">2. Overview of OpenGL ES Shading</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The OpenGL ES Shading Language is actually several closely related languages.
+These languages are used to create shaders for each of the programmable
+processors contained in the OpenGL ES processing pipeline.
+Currently, these processors are the vertex, tessellation control,
+tessellation evaluation, geometry, fragment, and compute processors.</p>
+</div>
+<div class="paragraph">
+<p>Compilation units for these processors are referred to as <em>shaders</em> and the
+processors themselves are also referred to as <em>shader stages</em>.
+Only one shader can be run on a processor at any one time; there is no
+support for linking multiple compilation units together for a single shader
+stage.
+One or more shaders are linked together to form a single program and each
+program contains shader <em>executables</em> for one or more consecutive shader
+stages.</p>
+</div>
+<div class="paragraph">
+<p>Unless otherwise noted in this paper, a language feature applies to all
+languages, and common usage will refer to these languages as a single
+language.
+The specific languages will be referred to by the name of the processor they
+target: vertex, tessellation control, tessellation evaluation, geometry,
+fragment, or compute.</p>
+</div>
+<div class="paragraph">
+<p>Most OpenGL ES state is not tracked or made available to shaders.
+Typically, user-defined variables will be used for communicating between
+different stages of the OpenGL ES pipeline.
+However, a small amount of state is still tracked and automatically made
+available to shaders, and there are a few built-in variables for interfaces
+between different stages of the OpenGL ES pipeline.</p>
+</div>
+<div class="sect2">
+<h3 id="vertex-processor">2.1. Vertex Processor</h3>
+<div class="paragraph">
+<p>The <em>vertex processor</em> is a programmable unit that operates on incoming
+vertices and their associated data.
+Compilation units written in the OpenGL ES Shading Language to run on this processor are called
+<em>vertex shaders</em>.</p>
+</div>
+<div class="paragraph">
+<p>The vertex processor operates on one vertex at a time.
+It does not replace graphics operations that require knowledge of several
+vertices at a time.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="tessellation-control-processor">2.2. Tessellation Control Processor</h3>
+<div class="paragraph">
+<p>The <em>tessellation control processor</em> is a programmable unit that operates on
+a patch of incoming vertices and their associated data, emitting a new
+output patch.
+Compilation units written in the OpenGL ES Shading Language to run on this processor are called
+tessellation control shaders.</p>
+</div>
+<div class="paragraph">
+<p>The tessellation control shader is invoked for each vertex of the output
+patch.
+Each invocation can read the attributes of any vertex in the input or output
+patches, but can only write per-vertex attributes for the corresponding
+output patch vertex.
+The shader invocations collectively produce a set of per-patch attributes
+for the output patch.</p>
+</div>
+<div class="paragraph">
+<p>After all tessellation control shader invocations have completed, the output
+vertices and per-patch attributes are assembled to form a patch to be used
+by subsequent pipeline stages.</p>
+</div>
+<div class="paragraph">
+<p>Tessellation control shader invocations run mostly independently, with
+undefined relative execution order.
+However, the built-in function <strong>barrier</strong>() can be used to control execution
+order by synchronizing invocations, effectively dividing tessellation
+control shader execution into a set of phases.
+Tessellation control shaders will get undefined results if one invocation
+reads from a per-vertex or per-patch attribute written by another invocation
+at any point during the same phase, or if two invocations attempt to write
+different values to the same per-patch output
+in a single phase.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="tessellation-evaluation-processor">2.3. Tessellation Evaluation Processor</h3>
+<div class="paragraph">
+<p>The <em>tessellation evaluation processor</em> is a programmable unit that
+evaluates the position and other attributes of a vertex generated by the
+tessellation primitive generator, using a patch of incoming vertices and
+their associated data.
+Compilation units written in the OpenGL ES Shading Language to run on this processor are called
+tessellation evaluation shaders.</p>
+</div>
+<div class="paragraph">
+<p>Each invocation of the tessellation evaluation executable computes the
+position and attributes of a single vertex generated by the tessellation
+primitive generator.
+The executable can read the attributes of any vertex in the input patch,
+plus the tessellation coordinate, which is the relative location of the
+vertex in the primitive being tessellated.
+The executable writes the position and other attributes of the vertex.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="geometry-processor">2.4. Geometry Processor</h3>
+<div class="paragraph">
+<p>The <em>geometry processor</em> is a programmable unit that operates on data for
+incoming vertices for a primitive assembled after vertex processing and
+outputs a sequence of vertices forming output primitives.
+Compilation units written in the OpenGL ES Shading Language to run on this processor are called
+<em>geometry shaders</em>.</p>
+</div>
+<div class="paragraph">
+<p>A single invocation of the geometry shader executable on the geometry
+processor will operate on a declared input primitive with a fixed number of
+vertices.
+This single invocation can emit a variable number of vertices that are
+assembled into primitives of a declared output primitive type and passed to
+subsequent pipeline stages.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="fragment-processor">2.5. Fragment Processor</h3>
+<div class="paragraph">
+<p>The <em>fragment processor</em> is a programmable unit that operates on fragment
+values and their associated data.
+Compilation units written in the OpenGL ES Shading Language to run on this processor are called
+<em>fragment shaders</em>.</p>
+</div>
+<div class="paragraph">
+<p>A fragment shader cannot change a fragment’s (<em>x</em>, <em>y</em>) position.
+Access to neighboring fragments is not allowed.
+The values computed by the fragment shader are ultimately used to update
+framebuffer memory or texture memory, depending on the current OpenGL ES
+state and the OpenGL ES command that caused the fragments to be generated.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="compute-processor">2.6. Compute Processor</h3>
+<div class="paragraph">
+<p>The <em>compute processor</em> is a programmable unit that operates independently
+from the other shader processors.
+Compilation units written in the OpenGL ES Shading Language to run on this processor are called
+<em>compute shaders</em>.</p>
+</div>
+<div class="paragraph">
+<p>A compute shader has access to many of the same resources as fragment and
+other shader processors, such as textures, buffers, image variables, and
+atomic counters.
+It does not have fixed-function outputs.
+It is not part of the graphics pipeline and its visible side effects are
+through changes to images, storage buffers, and atomic counters.</p>
+</div>
+<div class="paragraph">
+<p>A compute shader operates on a group of work items called a <em>workgroup</em>.
+A workgroup is a collection of shader invocations that execute the same
+code, potentially in parallel.
+An invocation within a workgroup may share data with other members of the
+same workgroup through shared variables and issue memory and control flow
+barriers to synchronize with other members of the same workgroup.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="basics">3. Basics</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="character-set">3.1. Character Set</h3>
+<div class="paragraph">
+<p>The source character set used for the OpenGL ES Shading Language is Unicode in the UTF-8
+encoding scheme.
+Invalid UTF-8 characters are ignored.
+During pre-processing, the following applies:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A byte with the value zero is always interpreted as the end of the
+string</p>
+</li>
+<li>
+<p>Backslash ('\'), is used to indicate line continuation when immediately
+preceding a new-line.</p>
+</li>
+<li>
+<p>White space consists of one or more of the following characters: the
+space character, horizontal tab, vertical tab, form feed,
+carriage-return, line-feed.</p>
+</li>
+<li>
+<p>The number sign (<strong>#</strong>) is used for preprocessor directives</p>
+</li>
+<li>
+<p>Macro names are restricted to:</p>
+<div class="ulist">
+<ul>
+<li>
+<p>The letters <strong>a-z</strong>, <strong>A-Z</strong>, and the underscore (<strong>_</strong>).</p>
+</li>
+<li>
+<p>The numbers <strong>0-9</strong>, except for the first character of a macro name.</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>After preprocessing, only the following characters are allowed in the
+resulting stream of GLSL tokens:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The letters <strong>a-z</strong>, <strong>A-Z</strong>, and the underscore (<strong>_</strong>).</p>
+</li>
+<li>
+<p>The numbers <strong>0-9</strong>.</p>
+</li>
+<li>
+<p>The symbols period (<strong>.</strong>), plus (<strong>+</strong>), dash (<strong>-</strong>), slash (<strong>/</strong>), asterisk
+(<strong>*</strong>), percent (<strong>%</strong>), angled brackets (<strong><</strong> and <strong>></strong>), square brackets
+(<strong>[</strong> and <strong>]</strong>), parentheses (<strong>(</strong> and <strong>)</strong>), braces (<strong>{</strong> and <strong>}</strong>), caret
+(<strong>^</strong>), vertical bar (<strong>|</strong>), ampersand (<strong>&</strong>), tilde (<strong>~</strong>), equals (<strong>=</strong>),
+exclamation point (<strong>!</strong>), colon (<strong>:</strong>), semicolon (<strong>;</strong>), comma (<strong>,</strong>), and
+question mark (<strong>?</strong>).</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>There are no digraphs or trigraphs.
+There are no escape sequences or other uses of the backslash beyond use as
+the line-continuation character.</p>
+</div>
+<div class="paragraph">
+<p>Lines are relevant for compiler diagnostic messages and the preprocessor.
+They are terminated by carriage-return or line-feed.
+If both are used together, it will count as only a single line termination.
+For the remainder of this document, any of these combinations is simply
+referred to as a new-line.
+Lines may be of arbitrary length.</p>
+</div>
+<div class="paragraph">
+<p>In general, the language’s use of this character set is case sensitive.</p>
+</div>
+<div class="paragraph">
+<p>There are no character or string data types, so no quoting characters are
+included.</p>
+</div>
+<div class="paragraph">
+<p>There is no end-of-file character.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="source-strings">3.2. Source Strings</h3>
+<div class="paragraph">
+<p>The source for a single shader is an array of strings of characters from the
+character set.
+A single shader is made from the concatenation of these strings.
+Each string can contain multiple lines, separated by new-lines.
+No new-lines need be present in a string; a single line can be formed from
+multiple strings.
+No new-lines or other characters are inserted by the implementation when it
+concatenates the strings to form a single shader.</p>
+</div>
+<div class="paragraph">
+<p>Diagnostic messages returned from compiling a shader must identify both the
+line number within a string and which source string the message applies to.
+Source strings are counted sequentially with the first string being string
+0.
+Line numbers are one more than the number of new-lines that have been
+processed, including counting the new-lines that will be removed by the
+line-continuation character (<strong>\</strong>).</p>
+</div>
+<div class="paragraph">
+<p>Lines separated by the line-continuation character preceding a new-line are
+concatenated together before either comment processing or preprocessing.
+This means that no white space is substituted for the line-continuation
+character.
+That is, a single token could be formed by the concatenation by taking the
+characters at the end of one line concatenating them with the characters at
+the beginning of the next line.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> f\
+oo;
+<span class="comment">// forms a single line equivalent to "float foo;"</span>
+<span class="comment">// (assuming '\' is the last character before the new-line and "oo" are</span>
+<span class="comment">// the first two characters of the next line)</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="version-declaration">3.3. Version Declaration</h3>
+<div class="paragraph">
+<p>Shaders must declare the version of the language they are written to.
+The version is specified in the first line of a shader by a character
+string:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#version</span> number es</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>number</em> must be a version of the language, following the same
+convention as __VERSION__ above.
+The directive “<strong>#version 320 es</strong>” is required in any shader that
+uses version 3.20 of the language.
+Any <em>number</em> representing a version of the language a compiler does not
+support will cause an error to be generated.
+Version 1.00 of the language does not require shaders to include this
+directive, and shaders that do not include a <strong>#version</strong> directive will be
+treated as targeting version 1.00.</p>
+</div>
+<div class="paragraph">
+<p>Shaders declaring version 3.20 of the shading language cannot be
+linked with shaders declaring a previous version.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>#version</strong> directive must be present in the first line of a shader and
+must be followed by a newline.
+It may contain optional white-space as specified below but no other
+characters are allowed.
+The directive is only permitted in the first line of a shader.</p>
+</div>
+<div class="paragraph">
+<p>Processing of the #version directive occurs before all other preprocessing,
+including line concatenation and comment processing.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>version-declaration</em> : </dt>
+<dd>
+<p><em>whitespace<sub>opt</sub></em> POUND <em>whitespace<sub>opt</sub></em> VERSION <em>whitespace</em> <em>number</em>
+<em>whitespace</em> ES <em>whitespace<sub>opt</sub></em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Tokens:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>POUND <strong>#</strong><br>
+VERSION <strong>version</strong><br>
+ES <strong>es</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+<div class="sect2">
+<h3 id="preprocessor">3.4. Preprocessor</h3>
+<div class="paragraph">
+<p>There is a preprocessor that processes the source strings as part of the
+compilation process.
+Except as noted below, it behaves as the C++ standard preprocessor (see
+“<a href="#references">Normative References</a>”).</p>
+</div>
+<div class="paragraph">
+<p>The complete list of preprocessor directives is as follows.</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#<br>
+#define<br>
+#undef<br></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#if<br>
+#ifdef<br>
+#ifndef<br>
+#else<br>
+#elif<br>
+#endif<br></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#error<br>
+#pragma<br></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#extension<br></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#line</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The following
+operator is
+also available:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>defined<br></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>Note that the version directive is not considered to be a preprocessor
+directive and so is not listed here.</p>
+</div>
+<div class="paragraph">
+<p>Each number sign (<strong>#</strong>) can be preceded in its line only by spaces or
+horizontal tabs.
+It may also be followed by spaces and horizontal tabs, preceding the
+directive.
+Each directive is terminated by a new-line.
+Preprocessing does not change the number or relative location of new-lines
+in a source string.</p>
+</div>
+<div class="paragraph">
+<p>The number sign (<strong>#</strong>) on a line by itself is ignored.
+Any directive not listed above will cause an error.</p>
+</div>
+<div class="paragraph">
+<p><strong>#define</strong> and <strong>#undef</strong> functionality are defined as is standard for C++
+preprocessors for macro definitions both with and without macro parameters.</p>
+</div>
+<div class="paragraph">
+<p>The following predefined macros are available:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>__LINE__<br>
+__FILE__<br>
+__VERSION__<br>
+GL_ES</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>__LINE__ will substitute a decimal integer constant that is one more than
+the number of preceding new-lines in the current source string.</p>
+</div>
+<div class="paragraph">
+<p>__FILE__ will substitute a decimal integer constant that says which source
+string number is currently being processed.</p>
+</div>
+<div class="paragraph">
+<p>__VERSION__ will substitute a decimal integer reflecting the version
+number of the OpenGL ES Shading Language.
+The version of the shading language described in this document will have
+__VERSION__ substitute the decimal integer 320.</p>
+</div>
+<div class="paragraph">
+<p>GL_ES will be defined and set to 1.
+This is not true for the non-ES OpenGL Shading Language, so it can be used
+to do a compile time test to determine if a shader is running on an ES
+system.</p>
+</div>
+<div class="paragraph">
+<p>By convention, all macro names containing two consecutive underscores (__)
+are reserved for use by underlying software layers.
+Defining
+such a name in a shader does not itself result in an error, but may
+result in unintended behaviors that stem from having multiple definitions of
+the same name.
+All macro names prefixed with “GL_” (“GL” followed by a single
+underscore) are also reserved, and defining
+such a name results in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>It is an error to undefine or to redefine a built-in (pre-defined) macro
+name.</p>
+</div>
+<div class="paragraph">
+<p>Implementations must support macro-name lengths of up to 1024 characters.
+It is an error to declare a name with a length greater than this.</p>
+</div>
+<div class="paragraph">
+<p><strong>#if</strong>, <strong>#ifdef</strong>, <strong>#ifndef</strong>, <strong>#else</strong>, <strong>#elif</strong>, and <strong>#endif</strong> are defined to
+operate as is standard for C++ preprocessors except for the following:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Expressions following <strong>#if</strong> and <strong>#elif</strong> are
+restricted to <em>pp-constant-expressions</em> as defined below.</p>
+</li>
+<li>
+<p>Undefined identifiers not consumed by the <strong>defined</strong> operator do not
+default to '0'.
+Use of such identifiers causes an error.</p>
+</li>
+<li>
+<p>Character constants are not supported.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>As in C++, a macro name defined with an empty replacement list does not
+default to '0' when used in a preprocessor expression.</p>
+</div>
+<div class="paragraph">
+<p>A <em>pp-constant-expression</em> is an integral expression, evaluated at
+compile-time during preprocessing and formed from literal integer constants
+and the following operators:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Precedence</th>
+<th class="tableblock halign-left valign-top">Operator class</th>
+<th class="tableblock halign-left valign-top">Operators</th>
+<th class="tableblock halign-left valign-top">Associativity</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1 (highest)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">parenthetical grouping</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">( )</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">NA</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">unary</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">defined<br>
+ + - ~ !</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Right to Left</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">multiplicative</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">* / %</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">additive</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">+ -</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise shift</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><< >></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">6</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">relational</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">< > <= >=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">7</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">equality</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">== !=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">8</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise and</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">&</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">9</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise exclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">^</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">10</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise inclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">|</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">11</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical and</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">&&</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">12 (lowest)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical inclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">||</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The <strong>defined</strong> operator can be used in either of the following ways:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">defined identifier
+defined ( identifier )</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>There are no number sign based operators (e.g. no <strong><mark></strong> or <strong></mark>@</strong>), no <strong>##</strong>
+operator, nor is there a <strong>sizeof</strong> operator.</p>
+</div>
+<div class="paragraph">
+<p>The semantics of applying operators in the preprocessor match those standard
+in the C++ preprocessor with the following exceptions:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The 2<sup>nd</sup> operand in a logical and ('&&') operation is evaluated if and
+only if the 1<sup>st</sup> operand evaluates to non-zero.</p>
+</li>
+<li>
+<p>The 2<sup>nd</sup> operand in a logical or ('||') operation is evaluated if and
+only if the 1<sup>st</sup> operand evaluates to zero.</p>
+</li>
+<li>
+<p>There is no boolean type and no boolean literals.
+A <em>true</em> or <em>false</em> result is returned as integer <em>one</em> or <em>zero</em>
+respectively.
+Wherever a boolean operand is expected, any non-zero integer is
+interpreted as <em>true</em> and a zero integer as <em>false</em>.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>If an operand is not evaluated, the presence of undefined identifiers in the
+operand will not cause an error.</p>
+</div>
+<div class="paragraph">
+<p><strong>#error</strong> will cause the implementation to put a compile-time diagnostic message
+into the shader object’s information log (see section 7.12 “Shader, Program
+and Program Pipeline Queries” of the <a href="#references">OpenGL ES Specification</a> for how to
+access a shader object’s information log).
+The message will be the tokens following the <strong>#error</strong> directive, up to the
+first new-line.
+The implementation must treat the presence of a <strong>#error</strong> directive as a
+compile-time error.</p>
+</div>
+<div class="paragraph">
+<p><strong>#pragma</strong> allows implementation-dependent compiler control.
+Tokens following <strong>#pragma</strong> are not subject to preprocessor macro expansion.
+If an implementation does not recognize the tokens following <strong>#pragma</strong>, then
+it will ignore that pragma.
+The following pragmas are defined as part of the language.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> STDGL</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>STDGL</strong> pragma is used to reserve pragmas for use by future revisions of
+this language.
+No implementation may use a pragma whose first token is <strong>STDGL</strong>.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> optimize(on)
+<span class="preprocessor">#pragma</span> optimize(off)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>can be used to turn off optimizations as an aid in developing and debugging
+shaders.
+It can only be used outside function definitions.
+By default, optimization is turned on for all shaders.
+The debug pragma</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> debug(on)
+<span class="preprocessor">#pragma</span> debug(off)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>can be used to enable compiling and annotating a shader with debug
+information, so that it can be used with a debugger.
+It can only be used outside function definitions.
+By default, debug is turned off.</p>
+</div>
+<div class="paragraph">
+<p>The scope as well as the effect of the optimize and debug pragmas is
+implementation-dependent except that their use must not generate an error.
+Incorrect use of predefined pragmas does not cause an error.</p>
+</div>
+<div class="paragraph">
+<p>By default, compilers of this language must issue compile-time
+syntactic, semantic,
+and
+grammatical errors for shaders that do not conform to this specification.
+Any extended behavior must first be enabled.
+Directives to control the behavior of the compiler with respect to
+extensions are declared with the <strong>#extension</strong> directive</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#extension</span> extension_name : behavior
+<span class="preprocessor">#extension</span> all : behavior</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>extension_name</em> is the name of an extension.
+Extension names are not documented in this specification.
+The token <strong>all</strong> means the behavior applies to all extensions supported by
+the compiler.
+The <em>behavior</em> can be one of the following:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Behavior</th>
+<th class="tableblock halign-left valign-top">Effect</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>require</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Behave as specified by the extension <em>extension_name</em>.<br>
+ Give a compile-time error on the <strong>#extension</strong> if the extension
+ <em>extension_name</em> is not supported, or if <strong>all</strong> is specified.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>enable</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Behave as specified by the extension <em>extension_name</em>.<br>
+ Warn on the <strong>#extension</strong> if the extension <em>extension_name</em> is
+ not supported.<br>
+ Give an error on the <strong>#extension</strong> if <strong>all</strong> is
+ specified.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>warn</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Behave as specified by the extension <em>extension_name</em>,
+ except issue warnings on any detectable use of that extension,
+ unless such use is supported by other enabled or required
+ extensions.<br>
+ If <strong>all</strong> is specified, then warn on all detectable uses of any
+ extension used.<br>
+ Warn on the <strong>#extension</strong> if the extension <em>extension_name</em> is
+ not supported.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>disable</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Behave (including issuing errors and warnings) as if the
+ extension <em>extension_name</em> is not part of the language
+ definition.<br>
+ If <strong>all</strong> is specified, then behavior must revert back to that
+ of the non-extended core version of the language being
+ compiled to.<br>
+ Warn on the <strong>#extension</strong> if the extension <em>extension_name</em> is
+ not supported.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The <strong>extension</strong> directive is a simple, low-level mechanism to set the
+behavior for each extension.
+It does not define policies such as which combinations are appropriate,
+those must be defined elsewhere.
+Order of directives matters in setting the behavior for each extension:
+Directives that occur later override those seen earlier.
+The <strong>all</strong> variant sets the behavior for all extensions, overriding all
+previously issued <strong>extension</strong> directives, but only for the <em>behaviors</em>
+<strong>warn</strong> and <strong>disable</strong>.</p>
+</div>
+<div class="paragraph">
+<p>The initial state of the compiler is as if the directive</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#extension</span> all : disable</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>was issued, telling the compiler that all error and warning reporting must
+be done according to this specification, ignoring any extensions.</p>
+</div>
+<div class="paragraph">
+<p>Each extension can define its allowed granularity of scope.
+If nothing is said, the granularity is a shader (that is, a single
+compilation unit), and the extension directives must occur before any
+non-preprocessor tokens.
+If necessary, the linker can enforce granularities larger than a single
+compilation unit, in which case each involved shader will have to contain
+the necessary extension directive.</p>
+</div>
+<div class="paragraph">
+<p>Macro expansion is not done on lines containing <strong>#extension</strong> and <strong>#version</strong>
+directives.</p>
+</div>
+<div class="paragraph">
+<p>For each extension there is an associated macro.
+The macro is always defined in an implementation that supports the
+extension.
+This allows the following construct to be used:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#ifdef</span> OES_extension_name
+ <span class="preprocessor">#extension</span> OES_extension_name : enable
+ <span class="comment">// code that requires the extension</span>
+<span class="preprocessor">#else</span>
+ <span class="comment">// alternative code</span>
+<span class="preprocessor">#endif</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p><strong>#line</strong> must have, after macro substitution, one of the following forms:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#line</span> line
+<span class="preprocessor">#line</span> line source-<span class="predefined-type">string</span>-number</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>line</em> and <em>source-string-number</em> are
+<em>pp-constant-expressions</em>.
+If these constant expressions are not integer literals then behavior is undefined.
+After processing this directive (including its new-line), the implementation
+will behave as if it is compiling at line number <em>line</em> and source string
+number <em>source-string-number</em>.
+Subsequent source strings will be numbered sequentially, until another
+<strong>#line</strong> directive overrides that numbering.</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="title">Note</div>
+<div class="paragraph">
+<p>Some implementations have allowed constant expressions in #line directives and
+some have not. Even where expressions are supported the grammar is ambiguous and so
+results are implementation dependent. For example,
++ #line +2 +2 // Line number set to 4, or file to 2 and line to 2</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="paragraph">
+<p>If during macro expansion a preprocessor directive is encountered, the
+results are undefined; the compiler may or may not report an error in such
+cases.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="comments">3.5. Comments</h3>
+<div class="paragraph">
+<p>Comments are delimited by <strong>/*</strong> and <strong>*/</strong>, or by <strong>//</strong> and a new-line.
+// style comments include the initial // marker and continue up to, but
+not including, the terminating newline.
+/*...*/ comments include both the start and end marker.
+The begin comment delimiters (/* or //) are not recognized as comment
+delimiters inside of a comment, hence comments cannot be nested.
+Comments are treated syntactically as a single space.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="tokens">3.6. Tokens</h3>
+<div class="paragraph">
+<p>The language, after preprocessing, is a sequence of tokens.
+A token can be</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>token</em> : </dt>
+<dd>
+<p><em>keyword</em><br>
+<em>identifier</em><br>
+<em>integer-constant</em><br>
+<em>floating-constant</em><br>
+<em>operator</em><br>
+<strong>;</strong> <strong>{</strong> <strong>}</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="keywords">3.7. Keywords</h3>
+<div class="paragraph">
+<p>The following are the keywords in the language and (after
+preprocessing) can only be used as described in this specification,
+or an error results:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>const</strong> <strong>uniform</strong> <strong>buffer</strong> <strong>shared</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>coherent</strong> <strong>volatile</strong> <strong>restrict</strong> <strong>readonly</strong> <strong>writeonly</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>atomic_uint</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>layout</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>centroid</strong> <strong>flat</strong> <strong>smooth</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>patch</strong> <strong>sample</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>invariant</strong> <strong>precise</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>break</strong> <strong>continue</strong> <strong>do</strong> <strong>for</strong> <strong>while</strong> <strong>switch</strong> <strong>case</strong> <strong>default</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>if</strong> <strong>else</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>in</strong> <strong>out</strong> <strong>inout</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>int</strong> <strong>void</strong> <strong>bool</strong> <strong>true</strong> <strong>false</strong> <strong>float</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>discard</strong> <strong>return</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>vec2</strong> <strong>vec3</strong> <strong>vec4</strong> <strong>ivec2</strong> <strong>ivec3</strong> <strong>ivec4</strong> <strong>bvec2</strong> <strong>bvec3</strong> <strong>bvec4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>uint</strong> <strong>uvec2</strong> <strong>uvec3</strong> <strong>uvec4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>mat2</strong> <strong>mat3</strong> <strong>mat4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>mat2x2</strong> <strong>mat2x3</strong> <strong>mat2x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>mat3x2</strong> <strong>mat3x3</strong> <strong>mat3x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>mat4x2</strong> <strong>mat4x3</strong> <strong>mat4x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>lowp</strong> <strong>mediump</strong> <strong>highp</strong> <strong>precision</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler2D</strong> <strong>sampler2DShadow</strong> <strong>sampler2DArray</strong> <strong>sampler2DArrayShadow</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>isampler2D</strong> <strong>isampler2DArray</strong> <strong>usampler2D</strong> <strong>usampler2DArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler2DMS</strong> <strong>isampler2DMS</strong> <strong>usampler2DMS</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler2DMSArray</strong> <strong>isampler2DMSArray</strong> <strong>usampler2DMSArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler3D</strong> <strong>isampler3D</strong> <strong>usampler3D</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>samplerCube</strong> <strong>samplerCubeShadow</strong> <strong>isamplerCube</strong> <strong>usamplerCube</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>samplerCubeArray</strong> <strong>samplerCubeArrayShadow</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>isamplerCubeArray</strong> <strong>usamplerCubeArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>samplerBuffer</strong> <strong>isamplerBuffer</strong> <strong>usamplerBuffer</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2D</strong> <strong>iimage2D</strong> <strong>uimage2D</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2DArray</strong> <strong>iimage2DArray</strong> <strong>uimage2DArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image3D</strong> <strong>iimage3D</strong> <strong>uimage3D</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>imageCube</strong> <strong>iimageCube</strong> <strong>uimageCube</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>imageCubeArray</strong> <strong>iimageCubeArray</strong> <strong>uimageCubeArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>imageBuffer</strong> <strong>iimageBuffer</strong> <strong>uimageBuffer</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>struct</strong></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The following are the keywords reserved for future use.
+Using them will result in an error:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>attribute</strong> <strong>varying</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>noperspective</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>subroutine</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>common</strong> <strong>partition</strong> <strong>active</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>asm</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>class</strong> <strong>union</strong> <strong>enum</strong> <strong>typedef</strong> <strong>template</strong> <strong>this</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>resource</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>goto</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>inline</strong> <strong>noinline</strong> <strong>public</strong> <strong>static</strong> <strong>extern</strong> <strong>external</strong> <strong>interface</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>long</strong> <strong>short</strong> <strong>half</strong> <strong>fixed</strong> <strong>unsigned</strong> <strong>superp</strong>
+<strong>double</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>input</strong> <strong>output</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>hvec2</strong> <strong>hvec3</strong> <strong>hvec4</strong> <strong>fvec2</strong> <strong>fvec3</strong> <strong>fvec4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dvec2</strong> <strong>dvec3</strong> <strong>dvec4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dmat2</strong> <strong>dmat3</strong> <strong>dmat4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dmat2x2</strong> <strong>dmat2x3</strong> <strong>dmat2x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dmat3x2</strong> <strong>dmat3x3</strong> <strong>dmat3x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dmat4x2</strong> <strong>dmat4x3</strong> <strong>dmat4x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>filter</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sizeof</strong> <strong>cast</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>namespace</strong> <strong>using</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler1D</strong> <strong>sampler1DShadow</strong> <strong>sampler1DArray</strong> <strong>sampler1DArrayShadow</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>isampler1D</strong> <strong>isampler1DArray</strong> <strong>usampler1D</strong> <strong>usampler1DArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler2DRect</strong> <strong>sampler2DRectShadow</strong> <strong>isampler2DRect</strong> <strong>usampler2DRect</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler3DRect</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image1D</strong> <strong>iimage1D</strong> <strong>uimage1D</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image1DArray</strong> <strong>iimage1DArray</strong> <strong>uimage1DArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2DRect</strong> <strong>iimage2DRect</strong> <strong>uimage2DRect</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2DMS</strong> <strong>iimage2DMS</strong> <strong>uimage2DMS</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2DMSArray</strong> <strong>iimage2DMSArray</strong> <strong>uimage2DMSArray</strong></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>In addition, all identifiers containing two consecutive underscores (__)
+are reserved for use by underlying software layers.
+Defining such a name in a shader does not itself result in an error, but may
+result in unintended behaviors that stem from having multiple definitions of
+the same name.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="identifiers">3.8. Identifiers</h3>
+<div class="paragraph">
+<p>Identifiers are used for variable names, function names, structure names,
+and field selectors (field selectors select components of
+<code><a href="#vector-components">vectors</a></code> and <code><a href="#matrix-components">matrices</a></code>,
+similarly to structure members).
+Identifiers have the form:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>identifier</em> : </dt>
+<dd>
+<p><em>nondigit</em><br>
+<em>identifier</em> <em>nondigit</em><br>
+<em>identifier</em> <em>digit</em></p>
+</dd>
+<dt class="hdlist1"><em>nondigit</em> : one of </dt>
+<dd>
+<p><strong>_</strong> <strong>a b c d e f g h i j k l m n o p q r s t u v w x y z</strong><br>
+<strong>A B C D E F G H I J K L M N O P Q R S T U V W X Y Z</strong></p>
+</dd>
+<dt class="hdlist1"><em>digit</em> : one of </dt>
+<dd>
+<p><strong>0 1 2 3 4 5 6 7 8 9</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Identifiers starting with “gl_” are reserved for use by OpenGL ES, and
+in general, may not be declared in a shader;
+this results in an error.
+However, as noted in the specification, there are some cases where
+previously declared variables can be redeclared, and predeclared “gl_”
+names are allowed to be redeclared in a shader only for these specific
+purposes.</p>
+</div>
+<div class="paragraph">
+<p>Implementations must support identifier lengths of up to 1024 characters.
+It is an error if the length exceeds this value.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="definitions">3.9. Definitions</h3>
+<div class="paragraph">
+<p>Some language rules described below depend on the following definitions.</p>
+</div>
+<div class="sect3">
+<h4 id="static-use">3.9.1. Static Use</h4>
+<div class="paragraph">
+<p>A shader contains a <em>static use</em> of a variable <em>x</em> if, after preprocessing,
+the shader contains a statement that would access any part of <em>x</em>,
+whether or not flow of control will cause that statement to be executed.
+Such a variable is referred to as being <em>statically used</em>. If the access is a
+write then <em>x</em> is further said to be <em>statically assigned</em>.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="uniform-and-non-uniform-control-flow">3.9.2. Uniform and Non-Uniform Control Flow</h4>
+<div class="paragraph">
+<p>When executing statements in a fragment shader, control flow starts as
+<em>uniform control flow</em>; all fragments enter the same control path into
+<em>main()</em>.
+Control flow becomes <em>non-uniform</em> when different fragments take different
+paths through control-flow statements (selection, iteration, and jumps).
+Control flow subsequently returns to being uniform after such divergent
+sub-statements or skipped code completes, until the next time different
+control paths are taken.</p>
+</div>
+<div class="paragraph">
+<p>For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">main()
+{
+ <span class="predefined-type">float</span> a = ...; <span class="comment">// this is uniform control flow</span>
+ <span class="keyword">if</span> (a < b) { <span class="comment">// this expression is true for some fragments, not all</span>
+ ...; <span class="comment">// non-uniform control flow</span>
+ } <span class="keyword">else</span> {
+ ...; <span class="comment">// non-uniform control flow</span>
+ }
+ ...; <span class="comment">// uniform control flow again</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Other examples of non-uniform control flow can occur within switch
+statements and after conditional breaks, continues, early returns, and after
+fragment discards, when the condition is true for some fragments but not
+others.
+Loop iterations that only some fragments execute are also non-uniform
+control flow.</p>
+</div>
+<div class="paragraph">
+<p>This is similarly defined for other shader stages, based on the per-instance
+data items they process.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="dynamically-uniform-expressions">3.9.3. Dynamically Uniform Expressions</h4>
+<div class="paragraph">
+<p>A fragment-shader expression is <em>dynamically uniform</em> if all fragments
+evaluating it get the same resulting value.
+When loops are involved, this refers to the expression’s value for the same
+loop iteration.
+When functions are involved, this refers to calls from the same call point.</p>
+</div>
+<div class="paragraph">
+<p>This is similarly defined for other shader stages, based on the per-instance
+data they process.</p>
+</div>
+<div class="paragraph">
+<p>Note that constant expressions are trivially dynamically uniform.
+It follows that typical loop counters based on these are also dynamically
+uniform.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="logical-phases-of-compilation">3.10. Logical Phases of Compilation</h3>
+<div class="paragraph">
+<p>The compilation units for the shader processors are processed separately
+before optionally being linked together in the final stage of compilation.
+The logical phases of compilation are:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Source strings are input as byte sequences.
+The value 'zero' is interpreted as a terminator.</p>
+</li>
+<li>
+<p>Source strings are concatenated to form a single input.
+Zero bytes are discarded but all other values are retained.</p>
+</li>
+<li>
+<p>Each string is interpreted according to the UTF-8 standard, with the
+exception that all invalid byte sequences are retained in their original
+form for subsequent processing.</p>
+</li>
+<li>
+<p>Each {carriage-return, line-feed} and {line-feed, carriage return}
+sequence is replaced by a single newline.
+All remaining carriage-return and line-feed characters are then each
+replaced by a newline.</p>
+</li>
+<li>
+<p>Line numbering for each character, which is equal to the number of
+preceding newlines plus one, is noted.
+Note this can only be subsequently changed by the #line directive and is
+not affected by the removal of newlines in phase 6 of compilation.</p>
+</li>
+<li>
+<p>Wherever a backslash ('\') occurs immediately before a newline, both are
+deleted.
+Note that no whitespace is substituted, thereby allowing a single
+preprocessing token to span a newline.
+This operation is not recursive; any new {backslash newline} sequences
+generated are not removed.</p>
+</li>
+<li>
+<p>All comments are replaced with a single space.
+All (non-zero) characters and invalid UTF-8 byte sequences are allowed
+within comments.
+'//' style comments include the initial '//' marker and continue up to,
+but not including, the terminating newline.
+'/<strong>…​</strong>/' comments include both the start and end marker.</p>
+</li>
+<li>
+<p>The source string is converted into a sequence of preprocessing tokens.
+These tokens include preprocessing numbers, identifiers and
+preprocessing operations.
+The line number associated with each token is copied from the line
+number of the first character of the token.</p>
+</li>
+<li>
+<p>The preprocessor is run.
+Directives are executed and macro expansion is performed.</p>
+</li>
+<li>
+<p>White space and newlines are discarded.</p>
+</li>
+<li>
+<p>Preprocessing tokens are converted into tokens.</p>
+</li>
+<li>
+<p>The syntax is analyzed according to the GLSL ES grammar.</p>
+</li>
+<li>
+<p>The result is checked according to the semantic rules of the language.</p>
+</li>
+<li>
+<p>Optionally, the shaders are linked together to form one or more programs
+or separable programs.
+When a pair of shaders from consecutive stages are linked into the same
+program, any outputs and corresponding inputs not used in both shaders
+may be discarded.</p>
+</li>
+<li>
+<p>The binary is generated.</p>
+</li>
+</ol>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="variables-and-types">4. Variables and Types</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>All variables and functions must be declared before being used.
+Variable and function names are identifiers.</p>
+</div>
+<div class="paragraph">
+<p>There are no default types.
+All variable and function declarations must have a declared type, and
+optionally qualifiers.
+A variable is declared by specifying its type followed by one or more names
+separated by commas.
+In many cases, a variable can be initialized as part of its declaration by
+using the assignment operator (<strong>=</strong>).</p>
+</div>
+<div class="paragraph">
+<p>User-defined types may be defined using <strong>struct</strong> to aggregate a list of
+existing types into a single name.</p>
+</div>
+<div class="paragraph">
+<p>The OpenGL ES Shading Language is type safe.
+There are no implicit conversions between types.</p>
+</div>
+<div class="sect2">
+<h3 id="basic-types">4.1. Basic Types</h3>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definition</dt>
+<dd>
+<p>A <em>basic type</em> is a type defined by a keyword in the language.</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The OpenGL ES Shading Language supports the following basic data types, grouped as follows.</p>
+</div>
+<div class="sect3">
+<h4 id="_transparent_types">4.1.1. Transparent Types</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>void</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for functions that do not return a value</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>bool</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a conditional type, taking on values of true or false</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>int</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a signed integer</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uint</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">an unsigned integer</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>float</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point scalar</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component single-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component single-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component single-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>bvec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component Boolean vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>bvec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component Boolean vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>bvec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component Boolean vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component signed integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component signed integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component signed integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component unsigned integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component unsigned integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component unsigned integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 2 × 2 single-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 3 × 3 single-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 4 × 4 single-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>mat2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 2 columns and 3 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 2 columns and 4 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 3 columns and 2 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>mat3</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 3 columns and 4 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 4 columns and 2 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 4 columns and 3 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>mat4</strong></p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Note that where the following tables say “accessing a texture”, the
+<strong>sampler*</strong> opaque types access textures, and the <strong>image*</strong> opaque types
+access images, of a specified type.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="_floating_point_opaque_types">4.1.2. Floating-Point Opaque Types</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2D</strong><br>
+ <strong>image2D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DArray</strong><br>
+ <strong>image2DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DArrayShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D array depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DMS</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D multisample texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DMSArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D multisample array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler3D</strong><br>
+ <strong>image3D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 3D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerCube</strong><br>
+ <strong>imageCube</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a cube mapped texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerCubeShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a cube map depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerCubeArray</strong><br>
+ <strong>imageCubeArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a cube map array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerCubeArrayShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a cube map array depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerBuffer</strong><br>
+ <strong>imageBuffer</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a buffer texture</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="_signed_integer_opaque_types">4.1.3. Signed Integer Opaque Types</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2D</strong><br>
+ <strong>iimage2D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2DArray</strong><br>
+ <strong>iimage2DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2DMS</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D multisample texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2DMSArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D multisample array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler3D</strong><br>
+ <strong>iimage3D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 3D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isamplerCube</strong><br>
+ <strong>iimageCube</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer cube mapped texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isamplerCubeArray</strong><br>
+ <strong>iimageCubeArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer cube map array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isamplerBuffer</strong><br>
+ <strong>iimageBuffer</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer buffer texture</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="_unsigned_integer_opaque_types">4.1.4. Unsigned Integer Opaque Types</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2D</strong><br>
+ <strong>uimage2D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 2D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2DArray</strong><br>
+ <strong>uimage2DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 2D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2DMS</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 2D multisample texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2DMSArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 2D multisample array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler3D</strong><br>
+ <strong>uimage3D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 3D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usamplerCube</strong><br>
+ <strong>uimageCube</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer cube mapped texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usamplerCubeArray</strong><br>
+ <strong>uimageCubeArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer cube map array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usamplerBuffer</strong><br>
+ <strong>uimageBuffer</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer buffer texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>atomic_uint</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer atomic counter</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>In addition, a shader can aggregate these basic types using arrays and
+structures to build more complex types.</p>
+</div>
+<div class="paragraph">
+<p>There are no pointer types.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="void">4.1.5. Void</h4>
+<div class="paragraph">
+<p>Functions that do not return a value must be declared as <strong>void</strong>.
+There is no default function return type.
+The keyword <strong>void</strong> cannot be used in any other declarations (except for
+empty formal or actual parameter lists), or an error results.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="booleans">4.1.6. Booleans</h4>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definition</dt>
+<dd>
+<p>A <em>boolean type</em> is any boolean scalar or vector type (<strong>bool</strong>, <strong>bvec2</strong>,
+<strong>bvec3</strong>, <strong>bvec4</strong>)</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>To make conditional execution of code easier to express, the type <strong>bool</strong> is
+supported.
+There is no expectation that hardware directly supports variables of this
+type.
+It is a genuine Boolean type, holding only one of two values meaning either
+true or false.
+Two keywords <strong>true</strong> and <strong>false</strong> can be used as literal Boolean constants.
+Booleans are declared and optionally initialized as in the follow example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">bool</span> success; <span class="comment">// declare "success" to be a Boolean</span>
+<span class="predefined-type">bool</span> done = <span class="predefined-constant">false</span>; <span class="comment">// declare and initialize "done"</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Expressions used for conditional jumps (<strong>if</strong>, <strong>for</strong>, <strong>?:</strong>, <strong>while</strong>,
+<strong>do</strong>-<strong>while</strong>) must evaluate to the type <strong>bool</strong>.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="integers">4.1.7. Integers</h4>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definitions</dt>
+<dd>
+<p>An <em>integral type</em> is any signed or unsigned, scalar or vector integer type.
+It excludes arrays and structures.</p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>A <em>scalar integral type</em> is a scalar signed or unsigned integer type:</p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>A <em>vector integral type</em> is a vector of signed or unsigned integers:</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>Signed and unsigned integer variables are fully supported.
+In this document, the term <em>integer</em> is meant to generally include both
+signed and unsigned integers.
+<strong>highp</strong> unsigned
+integers have exactly 32 bits of precision.
+<strong>highp</strong> signed
+integers use 32 bits, including a sign bit, in two’s complement form.</p>
+</div>
+<div class="paragraph">
+<p><strong>mediump</strong> and <strong>lowp</strong> integers have implementation-defined numbers of bits.
+See “<a href="#range-and-precision">Range and Precision</a>” for details.
+For all precisions,
+operations resulting in overflow or
+underflow will not cause any exception, nor will they saturate, rather they
+will “wrap” to yield the low-order
+n bits of the result where n is the size in bits of the integer.
+However, for the case where the minimum representable value is divided by
+-1, it is allowed to return either the minimum representable value or the
+maximum representable value.</p>
+</div>
+<div class="paragraph">
+<p>Integers are declared and optionally initialized with integer expressions,
+as in the following example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> i, j = <span class="integer">42</span>; <span class="comment">// default integer literal type is int</span>
+uint k = <span class="integer">3</span>u; <span class="comment">// "u" establishes the type as uint</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Literal integer constants can be expressed in decimal (base 10), octal (base
+8), or hexadecimal (base 16) as follows.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>integer-constant</em> : </dt>
+<dd>
+<p><em>decimal-constant</em> <em>integer-suffix<sub>opt</sub></em><br>
+<em>octal-constant</em> <em>integer-suffix<sub>opt</sub></em><br>
+<em>hexadecimal-constant</em> <em>integer-suffix<sub>opt</sub></em></p>
+</dd>
+<dt class="hdlist1"><em>integer-suffix</em> : one of </dt>
+<dd>
+<p><strong>u</strong> <strong>U</strong></p>
+</dd>
+<dt class="hdlist1"><em>decimal-constant</em> : </dt>
+<dd>
+<p><em>nonzero-digit</em><br>
+<em>decimal-constant</em> <em>digit</em></p>
+</dd>
+<dt class="hdlist1"><em>octal-constant</em> : </dt>
+<dd>
+<p><strong>0</strong><br>
+<em>octal-constant</em> <em>octal-digit</em></p>
+</dd>
+<dt class="hdlist1"><em>hexadecimal-constant</em> : </dt>
+<dd>
+<p><strong>0x</strong> <em>hexadecimal-digit</em><br>
+<strong>0X</strong> <em>hexadecimal-digit</em><br>
+<em>hexadecimal-constant</em> <em>hexadecimal-digit</em></p>
+</dd>
+<dt class="hdlist1"><em>digit</em> : </dt>
+<dd>
+<p><strong>0</strong><br>
+<em>nonzero-digit</em></p>
+</dd>
+<dt class="hdlist1"><em>nonzero-digit</em> : one of </dt>
+<dd>
+<p><strong>1 2 3 4 5 6 7 8 9</strong></p>
+</dd>
+<dt class="hdlist1"><em>octal-digit</em> : one of </dt>
+<dd>
+<p><strong>0 1 2 3 4 5 6 7</strong></p>
+</dd>
+<dt class="hdlist1"><em>hexadecimal-digit</em> : one of </dt>
+<dd>
+<p><strong>0 1 2 3 4 5 6 7 8 9<br>
+a b c d e f<br>
+A B C D E F</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>No white space is allowed between the digits of an integer constant,
+including after the leading <strong>0</strong> or after the leading <strong>0x</strong> or <strong>0X</strong> of a
+constant, or before the suffix <strong>u</strong> or <strong>U</strong>.
+When the suffix <strong>u</strong> or <strong>U</strong> is present, the literal has type <strong>uint</strong>,
+otherwise the type is <strong>int</strong>.
+A leading unary minus sign (-) is interpreted as an arithmetic unary
+negation, not as part of the constant.
+Hence, literals themselves are always expressed with non-negative syntax,
+though they could result in a negative value.</p>
+</div>
+<div class="paragraph">
+<p>It is an error to provide a literal integer whose bit pattern
+cannot fit in 32 bits.
+The bit pattern of the literal is always used unmodified.
+So a signed literal whose bit pattern includes a set sign bit creates a
+negative value.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="integer">1</span> <span class="comment">// OK. Signed integer, value 1</span>
+<span class="integer">1</span>u <span class="comment">// OK. Unsigned integer, value 1</span>
+-<span class="integer">1</span> <span class="comment">// OK. Unary minus applied to signed integer.</span>
+ <span class="comment">// result is a signed integer, value -1</span>
+-<span class="integer">1</span>u <span class="comment">// OK. Unary minus applies to unsigned integer.</span>
+ <span class="comment">// Result is an unsigned integer, value 0xffffffff</span>
+<span class="hex">0xA0000000</span> <span class="comment">// OK. 32-bit signed hexadecimal</span>
+<span class="hex">0xABcdEF00</span>u <span class="comment">// OK. 32-bit unsigned hexadecimal</span>
+<span class="hex">0xffffffff</span> <span class="comment">// OK. Signed integer, value -1</span>
+<span class="hex">0x80000000</span> <span class="comment">// OK. Evaluates to -2147483648</span>
+<span class="hex">0xffffffff</span>u <span class="comment">// OK. Unsigned integer, value 0xffffffff</span>
+<span class="hex">0xfffffffff</span> <span class="comment">// Error: needs more than 32 bits</span>
+<span class="integer">3000000000</span> <span class="comment">// OK. A signed decimal literal taking 32 bits.</span>
+ <span class="comment">// It evaluates to -1294967296</span>
+<span class="integer">2147483648</span> <span class="comment">// OK. Evaluates to -2147483648 (the literal set the sign bit)</span>
+<span class="integer">5000000000</span> <span class="comment">// Error: needs more than 32 bits</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="floats">4.1.8. Floats</h4>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definition</dt>
+<dd>
+<p>A <em>floating-point type</em> is any floating-point scalar, vector or matrix type.
+It excludes arrays and structures.</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>Floats are available for use in a variety of scalar calculations.
+Floating-point variables are defined as in the following example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a, b = <span class="float">1</span><span class="float">.5</span>;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>As an input value to one of the processing units, a floating-point variable
+is expected to match the IEEE 754 single precision floating-point definition
+for precision and dynamic range.
+<strong>highp</strong> floating-point variables within a shader are encoded according to
+the IEEE 754 specification for single-precision floating-point values
+(logically, not necessarily physically).
+While encodings are logically IEEE 754, operations (addition,
+multiplication, etc.) are not necessarily performed as required by IEEE 754.
+See “<a href="#range-and-precision">Range and Precision</a>” for more details on
+precision and usage of NaNs (Not a Number) and Infs (positive or negative
+infinities).</p>
+</div>
+<div class="paragraph">
+<p>Floating-point constants are defined as follows.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>floating-constant</em> : </dt>
+<dd>
+<p><em>fractional-constant</em> <em>exponent-part<sub>opt</sub></em> <em>floating-suffix<sub>opt</sub></em><br>
+<em>digit-sequence</em> <em>exponent-part</em> <em>floating-suffix<sub>opt</sub></em></p>
+</dd>
+<dt class="hdlist1"><em>fractional-constant</em> : </dt>
+<dd>
+<p><em>digit-sequence</em> <strong>.</strong> <em>digit-sequence</em><br>
+<em>digit-sequence</em> <strong>.</strong><br>
+<strong>.</strong> <em>digit-sequence</em></p>
+</dd>
+<dt class="hdlist1"><em>exponent-part</em> : </dt>
+<dd>
+<p><strong>e</strong> <em>sign<sub>opt</sub></em> <em>digit-sequence</em><br>
+<strong>E</strong> <em>sign<sub>opt</sub></em> <em>digit-sequence</em></p>
+</dd>
+<dt class="hdlist1"><em>sign</em> : one of </dt>
+<dd>
+<p><strong>+</strong> <strong>-</strong></p>
+</dd>
+<dt class="hdlist1"><em>digit-sequence</em> : </dt>
+<dd>
+<p><em>digit</em><br>
+<em>digit-sequence</em> <em>digit</em></p>
+</dd>
+<dt class="hdlist1"><em>floating-suffix</em> : one of </dt>
+<dd>
+<p><strong>f</strong> <strong>F</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>A decimal point (<strong>.</strong>) is not needed if the exponent part is present.
+No white space may appear anywhere within a floating-point constant,
+including before a suffix.
+A leading unary minus sign (<strong>-</strong>) is interpreted as a unary operator and is
+not part of the floating-point constant.</p>
+</div>
+<div class="paragraph">
+<p>There is no limit on the number of digits in any <em>digit-sequence</em>.
+If the value of the floating-point number is too large (small) to be stored
+as a single precision value, it is converted to positive (negative)
+infinity.
+A value with a magnitude too small to be represented as a mantissa and
+exponent is converted to zero.
+Implementations may also convert subnormal (denormalized) numbers to zero.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="vectors">4.1.9. Vectors</h4>
+<div class="paragraph">
+<p>The OpenGL ES Shading Language includes data types for generic 2-, 3-, and 4-component vectors
+of floating-point values, integers, and Booleans.
+Floating-point vector variables can be used to store colors, normals,
+positions, texture coordinates, texture lookup results and the like.
+Boolean vectors can be used for component-wise comparisons of numeric
+vectors.
+Some examples of vector declarations are:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec2 texcoord1, texcoord2;
+vec3 position;
+vec4 myRGBA;
+ivec2 textureLookup;
+bvec3 less;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Initialization of vectors can be done with constructors.
+See “<a href="#vector-and-matrix-constructors">Vector and Matrix Constructors</a>”.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="matrices">4.1.10. Matrices</h4>
+<div class="paragraph">
+<p>The OpenGL ES Shading Language has built-in types for 2 × 2, 2 × 3, 2 × 4, 3
+× 2, 3 × 3, 3 × 4, 4 × 2, 4 × 3, and 4 ×
+4 matrices of floating-point numbers.
+The first number in the type is the number of columns, the second is the
+number of rows.
+If there is only one number, the matrix is square.
+Example matrix declarations:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat2 mat2D;
+mat3 optMatrix;
+mat4 view, projection;
+mat4x4 view; <span class="comment">// an alternate way of declaring a mat4</span>
+mat3x2 m; <span class="comment">// a matrix with 3 columns and 2 rows</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Initialization of matrix values is done with constructors (described in
+“<a href="#vector-and-matrix-constructors">Vector and Matrix Constructors</a>”) in
+column-major order.</p>
+</div>
+<div class="paragraph">
+<p><strong>mat2</strong> is an alias for <strong>mat2x2</strong>, not a distinct type.
+Similarly for <strong>mat3</strong> and <strong>mat4.</strong> The following is legal:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat2 a;
+mat2x2 b = a;</code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="opaque-types">4.1.11. Opaque Types</h4>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definition</dt>
+<dd>
+<p>An <em>opaque type</em> is a type where the internal structure of the type is
+hidden from the language.</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The opaque types, as listed in the following sections, declare variables
+that are effectively opaque handles to other objects.
+These objects are accessed through built-in functions, not through direct
+reading or writing of the declared variable.
+They can only be declared as function parameters or in <strong>uniform</strong>-qualified
+variables (see “<a href="#uniform-variables">Uniform Variables</a>”).
+The only opaque types that take memory qualifiers are the image types.
+Except for array indexing, structure member selection, and parentheses,
+opaque variables are not allowed to be operands in expressions; such use
+results in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>When aggregated into arrays within a shader, opaque types can only be
+indexed with a dynamically uniform integral expression (see “Dynamically
+Uniform Expressions”) unless otherwise noted; otherwise, results are
+undefined.</p>
+</div>
+<div class="paragraph">
+<p>Opaque variables cannot be treated as l-values; hence cannot be used as
+<strong>out</strong> or <strong>inout</strong> function parameters, nor can they be assigned into.
+Any such use results in a compile-time error.
+However, they can be passed as <strong>in</strong> parameters with matching types and
+memory qualifiers.
+They cannot be declared with an initializer.</p>
+</div>
+<div class="paragraph">
+<p>Because a single opaque type declaration effectively declares two objects,
+the opaque handle itself and the object it is a handle to, there is room for
+both a storage qualifier and a memory qualifier.
+The storage qualifier will qualify the opaque handle, while the memory
+qualifier will qualify the object it is a handle to.</p>
+</div>
+<div class="sect4">
+<h5 id="samplers">Samplers</h5>
+<div class="paragraph">
+<p>Sampler types (e.g. <strong>sampler2D</strong>) are opaque types, declared and behaving as
+described above for opaque types.</p>
+</div>
+<div class="paragraph">
+<p>Sampler variables are handles to
+two-, and three- dimensional textures, cube maps, depth textures (shadowing),
+etc., as enumerated in the basic types tables.
+There are distinct sampler types for each texture target, and for each of
+float, integer, and unsigned integer data types.
+Texture accesses are done through built-in texture functions (described in
+“<a href="#texture-functions">Texture Functions</a>”) and samplers are used to
+specify which texture to access and how it is to be filtered.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="images">Images</h5>
+<div class="paragraph">
+<p>Image types are opaque types, declared and behaving as described above for
+opaque types.
+They can be further qualified with memory qualifiers.
+When aggregated into arrays within a shader, images can only be indexed with
+a constant integral expression.</p>
+</div>
+<div class="paragraph">
+<p>Image variables are handles to
+two-, or three-dimensional images
+corresponding to all or a portion of a single level of a texture image bound
+to an image unit.
+There are distinct image variable types for each texture target, and for
+each of float, integer, and unsigned integer data types.
+Image accesses should use an image type that matches the target of the
+texture whose level is bound to the image unit, or for non-layered bindings
+of 3D or array images should use the image type that matches the
+dimensionality of the layer of the image (i.e., a layer of 3D, 2DArray,
+Cube, or CubeArray should use
+<strong>image2D</strong>).
+If the image target type does not match the bound image in this manner, if
+the data type does not match the bound image, or if the format layout
+qualifier does not match the image unit format as described in section
+8.22
+“Texture Image Loads and Stores” of the <a href="#references">OpenGL ES Specification</a>, the
+results of image accesses are undefined but cannot include program
+termination.</p>
+</div>
+<div class="paragraph">
+<p>Image variables are used in the image load, store, and atomic functions
+described in “<a href="#image-functions">Image Functions</a>” to specify an image to
+access.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="atomic-counters">Atomic Counters</h5>
+<div class="paragraph">
+<p>Atomic counter types (e.g. <strong>atomic_uint</strong>) are opaque handles to counters,
+declared and behaving as described above for opaque types.
+The variables they declare specify which counter to access when using the
+built-in atomic counter functions as described in
+“<a href="#atomic-counter-functions">Atomic Counter Functions</a>”.
+They are bound to buffers as described in
+“<a href="#atomic-counter-layout-qualifiers">Atomic Counter Layout Qualifiers</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Atomic counters aggregated into arrays within a shader can only be indexed
+with dynamically uniform integral expressions, otherwise results are
+undefined.</p>
+</div>
+<div class="paragraph">
+<p>Members of structures cannot be declared as atomic counter types.</p>
+</div>
+<div class="paragraph">
+<p>The default precision of all atomic types is <strong>highp</strong>.
+It is an error to declare an atomic type with a different precision or to
+specify the default precision for an atomic type to be <strong>lowp</strong> or <strong>mediump</strong>.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="structures">4.1.12. Structures</h4>
+<div class="paragraph">
+<p>User-defined types can be created by aggregating other already defined types
+into a structure using the <strong>struct</strong> keyword.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> light {
+ <span class="predefined-type">float</span> intensity;
+ vec3 position;
+} lightVar;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>In this example, <em>light</em> becomes the name of the new type, and <em>lightVar</em>
+becomes a variable of type <em>light</em>.
+To declare variables of the new type, use its name (without the keyword
+<strong>struct</strong>).</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">light lightVar2;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>More formally, structures are declared as follows.
+However, the definitive grammar is as given in
+“<a href="#shading-language-grammar">Shading Language Grammar</a>”.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>struct-definition</em> : </dt>
+<dd>
+<p><em>qualifier<sub>opt</sub></em> <strong>struct</strong> name<sub>opt</sub>_ <strong>{</strong> <em>member-list</em> <strong>}</strong>
+<em>declarators<sub>opt</sub></em> <strong>;</strong></p>
+</dd>
+<dt class="hdlist1"><em>member-list</em> : </dt>
+<dd>
+<p><em>member-declaration</em> <strong>;</strong><br>
+<em>member-declaration</em> <em>member-list</em> <strong>;</strong></p>
+</dd>
+<dt class="hdlist1"><em>member-declaration</em> : </dt>
+<dd>
+<p><em>basic-type</em> <em>declarators</em> <strong>;</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>name</em> becomes the user-defined type, and can be used to declare
+variables to be of this new type.
+The <em>name</em> shares the same name space as other variables, types, and
+functions.
+All previously visible variables, types, constructors, or functions with
+that name are hidden.
+The optional <em>qualifier</em> only applies to any <em>declarators</em>, and is not part
+of the type being defined for <em>name</em>.</p>
+</div>
+<div class="paragraph">
+<p>Structures must have at least one member declaration.
+Member declarators may contain precision qualifiers, but use of any other
+qualifier results in an error.
+Bit fields are not supported.
+Member types must be already defined (there are no forward references).
+Member declarations cannot contain initializers.
+Member declarators can contain arrays.
+Such arrays must have a size specified, and the size must be a constant
+integral expression that’s greater than zero (see
+“<a href="#constant-expressions">Constant Expressions</a>”).
+Each level of structure has its own name space for names given in member
+declarators; such names need only be unique within that name space.</p>
+</div>
+<div class="paragraph">
+<p>Anonymous structures are not supported.
+Embedded structure definitions are not supported.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> S { <span class="predefined-type">float</span> f; }; <span class="comment">// Allowed: S is defined as a structure.</span>
+
+<span class="keyword">struct</span> T {
+ S; <span class="comment">// Error: anonymous structures disallowed</span>
+ <span class="keyword">struct</span> { ... }; <span class="comment">// Error: embedded structures disallowed</span>
+ S s; <span class="comment">// Allowed: nested structure with a name.</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Structures can be initialized at declaration time using constructors, as
+discussed in “<a href="#structure-constructors">Structure Constructors</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Any restrictions on the usage of a type or qualifier also apply to any
+structure that contains a member of that type or qualifier.
+This also applies to structure members that are structures, recursively.</p>
+</div>
+<div class="paragraph">
+<p>Structures can contain variables of any type except:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>atomic_uint</strong> (since there is no mechanism to specify the binding)</p>
+</li>
+<li>
+<p>image types (since there is no mechanism to specify the format
+qualifier)</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="arrays">4.1.13. Arrays</h4>
+<div class="paragraph">
+<p>Variables of the same type can be aggregated into arrays by declaring a name
+followed by brackets (<strong>[ ]</strong>) enclosing an optional size.</p>
+</div>
+<div class="paragraph">
+<p>When present, the array size must be a constant integral expression (see
+“<a href="#constant-expressions">Constant Expressions</a>”) greater than zero.
+The type of the size parameter can be a signed or unsigned integer and the
+choice of type does not affect the type of the resulting array.
+Arrays only have a single dimension (a single number within “[ ]”),
+however, arrays of arrays can be declared.
+Any type can be formed into an array.</p>
+</div>
+<div class="paragraph">
+<p>Arrays are sized either at compile-time or at run-time.
+To size an array at compile-time, either the size must be specified within
+the brackets as above or must be inferred from the type of the initializer.</p>
+</div>
+<div class="paragraph">
+<p>If an array is declared as the last member of a shader storage block and the
+size is not specified at compile-time, it is sized at run-time.
+In all other cases, arrays are sized only at compile-time.
+An array declaration sized at compile-time which leaves the size of the
+array unspecified is an error.</p>
+</div>
+<div class="paragraph">
+<p>For compile-time sized arrays, it is illegal to index an array with a
+constant integral expression greater than or equal to the declared size or
+with a negative constant expression.
+Arrays declared as formal parameters in a function declaration must also
+specify a size.
+Undefined behavior results from indexing an array with a non-constant
+expression that’s greater than or equal to the array’s size or less than 0.
+If robust buffer access is enabled (see section 10.3.5 “Robust Buffer
+Access” of the <a href="#references">OpenGL ES Specification</a>), such indexing must not result in
+abnormal program termination.
+The results are still undefined, but implementations are encouraged to
+produce zero values for such accesses.</p>
+</div>
+<div class="paragraph">
+<p>Some examples are:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> frequencies[<span class="integer">3</span>];
+uniform vec4 lightPosition[<span class="integer">4</span>u];
+<span class="directive">const</span> <span class="predefined-type">int</span> numLights = <span class="integer">2</span>;
+light lights[numLights];
+vec4 a[<span class="integer">3</span>][<span class="integer">2</span>];
+a.length() <span class="comment">// this is 3</span>
+a[x].length() <span class="comment">// this is 2</span>
+<span class="comment">// a shader storage block, introduced in section 4.3.7 "Buffer Variables"</span>
+buffer b {
+ <span class="predefined-type">float</span> u[]; <span class="comment">// an error</span>
+ vec4 v[]; <span class="comment">// okay, v will be sized at run-time</span>
+} name[<span class="integer">3</span>]; <span class="comment">// when the block is arrayed, all u will be the same size,</span>
+ <span class="comment">// but not necessarily all v, if sized dynamically</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When the <strong>length</strong>() method will return a compile-time constant, the
+expression in brackets (x above) will be evaluated and subject to the rules
+required for array indices, but the array will not be dereferenced.
+Thus, behavior is well defined even if the run-time value of the expression
+is out of bounds.</p>
+</div>
+<div class="paragraph">
+<p>An array type can be formed by specifying a non-array type
+(<a href="#type_specifier_nonarray">[type_specifier_nonarray]</a>) followed by an <a href="#array_specifier">[array_specifier]</a>.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">5</span>]</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Note that the construct <em>type [size]</em> does not always result in an array of
+length <em>size</em> of type <em>type</em>:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">2</span>][<span class="integer">3</span>] <span class="comment">// an array of size [2] of array of size [3] of float,</span>
+ <span class="comment">// not size [3] of float[2]</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This type can be used anywhere any other type can be used, including as the
+return value from a function</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">5</span>] foo() { }</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>as a constructor of an array:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">5</span>](<span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span>, <span class="float">5</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>as an unnamed parameter:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> foo(<span class="predefined-type">float</span>[<span class="integer">5</span>])</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and as an alternate way of declaring a variable or function parameter:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">5</span>] a;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>An array type can also be formed without specifying a size if the definition
+includes an initializer:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> x[] = <span class="predefined-type">float</span>[<span class="integer">2</span>] (<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>); <span class="comment">// declares an array of size 2</span>
+<span class="predefined-type">float</span> y[] = <span class="predefined-type">float</span>[] (<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>); <span class="comment">// declares an array of size 3</span>
+<span class="predefined-type">float</span> a[<span class="integer">5</span>];
+<span class="predefined-type">float</span> b[] = a;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Note that the initializer itself does not need to be a constant expression
+but the length of the initializer will be a constant expression.</p>
+</div>
+<div class="paragraph">
+<p>Arrays can have initializers formed from array constructors:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a[<span class="integer">5</span>] = <span class="predefined-type">float</span>[<span class="integer">5</span>](<span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span>, <span class="float">5</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>);
+<span class="predefined-type">float</span> a[<span class="integer">5</span>] = <span class="predefined-type">float</span>[](<span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span>, <span class="float">5</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>); <span class="comment">// same thing</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>An array of arrays can be declared as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 a[<span class="integer">3</span>][<span class="integer">2</span>]; <span class="comment">// size-3 array of size-2 array of vec4</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>which declares a one-dimensional array of size 3 of one-dimensional arrays
+of size 2 of <strong>vec4</strong>.
+The following declarations do the same thing:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4[<span class="integer">2</span>] a[<span class="integer">3</span>]; <span class="comment">// size-3 array of size-2 array of vec4</span>
+vec4[<span class="integer">3</span>][<span class="integer">2</span>] a; <span class="comment">// size-3 array of size-2 array of vec4</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When in transparent memory (like in a uniform block), the layout is that the
+inner-most (right-most in declaration) dimensions iterate faster than the
+outer dimensions.
+That is, for the above, the order in memory would be:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>Low address : a[0][0] : a[0][1] : a[1][0] : a[1][1] : a[2][0] : a[2][1]
+: High address</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The last member of a shader storage block (see “<a href="#buffer-variables">Buffer
+Variables</a>”), may be declared without specifying a size.
+For such arrays, the effective array size is inferred at run-time from the
+size of the data store backing the shader storage block.
+Such runtime-sized arrays may be indexed with general integer expressions,
+but may not be passed as an argument to a function or indexed with a
+negative constant expression.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> S { <span class="predefined-type">float</span> f; };
+buffer ShaderStorageBlock1
+{
+ vec4 a[]; <span class="comment">// illegal</span>
+ vec4 b[]; <span class="comment">// legal, runtime-sized arrays are last member</span>
+};
+buffer ShaderStorageBlock2
+{
+ vec4 a[<span class="integer">4</span>]; <span class="comment">// legal, size declared</span>
+ S b[]; <span class="comment">// legal, runtime-sized arrays are allowed,</span>
+ <span class="comment">// including arrays of structures</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>However, it is a compile-time error to assign to a runtime-sized array.
+Assignments to individual elements of the array is allowed.</p>
+</div>
+<div class="paragraph">
+<p>Arrays have a fixed number of elements.
+This can be obtained by using the <strong>length</strong>() method:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a[<span class="integer">5</span>];
+a.length(); <span class="comment">// returns 5</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The return value is a signed integral expression.
+For compile-time sized arrays, the value returned by the length method is a
+constant expression.
+For run-time sized arrays , the value returned will not be constant
+expression and will be determined at run time based on the size of the
+buffer object providing storage for the block.</p>
+</div>
+<div class="paragraph">
+<p>The precision is determined using the same rules as for other cases where
+there is no intrinsic precision.
+See “<a href="#precision-qualifiers">Precision Qualifiers</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Any restrictions on the usage of a type also apply to arrays of that type.
+This applies recursively.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="scoping">4.2. Scoping</h3>
+<div class="paragraph">
+<p>The scope of a declaration determines where the declaration is visible.
+GLSL ES uses a system of statically nested scopes.
+This allows names to be redefined within a shader.</p>
+</div>
+<div class="sect3">
+<h4 id="definition-of-terms">4.2.1. Definition of Terms</h4>
+<div class="paragraph">
+<p>The term <em>scope</em> refers to a specified region of the program where names are
+guaranteed to be visible.
+For example, a <em>compound_statement_with_scope</em> ('{' <em>statement</em> <em>statement</em>
+…​
+'}') defines a scope.</p>
+</div>
+<div class="paragraph">
+<p>A <em>nested scope</em> is a scope defined within an outer scope.</p>
+</div>
+<div class="paragraph">
+<p>The terms '<em>same scope'</em> and '<em>current scope</em>' are equivalent to the term
+'<em>scope</em>' but used to emphasize that nested scopes are excluded.</p>
+</div>
+<div class="paragraph">
+<p>The <em>scope of a declaration</em> is the region or regions of the program where
+that declaration is visible.</p>
+</div>
+<div class="paragraph">
+<p>A <em>name space</em> defines where names may be defined.
+Within a single name space, a name has at most one entry, specifying it to
+be one of: structure, variable, or function.</p>
+</div>
+<div class="paragraph">
+<p>In general, each scope has an associated name space.
+However, in certain cases e.g. for uniforms, multiple scopes share the same
+name space.
+In these cases, conflicting declarations are an error, even though the name
+is only visible in the scopes where it is declared.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="types-of-scope">4.2.2. Types of Scope</h4>
+<div class="paragraph">
+<p>The scope of a variable is determined by where it is declared.
+If it is declared outside all function definitions, it has global scope,
+which starts from where it is declared and persists to the end of the shader
+it is declared in.
+If it is declared in a <strong>while</strong> test or a <strong>for</strong> statement, then it is scoped
+to the end of the following sub-statement (specified as
+<em>statement-no-new-scope</em> in the grammar).
+Otherwise, if it is declared as a statement within a compound statement, it
+is scoped to the end of that compound statement.
+If it is declared as a parameter in a function definition, it is scoped
+until the end of that function definition.
+A function’s parameter declarations and body together form a single scope.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> f( <span class="comment">/* nested scope begins here */</span> <span class="predefined-type">int</span> k)
+{
+ <span class="predefined-type">int</span> k = k + <span class="integer">3</span>; <span class="comment">// redeclaration error of the name k</span>
+ ...
+}
+<span class="predefined-type">int</span> f(<span class="predefined-type">int</span> k)
+{
+ {
+ <span class="predefined-type">int</span> k = k + <span class="integer">3</span>; <span class="comment">// 2nd k is parameter, initializing nested first k</span>
+ <span class="predefined-type">int</span> m = k <span class="comment">// use of new k, which is hiding the parameter</span>
+ }
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For both for and while loops, the sub-statement itself does not introduce a
+new scope for variable names, so the following has a redeclaration
+compile-time error:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">for</span> ( <span class="comment">/* nested scope begins here */</span> <span class="predefined-type">int</span> i = <span class="integer">0</span>; i < <span class="integer">10</span>; i++)
+{
+ <span class="predefined-type">int</span> i; <span class="comment">// redeclaration error</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The body of a <strong>do</strong>-<strong>while</strong> loop introduces a new scope lasting only between
+the do and while (not including the while test expression), whether or not
+the body is simple or compound:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> _i_ = <span class="integer">17</span>;
+<span class="keyword">do</span>
+ <span class="predefined-type">int</span> i = <span class="integer">4</span>; <span class="comment">// okay, in nested scope</span>
+<span class="keyword">while</span> (i == <span class="integer">0</span>); <span class="comment">// i is 17, scoped outside the do-while body</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The statement following a <strong>switch</strong> (…​) forms a nested scope.</p>
+</div>
+<div class="paragraph">
+<p>Representing the if construct as:</p>
+</div>
+<div class="paragraph">
+<p><strong>if</strong> if-expression <strong>then</strong> if-statement <strong>else</strong> else-statement,</p>
+</div>
+<div class="paragraph">
+<p>a variable declared in the if-statement is scoped to the end of the
+if-statement.
+A variable declared in the else-statement is scoped to the end of the
+else-statement.
+This applies both when these statements are simple statements and when they
+are compound statements.
+The if-expression does not allow new variables to be declared, hence does
+not form a new scope.</p>
+</div>
+<div class="paragraph">
+<p>Within a declaration, the scope of a name starts immediately after the
+initializer if present or immediately after the name being declared if not.
+Several examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> x = <span class="integer">1</span>;
+{
+ <span class="predefined-type">int</span> x = <span class="integer">2</span>,<span class="comment">/* 2nd x visible here */</span> y = x; <span class="comment">// y is initialized to 2</span>
+ <span class="predefined-type">int</span> z = z; <span class="comment">// error if z not previously defined.</span>
+}
+{
+<span class="predefined-type">int</span> x = x; <span class="comment">// x is initialized to '1'</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A structure name declaration is visible at the end of the <em>struct_specifier</em>
+in which it was declared:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> S
+{
+ <span class="predefined-type">int</span> x;
+};
+{
+ S S = S(<span class="integer">0</span>); <span class="comment">// 'S' is only visible as a struct and constructor</span>
+ S; <span class="comment">// 'S' is now visible as a variable</span>
+}
+<span class="predefined-type">int</span> x = x; <span class="comment">// Error if x has not been previously defined.</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="redeclaring-names">4.2.3. Redeclaring Names</h4>
+<div class="paragraph">
+<p>All variable names, structure type names, and function names in a given
+scope share the same name space.
+Function names can be redeclared in the same scope, with the same or
+different parameters, without error.
+Otherwise, within a shader, a declared name cannot be redeclared in the same
+scope; doing so results in a redeclaration error.
+If a nested scope redeclares a name used in an outer scope, it hides all
+existing uses of that name.
+There is no way to access the hidden name or make it unhidden, without
+exiting the scope that hid it.</p>
+</div>
+<div class="paragraph">
+<p>Names of built-in functions cannot be redeclared as functions.
+Therefore overloading or redefining built-in functions is an error.</p>
+</div>
+<div class="paragraph">
+<p>A <em>declaration</em> is considered to be a statement that adds a name or
+signature to the symbol table.
+A <em>definition</em> is a statement that fully defines that name or signature.
+E.g.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> f();<span class="comment">// declaration;</span>
+<span class="predefined-type">int</span> f() {<span class="keyword">return</span> <span class="integer">0</span>;}<span class="comment">// declaration and definition</span>
+<span class="predefined-type">int</span> x; <span class="comment">// declaration and definition</span>
+<span class="predefined-type">int</span> a[<span class="integer">4</span>];<span class="comment">// array declaration and definition</span>
+<span class="keyword">struct</span> S {<span class="predefined-type">int</span> x;};<span class="comment">// structure declaration and definition</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The determination of equivalence of two declarations depends on the type of
+declaration.
+For functions, the whole function signature must be considered (see
+“Function Definitions”).
+For variables (including arrays) and structures only the names must match.</p>
+</div>
+<div class="paragraph">
+<p>Within each scope, a name may be declared either as a variable declaration
+<em>or</em> as function declarations <em>or</em> as a structure.</p>
+</div>
+<div class="paragraph">
+<p>Examples of combinations that are allowed:</p>
+</div>
+<div class="paragraph">
+<p>1.</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> f(<span class="predefined-type">int</span>) {...}
+<span class="directive">void</span> f(<span class="predefined-type">float</span>) {...}<span class="comment">// function overloading allowed</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>2.</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> f(<span class="predefined-type">int</span>);<span class="comment">// 1^st^ declaration (allowed)</span>
+<span class="directive">void</span> f(<span class="predefined-type">int</span>);<span class="comment">// repeated declaration (allowed)</span>
+<span class="directive">void</span> f(<span class="predefined-type">int</span>) {...}<span class="comment">// single definition (allowed)</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Examples of combinations that are disallowed:</p>
+</div>
+<div class="paragraph">
+<p>1.</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> f(<span class="predefined-type">int</span>) {...}
+<span class="directive">void</span> f(<span class="predefined-type">int</span>) {...}<span class="comment">// Error: repeated definition</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>2.</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> f(<span class="predefined-type">int</span>);
+<span class="keyword">struct</span> f {<span class="predefined-type">int</span> x;};<span class="comment">// Error: type 'f' conflicts with function 'f'</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>3.</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> f {<span class="predefined-type">int</span> x;};
+<span class="predefined-type">int</span> f;<span class="comment">// Error: conflicts with the type 'f'</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>4.</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> a[<span class="integer">3</span>];
+<span class="predefined-type">int</span> a[<span class="integer">3</span>];<span class="comment">// Error: repeated array definition</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>5.</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> x;
+<span class="predefined-type">int</span> x;<span class="comment">// Error: repeated variable definition</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="global-scope">4.2.4. Global Scope</h4>
+<div class="paragraph">
+<p>The built-in functions are scoped in the global scope users declare global
+variables in.
+That is, a shader’s global scope, available for user-defined functions and
+global variables, is the same as the scope containing the built-in
+functions.
+Function declarations (prototypes) cannot occur inside of functions; they
+must be at global scope.
+Hence it is not possible to hide a name with a function.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="shared-globals">4.2.5. Shared Globals</h4>
+<div class="paragraph">
+<p>Shared globals are variables that can be accessed by multiple compilation
+units.
+In GLSL ES the only shared globals are uniforms.
+Vertex shader outputs are not considered to be shared globals since they
+must pass through the rasterization stage before they are used as input by
+the fragment shader.</p>
+</div>
+<div class="paragraph">
+<p>Shared globals share the same name space, and must be declared with the same
+type and precision.
+They will share the same storage.
+Shared global arrays must have the same base type and the same explicit
+size.
+Scalars must have exactly the same precision, type name and type definition.
+Structures must have the same name, sequence of type names, and type
+definitions, and member names to be considered the same type.
+This rule applies recursively for nested or embedded types.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="storage-qualifiers">4.3. Storage Qualifiers</h3>
+<div class="paragraph">
+<p>Variable declarations may have at most one storage qualifier specified in
+front of the type.
+These are summarized as</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Storage Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><none: default></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">local read/write memory, or an input parameter to a
+ function</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>const</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a compile-time constant</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>in</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">linkage into a shader from a previous stage, variable
+ is copied in</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>out</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">linkage out of a shader to a subsequent stage,
+ variable is copied out</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uniform</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">value does not change across the primitive being
+ processed, uniforms form the linkage between a shader,
+ OpenGL ES, and the application</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>buffer</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">value is stored in a buffer object, and can be read or
+ written both by shader invocations and the OpenGL ES
+ API</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>shared</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">compute shader only; variable storage is shared across
+ all work items in a workgroup</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Some input and output qualified variables can be qualified with at most one
+additional auxiliary storage qualifier:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Auxiliary Storage Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>centroid</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">centroid-based interpolation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sample</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">per-sample interpolation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>patch</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">per-tessellation-patch attributes</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Local variables can only use the <strong>const</strong> storage qualifier (or use no
+storage qualifier).</p>
+</div>
+<div class="paragraph">
+<p>Note that function parameters can use <strong>const</strong>, <strong>in</strong>, and <strong>out</strong> qualifiers,
+but as <em>parameter qualifiers</em>.
+Parameter qualifiers are discussed in
+“<a href="#function-calling-conventions">Function Calling Conventions</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Function return types and structure members do not use storage qualifiers.</p>
+</div>
+<div class="paragraph">
+<p>Data types for communication from one run of a shader executable to its next
+run (to communicate between fragments or between vertices) do not exist.
+This would prevent parallel execution of the same shader executable on
+multiple vertices or fragments.</p>
+</div>
+<div class="paragraph">
+<p>In declarations of global variables with no storage qualifier or with a
+const qualifier, any initializer must be a constant expression.
+Declarations of global variables with other storage qualifiers may not
+contain initializers.
+Global variables without storage qualifiers that are not initialized in
+their declaration or by the application will not be initialized by
+OpenGL ES, but rather will enter <em>main()</em> with undefined values.</p>
+</div>
+<div class="sect3">
+<h4 id="default-storage-qualifier">4.3.1. Default Storage Qualifier</h4>
+<div class="paragraph">
+<p>If no qualifier is present on a global variable, then the variable has no
+linkage to the application or shaders running on other pipeline stages.
+For either global or local unqualified variables, the declaration will
+appear to allocate memory associated with the processor it targets.
+This variable will provide read/write access to this allocated memory.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="constant-qualifier">4.3.2. Constant Qualifier</h4>
+<div class="paragraph">
+<p>Named compile-time constants
+can be declared using
+the <strong>const</strong> qualifier.
+Any variables qualified as constant are read-only variables for that shader.
+Declaring variables as constant allows more descriptive shaders than using
+hard-wired numerical constants.
+The <strong>const</strong> qualifier can be used with any of the non-void transparent basic
+data types, as well as with structures and arrays of these.
+It is an error to write to a <strong>const</strong> variable outside of its
+declaration, so they must be initialized when declared.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">const</span> vec3 zAxis = vec3 (<span class="float">0</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span>);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Structure members may not be qualified with <strong>const</strong>.
+Structure variables can be declared as <strong>const</strong>, and initialized with a
+structure
+constructor.</p>
+</div>
+<div class="paragraph">
+<p>Initializers for <strong>const</strong> declarations
+must be constant expressions, as defined in
+“<a href="#constant-expressions">Constant Expressions</a>”.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="constant-expressions">4.3.3. Constant Expressions</h4>
+<div class="paragraph">
+<p>A <em>constant expression</em> is one of</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A literal value (e.g. <strong>5</strong> or <strong>true</strong>).</p>
+</li>
+<li>
+<p>A global or local variable qualified as <strong>const</strong> (i.e., not including
+function parameters).</p>
+</li>
+<li>
+<p>An expression formed by an operator on operands that are all constant
+expressions, including getting an element of a constant array, or a
+member of a constant structure, or components of a constant vector.
+However, the lowest precedence operators of the sequence operator (<strong>,</strong>)
+and the assignment operators (<strong>=</strong>, <strong>+=</strong>, <strong>…​</strong>) are not included in the
+operators that can create a constant expression.</p>
+</li>
+<li>
+<p>The <strong>length</strong>() method on a compile-time sized array, whether or not the
+object itself is constant.</p>
+</li>
+<li>
+<p>A constructor whose arguments are all constant expressions.</p>
+</li>
+<li>
+<p>A built-in function call whose arguments are all constant expressions,
+with the exception of the texture lookup functions.
+This rule excludes functions with a <strong>void</strong> return or functions that have
+an <strong>out</strong> parameter.
+The built-in functions <strong>dFdx</strong>, <strong>dFdy</strong>, and <strong>fwidth</strong> must return 0 when
+evaluated inside an initializer with an argument that is a constant
+expression.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Function calls to user-defined functions (non-built-in functions) cannot be
+used to form constant expressions.</p>
+</div>
+<div class="paragraph">
+<p>Scalar, vector, matrix, array and structure variables are constant
+expressions if qualified as <strong>const</strong>.
+Opaque types cannot be constant expressions.</p>
+</div>
+<div class="paragraph">
+<p>A <em>constant integral expression</em> is a constant expression that evaluates to
+a scalar signed or unsigned integer.</p>
+</div>
+<div class="paragraph">
+<p>Constant expressions will be evaluated in an invariant way so as to create
+the same value in multiple shaders when the same constant expressions appear
+in those shaders.
+See “<a href="#the-invariant-qualifier">The Invariant Qualifier</a>” for more details
+on how to create invariant expressions and
+“<a href="#precision-qualifiers">Precision Qualifiers</a>” for detail on how
+expressions are evaluated.</p>
+</div>
+<div class="paragraph">
+<p>Constant expressions respect the <strong>precise</strong> and <strong>invariant</strong> qualifiers but
+will be always be evaluated in an invariant way, independent of the use of
+such qualification, so as to create the same value in multiple shaders when
+the same constant expressions appear in those shaders.
+See “<a href="#the-invariant-qualifier">The Invariant Qualifier</a>” and
+“<a href="#the-precise-qualifier">The Precise Qualifier</a>” for more details on how
+to create invariant expressions.</p>
+</div>
+<div class="paragraph">
+<p>Constant
+expressions may be evaluated by the compiler’s
+host platform, and are therefore not required to compute the same value that
+the same expression would evaluate to on the shader execution target.
+However, the host must use the same or greater precision than the target
+would use.
+When the precision qualification cannot be determined, the expression is
+evaluated at <strong>highp</strong>.
+See “<a href="#default-precision-qualifiers">Default Precision Qualifiers</a>”.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="input-variables">4.3.4. Input Variables</h4>
+<div class="paragraph">
+<p>Shader input variables are declared with the <strong>in</strong> storage qualifier.
+They form the input interface between previous stages of the OpenGL ES
+pipeline and the declaring shader.
+Input variables must be declared at global scope.
+Values from the previous pipeline stage are copied into input variables at
+the beginning of shader execution.
+It is an error to write to a variable declared as an input.</p>
+</div>
+<div class="paragraph">
+<p>Only the input variables that are
+actually
+read need to be written by the
+previous stage; it is allowed to have superfluous declarations of input
+variables.</p>
+</div>
+<div class="paragraph">
+<p>See “<a href="#built-in-variables">Built-In Variables</a>” for a list of the built-in
+input names.</p>
+</div>
+<div class="paragraph">
+<p>Vertex shader input variables (or attributes) receive per-vertex data.
+It is an error to use auxiliary storage or interpolation qualifiers
+on a vertex shader input.
+The values copied in are established by the OpenGL ES API or through the use
+of the layout identifier <strong>location</strong>.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to declare a vertex shader input with, or that
+contains, any of the following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+<li>
+<p>An array</p>
+</li>
+<li>
+<p>A structure</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Example declarations in a vertex shader:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 position;
+in vec3 normal;
+in vec2 texCoord[<span class="integer">4</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It is expected that graphics hardware will have a small number of fixed
+vector locations for passing vertex inputs.
+Therefore, the OpenGL ES Shading Language defines each non-matrix input variable as taking up
+one such vector location.
+There is an implementation-dependent limit on the number of locations that
+can be used, and if this is exceeded it will cause a link-time error.
+(Declared input variables that are not statically used do not count against
+this limit.) A scalar input counts the same amount against this limit as a
+<strong>vec4</strong>, so applications may want to consider packing groups of four
+unrelated float inputs together into a vector to better utilize the
+capabilities of the underlying hardware.
+A matrix input will use up multiple locations.
+The number of locations used will equal the number of columns in the matrix.</p>
+</div>
+<div class="paragraph">
+<p>Tessellation control, evaluation, and geometry shader input variables get
+the per-vertex values written out by output variables of the same names in
+the previous active shader stage.
+For these inputs, <strong>centroid</strong> and interpolation qualifiers are allowed, but
+have no effect.
+Since tessellation control, tessellation evaluation, and geometry shaders
+operate on a set of vertices, each input variable (or input block, see
+interface blocks below) needs to be declared as an array.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in <span class="predefined-type">float</span> foo[]; <span class="comment">// geometry shader input for vertex "out float foo"</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Each element of such an array corresponds to one vertex of the primitive
+being processed.
+Each array can optionally have a size declared.
+For geometry shaders, the array size will be set by, (or if provided must be
+consistent with) the input <strong>layout</strong> declaration(s) establishing the type of
+input primitive, as described later in “<a href="#input-layout-qualifiers">Input
+Layout Qualifiers</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Some inputs and outputs are <em>arrayed</em>, meaning that for an interface between
+two shader stages either the input or output declaration requires an extra
+level of array indexing for the declarations to match.
+For example, with the interface between a vertex shader and a geometry
+shader, vertex shader output variables and geometry shader input variables
+of the same name must have matching types, except that the geometry shader
+will have one more array dimension than the vertex shader, to allow for
+vertex indexing.
+If such an arrayed interface variable is not declared with the necessary
+additional input or output array dimension, a link-time error will result.
+Geometry shader inputs, tessellation control shader inputs and outputs, and
+tessellation evaluation inputs all have an additional level of arrayness
+relative to other shader inputs and outputs.
+These inputs and outputs are known as <em>per-vertex-arrayed</em> inputs and
+outputs.
+Component limits for arrayed interfaces (e.g.
+<em>gl_MaxTessControlInputComponents</em>) are limits per vertex, not limits for
+the entire interface.</p>
+</div>
+<div class="paragraph">
+<p>For non-arrayed interfaces (meaning array dimensionally stays the same
+between stages), it is a link-time error if the input variable is not
+declared with the same type, including array dimensionality, as the matching
+output variable.</p>
+</div>
+<div class="paragraph">
+<p>The link-time type-matching rules apply to all declared input and output
+variables, whether or not they are used.</p>
+</div>
+<div class="paragraph">
+<p>Additionally, tessellation evaluation shaders support per-patch input
+variables declared with the <strong>patch</strong> and <strong>in</strong> qualifiers.
+Per-patch input variables are filled with the values of per-patch output
+variables written by the tessellation control shader.
+Per-patch inputs may be declared as one-dimensional arrays, but are not
+indexed by vertex number.
+Applying the <strong>patch</strong> qualifier to inputs can only be done in tessellation
+evaluation shaders.
+As with other input variables, per-patch inputs must be declared using the
+same type and qualification as per-patch outputs from the previous
+(tessellation control) shader stage.
+It is a compile-time error to use <strong>patch</strong> with inputs in any other stage.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to declare a tessellation control, tessellation
+evaluation or geometry shader input with, or that contains, any of the
+following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+<li>
+<p>A structure containing an array</p>
+</li>
+<li>
+<p>A structure containing a structure</p>
+</li>
+<li>
+<p>For per-vertex-arrayed variables:</p>
+<div class="ulist">
+<ul>
+<li>
+<p>Per-vertex-arrayed arrays of arrays</p>
+</li>
+<li>
+<p>Per-vertex-arrayed arrays of structures</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>For non-per-vertex-arrayed variables:</p>
+<div class="ulist">
+<ul>
+<li>
+<p>An array of arrays</p>
+</li>
+<li>
+<p>An array of structures</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Fragment shader inputs get per-fragment values, typically interpolated from
+a previous stage’s outputs.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to declare a fragment shader input with, or that
+contains, any of the following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+<li>
+<p>An array of arrays</p>
+</li>
+<li>
+<p>An array of structures</p>
+</li>
+<li>
+<p>A structure containing an array</p>
+</li>
+<li>
+<p>A structure containing a structure</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Fragment shader inputs that are, or contain, integral
+types must be
+qualified with the interpolation qualifier <strong>flat</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Fragment inputs are declared as in the following examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec3 normal;
+centroid in vec2 TexCoord;
+flat in vec3 myColor;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The fragment shader inputs form an interface with the last active shader in
+the vertex processing pipeline.
+For this interface, the last active shader stage output variables and
+fragment shader input variables of the same name must match in type and
+qualification, with a few exceptions: The storage qualifiers must, of
+course, differ (one is <strong>in</strong> and one is <strong>out</strong>).
+Also,
+auxiliary qualification (e.g. <strong>centroid</strong>) may differ.
+When
+auxiliary qualifiers do not match, those provided in
+the fragment shader supersede those provided in previous stages.
+If any such qualifiers are completely missing in the fragment shaders, then
+the default is used, rather than any qualifiers that may have been declared
+in previous stages.
+That is, what matters is what is declared in the fragment shaders, not what
+is declared in shaders in previous stages.</p>
+</div>
+<div class="paragraph">
+<p>When an interface between shader stages is formed using shaders from two
+separate program objects, it is not possible to detect mismatches between
+inputs and outputs when the programs are linked.
+When there are mismatches between inputs and outputs on such interfaces,
+attempting to use the two programs in the same program pipeline will result
+in program pipeline validation failures, as described in section 7.4.1
+“Shader Interface Matching” of the <a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>Shaders can ensure matches across such interfaces either by using input and
+output layout qualifiers (sections “<a href="#input-layout-qualifiers">Input Layout
+Qualifiers</a>” and “<a href="#output-layout-qualifiers">Output Layout
+Qualifiers</a>”) or by using identical input and output declarations of
+blocks or variables.
+Complete rules for interface matching are found in section 7.4.1 “Shader
+Interface Matching” of the <a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>Compute shaders do not permit user-defined input variables and do not form a
+formal interface with any other shader stage.
+See “<a href="#compute-shader-special-variables">Compute Shader Special
+Variables</a>” for a description of built-in compute shader input variables.
+All other input to a compute shader is retrieved explicitly through image
+loads, texture fetches, loads from uniforms or uniform buffers, or other
+user supplied code.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="uniform-variables">4.3.5. Uniform Variables</h4>
+<div class="paragraph">
+<p>The <strong>uniform</strong> qualifier is used to declare global variables whose values are
+the same across the entire primitive being processed.
+All <strong>uniform</strong> variables are read-only.
+Except for variables declared within a uniform block, all uniform variables
+are initialized to 0 at link time and may be updated through the API.</p>
+</div>
+<div class="paragraph">
+<p>Example declarations are:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">uniform vec4 lightPosition;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>uniform</strong> qualifier can be used with any of the basic data types, or
+when declaring a variable whose type is a structure, or an array of any of
+these.</p>
+</div>
+<div class="paragraph">
+<p>There is an implementation-dependent limit on the amount of storage for
+uniforms that can be used for each type of shader and if this is exceeded it
+will cause a compile-time or link-time error.
+Uniform variables that are declared but not
+statically
+used do not count against this limit.
+The number of user-defined uniform variables and the number of built-in
+uniform variables that are used within a shader are added together to
+determine whether available uniform storage has been exceeded.</p>
+</div>
+<div class="paragraph">
+<p>Uniforms in shaders all share a single global name space when linked into a
+program or separable program.
+Hence, the types,
+precisions,
+and any location specifiers of all statically used uniform variables with the
+same name must match across all shaders that are linked into a single program.
+However it is not required to repeat the
+location specifier in all the linked shaders.
+While this single uniform name space is cross stage, a uniform variable
+name’s scope is per stage: If a uniform variable name is declared in one
+stage (e.g. a vertex shader) but not in another (e.g. a fragment shader),
+then that name is still available in the other stage for a different use.</p>
+</div>
+<div class="paragraph">
+<p>A compile or link-time error is generated if any of the explicitly given or
+compiler generated uniform locations is greater than the
+implementation-defined maximum number of uniform locations minus one.</p>
+</div>
+<div class="paragraph">
+<p>Unlike locations for inputs and outputs, uniform locations are logical
+values, not register locations, and there is no concept of overlap.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">2</span>) uniform mat4 x;
+layout(location = <span class="integer">3</span>) uniform mat4 y; <span class="comment">// No overlap with x</span>
+layout(location = <span class="integer">2</span>) in mat4 x;
+layout(location = <span class="integer">3</span>) in mat4 y; <span class="comment">// Error, locations conflict with x</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="output-variables">4.3.6. Output Variables</h4>
+<div class="paragraph">
+<p>Shader output variables are declared with the <strong>out</strong> storage qualifier.
+They form the output interface between the declaring shader and the
+subsequent stages of the OpenGL ES pipeline.
+Output variables must be declared at global scope.
+During shader execution they will behave as normal unqualified global
+variables.
+Their values are copied out to the subsequent pipeline stage on shader exit.
+Only output variables that are read by the subsequent pipeline stage need to
+be written; it is allowed to have superfluous declarations of output
+variables.</p>
+</div>
+<div class="paragraph">
+<p>There is <em>not</em> an <strong>inout</strong> storage qualifier for declaring a single variable
+name as both input and output to a shader.
+Also, a variable cannot be declared with both the <strong>in</strong> and the <strong>out</strong>
+qualifiers, this will result in a compile-time or link-time error.
+Output variables must be declared with different names than input variables.
+However, nesting an input or output inside an interface block with an
+instance name allows the same names with one referenced through a block
+instance name.</p>
+</div>
+<div class="paragraph">
+<p>Vertex, tessellation evaluation, and geometry output variables output
+per-vertex data and are declared using the <strong>out</strong> storage qualifier.
+Applying <strong>patch</strong> to an output can only be done in a tessellation control
+shader.
+It is a compile-time error to use <strong>patch</strong> on outputs in any other stage.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to declare a vertex, tessellation evaluation,
+tessellation control, or geometry shader output with, or that contains, any
+of the following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+<li>
+<p>A structure containing an array</p>
+</li>
+<li>
+<p>A structure containing a structure</p>
+</li>
+<li>
+<p>For per-vertex-arrayed variables (applies to tessellation control,
+tessellation evaluation and geometry shaders):</p>
+<div class="ulist">
+<ul>
+<li>
+<p>Per-vertex-arrayed arrays of arrays</p>
+</li>
+<li>
+<p>Per-vertex-arrayed arrays of structures</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>For non-per-vertex-arrayed variables:</p>
+<div class="ulist">
+<ul>
+<li>
+<p>An array of arrays</p>
+</li>
+<li>
+<p>An array of structures</p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Vertex shader outputs may be qualified with the interpolation qualifier
+<strong>flat</strong><sup>1</sup>.</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">1</dt>
+<dd>
+<p>Unlike previous versions of the OpenGL ES Shading Language, there is no requirement for
+outputs containing integers to be qualified as <strong>flat</strong>, since the vertex
+shader may interface with the tessellation control shader.
+However, in all cases, the qualifier must match across interfaces.</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>Individual outputs are declared as in the following examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out vec3 normal;
+centroid out vec2 TexCoord;
+invariant centroid out vec4 Color;
+flat out vec3 myColor;
+sample out vec4 perSampleColor;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>These can also appear in interface blocks, as described in
+“<a href="#interface-blocks">Interface Blocks</a>”.
+Interface blocks allow simpler addition of arrays to the interface from
+vertex to geometry shader.
+They also allow a fragment shader to have the same input interface as a
+geometry shader for a given vertex shader.</p>
+</div>
+<div class="paragraph">
+<p>Tessellation control shader output variables are used to output
+per-vertex and per-patch data.
+Per-vertex output variables are arrayed (see <em>arrayed</em> under
+“<a href="#input-variables">Input Variables</a>”) and declared using the <strong>out</strong>
+qualifier without the <strong>patch</strong> qualifier.
+Per-patch output variables are declared using the <strong>patch</strong> and <strong>out</strong>
+qualifiers.</p>
+</div>
+<div class="paragraph">
+<p>Since tessellation control shaders produce an arrayed primitive comprising
+multiple vertices, each per-vertex output variable (or output block, see
+interface blocks below) needs to be declared as an array.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out <span class="predefined-type">float</span> foo[]; <span class="comment">// feeds next stage input "in float foo[]"</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Each element of such an array corresponds to one vertex of the primitive
+being produced.
+Each array can optionally have a size declared.
+The array size will be set by (or if provided must be consistent with) the
+output layout declaration(s) establishing the number of vertices in the
+output patch, as described later in
+“<a href="#tessellation-control-outputs">Tessellation Control Outputs</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Each tessellation control shader invocation has a corresponding output patch
+vertex, and may assign values to per-vertex outputs only if they belong to
+that corresponding vertex.
+If a per-vertex output variable is used as an l-value, it is a compile-time
+or link-time error if the expression indicating the vertex index is not the
+identifier <em>gl_InvocationID</em>.</p>
+</div>
+<div class="paragraph">
+<p>The order of execution of a tessellation control shader invocation relative
+to the other invocations for the same input patch is undefined unless the
+built-in function <strong>barrier</strong>() is used.
+This provides some control over relative execution order.
+When a shader invocation calls <strong>barrier</strong>(), its execution pauses until all
+other invocations have reached the same point of execution.
+Output variable assignments performed by any invocation executed prior to
+calling <strong>barrier</strong>() will be visible to any other invocation after the call
+to <strong>barrier</strong>() returns.</p>
+</div>
+<div class="paragraph">
+<p>Because tessellation control shader invocations execute in undefined order
+between barriers, the values of per-vertex or per-patch output variables
+will sometimes be undefined.
+Consider the beginning and end of shader execution and each call to
+<strong>barrier</strong>() as synchronization points.
+The value of an output variable will be undefined in any of the three
+following cases:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>At the beginning of execution.</p>
+</li>
+<li>
+<p>At each synchronization point, unless</p>
+<div class="openblock">
+<div class="content">
+<div class="ulist">
+<ul>
+<li>
+<p>the value was well-defined after the previous synchronization point and
+was not written by any invocation since, or</p>
+</li>
+<li>
+<p>the value was written by exactly one shader invocation since the previous
+synchronization point, or</p>
+</li>
+<li>
+<p>the value was written by multiple shader invocations since the previous
+synchronization point, and the last write performed by all such
+invocations wrote the same value.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</li>
+<li>
+<p>When read by a shader invocation, if</p>
+<div class="openblock">
+<div class="content">
+<div class="ulist">
+<ul>
+<li>
+<p>the value was undefined at the previous synchronization point and has not
+been written by the same shader invocation since, or</p>
+</li>
+<li>
+<p>the output variable is written to by any other shader invocation between
+the previous and next synchronization points, even if that assignment
+occurs in code following the read.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Fragment outputs output per-fragment data and are declared using the <strong>out</strong>
+storage qualifier.
+It is an error to use auxiliary storage qualifiers or
+interpolation qualifiers in a fragment shader output declaration.
+It is a compile-time error to declare a fragment shader output with, or that
+contains, any of the following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+<li>
+<p>A matrix type</p>
+</li>
+<li>
+<p>A structure</p>
+</li>
+<li>
+<p>An array of arrays</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Fragment shader outputs declared as arrays may only be indexed by a constant
+integral expression.</p>
+</div>
+<div class="paragraph">
+<p>Fragment outputs are declared as in the following examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out vec4 FragmentColor;
+out uint Luminosity;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Compute shaders have no built-in output variables, do not support
+user-defined output variables and do not form a formal interface with any
+other shader stage.
+All outputs from a compute shader take the form of the side effects such as
+image stores and operations on atomic counters.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="buffer-variables">4.3.7. Buffer Variables</h4>
+<div class="paragraph">
+<p>The <strong>buffer</strong> qualifier is used to declare global variables whose values are
+stored in the data store of a buffer object bound through the OpenGL ES API.
+Buffer variables can be read and written, with the underlying storage shared
+among all active shader invocations.
+Buffer variable memory reads and writes within a single shader invocation
+are processed in order.
+However, the order of reads and writes performed in one invocation relative
+to those performed by another invocation is largely undefined.
+Buffer variables may be qualified with memory qualifiers affecting how the
+underlying memory is accessed, as described in “<a href="#memory-qualifiers">Memory
+Qualifiers</a>”.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>buffer</strong> qualifier can be used to declare interface blocks (see
+“<a href="#interface-blocks">Interface Blocks</a>”), which are then referred to as
+shader storage blocks.
+It is a compile-time error to declare buffer variables outside a block.
+Buffer variables cannot have initializers.</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="title">Note</div>
+<div class="paragraph">
+<p>The terms <em>shader storage buffer (object)</em>, and <em>shader storage block</em> are
+often used interchangeably.
+The former generally refers to the underlying storage whereas the latter
+refers only to the interface definition in the shader.
+Similarly for <em>uniform buffer objects</em> and _uniform blocks.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// use buffer to create a buffer block (shader storage block)</span>
+buffer BufferName { <span class="comment">// externally visible name of buffer</span>
+ <span class="predefined-type">int</span> count; <span class="comment">// typed, shared memory...</span>
+ ... <span class="comment">// ...</span>
+ vec4 v[]; <span class="comment">// last member may be an array that is not sized</span>
+ <span class="comment">// until after link time (dynamically sized)</span>
+} Name; <span class="comment">// name of block within the shader</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>There are implementation-dependent limits on the number of shader storage
+blocks used for each type of shader, the combined number of shader storage
+blocks used for a program, and the amount of storage required by each
+individual shader storage block.
+If any of these limits are exceeded, it will cause a compile-time or
+link-time error.</p>
+</div>
+<div class="paragraph">
+<p>If multiple shaders are linked together, then they will share a single
+global buffer variable name space.
+Hence, the types of all declared buffer variables with the same name must
+match across all shaders that are linked into a single program.</p>
+</div>
+<div class="paragraph">
+<p>Precision qualifiers for such variables need not match.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="shared-variables">4.3.8. Shared Variables</h4>
+<div class="paragraph">
+<p>The <strong>shared</strong> qualifier is used to declare global variables that have storage
+shared between all work items in a compute shader workgroup.
+Variables declared as <strong>shared</strong> may only be used in compute shaders (see
+“<a href="#compute-processor">Compute Processor</a>”).
+Any other declaration of a <strong>shared</strong> variable is an error.
+Shared variables are implicitly coherent (see
+“<a href="#memory-qualifiers">Memory Qualifiers</a>”).</p>
+</div>
+<div class="paragraph">
+<p>Variables declared as <strong>shared</strong> may not have initializers and their contents
+are undefined at the beginning of shader execution.
+Any data written to <strong>shared</strong> variables will be visible to other work items
+(executing the same shader) within the same workgroup.</p>
+</div>
+<div class="paragraph">
+<p>In the absence of synchronization, the order of reads and writes to the same
+<strong>shared</strong> variable by different invocations of a shader is not defined.</p>
+</div>
+<div class="paragraph">
+<p>In order to achieve ordering with respect to reads and writes to <strong>shared</strong>
+variables, control flow barriers must be employed using the <strong>barrier</strong>() function
+(see “<a href="#shader-invocation-control-functions">Shader Invocation Control
+Functions</a>”).</p>
+</div>
+<div class="paragraph">
+<p>There is a limit to the total size of all variables declared as <strong>shared</strong> in a
+single program.
+This limit, expressed in units of basic machine units may be determined by
+using the OpenGL ES API to query the value of
+MAX_COMPUTE_SHARED_MEMORY_SIZE.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="interface-blocks">4.3.9. Interface Blocks</h4>
+<div class="paragraph">
+<p>Input, output, uniform, and buffer variable declarations can be grouped into
+named interface blocks to provide coarser granularity backing than is
+achievable with individual declarations.
+They can have an optional instance name, used in the shader to reference
+their members.
+An output block of one programmable stage is backed by a corresponding input
+block in the subsequent programmable stage.
+A <em>uniform block</em> is backed by the application with a buffer object.
+A block of buffer variables, called a <em>shader storage block</em>, is also backed
+by the application with a buffer object.
+It is a compile-time error to have an input block in a vertex shader or an
+output block in a fragment shader.
+These uses are reserved for future use.</p>
+</div>
+<div class="paragraph">
+<p>An interface block is started by an <strong>in</strong>, <strong>out</strong>, <strong>uniform</strong>, or <strong>buffer</strong>
+keyword, followed by a block name, followed by an open curly brace (<strong>{</strong>) as
+follows:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>interface-block</em> : </dt>
+<dd>
+<p><em>layout-qualifier<sub>opt</sub></em> <em>interface-qualifier</em> <em>block-name</em> <strong>{</strong>
+<em>member-list</em> <strong>}</strong> <em>instance-name<sub>opt</sub></em> <strong>;</strong></p>
+</dd>
+</dl>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>interface-qualifier</em> : </dt>
+<dd>
+<p><strong>in</strong><br>
+<strong>out</strong><br>
+<strong>patch</strong> <strong>in</strong> // Note: Qualifiers can be in any order.<br>
+<strong>patch</strong> <strong>out</strong><br>
+<strong>uniform</strong><br>
+<strong>buffer</strong></p>
+</dd>
+</dl>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>member-list</em> : </dt>
+<dd>
+<p><em>member-declaration</em><br>
+<em>member-declaration</em> <em>member-list</em></p>
+</dd>
+<dt class="hdlist1"><em>member-declaration</em> : </dt>
+<dd>
+<p><em>layout-qualifier<sub>opt</sub></em> <em>qualifiers<sub>opt</sub></em> <em>type</em> <em>declarators</em> <strong>;</strong></p>
+</dd>
+</dl>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>instance-name</em> : </dt>
+<dd>
+<p><em>identifier</em><br>
+<em>identifier</em> <strong>[</strong> <strong>]</strong><br>
+<em>identifier</em> <strong>[</strong> <em>constant-integral-expression</em> <strong>]</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Each of the above elements is discussed below, with the exception of layout
+qualifiers (<em>layout-qualifier</em>), which are defined in the next section.</p>
+</div>
+<div class="paragraph">
+<p>First, an example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">uniform Transform {
+ mat4 ModelViewMatrix;
+ mat4 ModelViewProjectionMatrix;
+ uniform mat3 NormalMatrix; <span class="comment">// allowed restatement of qualifier</span>
+ <span class="predefined-type">float</span> Deformation;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The above establishes a uniform block named “Transform” with four uniforms
+grouped inside it.</p>
+</div>
+<div class="paragraph">
+<p>Types and declarators are the same as for other input, output, uniform, and
+buffer variable declarations outside blocks, with these exceptions:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Opaque types are not allowed</p>
+</li>
+<li>
+<p>Structure definitions cannot be nested inside a block</p>
+</li>
+<li>
+<p>Arrays of arrays of blocks are not allowed, except for the case in the
+tessellation pipe where the declaration is a per-vertex-array of arrays
+of blocks.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>If no optional qualifier is used in a member-declaration, the qualification
+of the member includes all <strong>in</strong>, <strong>out</strong>, <strong>patch</strong>, <strong>uniform</strong>, or <strong>buffer</strong> as
+determined by <em>interface-qualifier</em>.
+If optional qualifiers are used, they can include interpolation qualifiers,
+auxiliary storage qualifiers,
+precision qualifiers,
+and storage qualifiers and they must declare
+an input, output, or uniform member consistent with the interface qualifier
+of the block: Input variables, output variables, uniform variables, and
+<strong>buffer</strong> members can only be in <strong>in</strong> blocks, <strong>out</strong> blocks, <strong>uniform</strong> blocks,
+and shader storage blocks, respectively.</p>
+</div>
+<div class="paragraph">
+<p>Repeating the <strong>in</strong>, <strong>out</strong>, <strong>patch</strong>, <strong>uniform</strong>, or <strong>buffer</strong> interface
+qualifier for a member’s storage qualifier is optional.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in Material {
+ smooth in vec4 Color1; <span class="comment">// legal, input inside in block</span>
+ smooth vec4 Color2; <span class="comment">// legal, 'in' inherited from 'in Material'</span>
+ vec2 TexCoord; <span class="comment">// legal, TexCoord is an input</span>
+ uniform <span class="predefined-type">float</span> Atten; <span class="comment">// illegal, mismatched storage qualifier</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A <em>shader interface</em> is defined to be one of these:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>All the uniform variables and uniform blocks declared in a program.
+This spans all compilation units linked together within one program.</p>
+</li>
+<li>
+<p>All the <strong>buffer</strong> blocks declared in a program.</p>
+</li>
+<li>
+<p>The boundary between adjacent programmable pipeline stages: This spans
+all the outputs declared in all compilation units of the first stage and
+all the inputs declared in all compilation units of the second stage.
+Note that for the purposes of this definition, the fragment shader and
+the preceding shader are considered to have a shared boundary even
+though in practice, all values passed to the fragment shader first pass
+through the rasterizer and interpolator.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>The block name (<em>block-name</em>) is used to match within shader interfaces: an
+output block of one pipeline stage will be matched to an input block with
+the same name in the subsequent pipeline stage.
+For uniform or shader storage blocks, the application uses the block name to
+identify the block.
+Block names have no other use within a shader beyond interface matching; it
+is an error
+to use a block name at global scope for anything other than as a
+block name (e.g. use of a block name for a global variable name or function
+name is currently reserved).
+It is a compile-time error to use the same block name for more than one
+block declaration in the same shader interface (as defined above) within one
+shader, even if the block contents are identical.</p>
+</div>
+<div class="paragraph">
+<p>Matched block names within a shader interface (as defined above) must match
+in terms of having the same number of declarations with the same sequence of
+types and the same sequence of member names, as well as having matching
+member-wise layout qualification
+as defined in “<a href="#matching-of-qualifiers">Matching of Qualifiers</a>”.
+Matched uniform or shader storage block names (but not input or output block
+names) must also either all be lacking an instance name or all having an
+instance name, putting their members at the same scoping level.
+When instance names are present on matched block names, it is allowed for
+the instance names to differ; they need not match for the blocks to match.
+Furthermore, if a matching block is declared as an array, then the array
+sizes must also match (or follow array matching rules for the shader
+interface between consecutive shader stages).
+Any mismatch will generate a link-time error.
+A block name is allowed to have different definitions in different shader
+interfaces within the same shader, allowing, for example, an input block and
+output block to have the same name.</p>
+</div>
+<div class="paragraph">
+<p>If an instance name (<em>instance-name</em>) is not used, the names declared inside
+the block are scoped at the global level and accessed as if they were
+declared outside the block.
+If an instance name (<em>instance-name</em>) is used, then it puts all the members
+inside a scope within its own name space, accessed with the field selector
+(<strong>.</strong>) operator (analogously to structures).
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in Light {
+ vec4 LightPos;
+ vec3 LightColor;
+};
+in ColoredTexture {
+ vec4 Color;
+ vec2 TexCoord;
+} Material; <span class="comment">// instance name</span>
+vec3 Color; <span class="comment">// different Color than Material.Color</span>
+vec4 LightPos; <span class="comment">// illegal, already defined</span>
+...
+... = LightPos; <span class="comment">// accessing LightPos</span>
+... = Material.Color; <span class="comment">// accessing Color in ColoredTexture block</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Outside the shading language (i.e., in the API), members are similarly
+identified except the block name is always used in place of the instance
+name (API accesses are to shader interfaces, not to shaders).
+If there is no instance name, then the API does not use the block name to
+access a member, just the member name.</p>
+</div>
+<div class="paragraph">
+<p>Within a shader interface, all declarations of the same global name must be
+for the same object and must match in type and in whether they declare a
+variable or member of a block with no instance name.
+The API also needs this name to uniquely identify an object in the shader
+interface.
+It is a link-time error if any particular shader interface contains</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>two different blocks, each having no instance name, and each having a
+member of the same name, or</p>
+</li>
+<li>
+<p>a variable outside a block, and a block with no instance name, where the
+variable has the same name as a member in the block.</p>
+</li>
+</ul>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out Vertex {
+ vec4 Position; <span class="comment">// API transform/feedback will use "Vertex.Position"</span>
+ vec2 Texture;
+} Coords; <span class="comment">// shader will use "Coords.Position"</span>
+out Vertex2 {
+ vec4 Color; <span class="comment">// API will use "Color"</span>
+ <span class="predefined-type">float</span> Color2;
+};
+
+<span class="comment">// in same program as Vertex2 above:</span>
+out Vertex3 {
+ <span class="predefined-type">float</span> Intensity;
+ vec4 Color; <span class="comment">// ERROR, name collision with Color in Vertex2</span>
+};
+<span class="predefined-type">float</span> Color2; <span class="comment">// ERROR, collides with Color2 in Vertex2</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For blocks declared as arrays, the array index must also be included when
+accessing members, as in this example</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">uniform Transform { <span class="comment">// API uses "Transform[2]" to refer to instance 2</span>
+ mat4 ModelViewMatrix;
+ mat4 ModelViewProjectionMatrix;
+ <span class="predefined-type">float</span> Deformation;
+} transforms[<span class="integer">4</span>];
+...
+... = transforms[<span class="integer">2</span>].ModelViewMatrix; <span class="comment">// shader access of instance 2</span>
+<span class="comment">// API uses "Transform.ModelViewMatrix" to query an offset or other query</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For uniform or shader storage blocks declared as an array, each individual
+array element corresponds to a separate buffer object bind range, backing
+one instance of the block.
+As the array size indicates the number of buffer objects needed, uniform and
+shader storage block array declarations must specify an array size.
+All indices used to index a shader storage block array must be constant
+integral expressions.
+A uniform block array can only be indexed with a dynamically uniform
+integral expression, otherwise results are undefined.</p>
+</div>
+<div class="paragraph">
+<p>When using OpenGL ES API entry points to identify the name of an individual
+block in an array of blocks, the name string may include an array index
+(e.g. <em>Transform[2]</em>).
+When using OpenGL ES API entry points to refer to offsets or other
+characteristics of a block member, an array index must not be specified
+(e.g. <em>Transform.ModelViewMatrix</em>).
+See section 7.3.1 “Program Interfaces” of the <a href="#references">OpenGL ES Specification</a> for
+details.</p>
+</div>
+<div class="paragraph">
+<p>Tessellation control, tessellation evaluation and geometry shader input
+blocks must be declared as arrays and follow the array declaration and
+linking rules for all shader inputs for the respective stages.
+All other input and output block arrays must specify an array size.</p>
+</div>
+<div class="paragraph">
+<p>There are implementation-dependent limits on the number of uniform blocks
+and the number of shader storage blocks that can be used per stage.
+If either limit is exceeded, it will cause a link-time error.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="layout-qualifiers">4.4. Layout Qualifiers</h3>
+<div class="paragraph">
+<p>Layout qualifiers can appear in several forms of declaration.
+They can appear as part of an interface block definition or block member, as
+shown in the grammar in the previous section.
+They can also appear with just an <em>interface-qualifier</em> to establish layouts
+of other declarations made with that qualifier:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="paragraph">
+<p><em>layout-qualifier</em> <em>interface-qualifier</em> <strong>;</strong></p>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Or, they can appear with an individual variable declared with an interface
+qualifier:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="paragraph">
+<p><em>layout-qualifier</em> <em>interface-qualifier</em> <em>declaration</em> <strong>;</strong></p>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Declarations of layouts can only be made at global scope or block members,
+and only where indicated in the following subsections; their details are
+specific to what the interface qualifier is, and are discussed individually.</p>
+</div>
+<div class="paragraph">
+<p>The <em>layout-qualifier</em> expands to:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier</em> : </dt>
+<dd>
+<p><strong>layout</strong> <strong>(</strong> <em>layout-qualifier-id-list</em> <strong>)</strong></p>
+</dd>
+<dt class="hdlist1"><em>layout-qualifier-id-list</em> : </dt>
+<dd>
+<p><em>layout-qualifier-id</em><br>
+<em>layout-qualifier-id</em> <strong>,</strong> <em>layout-qualifier-id-list</em></p>
+</dd>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><em>layout-qualifier-name</em><br>
+<em>layout-qualifier-name</em> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>shared</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-value</em> : </dt>
+<dd>
+<p><em>integer-constant</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The tokens used for <em>layout-qualifier-name</em> are identifiers, not keywords,
+however, the <strong>shared</strong> keyword is allowed as a <em>layout-qualifier-id</em>.
+Generally, they can be listed in any order.
+Order-dependent meanings exist only if explicitly called out below.
+As for other identifiers, they are case sensitive.</p>
+</div>
+<div class="paragraph">
+<p>The set of allowed layout qualifiers depends on the shader, the interface
+and the variable type as specified in the following sections.
+For example, a sampler in the default uniform block in a fragment shader can
+have <strong>location</strong> and <strong>binding</strong> layout qualifiers but no others.
+Invalid use of layout qualifiers is an error.</p>
+</div>
+<div class="paragraph">
+<p>The following table summarizes the use of layout qualifiers.
+It shows for each one what kinds of declarations it may be applied to.
+These are all discussed in detail in the following sections.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Layout Qualifier</th>
+<th class="tableblock halign-left valign-top">Qualifier Only</th>
+<th class="tableblock halign-left valign-top">Individual Variable</th>
+<th class="tableblock halign-left valign-top">Block</th>
+<th class="tableblock halign-left valign-top">Block Member</th>
+<th class="tableblock halign-left valign-top">Allowed Interfaces</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>shared</strong><br>
+ <strong>packed</strong><br>
+ <strong>std140</strong><br>
+ <strong>std430</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top" rowspan="4"><p class="tableblock"><strong>uniform</strong> / <strong>buffer</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>row_major</strong><br>
+ <strong>column_major</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>binding</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">opaque types only</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>offset</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">atomic counters only</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>location</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uniform</strong> / <strong>buffer</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>location</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X<sup>1</sup></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">all <strong>in</strong> / <strong>out</strong>, except for compute</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>triangles</strong><br>
+ <strong>quads</strong><br>
+ <strong>isolines</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation evaluation <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>equal_spacing</strong><br>
+ <strong>fractional_even_spacing</strong><br>
+ <strong>fractional_odd_spacing</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation evaluation <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>cw</strong><br>
+ <strong>ccw</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation evaluation <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>point_mode</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation evaluation <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>points</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">geometry <strong>in</strong>/<strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">[ <strong>points</strong> ]<br>
+ <strong>lines</strong><br>
+ <strong>lines_adjacency</strong><br>
+ <strong>triangles</strong><br>
+ <strong>triangles_adjacency</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">geometry <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>invocations</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">geometry <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>early_fragment_tests</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">fragment <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>local_size_x</strong> =<br>
+ <strong>local_size_y</strong> =<br>
+ <strong>local_size_z</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">compute <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vertices</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation control <strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">[ <strong>points</strong> ]<br>
+ <strong>line_strip</strong><br>
+ <strong>triangle_strip</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">geometry <strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>max_vertices</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">geometry <strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>rgba32f</strong><br>
+ <strong>rgba16f</strong><br>
+ <strong>r32f</strong><br>
+ <strong>rgba8</strong><br>
+ <strong>rgba8_snorm</strong><br>
+ <strong>rgba32i</strong><br>
+ <strong>rgba16i</strong><br>
+ <strong>rgba8i</strong><br>
+ <strong>r32i</strong><br>
+ <strong>rgba32ui</strong><br>
+ <strong>rgba16ui</strong><br>
+ <strong>rgba8ui</strong><br>
+ <strong>r32ui</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">image types only</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uniform</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_multiply</strong><br>
+ <strong>blend_support_screen</strong><br>
+ <strong>blend_support_overlay</strong><br>
+ <strong>blend_support_darken</strong><br>
+ <strong>blend_support_lighten</strong><br>
+ <strong>blend_support_colordodge blend_support_colorburn</strong><br>
+ <strong>blend_support_hardlight</strong><br>
+ <strong>blend_support_softlight</strong><br>
+ <strong>blend_support_difference</strong><br>
+ <strong>blend_support_exclusion</strong><br>
+ <strong>blend_support_hsl_hue</strong><br>
+ <strong>blend_support_hsl_saturation</strong><br>
+ <strong>blend_support_hsl_color blend_support_hsl_luminosity</strong><br>
+ <strong>blend_support_all_equations</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">fragment <strong>out</strong></p></td>
+</tr>
+</tbody>
+</table>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">1</dt>
+<dd>
+<p>Location qualifiers are not allowed for members of an arrayed block,
+except for per-vertex-arrays (see “<a href="#interface-blocks">Interface
+Blocks</a>”).</p>
+</dd>
+</dl>
+</div>
+<div class="sect3">
+<h4 id="input-layout-qualifiers">4.4.1. Input Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Layout qualifiers specific to a particular shader language are discussed in
+separate sections below.</p>
+</div>
+<div class="paragraph">
+<p>All shaders except compute shaders allow <strong>location</strong> layout qualifiers on
+input variable declarations, input block declarations, and input block
+member declarations.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>location</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">3</span>) in vec4 normal;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the shader input <em>normal</em> is assigned to vector location
+number
+3.
+For vertex shader inputs, the location specifies the number of the
+vertex attribute from which input values are taken.
+For inputs of all other shader types, the location specifies a vector number
+that can be used to match against outputs from a previous shader stage, even
+if that shader is in a different program object.</p>
+</div>
+<div class="paragraph">
+<p>The following language describes how many locations are consumed by a given
+type.
+However, geometry shader inputs, tessellation control shader inputs and
+outputs, and tessellation evaluation inputs all have an additional level of
+arrayness relative to other shader inputs and outputs.
+This outer array level is removed from the type before considering how many
+locations the type consumes.</p>
+</div>
+<div class="paragraph">
+<p>If a shader input is any scalar or vector type, it will consume a single
+location.</p>
+</div>
+<div class="paragraph">
+<p>If the declared input (after potentially removing an outer array level as
+just described above) is an array of size <em>n</em> and each of the elements takes
+<em>m</em> locations, it will be assigned <em>m</em> * <em>n</em> consecutive locations starting
+with the location specified.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">6</span>) in vec4 colors[<span class="integer">3</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the shader input <em>colors</em> is assigned to vector location
+numbers 6, 7, and 8.</p>
+</div>
+<div class="paragraph">
+<p>If the declared input is an <em>n</em> × <em>m</em>
+matrix, it will be assigned multiple locations starting with the location
+specified.
+The number of locations assigned for each matrix will be the same as for an
+<em>n</em>-element array of <em>m</em>-component vectors.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">9</span>) in mat4 transforms[<span class="integer">2</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that shader input <em>transforms</em> is assigned to vector
+locations 9-16, with <em>transforms[0]</em> being assigned to locations 9-12, and
+<em>transforms[1]</em> being assigned to locations 13-16.</p>
+</div>
+<div class="paragraph">
+<p>If the declared input is a structure or block, its members will be assigned
+consecutive locations in their order of declaration, with the first member
+assigned the location provided in the layout qualifier.
+For a structure, this process applies to the entire structure.
+It is a compile-time error to use a <strong>location</strong> qualifier on a member of a
+structure.
+For a block, this process applies to the entire block, or until the first
+member is reached that has a <strong>location</strong> layout qualifier.</p>
+</div>
+<div class="paragraph">
+<p>When a block member is declared with a <strong>location</strong> qualifier, its location
+comes from that qualifier; the member’s <strong>location</strong> qualifier overrides the
+block-level declaration.
+Subsequent members are again assigned consecutive locations, based on the
+newest location, until the next member declared with a <strong>location</strong> qualifier.
+The values used for locations do not have to be declared in increasing
+order.</p>
+</div>
+<div class="paragraph">
+<p>If a block has no block-level <strong>location</strong> layout qualifier, it is required
+that either all or none of its members have a <strong>location</strong> layout qualifier,
+or a compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>If an input is declared as an array of blocks, excluding per-vertex-arrays
+as required for tessellation, it is an error to declare a member of the
+block with a <strong>location</strong> qualifier.</p>
+</div>
+<div class="paragraph">
+<p>The locations consumed by block and structure members are determined by
+applying the rules above recursively as though the structure member were
+declared as an input variable of the same type.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">3</span>) in <span class="keyword">struct</span> S
+{
+ vec3 a; <span class="comment">// gets location 3</span>
+ mat2 b; <span class="comment">// gets locations 4 and 5</span>
+ vec4 c[<span class="integer">2</span>]; <span class="comment">// gets locations 6 and 7</span>
+ layout(location = <span class="integer">8</span>) vec2 A; <span class="comment">// ERROR, can't use on struct member</span>
+} s;
+layout(location = <span class="integer">4</span>) in block
+{
+ vec4 d; <span class="comment">// gets location 4</span>
+ vec4 e; <span class="comment">// gets location 5</span>
+ layout(location = <span class="integer">7</span>) vec4 f; <span class="comment">// gets location 7</span>
+ vec4 g; <span class="comment">// gets location 8</span>
+ layout(location = <span class="integer">1</span>) vec4 h; <span class="comment">// gets location 1</span>
+ vec4 i; <span class="comment">// gets location 2</span>
+ vec4 j; <span class="comment">// gets location 3</span>
+ vec4 k; <span class="comment">// ERROR, location 4 already used</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The number of input locations available to a shader is limited.
+For vertex shaders, the limit is the advertised number of vertex attributes.
+For all other shaders, the limit is implementation-dependent and must be no
+less than one fourth of the advertised maximum input component count.</p>
+</div>
+<div class="paragraph">
+<p>A program will fail to link if any attached shader uses a location greater
+than or equal to the number of supported locations, unless device-dependent
+optimizations are able to make the program fit within available hardware
+resources.</p>
+</div>
+<div class="paragraph">
+<p>A program will fail to link if explicit location assignments leave the
+linker unable to find space for other variables without explicit
+assignments.</p>
+</div>
+<div class="paragraph">
+<p>For the purposes of determining if a non-vertex input matches an output from
+a previous shader stage, the <strong>location</strong> layout qualifier (if any) must
+match.</p>
+</div>
+<div class="paragraph">
+<p>If a vertex shader input variable with no location assigned in the shader
+text has a location specified through the OpenGL ES API, the API-assigned
+location will be used.
+Otherwise, such variables will be assigned a location by the linker.
+See section 11.1.1 “Vertex Attributes” of the <a href="#references">OpenGL ES Specification</a> for
+more details.</p>
+</div>
+<div class="paragraph">
+<p>It is an error if more than one input or element of a matrix input is bound
+to the same location.</p>
+</div>
+<div class="sect4">
+<h5 id="tessellation-evaluation-inputs">Tessellation Evaluation Inputs</h5>
+<div class="paragraph">
+<p>Additional input layout qualifier identifiers allowed for tessellation
+evaluation shaders are described below.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><em>primitive_mode</em><br>
+<em>vertex_spacing</em><br>
+<em>ordering</em><br>
+<em>point_mode</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>primitive-mode</strong> is used to specify a tessellation primitive mode to be
+used by the tessellation primitive generator.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>primitive-mode</em>: </dt>
+<dd>
+<p><strong>triangles</strong><br>
+<strong>quads</strong><br>
+<strong>isolines</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>If present, the <em>primitive-mode</em> specifies that the tessellation primitive
+generator should subdivide a triangle into smaller triangles, a quad into
+triangles, or a quad into a collection of lines, respectively.</p>
+</div>
+<div class="paragraph">
+<p>A second group of layout identifiers, <em>vertex spacing</em>, is used to specify
+the spacing used by the tessellation primitive generator when subdividing an
+edge.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>vertex-spacing</em>: </dt>
+<dd>
+<p><strong>equal_spacing</strong><br>
+<strong>fractional_even_spacing</strong><br>
+<strong>fractional_odd_spacing</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p><strong>equal_spacing</strong> specifies that edges should be divided into a collection of
+equal-sized segments;</p>
+</div>
+<div class="paragraph">
+<p><strong>fractional_even_spacing</strong> specifies that edges should be divided into an
+even number of equal-length segments plus two additional shorter
+“fractional” segments; or</p>
+</div>
+<div class="paragraph">
+<p><strong>fractional_odd_spacing</strong> specifies that edges should be divided into an odd
+number of equal-length segments plus two additional shorter “fractional”
+segments.</p>
+</div>
+<div class="paragraph">
+<p>A third group of layout identifiers, <em>ordering</em>, specifies whether the
+tessellation primitive generator produces triangles in clockwise or
+counter-clockwise order, according to the coordinate system depicted in the
+<a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>ordering</em>: </dt>
+<dd>
+<p><strong>cw</strong><br>
+<strong>ccw</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The identifiers <strong>cw</strong> and <strong>ccw</strong> indicate clockwise and counter-clockwise
+triangles, respectively.
+If the tessellation primitive generator does not produce triangles, the
+order is ignored.</p>
+</div>
+<div class="paragraph">
+<p>Finally, <em>point mode</em> indicates that the tessellation primitive generator
+should produce one point for each distinct vertex in the subdivided
+primitive, rather than generating lines or triangles.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>point-mode</em>: </dt>
+<dd>
+<p><strong>point_mode</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Any or all of these identifiers may be specified one or more times in a
+single input layout declaration.</p>
+</div>
+<div class="paragraph">
+<p>The tessellation evaluation shader object in a program must declare a
+primitive mode in its input layout.
+Declaring vertex spacing, ordering, or point mode identifiers is optional.
+If spacing or vertex ordering declarations are omitted, the tessellation
+primitive generator will use equal spacing or counter-clockwise vertex
+ordering, respectively.
+If a point mode declaration is omitted, the tessellation primitive generator
+will produce lines or triangles according to the primitive mode.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="geometry-shader-inputs">Geometry Shader Inputs</h5>
+<div class="paragraph">
+<p>Additional layout qualifier identifiers for geometry shader inputs include
+<em>primitive</em> identifiers and an <em>invocation count</em> identifier:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>points</strong><br>
+<strong>lines</strong><br>
+<strong>lines_adjacency</strong><br>
+<strong>triangles</strong><br>
+<strong>triangles_adjacency</strong><br>
+<strong>invocations</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The identifiers <strong>points</strong>, <strong>lines</strong>, <strong>lines_adjacency</strong>, <strong>triangles</strong>, and
+<strong>triangles_adjacency</strong> are used to specify the type of input primitive
+accepted by the geometry shader, and only one of these is accepted.
+The geometry shader must declare this input primitive layout.</p>
+</div>
+<div class="paragraph">
+<p>The identifier <strong>invocations</strong> is used to specify the number of times the
+geometry shader executable is invoked for each input primitive received.
+Invocation count declarations are optional.
+If no invocation count is declared in the geometry shader, it will be run
+once for each input primitive.
+If an invocation count is declared, all such declarations must specify the
+same count.
+If a shader specifies an invocation count greater than the
+implementation-dependent maximum, or less than or equal to zero,
+a compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(triangles, invocations = <span class="integer">6</span>) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that all inputs to the geometry shader are triangles and that
+the geometry shader executable is run six times for each triangle processed.</p>
+</div>
+<div class="paragraph">
+<p>All geometry shader input unsized array declarations will be sized by an
+earlier input primitive layout qualifier, when present, as per the following
+table.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Layout</th>
+<th class="tableblock halign-left valign-top">Size of Input Arrays</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>points</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lines</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lines_adjacency</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>triangles</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>triangles_adjacency</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">6</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The intrinsically declared input array <em>gl_in[]</em> will also be sized by any
+input primitive-layout declaration.
+Hence, the expression</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_in.length()</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will return the value from the table above.</p>
+</div>
+<div class="paragraph">
+<p>An input can be declared without an array size if there is a previous layout
+which specifies the size.
+For built-in inputs (e.g. <em>gl_in[]</em>), a layout must be declared before any
+use.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error if a layout declaration’s array size (from the
+table above) does not match all the explicit array sizes specified in
+declarations of an input variables in the same shader.
+The following includes examples of compile-time errors:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// code sequence within one shader...</span>
+in vec4 Color2[<span class="integer">2</span>]; <span class="comment">// legal, size is 2</span>
+in vec4 Color3[<span class="integer">3</span>]; <span class="comment">// illegal, input sizes are inconsistent</span>
+layout(lines) in; <span class="comment">// legal for Color2, input size is 2, matching Color2</span>
+in vec4 Color4[<span class="integer">3</span>]; <span class="comment">// illegal, contradicts layout of lines</span>
+layout(lines) in; <span class="comment">// legal, matches other layout() declaration</span>
+layout(triangles) in; <span class="comment">// illegal, does not match earlier layout() declaration</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It is a link-time error if not all provided sizes (sized input arrays and
+layout size) match in the geometry shader of a program.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="fragment-shader-inputs">Fragment Shader Inputs</h5>
+<div class="paragraph">
+<p>Fragment shaders allow the following layout qualifier on <strong>in</strong> only (not with
+variable declarations):</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>early_fragment_tests</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>to request that fragment tests be performed before fragment shader
+execution, as described in section 13.8.4 “Early Fragment Tests” of the
+<a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(early_fragment_tests) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Specifying this will make per-fragment tests be performed before fragment
+shader execution.
+In addition it is an error to statically write to <em>gl_FragDepth</em> in the
+fragment shader.
+If this is not declared, per-fragment tests will be performed after fragment
+shader execution.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="compute-shader-inputs">Compute Shader Inputs</h5>
+<div class="paragraph">
+<p>There are no layout location qualifiers for compute shader inputs.</p>
+</div>
+<div class="paragraph">
+<p>Layout qualifier identifiers for compute shader inputs are the workgroup
+size qualifiers:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>local_size_x</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>local_size_y</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>local_size_z</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>local_size_x</strong>, <strong>local_size_y</strong>, and <strong>local_size_z</strong> qualifiers are used
+to declare a fixed workgroup size by the compute shader in the first,
+second, and third dimension, respectively.
+If a shader does not specify a size for one of the dimensions, that
+dimension will have a size of 1.</p>
+</div>
+<div class="paragraph">
+<p>For example, the following declaration in a compute shader</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(local_size_x = <span class="integer">32</span>, local_size_y = <span class="integer">32</span>) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is used to declare a two-dimensional compute shader with a workgroup size of 32
+X 32 elements, which is equivalent to a three-dimensional compute shader
+where the third dimension has size one.</p>
+</div>
+<div class="paragraph">
+<p>As another example, the declaration</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(local_size_x = <span class="integer">8</span>) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>effectively specifies that a one-dimensional compute shader is being
+compiled, and its size is 8 elements.</p>
+</div>
+<div class="paragraph">
+<p>If the fixed workgroup size of the shader in any dimension is less than
+or equal to zero or greater than the maximum size supported by the
+implementation for that dimension, a compile-time error results.
+Also, if such a layout qualifier is declared more than once in the same
+shader, all those declarations must set the same set of workgroup
+sizes and set them to the same values; otherwise a compile-time error
+results.</p>
+</div>
+<div class="paragraph">
+<p>Furthermore, if a program object contains a compute shader, that shader must
+contain an input layout qualifier specifying a fixed workgroup size for
+the program, or a link-time error will occur.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="output-layout-qualifiers">4.4.2. Output Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Some output layout qualifiers apply to all shader stages and some apply only
+to specific stages.
+The latter are discussed in separate sections below.</p>
+</div>
+<div class="paragraph">
+<p>As with input layout qualifiers, all shaders except compute shaders allow
+<strong>location</strong> layout qualifiers on output variable declarations, output block
+declarations, and output block member declarations.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>location</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The usage and rules for applying the <strong>location</strong> qualifier
+to blocks and structures are exactly as described in
+“<a href="#input-layout-qualifiers">Input Layout Qualifiers</a>”.</p>
+</div>
+<div class="paragraph">
+<p>The qualifier may appear at most once within a declaration.
+For example, in a fragment shader,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">3</span>) out vec4 color;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the fragment shader output <em>color</em> is assigned to</p>
+</div>
+<div class="paragraph">
+<p>fragment color 3.</p>
+</div>
+<div class="paragraph">
+<p>For fragment shader outputs, the location
+specifies
+the color output number
+receiving the values of the output.
+For outputs of all other shader stages, the location specifies a vector
+number that can be used to match against inputs in a subsequent shader
+stage, even if that shader is in a different program object.</p>
+</div>
+<div class="paragraph">
+<p>Declared outputs of scalar or vector type consume a single location.</p>
+</div>
+<div class="paragraph">
+<p>If the declared output is an array, it will be assigned consecutive
+locations starting with the location specified.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">2</span>) out vec4 colors[<span class="integer">3</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that <em>colors</em> is assigned to vector location numbers 2, 3,
+and 4.</p>
+</div>
+<div class="paragraph">
+<p>If the declared output is an <em>n</em> × <em>m</em>
+matrix, it will be assigned multiple locations starting with the location
+specified.
+The number of locations assigned will be the same as for an <em>n</em>-element
+array of <em>m</em>-component vectors.</p>
+</div>
+<div class="paragraph">
+<p>If the declared output is a structure, its members will be assigned
+consecutive locations in the order of declaration, with the first member
+assigned the location specified for the structure.
+The number of locations consumed by a structure member is determined by
+applying the rules above recursively as though the structure member were
+declared as an output variable of the same type.</p>
+</div>
+<div class="paragraph">
+<p><strong>location</strong> layout qualifiers may be used on output variables declared as
+structures.
+However, it is a compile-time error to use a <strong>location</strong> qualifier on a
+structure member.
+Location layout qualifiers may be used on output blocks and output block
+members.</p>
+</div>
+<div class="paragraph">
+<p>If an output is declared as an array of blocks, excluding per-vertex-arrays
+as required for tessellation, it is an error to declare a member of the
+block with a <strong>location</strong> qualifier.</p>
+</div>
+<div class="paragraph">
+<p>The number of output locations available to a shader is limited.
+For fragment shaders, the limit is the advertised number of draw buffers.</p>
+</div>
+<div class="paragraph">
+<p>For all other shaders, the limit is implementation-dependent and must be no
+less than one fourth of the advertised maximum output component count
+(compute shaders have no outputs).
+A program will fail to link if any attached shader uses a location greater
+than or equal to the number of supported locations, unless device-dependent
+optimizations are able to make the program fit within available hardware
+resources.</p>
+</div>
+<div class="paragraph">
+<p>Compile-time errors may also be given if at compile time it is known the
+link will fail.
+A negative output location will result in an error.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time or link-time error if any of the following occur:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>any two fragment shader output variables are assigned to the same
+location.</p>
+</li>
+<li>
+<p>if any two output variables from the same vertex, tessellation or
+geometry shader stage are assigned to the same location.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>For all shader types, a program will fail to link if explicit location
+assignments leave the linker unable to find space for other variables
+without explicit assignments.</p>
+</div>
+<div class="paragraph">
+<p>If an output variable has no location assigned in the shader text, it will
+be assigned a location by the linker.
+See section 11.1.3 “Shader Execution” of the <a href="#references">OpenGL ES Specification</a> for
+more details.</p>
+</div>
+<div class="paragraph">
+<p>If there is only a single output, the location does not need to be
+specified, in which case it defaults to zero.
+This applies for all output types, including arrays.
+If there is more than one fragment output, the location must be specified
+for all outputs.</p>
+</div>
+<div class="paragraph">
+<p>For the purposes of determining if a non-fragment output matches an input
+from a subsequent shader stage, the <strong>location</strong> layout qualifier (if any)
+must match.</p>
+</div>
+<div class="sect4">
+<h5 id="tessellation-control-outputs">Tessellation Control Outputs</h5>
+<div class="paragraph">
+<p>Tessellation control shaders allow output layout qualifiers only on the
+interface
+qualifier <strong>out</strong>, not on an output block, block member, or variable
+declaration.
+The output layout qualifier identifiers allowed for tessellation control
+shaders are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>vertices</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The identifier <strong>vertices</strong> specifies the number of vertices in the output
+patch produced by the tessellation control shader, which also specifies the
+number of times the tessellation control shader is invoked.
+It is a compile- or link-time error for the output vertex count to be less
+than or equal to zero, or greater than the implementation-dependent maximum
+patch size.</p>
+</div>
+<div class="paragraph">
+<p>The intrinsically declared tessellation control output array <em>gl_out[]</em> will
+also be sized by any output layout declaration.
+Hence, the expression</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_out.length()</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will return the output patch vertex count specified in a previous output
+layout qualifier.
+For outputs declared without an array size, including intrinsically declared
+outputs (i.e., <em>gl_out</em>), a layout must be declared before any use of the
+method <strong>length</strong>() or other array use that requires its size to be known.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error if the output patch vertex count specified in an
+output layout qualifier does not match the array size specified in any
+output variable declaration in the same shader.</p>
+</div>
+<div class="paragraph">
+<p>All tessellation control shader layout declarations in a program must
+specify the same output patch vertex count.
+There must be at least one layout qualifier specifying an output patch
+vertex count in any program containing a tessellation control shader.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="geometry-outputs">Geometry Outputs</h5>
+<div class="paragraph">
+<p>Geometry shaders can have two additional types of output layout identifiers:
+an output primitive type and a maximum output vertex count.
+The primitive type and vertex count identifiers are allowed only on the
+interface qualifier <strong>out</strong>, not on an output block, block member, or variable
+declaration.</p>
+</div>
+<div class="paragraph">
+<p>The layout qualifier identifiers for geometry shader outputs are</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>points</strong><br>
+<strong>line_strip</strong><br>
+<strong>triangle_strip</strong><br>
+<strong>max_vertices</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The primitive type identifiers <strong>points</strong>, <strong>line_strip</strong>, and <strong>triangle_strip</strong>
+are used to specify the type of output primitive produced by the geometry
+shader, and only one of these is accepted.
+The geometry shader object in a program must declare an output primitive
+type, and all geometry shader output primitive type declarations in a
+program must declare the same primitive type.</p>
+</div>
+<div class="paragraph">
+<p>The vertex count identifier <strong>max_vertices</strong> is used to specify the maximum
+number of vertices the shader will ever emit in a single invocation.
+The geometry shader object in a program must declare a maximum output vertex
+count, and all geometry shader output vertex count declarations in a program
+must declare the same count.</p>
+</div>
+<div class="paragraph">
+<p>In this example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(triangle_strip, max_vertices = <span class="integer">60</span>) out; <span class="comment">// order does not matter</span>
+layout(max_vertices = <span class="integer">60</span>) out; <span class="comment">// redeclaration okay</span>
+layout(triangle_strip) out; <span class="comment">// redeclaration okay</span>
+layout(points) out; <span class="comment">// error, contradicts triangle_strip</span>
+layout(max_vertices = <span class="integer">30</span>) out; <span class="comment">// error, contradicts 60</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>all outputs from the geometry shader are triangles and at most 60 vertices
+will be emitted by the shader.
+It is an error for the maximum number of vertices to be greater than
+<em>gl_MaxGeometryOutputVertices</em>.</p>
+</div>
+<div class="paragraph">
+<p>All geometry shader output layout declarations in a program must declare the
+same layout and same value for <strong>max_vertices</strong>.
+If geometry shaders are in a program, there must be at least one geometry
+output layout declaration somewhere in that
+program.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="fragment-outputs">Fragment Outputs</h5>
+<div class="paragraph">
+<p>Fragment shaders can have an output layout for redeclaring the built-in
+variable <em>gl_FragDepth</em>:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// redeclaration that changes nothing is allowed +</span>
+out <span class="predefined-type">float</span> gl_FragDepth;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The built-in <em>gl_FragDepth</em> is only predeclared in fragment shaders, so
+redeclaring it in any other shader language results in an error.</p>
+</div>
+<div class="paragraph">
+<p>Fragment shaders additionally support layout qualifiers specifying a set of
+advanced blend equations supported when the fragment shader is used.
+These layout qualifiers are only permitted on the interface qualifier <strong>out</strong>,
+and use the identifiers specified in the “Layout Qualifier” column of the
+table below.</p>
+</div>
+<div class="paragraph">
+<p>If a layout qualifier in the table below is specified in the fragment
+shader, the fragment shader may be used with the corresponding advanced
+blend equation in the “Blend Equation(s) Supported” column.
+Additionally, the special qualifier <strong>blend_support_all_equations</strong> indicates
+that the shader may be used with any advanced blending equation supported by
+the <a href="#references">OpenGL ES Specification</a>.
+It is not an error to specify more than one of these identifiers in any
+fragment shader.
+Specifying more than one qualifier or <strong>blend_support_all_equations</strong> means
+that the fragment shader may be used with multiple advanced blend equations.
+Additionally, it is not an error to specify any single one of these layout
+qualifiers more than once.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Layout Qualifier</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Blend Equations(s) Supported</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_multiply</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">MULTIPLY</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_screen</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">SCREEN</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_overlay</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">OVERLAY</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_darken</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">DARKEN</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_lighten</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">LIGHTEN</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_colordodge</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">COLORDODGE</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_colorburn</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">COLORBURN</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_hardlight</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">HARDLIGHT</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_softlight</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">SOFTLIGHT</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_difference</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">DIFFERENCE</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_exclusion</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">EXCLUSION</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_hsl_hue</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">HSL_HUE</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_hsl_saturation</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">HSL_SATURATION</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_hsl_color</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">HSL_COLOR</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_hsl_luminosity</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">HSL_LUMINOSITY</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>blend_support_all_equations</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">all blend equations</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect3">
+<h4 id="uniform-variable-layout-qualifiers">4.4.3. Uniform Variable Layout Qualifiers</h4>
+<div class="paragraph">
+<p>The following layout qualifier can be used for all default-block uniform
+variables but not for variables in uniform or shader storage blocks.
+The layout qualifier identifier for uniform variables is:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>location</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The location specifies the location by which the OpenGL ES API can reference
+the uniform and update its value.
+Individual elements of a uniform array are assigned consecutive locations
+with the first element taking location <strong>location</strong>.
+No two default-block uniform variables in the program can have the same
+location, even if they are unused, otherwise a compile-time or link-time
+error will be generated.
+Valid locations for default-block uniform variable locations are in the
+range of 0 to the implementation-defined maximum number of uniform locations
+minus one.</p>
+</div>
+<div class="paragraph">
+<p>Locations can be assigned to default-block uniform arrays and structures.
+The first inner-most scalar, vector or matrix member or element takes the
+specified <strong>location</strong> and the compiler assigns the next inner-most member or
+element the next incremental location value.
+Each subsequent inner-most member or element gets incremental locations for
+the entire structure or array.
+This rule applies to nested structures and arrays and gives each inner-most
+scalar, vector, or matrix member a unique location.
+When the linker generates locations for uniforms without an explicit
+location, it assumes for all uniforms with an explicit location all their
+array elements and structure members are used and the linker will not
+generate a conflicting location, even if that element or member is deemed
+unused.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="uniform-and-shader-storage-block-layout-qualifiers">4.4.4. Uniform and Shader Storage Block Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Layout qualifiers can be used for uniform and shader storage blocks, but not
+for non-block uniform declarations.
+The layout qualifier identifiers (and <strong>shared</strong> keyword) for uniform and
+shader storage blocks are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>shared</strong><br>
+<strong>packed</strong><br>
+<strong>std140</strong><br>
+<strong>std430</strong><br>
+<strong>row_major</strong><br>
+<strong>column_major</strong><br>
+<strong>binding</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>None of these have any semantic effect at all on the usage of the variables
+being declared; they only describe how data is laid out in memory.
+For example, matrix semantics are always column-based, as described in the
+rest of this specification, no matter what layout qualifiers are being used.</p>
+</div>
+<div class="paragraph">
+<p>Uniform and shader storage block layout qualifiers can be declared for
+global scope, on a single uniform or shader storage block, or on a single
+block member declaration.</p>
+</div>
+<div class="paragraph">
+<p>Default layouts are established at global scope for uniform blocks as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(layout-qualifier-id-list) uniform;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and for shader storage blocks as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(layout-qualifier-id-list) buffer;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When this is done, the previous default qualification is first inherited and
+then overridden as per the override rules listed below for each qualifier
+listed in the declaration.
+The result becomes the new default qualification scoped to subsequent
+uniform or shader storage block definitions.</p>
+</div>
+<div class="paragraph">
+<p>The initial state of compilation
+is as if the following were declared:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(shared, column_major) uniform;
+layout(shared, column_major) buffer;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Uniform and shader storage blocks can be declared with optional layout
+qualifiers, and so can their individual member declarations.
+Such block layout qualification is scoped only to the content of the block.
+As with global layout declarations, block layout qualification first
+inherits from the current default qualification and then overrides it.
+Similarly, individual member layout qualification is scoped just to the
+member declaration, and inherits from and overrides the block’s
+qualification.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>shared</strong> qualifier overrides only the <strong>std140</strong>, <strong>std430</strong>, and <strong>packed</strong>
+qualifiers; other qualifiers are inherited.
+The compiler/linker will ensure that multiple programs and programmable
+stages containing this definition will share the same memory layout for this
+block, as long as they also matched in their <strong>row_major</strong> and/or
+<strong>column_major</strong> qualifications.
+This allows use of the same buffer to back the same block definition across
+different programs.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>packed</strong> qualifier overrides only <strong>std140</strong>, <strong>std430</strong>, and <strong>shared</strong>;
+other qualifiers are inherited.
+When <strong>packed</strong> is used, no shareable layout is guaranteed.
+The compiler and linker can optimize memory use based on what variables
+actively get used and on other criteria.
+Offsets must be queried, as there is no other way of guaranteeing where (and
+which) variables reside within the block.</p>
+</div>
+<div class="paragraph">
+<p>It is a link-time error to access the same packed uniform or shader storage
+block in multiple stages within a program.
+Attempts to access the same packed uniform or shader storage block across
+programs can result in conflicting member offsets and in undefined values
+being read.
+However, implementations may aid application management of packed blocks by
+using canonical layouts for packed blocks.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>std140</strong> and <strong>std430</strong> qualifiers override only the <strong>packed</strong>, <strong>shared</strong>,
+<strong>std140</strong>, and <strong>std430</strong> qualifiers; other qualifiers are inherited.
+The <strong>std430</strong> qualifier is supported only for shader storage blocks; a shader
+using the <strong>std430</strong> qualifier on a uniform block will fail to compile.</p>
+</div>
+<div class="paragraph">
+<p>The layout is explicitly determined by this, as described in section 7.6.2.2
+“Standard Uniform Block Layout” of the <a href="#references">OpenGL ES Specification</a>.
+Hence, as in <strong>shared</strong> above, the resulting layout is shareable across
+programs.</p>
+</div>
+<div class="paragraph">
+<p>Layout qualifiers on member declarations cannot use the <strong>shared</strong>, <strong>packed</strong>,
+<strong>std140</strong>, or <strong>std430</strong> qualifiers.
+These can only be used at global scope (without an object) or on a block
+declaration, or an error results.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>row_major</strong> and <strong>column_major</strong> qualifiers only affect the layout of
+matrices, including all matrices contained in structures and arrays they are
+applied to, to all depths of nesting.
+These qualifiers can be applied to other types, but will have no effect.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>row_major</strong> qualifier overrides only the <strong>column_major</strong> qualifier; other
+qualifiers are inherited.
+Elements within a matrix row will be contiguous in memory.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>column_major</strong> qualifier overrides only the <strong>row_major</strong> qualifier; other
+qualifiers are inherited.
+Elements within a matrix column will be contiguous in memory.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>binding</strong> qualifier specifies the uniform buffer binding point
+corresponding to the uniform or shader storage block, which will be used to
+obtain the values of the member variables of the block.
+It is a compile-time error to specify the <strong>binding</strong> qualifier for the global
+scope or for block member declarations.
+Any uniform or shader storage block declared without a <strong>binding</strong> qualifier
+is initially assigned to block binding point zero.
+After a program is linked, the binding points used for uniform
+(but not shader storage) blocks
+declared with or without a <strong>binding</strong> qualifier can be updated
+by the OpenGL ES API.</p>
+</div>
+<div class="paragraph">
+<p>If the <strong>binding</strong> qualifier is used with a uniform block or shader storage
+block instanced as an array, the first element of the array takes the
+specified block binding and each subsequent element takes the next
+consecutive binding point.
+For an array of arrays, each element (e.g. 6 elements for a[2][3]) gets a
+binding point, and they are ordered per the array of array ordering
+described in “<a href="#arrays.">Arrays.</a>”</p>
+</div>
+<div class="paragraph">
+<p>If the binding point for any uniform or shader storage block instance is
+less than zero, or greater than or equal to the corresponding
+implementation-dependent maximum number of buffer bindings, a compile-time
+error will occur.
+When the <strong>binding</strong> qualifier is used with a uniform or shader storage block
+instanced as an array of size <em>N</em>, all elements of the array from <strong>binding</strong>
+through <em>binding + N - 1</em> must be within this range.</p>
+</div>
+<div class="paragraph">
+<p>When multiple arguments are listed in a <strong>layout</strong> declaration, the effect
+will be the same as if they were declared one at a time, in order from left
+to right, each in turn inheriting from and overriding the result from the
+previous qualification.</p>
+</div>
+<div class="paragraph">
+<p>For example</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(row_major, column_major)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>results in the qualification being <strong>column_major</strong>.
+Other examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(shared, row_major) uniform; <span class="comment">// default is now shared and row_major</span>
+
+layout(std140) uniform Transform { <span class="comment">// layout of this block is std140</span>
+ mat4 M1; <span class="comment">// row major</span>
+ layout(column_major) mat4 M2; <span class="comment">// column major</span>
+ mat3 N1; <span class="comment">// row major</span>
+};
+
+uniform T2 { <span class="comment">// layout of this block is shared</span>
+ ...
+};
+
+layout(column_major) uniform T3 { <span class="comment">// shared and column major</span>
+ mat4 M3; <span class="comment">// column major</span>
+ layout(row_major) mat4 m4; <span class="comment">// row major</span>
+ mat3 N2; <span class="comment">// column major</span>
+};</code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="opaque-uniform-layout-qualifiers">4.4.5. Opaque Uniform Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Uniform layout qualifiers can be used to bind opaque uniform variables to
+specific buffers or units.
+Samplers can be bound to texture image units, images can be bound to image
+units, and atomic counters can be bound to buffers.</p>
+</div>
+<div class="paragraph">
+<p>Sampler, image and atomic counter types take the uniform layout qualifier
+identifier for binding:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>binding</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The identifier <strong>binding</strong> specifies which unit will be bound.
+Any uniform sampler, image or atomic counter
+variable declared without a binding qualifier
+is initially bound to unit zero.
+After a program is linked, the unit referenced by a sampler
+uniform variable declared with or without a <strong>binding</strong> qualifier can be
+updated by the OpenGL ES API.</p>
+</div>
+<div class="paragraph">
+<p>If the <strong>binding</strong> qualifier is used with an array, the first element of the
+array takes the specified unit and each subsequent element takes the next
+consecutive unit.
+For an array of arrays, each element (e.g. 6 elements for a[2][3]) gets a
+binding point, and they are ordered per the array of array ordering
+described in “<a href="#arrays.">Arrays.</a>”</p>
+</div>
+<div class="paragraph">
+<p>If the <strong>binding</strong> is less than zero, or greater than or equal to the
+implementation-dependent maximum supported number of units, a compile-time
+error will occur.
+When the <strong>binding</strong> qualifier is used with an array of size <em>N</em>, all elements
+of the array from <strong>binding</strong> through <em>binding + N - 1</em> must be within this
+range.</p>
+</div>
+<div class="paragraph">
+<p>A link-time error will result if two shaders in a program specify different
+<em>layout-qualifier-value</em> bindings for the same opaque-uniform name.
+However, it is not an error to specify a binding on some but not all
+declarations for the same name, as shown in the examples below.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// in one shader...</span>
+layout(binding=<span class="integer">3</span>) uniform sampler2D s; <span class="comment">// s bound to unit 3</span>
+
+<span class="comment">// in another shader...</span>
+uniform sampler2D s; <span class="comment">// okay, s still bound at 3</span>
+
+<span class="comment">// in another shader...</span>
+layout(binding=<span class="integer">4</span>) uniform sampler2D s; <span class="comment">// ERROR: contradictory bindings</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="atomic-counter-layout-qualifiers">4.4.6. Atomic Counter Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Atomic counter layout qualifiers can be used on atomic counter declarations.
+The atomic counter qualifiers are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>binding</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>offset</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding = <span class="integer">2</span>, offset = <span class="integer">4</span>) uniform atomic_uint a;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the opaque handle to the atomic counter <em>a</em> will be
+bound to atomic counter buffer binding point 2 at an offset of 4 basic
+machine units into that buffer.
+The default <em>offset</em> for binding point 2 will be post incremented by 4 (the
+size of an atomic counter).</p>
+</div>
+<div class="paragraph">
+<p>A subsequent atomic counter declaration will inherit the previous (post
+incremented) offset.
+For example, a subsequent declaration of</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding = <span class="integer">2</span>) uniform atomic_uint bar;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the atomic counter <em>bar</em> has a binding to buffer binding
+point 2 at an offset of 8 basic machine units into that buffer.
+The offset for binding point 2 will again be post-incremented by 4 (the size
+of an atomic counter).</p>
+</div>
+<div class="paragraph">
+<p>When multiple variables are listed in a layout declaration, the effect will
+be the same as if they were declared one at a time, in order from left to
+right.</p>
+</div>
+<div class="paragraph">
+<p>Binding points are not inherited, only offsets.
+Each binding point tracks its own current default <em>offset</em> for inheritance
+of subsequent variables using the same <strong>binding</strong>.
+The initial state of compilation is that all <strong>binding</strong> points have an
+<em>offset</em> of 0.
+The <em>offset</em> can be set per binding point at global scope (without declaring
+a variable).
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding = <span class="integer">2</span>, offset = <span class="integer">4</span>) uniform atomic_uint;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Establishes that the next <strong>atomic_uint</strong> declaration for binding point 2 will
+inherit <em>offset</em> 4 (but does not establish a default <strong>binding</strong>):</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding = <span class="integer">2</span>) uniform atomic_uint bar; <span class="comment">// offset is 4</span>
+layout(offset = <span class="integer">8</span>) uniform atomic_uint bar; <span class="comment">// error, no default binding</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Atomic counters may share the same binding point, but if a binding is
+shared, their offsets must be either explicitly or implicitly (from
+inheritance) unique and non overlapping.</p>
+</div>
+<div class="paragraph">
+<p>Example valid uniform declarations, assuming top of shader:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding=<span class="integer">3</span>, offset=<span class="integer">4</span>) uniform atomic_uint a; <span class="comment">// offset = 4</span>
+layout(binding=<span class="integer">2</span>) uniform atomic_uint b; <span class="comment">// offset = 0</span>
+layout(binding=<span class="integer">3</span>) uniform atomic_uint c; <span class="comment">// offset = 8</span>
+layout(binding=<span class="integer">2</span>) uniform atomic_uint d; <span class="comment">// offset = 4</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Example of an invalid uniform declaration:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(offset=<span class="integer">4</span>) ... <span class="comment">// error, must include binding</span>
+layout(binding=<span class="integer">1</span>, offset=<span class="integer">0</span>) ... a; <span class="comment">// okay</span>
+layout(binding=<span class="integer">2</span>, offset=<span class="integer">0</span>) ... b; <span class="comment">// okay</span>
+layout(binding=<span class="integer">1</span>, offset=<span class="integer">0</span>) ... c; <span class="comment">// error, offsets must not be shared</span>
+ <span class="comment">// between a and c</span>
+layout(binding=<span class="integer">1</span>, offset=<span class="integer">2</span>) ... d; <span class="comment">// error, overlaps offset 0 of a</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to bind an atomic counter with a binding value
+greater than or equal to <em>gl_MaxAtomicCounterBindings</em>.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="format-layout-qualifiers">4.4.7. Format Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Format layout qualifiers can be used on image variable declarations (those
+declared with a basic type having “<strong>image</strong>” in its keyword).
+The format layout qualifier identifiers for image variable declarations are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><em>float-image-format-qualifier</em><br>
+<em>int-image-format-qualifier</em><br>
+<em>uint-image-format-qualifier</em><br>
+<strong>binding</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>float-image-format-qualifier</em> : </dt>
+<dd>
+<p><strong>rgba32f</strong><br>
+<strong>rgba16f</strong><br>
+<strong>r32f</strong><br>
+<strong>rgba8</strong><br>
+<strong>rgba8_snorm</strong></p>
+</dd>
+<dt class="hdlist1"><em>int-image-format-qualifier</em> : </dt>
+<dd>
+<p><strong>rgba32i</strong><br>
+<strong>rgba16i</strong><br>
+<strong>rgba8i</strong><br>
+<strong>r32i</strong></p>
+</dd>
+<dt class="hdlist1"><em>uint-image-format-qualifier</em> : </dt>
+<dd>
+<p><strong>rgba32ui</strong><br>
+<strong>rgba16ui</strong><br>
+<strong>rgba8ui</strong><br>
+<strong>r32ui</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>A format layout qualifier specifies the image format associated with a
+declared image variable.
+Only one format qualifier may be specified for any image variable
+declaration.
+For image variables with floating-point component types (keywords starting
+with “<strong>image</strong>”), signed integer component types (keywords starting with
+“<strong>iimage</strong>”), or unsigned integer component types (keywords starting with
+“<strong>uimage</strong>”), the format qualifier used must match the
+<em>float-image-format-qualifier</em>, <em>int-image-format-qualifier</em>, or
+<em>uint-image-format-qualifier</em> grammar rules, respectively.
+It is an error to declare an image variable where the format
+qualifier does not match the image variable type.</p>
+</div>
+<div class="paragraph">
+<p>Any image variable must specify a format layout qualifier.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>binding</strong> qualifier was described in
+“<a href="#opaque-uniform-layout-qualifiers">Opaque Uniform Layout Qualifiers</a>”.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="interpolation-qualifiers">4.5. Interpolation Qualifiers</h3>
+<div class="paragraph">
+<p>Inputs and outputs that could be interpolated can be further qualified by at
+most one of the following interpolation qualifiers:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>smooth</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">perspective correct interpolation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>flat</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">no interpolation</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The presence of and type of interpolation is controlled by the above
+interpolation qualifiers as well as the auxiliary storage qualifiers
+<strong>centroid</strong> and <strong>sample</strong>.
+When no interpolation qualifier is present, smooth interpolation is used.
+It is a compile-time error to use more than one interpolation qualifier.
+The auxiliary storage qualifier <strong>patch</strong> is not used for interpolation; it is
+a compile-time error to use interpolation qualifiers with <strong>patch</strong>.</p>
+</div>
+<div class="paragraph">
+<p>A variable qualified as <strong>flat</strong> will not be interpolated.
+Instead, it will have the same value for every fragment within a primitive.
+This value will come from a single provoking vertex, as described by the
+<a href="#references">OpenGL ES Specification</a>.
+A variable qualified as <strong>flat</strong> may also be qualified as <strong>centroid</strong> or
+<strong>sample</strong>, which will mean the same thing as qualifying it only as <strong>flat</strong>.</p>
+</div>
+<div class="paragraph">
+<p>A variable qualified as <strong>smooth</strong> will be interpolated in a
+perspective-correct manner over the primitive being rendered.</p>
+</div>
+<div class="paragraph">
+<p>Interpolation in a perspective correct manner is specified in equations 13.4
+of the <a href="#references">OpenGL ES Specification</a>, section 13.4.1 “Line Segments” and
+equation 13.7, section 13.5.1 “Polygon Interpolation”.</p>
+</div>
+<div class="paragraph">
+<p>When multisample rasterization is disabled, or for fragment shader input
+variables qualified with neither <strong>centroid</strong> nor <strong>sample</strong>, the value of the
+assigned variable may be interpolated anywhere within the pixel and a single
+value may be assigned to each sample within the pixel, to the extent
+permitted by the <a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>When multisample rasterization is enabled, <strong>centroid</strong> and <strong>sample</strong> may be
+used to control the location and frequency of the sampling of the qualified
+fragment shader input.
+If a fragment shader input is qualified with <strong>centroid</strong>, a single value may
+be assigned to that variable for all samples in the pixel, but that value
+must be interpolated at a location that lies in both the pixel and in the
+primitive being rendered, including any of the pixel’s samples covered by
+the primitive.
+Because the location at which the variable is interpolated may be different
+in neighboring pixels, and derivatives may be computed by computing
+differences between neighboring pixels, derivatives of centroid-sampled
+inputs may be less accurate than those for non-centroid interpolated
+variables.
+If a fragment shader input is qualified with <strong>sample</strong>, a separate value must
+be assigned to that variable for each covered sample in the pixel, and that
+value must be sampled at the location of the individual sample.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="parameter-qualifiers">4.6. Parameter Qualifiers</h3>
+<div class="paragraph">
+<p>In addition to precision qualifiers and memory qualifiers, parameters can
+have these parameter qualifiers.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><none: default></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>in</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for function parameters passed into a function</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>out</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for function parameters passed back out of a function,
+ but not initialized for use when passed in</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>inout</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for function parameters passed both into and out of a
+ function</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Parameter qualifiers are discussed in more detail in
+“<a href="#function-calling-conventions">Function Calling Conventions</a>”.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="precision-and-precision-qualifiers">4.7. Precision and Precision Qualifiers</h3>
+<div class="sect3">
+<h4 id="range-and-precision">4.7.1. Range and Precision</h4>
+<div class="paragraph">
+<p>The precision of <strong>highp</strong> floating-point variables is defined by the IEEE 754
+standard for 32-bit floating-point numbers.
+This includes support for NaNs (Not a Number) and Infs (positive or negative
+infinities) and positive and negative zeros.</p>
+</div>
+<div class="paragraph">
+<p>The following rules apply to
+<strong>highp</strong>
+operations:
+Signed infinities and zeros are generated as dictated by IEEE, but subject
+to the precisions allowed in the following table.
+Any subnormal (denormalized) value input into a shader or potentially
+generated by any operation in a shader can be flushed to 0.
+The rounding mode cannot be set and is undefined but must not affect the
+result by more than 1 ULP.
+NaNs are not required to be generated.
+Support for signaling NaNs is not required and exceptions are never raised.
+Operations including built-in functions that operate on a NaN are not
+required to return a NaN as the result.
+However if NaNs are generated, <strong>isnan</strong>() must return the correct value.</p>
+</div>
+<div class="paragraph">
+<p>Precisions are expressed in terms of maximum relative error in units of ULP
+(units in the last place), unless otherwise noted.</p>
+</div>
+<div class="paragraph">
+<p>For single precision operations, precisions are required as follows:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Operation</th>
+<th class="tableblock halign-left valign-top">Precision</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>a</em> + <em>b</em>, <em>a</em> - <em>b</em>, <em>a</em> * <em>b</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Correctly rounded.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><, <=, ==, >, >=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Correct result.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>a</em> / <em>b</em>, 1.0 / <em>b</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2.5 ULP for <span class="eq">|b|</span> in the range <span class="eq">[2<sup>-126</sup>, 2<sup>126</sup>]</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>a</em> * <em>b</em> + <em>c</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Correctly rounded single operation or
+ sequence of two correctly rounded operations.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>fma</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inherited from <em>a</em> * <em>b</em> + <em>c</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>pow</strong>(<em>x</em>, <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inherited from <strong>exp2</strong>(<em>y</em> * <strong>log2</strong>(<em>x</em>)).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>exp</strong>(<em>x</em>), <strong>exp2</strong>(<em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">(3 + 2 · |x|)</span> ULP.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>log</strong>(), <strong>log2</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3 ULP outside the range <span class="eq">[0.5,2.0]</span>.<br>
+ Absolute error < 2<sup>-21</sup> inside the range
+ <span class="eq">[0.5,2.0]</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sqrt</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inherited from 1.0 / <strong>inversesqrt</strong>().</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>inversesqrt</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2 ULP.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">implicit and explicit
+ conversions between types</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Correctly rounded.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Built-in functions defined in the specification with an equation built from
+the above operations inherit the above errors.
+These include, for example, the geometric functions, the common functions,
+and many of the matrix functions.
+Built-in functions not listed above and not defined as equations of the
+above have undefined precision.
+These include, for example, the trigonometric functions and determinant.</p>
+</div>
+<div class="paragraph">
+<p>Storage requirements are declared through use of <em>precision qualifiers</em>.
+The precision of operations must preserve the storage precisions of the
+variables involved.</p>
+</div>
+<div class="paragraph">
+<p><strong>highp</strong> floating-point values are stored in IEEE 754 single precision
+floating-point format.
+<strong>mediump</strong> and <strong>lowp</strong> floating-point values have minimum range and precision
+requirements as detailed below and have maximum range and precision as
+defined by IEEE 754.</p>
+</div>
+<div class="paragraph">
+<p>All integral types are assumed to be implemented as integers and so may not
+be emulated by floating-point values.
+<strong>highp</strong> signed integers are represented as twos-complement 32-bit signed
+integers.
+<strong>highp</strong> unsigned integers are represented as unsigned 32-bit integers.
+<strong>mediump</strong> integers (signed and unsigned) must be represented as an integer
+with between 16 and 32 bits inclusive.
+<strong>lowp</strong> integers (signed and unsigned) must be represented as an integer with
+between 9 and 32 bits inclusive.</p>
+</div>
+<div class="paragraph">
+<p>The required ranges and precisions for precision qualifiers are:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top">Floating-Point Range</th>
+<th class="tableblock halign-left valign-top">Floating-Point Magnitude Range</th>
+<th class="tableblock halign-left valign-top">Floating-Point Precision</th>
+<th class="tableblock halign-left valign-top">Signed Integer Range</th>
+<th class="tableblock halign-left valign-top">Unsigned Integer Range</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>highp</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Subset of IEEE-754<br>
+ <span class="eq">(-2<sup>128</sup>,2<sup>128</sup>)</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Subset of IEEE-754<br>
+ 0.0, <span class="eq">[2<sup>-126</sup>,2<sup>128</sup>)</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Subset of IEEE-754 relative:<br>
+ <span class="eq">2<sup>-24</sup></span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">[-2<sup>31</sup>,2<sup>31</sup>-1]</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">[0,2<sup>32</sup>-1]</span></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mediump</strong> (minimum requirements)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">(-2<sup>14</sup>,2<sup>14</sup>)</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">(2<sup>14</sup>,2<sup>14</sup>)</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Relative:<br>
+ <span class="eq">2<sup>-10</sup></span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">[-2<sup>15</sup>,2<sup>15</sup>-1]</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">[0,2<sup>16</sup>-1]</span></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lowp</strong> (minimum requirements)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">(-2,2)</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">(2<sup>-8</sup>,2)</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Absolute:<br>
+ <span class="eq">2<sup>-8</sup></span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">[-2<sup>8</sup>,2<sup>8</sup>-1]</span></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">[0,2<sup>9</sup>-1]</span></p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p><em>Relative precision</em> is defined as the worst case (i.e. largest) ratio of
+the smallest step in relation to the value for all non-zero values:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+ \mathit{Precision}_{relative} = {
+ \left| {
+ { \left| v_1 - v_2 \right| }_{min} } \over { v_1 }
+ \right|
+ }_{max}, &
+ v_1,v_2 \neq 0, v_1 \neq v_2
+\end{aligned}\]
+</div>
+</div>
+<div class="paragraph">
+<p>It is therefore twice the maximum rounding error when converting from a real
+number.</p>
+</div>
+<div class="paragraph">
+<p>In addition, the range and precision of a <strong>mediump</strong> floating-point value
+must be the same as or greater than the range and precision of a <strong>lowp</strong>
+floating-point value.
+The range and precision of a <strong>highp</strong> floating-point value must be the same
+as or greater than the range and precision of a <strong>mediump</strong> floating-point
+value.</p>
+</div>
+<div class="paragraph">
+<p>The range of a <strong>mediump</strong> integer value must be the same as or greater than
+the range of a <strong>lowp</strong> integer value.
+The range of a <strong>highp</strong> integer value must be the same as or greater than the
+range of a <strong>mediump</strong> integer value.</p>
+</div>
+<div class="paragraph">
+<p>Within the above specification, an implementation is allowed to vary the
+representation of numeric values, both within a shader and between different
+shaders.
+If necessary, this variance can be controlled using the invariance
+qualifier.</p>
+</div>
+<div class="paragraph">
+<p>The actual ranges and precisions provided by an implementation can be
+queried through the API.
+See the <a href="#references">OpenGL ES Specification</a> specification for details on how to do
+this.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="conversion-between-precisions">4.7.2. Conversion between precisions</h4>
+<div class="paragraph">
+<p>Within the same type, conversion from a lower to a higher precision must be
+exact.
+When converting from a higher precision to a lower precision, if the value
+is representable by the implementation of the target precision, the
+conversion must also be exact.
+If the value is not representable, the behavior is dependent on the type:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>For signed and unsigned integers, the value is truncated; bits in
+positions not present in the target precision are set to zero.
+(Positions start at zero and the least significant bit is considered to
+be position zero for this purpose.)</p>
+</li>
+<li>
+<p>For floating-point values, the value should either clamp to +Inf or
+-Inf, or to the maximum or minimum value that the implementation
+supports.
+While this behavior is implementation-dependent, it should be consistent
+for a given implementation.</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="precision-qualifiers">4.7.3. Precision Qualifiers</h4>
+<div class="paragraph">
+<p>Any
+floating-point, integer, or opaque-type declaration can have the type
+preceded by one of these precision qualifiers:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Qualifier</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Meaning</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>highp</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The variable satisfies the minimum requirements for <strong>highp</strong>
+ described above. <strong>highp</strong> variables have the maximum range and
+ precision available but may cause operations to run more
+ slowly on some implementations.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mediump</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The variable satisfies the minimum requirements for <strong>mediump</strong>
+ described above. <strong>mediump</strong> variables may typically be used to
+ store high dynamic range colors and low precision geometry.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lowp</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The variable satisfies the minimum requirements for <strong>lowp</strong>
+ described above. <strong>lowp</strong> variables may typically be used to
+ store 8-bit color values.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">lowp <span class="predefined-type">float</span> color;
+out mediump vec2 P;
+lowp ivec2 foo(lowp mat3);
+highp mat4 m;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Literal constants do not have precision qualifiers.
+Neither do Boolean variables.
+Neither do constructors.</p>
+</div>
+<div class="paragraph">
+<p>For this paragraph, “operation” includes operators, built-in functions,
+and constructors, and “operand” includes function arguments and
+constructor arguments.
+The precision used to internally evaluate an operation, and the precision
+qualification subsequently associated with any resulting intermediate
+values, must be at least as high as the highest precision qualification of
+the operands consumed by the operation.</p>
+</div>
+<div class="paragraph">
+<p>In cases where operands do not have a precision qualifier, the precision
+qualification will come from the other operands.
+If no operands have a precision qualifier, then the precision qualifications
+of the operands of the next consuming operation in the expression will be
+used.
+This rule can be applied recursively until a precision qualified operand is
+found.
+If necessary, it will also include the precision qualification of l-values
+for assignments, of the declared variable for initializers, of formal
+parameters for function call arguments, or of function return types for
+function return values.
+If the precision cannot be determined by this method e.g. if an entire
+expression is composed only of operands with no precision qualifier, and the
+result is not assigned or passed as an argument, then it is evaluated at the
+default precision of the type or greater.
+When this occurs in the fragment shader, the default precision must be
+defined.</p>
+</div>
+<div class="paragraph">
+<p>For example, consider the statements:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">uniform highp <span class="predefined-type">float</span> h1;
+highp <span class="predefined-type">float</span> h2 = <span class="float">2</span><span class="float">.3</span> * <span class="float">4</span><span class="float">.7</span>; <span class="comment">// operation and result are highp</span>
+precision
+mediump <span class="predefined-type">float</span> m;
+m = <span class="float">3</span><span class="float">.7</span> * h1 * h2; <span class="comment">// all operations are highp precision</span>
+h2 = m * h1; <span class="comment">// operation is highp precision</span>
+m = h2 - h1; <span class="comment">// operation is highp precision</span>
+h2 = m + m; <span class="comment">// addition and result at mediump precision</span>
+<span class="directive">void</span> f(highp <span class="predefined-type">float</span> p);
+f(<span class="float">3</span><span class="float">.3</span>); <span class="comment">// 3.3 will be passed in at highp precision</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Precision qualifiers, as with other qualifiers, do not affect the basic type
+of the variable.
+In particular, there are no constructors for precision conversions;
+constructors only convert types.
+Similarly, precision qualifiers, as with other qualifiers, do not contribute
+to function overloading based on parameter types.
+As discussed in “<a href="#function-calling-conventions">Function Calling
+Conventions</a>”, function input and output is done through copies, and
+therefore qualifiers do not have to match.</p>
+</div>
+<div class="paragraph">
+<p>Precision qualifiers for outputs in one shader matched to inputs in another
+shader need not match when both shaders are linked into the same program.
+When both shaders are in separate programs, mismatched precision qualifiers
+will result in a program interface mismatch that will result in program
+pipeline validation failures, as described in section 7.4.1 “Shader
+Interface Matching” of the <a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>The precision of a variable is determined when the variable is declared and
+cannot be subsequently changed.</p>
+</div>
+<div class="paragraph">
+<p>Where the precision of a constant integral or constant floating-point
+expression is not specified, evaluation is performed at <strong>highp</strong>.
+This rule does not affect the precision qualification of the expression.</p>
+</div>
+<div class="paragraph">
+<p>The evaluation of constant expressions must be invariant and will usually be
+performed at compile time.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="default-precision-qualifiers">4.7.4. Default Precision Qualifiers</h4>
+<div class="paragraph">
+<p>The precision statement</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precision precision-qualifier type;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>can be used to establish a default precision qualifier.
+The <em>type</em> field can be either <strong>int</strong>, <strong>float</strong>, or any of the opaque types,
+and the <em>precision-qualifier</em> can be <strong>lowp</strong>, <strong>mediump</strong>, or <strong>highp</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Any other types or qualifiers will result in an error.
+If <em>type</em> is <strong>float</strong>, the directive applies to non-precision-qualified
+floating-point type (scalar, vector, and matrix) declarations.
+If <em>type</em> is <strong>int</strong>, the directive applies to all non-precision-qualified
+integer type (scalar, vector, signed, and unsigned) declarations.
+This includes global variable declarations, function return declarations,
+function parameter declarations, and local variable declarations.</p>
+</div>
+<div class="paragraph">
+<p>Non-precision qualified declarations will use the precision qualifier
+specified in the most recent <strong>precision</strong> statement that is still in scope.
+The <strong>precision</strong> statement has the same scoping rules as variable
+declarations.
+If it is declared inside a compound statement, its effect stops at the end
+of the inner-most statement it was declared in.
+Precision statements in nested scopes override precision statements in outer
+scopes.
+Multiple precision statements for the same basic type can appear inside the
+same scope, with later statements overriding earlier statements within that
+scope.</p>
+</div>
+<div class="paragraph">
+<p>All languages except for the fragment language have the following
+predeclared globally scoped default precision statements:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precision highp <span class="predefined-type">float</span>;
+precision highp <span class="predefined-type">int</span>;
+precision lowp sampler2D;
+precision lowp samplerCube;
+precision highp atomic_uint;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The fragment language has the following predeclared globally scoped default
+precision statements:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precision mediump <span class="predefined-type">int</span>;
+precision lowp sampler2D;
+precision lowp samplerCube;
+precision highp atomic_uint;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The fragment language has no default precision qualifier for floating-point
+types.
+Hence for <strong>float</strong>, floating-point vector and matrix variable declarations,
+either the declaration must include a precision qualifier or the default
+float precision must have been previously declared.
+Similarly, there is no default precision qualifier in any of the languages
+for any type not listed above.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="available-precision-qualifiers">4.7.5. Available Precision Qualifiers</h4>
+<div class="paragraph">
+<p>The built-in macro GL_FRAGMENT_PRECISION_HIGH is defined to one:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#define</span> GL_FRAGMENT_PRECISION_HIGH <span class="integer">1</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This macro is available in all languages except compute.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="variance-and-the-invariant-qualifier">4.8. Variance and the Invariant Qualifier</h3>
+<div class="paragraph">
+<p>In this section, <em>variance</em> refers to the possibility of getting different
+values from the same expression in different programs.
+For example, consider the situation where two vertex shaders, in different
+programs, each set <em>gl_Position</em> with the same expression, and the input
+values into that expression are the same when both shaders run.
+It is possible, due to independent compilation of the two shaders, that the
+values assigned to <em>gl_Position</em> are not exactly the same when the two
+shaders run.
+In this example, this can cause problems with alignment of geometry in a
+multi-pass algorithm.</p>
+</div>
+<div class="paragraph">
+<p>In general, such variance between shaders is allowed.
+When such variance does not exist for a particular output variable, that
+variable is said to be <em>invariant</em>.</p>
+</div>
+<div class="sect3">
+<h4 id="the-invariant-qualifier">4.8.1. The Invariant Qualifier</h4>
+<div class="paragraph">
+<p>To ensure that a particular output variable is invariant, it is necessary to
+use the <strong>invariant</strong> qualifier.
+It can either be used to qualify a previously declared variable as being
+invariant:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">invariant gl_Position; <span class="comment">// make existing gl_Position be invariant</span>
+out vec3 Color;
+invariant Color; <span class="comment">// make existing Color be invariant</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>or as part of a declaration when a variable is declared:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">invariant centroid out vec3 Color;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Only variables output from a shader can be candidates for invariance.
+This includes user-defined output variables and the built-in output
+variables.
+As only outputs can be declared as invariant, an output from one shader
+stage will still match an input of a subsequent stage without the input
+being declared as invariant.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>invariant</strong> keyword can be followed by a comma separated list of
+previously declared identifiers.
+All uses of <strong>invariant</strong> must be at global scope or on block
+members, and before any use of the variables being declared as invariant.</p>
+</div>
+<div class="paragraph">
+<p>To guarantee invariance of a particular output variable across two programs,
+the following must also be true:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The output variable is declared as invariant in both programs.</p>
+</li>
+<li>
+<p>The same values must be input to all shader input variables consumed by
+expressions and control flow contributing to the value assigned to the
+output variable.</p>
+</li>
+<li>
+<p>The texture formats, texel values, and texture filtering are set the
+same way for any texture function calls contributing to the value of the
+output variable.</p>
+</li>
+<li>
+<p>All input values are all operated on in the same way.
+ All operations in the consuming expressions and any intermediate
+ expressions must be the same, with the same order of operands and same
+ associativity, to give the same order of evaluation.
+ Intermediate variables and functions must be declared as the same type
+ with the same explicit or implicit precision
+qualifiers and the same constant qualifiers.
+ Any control flow affecting the output value must be the same, and any
+ expressions consumed to determine this control flow must also follow
+ these invariance rules.</p>
+</li>
+<li>
+<p>All the data flow and control flow leading to setting the invariant
+output variable reside in a single compilation unit.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Essentially, all the data flow and control flow leading to an invariant
+output must match.</p>
+</div>
+<div class="paragraph">
+<p>Initially, by default, all output variables are allowed to be variant.
+To force all output variables to be invariant, use the pragma</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> STDGL invariant(all)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>before all declarations in a shader.
+If this pragma is used after the declaration of any variables or functions,
+then the set of outputs that behave as invariant is undefined.</p>
+</div>
+<div class="paragraph">
+<p>Generally, invariance is ensured at the cost of flexibility in optimization,
+so performance can be degraded by use of invariance.
+Hence, use of this pragma is intended as a debug aid, to avoid individually
+declaring all output variables as invariant.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="invariance-within-a-shader">4.8.2. Invariance Within a Shader</h4>
+<div class="paragraph">
+<p>When a value is stored in a variable, it is usually assumed it will remain
+constant unless explicitly changed.
+However, during the process of optimization, it is possible that the
+compiler may choose to recompute a value rather than store it in a register.
+Since the precision of operations is not completely specified (e.g. a low
+precision operation may be done at medium or high precision), it would be
+possible for the recomputed value to be different from the original value.</p>
+</div>
+<div class="paragraph">
+<p>Values are allowed to be variant within a shader.
+To prevent this, the invariant qualifier or invariant pragma must be used.</p>
+</div>
+<div class="paragraph">
+<p>Within a shader, there is no invariance for values generated by different
+non-constant expressions, even if those expressions are identical.</p>
+</div>
+<div class="paragraph">
+<p>Example 1:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precision mediump;
+vec4 col;
+vec2 a = ...
+...
+col = texture(tex, a);<span class="comment">// a has a value _a1 _</span>
+...
+col = texture(tex, a); <span class="comment">// a has a value a2 where possibly a1 != a2</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>To enforce invariance in this example use:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> STDGL invariant(all)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Example 2:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec2 m = ...;
+vec2 n = ...;
+vec2 a = m + n;
+vec2 b = m + n; <span class="comment">// a and b are not guaranteed to be exactly equal</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>There is no mechanism to enforce invariance between a and b.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="invariance-of-constant-expressions">4.8.3. Invariance of Constant Expressions</h4>
+<div class="paragraph">
+<p>Invariance must be guaranteed for constant expressions.
+A particular constant expression must evaluate to the same result if it
+appears again in the same shader or a different shader.
+This includes the same expression appearing in two shaders of the same
+language or shaders of two different languages.</p>
+</div>
+<div class="paragraph">
+<p>Constant expressions must evaluate to the same result when operated on as
+already described above for invariant variables.
+Constant expressions are not invariant with respect to equivalent
+non-constant expressions, even when the <strong>invariant</strong> qualifier or pragma is
+used.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="invariance-of-undefined-values">4.8.4. Invariance of Undefined Values</h4>
+<div class="paragraph">
+<p>Undefined values are not invariant nor can they be made invariant by use of
+the invariant qualifier or pragma.
+In some implementations, undefined values may cause unexpected behavior if
+they are used in control-flow expressions e.g. in the following case, one,
+both or neither functions may be executed and this may not be consistent
+over multiple invocations of the shader:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> x; <span class="comment">// undefined value</span>
+<span class="keyword">if</span> (x == <span class="integer">1</span>)
+{
+ f(); <span class="comment">// Undefined whether f() is executed</span>
+}
+<span class="keyword">if</span> (x == <span class="integer">2</span>)
+{
+ g(); <span class="comment">// Undefined whether g() is executed.</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Note that an undefined value is a value that has not been specified.
+A value that has been specified but has a potentially large error due to,
+for example, lack of precision in an expression, is not undefined and so can
+be made invariant.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="the-precise-qualifier">4.9. The Precise Qualifier</h3>
+<div class="paragraph">
+<p>Some algorithms require floating-point computations to exactly follow the
+order of operations specified in the source code and to treat all operations
+consistently, even if the implementation supports optimizations that could
+produce nearly equivalent results with higher performance.
+For example, many GL implementations support a “multiply-add” instruction
+that can compute a floating-point expression such as</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">result = (a * b) + (c * d);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>in two operations instead of three operations; one multiply and one
+multiply-add instead of two multiplies and one add.
+The result of a floating-point multiply-add might not always be identical to
+first doing a multiply yielding a floating-point result and then doing a
+floating-point add.
+Hence, in this example, the two multiply operations would not be treated
+consistently; the two multiplies could effectively appear to have differing
+precisions.</p>
+</div>
+<div class="paragraph">
+<p>The key computation that needs to be made consistent appears when
+tessellating, where intermediate points for subdivision are synthesized in
+different directions, yet need to yield the same result, as shown in the
+diagram below.</p>
+</div>
+<div id="img-precise" class="imageblock">
+<div class="content">
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</svg>" alt="precise">
+</div>
+</div>
+<div class="paragraph">
+<p>As stated in “<a href="#evaluation-of-expressions">Evaluation of Expressions</a>”,
+the compiler may transform expressions even if this changes the resulting
+value.
+Without any qualifiers, implementations are permitted to perform
+optimizations that effectively modify the order or number of operations used
+to evaluate an expression, even if those optimizations may produce slightly
+different results relative to unoptimized code.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>precise</strong> qualifier ensures that operations contributing to a
+variable’s value are done in their stated order and with operator consistency.
+The order is determined by operator precedence and parenthesis, as described in
+“<a href="#operators">Operators</a>”.
+Operator consistency means for each particular operator, for example the
+multiply operator (<strong>*</strong>), its operation is always computed with the same
+precision.
+Specifically, values computed by compiler-generated code must adhere to the
+following identities:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>a + b = b + a</p>
+</li>
+<li>
+<p>a * b = b * a</p>
+</li>
+<li>
+<p>a * b + c * d = b * a + c* d = d * c + b * a = <any other mathematically
+valid combination></p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>While the following are prevented:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>a + (b + c) is not allowed to become (a + b) + c</p>
+</li>
+<li>
+<p>a * (b * c) is not allowed to become (a * b) * c</p>
+</li>
+<li>
+<p>a * b + c is not allowed to become a single operation <strong>fma</strong>(a, b, c)</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Where <em>a</em>, <em>b</em>, <em>c</em>, and <em>d</em>, are scalars or vectors, not matrices.
+(Matrix multiplication generally does not commute.) It is the shader
+writer’s responsibility to express the computation in terms of these rules
+and the compiler’s responsibility to follow these rules.
+See the description of <em>gl_TessCoord</em> for the rules the tessellation stages
+are responsible for following, which in conjunction with the above allow
+avoiding cracking when subdividing.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precise out vec4 position;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>declares that operations used to produce the value of <em>position</em> must be
+performed in exactly the order specified in the source code and with all
+operators being treated consistently.
+As with the <strong>invariant</strong> qualifier (see “<a href="#the-invariant-qualifier">The
+Invariant Qualifier</a>”), the <strong>precise</strong> qualifier may be used to qualify a
+built-in or previously declared user-defined variable as being precise:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out vec3 Color;
+precise Color; <span class="comment">// make existing Color be precise</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When applied to a block, a structure type, or a variable of structure type,
+<strong>precise</strong> applies to each contained member, recursively.</p>
+</div>
+<div class="paragraph">
+<p>This qualifier will affect the evaluation of an r-value in a particular
+function if and only if the result is eventually consumed in the same
+function by an l-value qualified as <strong>precise</strong>.
+Any other expressions within a function are not affected, including return
+values and output parameters not declared as <strong>precise</strong> but that are
+eventually consumed outside the function by a variable qualified as
+<strong>precise</strong>. Unaffected expressions also include the controlling expressions
+in selection and iteration statements and the condition in ternary
+operators (<strong>?:</strong>).</p>
+</div>
+<div class="paragraph">
+<p>Some examples of the use of <strong>precise</strong>:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 a, b, c, d;
+precise out vec4 v;
+
+<span class="predefined-type">float</span> func(<span class="predefined-type">float</span> e, <span class="predefined-type">float</span> f, <span class="predefined-type">float</span> g, <span class="predefined-type">float</span> h)
+{
+ <span class="keyword">return</span> (e*f) + (g*h); <span class="comment">// no constraint on order or operator consistency</span>
+}
+
+<span class="predefined-type">float</span> func2(<span class="predefined-type">float</span> e, <span class="predefined-type">float</span> f, <span class="predefined-type">float</span> g, <span class="predefined-type">float</span> h)
+{
+ precise <span class="predefined-type">float</span> result = (e*f) + (g*h); <span class="comment">// ensures same precision for the two multiplies</span>
+ <span class="keyword">return</span> result;
+}
+
+<span class="predefined-type">float</span> func3(<span class="predefined-type">float</span> i, <span class="predefined-type">float</span> j, precise out <span class="predefined-type">float</span> k)
+{
+ k = i * i + j; <span class="comment">// precise, due to <k> declaration</span>
+}
+
+<span class="directive">void</span> main()
+{
+ vec3 r = vec3(a * b); <span class="comment">// precise, used to compute v.xyz</span>
+ vec3 s = vec3(c * d); <span class="comment">// precise, used to compute v.xyz</span>
+ v.xyz = r + s; <span class="comment">// precise</span>
+ v.w = (a.w * b.w) + (c.w * d.w); <span class="comment">// precise</span>
+ v.x = func(a.x, b.x, c.x, d.x); <span class="comment">// values computed in func() are NOT precise</span>
+ v.x = func2(a.x, b.x, c.x, d.x); <span class="comment">// precise!</span>
+ func3(a.x * b.x, c.x * d.x, v.x); <span class="comment">// precise!</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For the purposes of determining if an output from one shader stage matches
+an input of the next stage, the <strong>precise</strong> qualifier need not match between
+the input and the output.</p>
+</div>
+<div class="paragraph">
+<p>All constant expressions are evaluated as if <strong>precise</strong> was present, whether
+or not it is present.
+However, as described in “<a href="#constant-expressions">Constant Expressions</a>”,
+there is no requirement that a compile-time constant expression evaluates to
+the same value as a corresponding non-constant expression.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="memory-qualifiers">4.10. Memory Qualifiers</h3>
+<div class="paragraph">
+<p>Shader storage blocks, variables declared within shader storage blocks and
+variables declared as image types (the basic opaque types with “<strong>image</strong>”
+in their keyword), can be further qualified with one or more of the
+following memory qualifiers:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>coherent</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable where reads and writes are coherent with
+ reads and writes from other shader invocations</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>volatile</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable whose underlying value may be changed at any
+ point during shader execution by some source other than the
+ current shader invocation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>restrict</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable where use of that variable is the only way
+ to read and write the underlying memory in the relevant
+ shader stage</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>readonly</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable that can be used to read the underlying
+ memory, but cannot be used to write the underlying memory</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>writeonly</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable that can be used to write the underlying
+ memory, but cannot be used to read the underlying memory</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>coherent</strong> qualifier
+are performed coherently with accesses to the same location from other
+shader invocations.</p>
+</div>
+<div class="paragraph">
+<p>As described in section
+7.11
+“Shader Memory Access” of the
+<a href="#references">OpenGL ES Specification</a>, shader memory reads and writes complete in a
+largely undefined order.
+The built-in function <strong>memoryBarrier</strong>() can be used if needed to guarantee
+the completion and relative ordering of memory accesses performed by a
+single shader invocation.</p>
+</div>
+<div class="paragraph">
+<p>When accessing memory using variables not declared as <strong>coherent</strong>, the memory
+accessed by a shader may be cached by the implementation to service future
+accesses to the same address.
+Memory stores may be cached in such a way that the values written may not be
+visible to other shader invocations accessing the same memory.
+The implementation may cache the values fetched by memory reads and return
+the same values to any shader invocation accessing the same memory, even if
+the underlying memory has been modified since the first memory read.
+While variables not declared as <strong>coherent</strong> may not be useful for
+communicating between shader invocations, using non-coherent accesses may
+result in higher performance.</p>
+</div>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>volatile</strong> qualifier
+must treat the underlying memory as though it could be read or written at
+any point during shader execution by some source other than the executing
+shader invocation.
+When a volatile variable is read, its value must be re-fetched from the
+underlying memory, even if the shader invocation performing the read had
+previously fetched its value from the same memory.
+When a volatile variable is written, its value must be written to the
+underlying memory, even if the compiler can conclusively determine that its
+value will be overwritten by a subsequent write.
+Since the external source reading or writing a <strong>volatile</strong> variable may be
+another shader invocation, variables declared as <strong>volatile</strong> are
+automatically treated as coherent.</p>
+</div>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>restrict</strong> qualifier
+may be compiled assuming that the variable used to perform the memory access
+is the only way to access the underlying memory using the shader stage in
+question.
+This allows the compiler to coalesce or reorder loads and stores using
+<strong>restrict</strong>-qualified image variables in ways that wouldn’t be permitted
+for image variables not so qualified, because the compiler can assume that
+the underlying image won’t be read or written by other code.
+Applications are responsible for ensuring that image memory referenced by
+variables qualified with <strong>restrict</strong> will not be referenced using other
+variables in the same scope; otherwise, accesses to <strong>restrict</strong>-qualified
+variables will have undefined results.</p>
+</div>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>readonly</strong> qualifier
+may only read the underlying memory, which is treated as read-only memory
+and cannot be written to.
+It is an error to pass an image variable qualified with
+<strong>readonly</strong> to <strong>imageStore</strong>() or other built-in functions that modify image
+memory.</p>
+</div>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>writeonly</strong> qualifier
+may only write the underlying memory; the underlying memory cannot be read.
+It is an error to pass an image variable qualified with
+<strong>writeonly</strong> to <strong>imageLoad</strong>() or other built-in functions that read image
+memory.</p>
+</div>
+<div class="paragraph">
+<p>A variable could be qualified as both <strong>readonly</strong> and <strong>writeonly</strong>, disallowing
+both read and write. Such variables can still be used with some queries, for
+example <strong>imageSize</strong>() and <strong>.length</strong>().</p>
+</div>
+<div class="paragraph">
+<p>Except for image variables qualified with the format qualifiers <strong>r32f</strong>,
+<strong>r32i</strong>, and <strong>r32ui</strong>, image variables must specify a memory qualifier
+(<strong>readonly</strong>, <strong>writeonly</strong>, or both).</p>
+</div>
+<div class="paragraph">
+<p>The memory qualifiers <strong>coherent</strong>, <strong>volatile</strong>, <strong>restrict</strong>, <strong>readonly</strong>, and
+<strong>writeonly</strong> may be used in the declaration of buffer variables (i.e.,
+members of shader storage blocks).
+When a buffer variable is declared with a memory qualifier, the behavior
+specified for memory accesses involving image variables described above
+applies identically to memory accesses involving that buffer variable.
+It is a compile-time error to assign to a buffer variable qualified with
+<strong>readonly</strong> or to read from a buffer variable qualified with <strong>writeonly</strong>.
+The combination <strong>readonly</strong> <strong>writeonly</strong> is allowed.</p>
+</div>
+<div class="paragraph">
+<p>Additionally, memory qualifiers may be used at the block-level declaration
+of a shader storage block, including the combination <strong>readonly</strong> <strong>writeonly</strong>.
+When a block declaration is qualified with a memory qualifier, it is as if
+all of its members were declared with the same memory qualifier.
+For example, the block declaration</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">coherent buffer Block {
+ readonly vec4 member1;
+ vec4 member2;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">buffer Block {
+ coherent readonly vec4 member1;
+ coherent vec4 member2;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Memory qualifiers are only supported in the declarations of image variables,
+buffer variables, and shader storage blocks; it is an error to use such
+qualifiers in any other declarations.</p>
+</div>
+<div class="paragraph">
+<p>When calling user-defined functions, variables qualified with <strong>coherent</strong>,
+<strong>volatile</strong>, <strong>readonly</strong>, or <strong>writeonly</strong> may not be passed to functions whose
+formal parameters lack such qualifiers.
+(See “<a href="#function-definitions">Function Definitions</a>” for more detail on
+function calling.) It is legal to have any additional memory qualifiers on a
+formal parameter, but only <strong>restrict</strong> can be taken away from a calling
+argument, by a formal parameter that lacks the <strong>restrict</strong> qualifier.</p>
+</div>
+<div class="paragraph">
+<p>When a built-in function is called, the code generated is to be based on the
+actual qualification of the calling argument, not on the list of memory
+qualifiers specified on the formal parameter in the prototype.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 funcA(restrict image2D a) { ... }
+vec4 funcB(image2D a) { ... }
+layout(rgba32f) uniform image2D img1;
+layout(rgba32f) coherent uniform image2D img2;
+
+funcA(img1); <span class="comment">// OK, adding "restrict" is allowed</span>
+funcB(img2); <span class="comment">// illegal, stripping "coherent" is not</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Layout qualifiers cannot be used on formal function parameters, and layout
+qualification is not included in parameter matching.</p>
+</div>
+<div class="paragraph">
+<p>Note that the use of <strong>const</strong> in an image variable declaration is qualifying
+the const-ness of the variable being declared, not the image it refers to.
+The qualifier <strong>readonly</strong> qualifies the image memory (as accessed through
+that variable) while <strong>const</strong> qualifies the variable itself.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="order-of-qualification">4.11. Order and Repetition of Qualification</h3>
+<div class="paragraph">
+<p>When multiple qualifiers are present in a declaration, they may appear in
+any order, but they must all appear before the type.
+The <strong>layout</strong> qualifier is the only qualifier that can appear more than once.
+Further, a declaration can have at most one storage qualifier, at most one
+auxiliary storage qualifier, and at most one interpolation qualifier.
+Multiple memory qualifiers can be used.
+Any violation of these rules will cause a compile-time error.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="empty-declarations">4.12. Empty Declarations</h3>
+<div class="paragraph">
+<p><em>Empty declarations</em> are declarations without a variable name, meaning no
+object is instantiated by the declaration.
+Generally, empty declarations are allowed.
+Some are useful when declaring structures, while many others have no effect.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span>; <span class="comment">// No effect</span>
+<span class="keyword">struct</span> S {<span class="predefined-type">int</span> x;}; <span class="comment">// Defines a struct S</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The combinations of qualifiers that cause compile-time or link-time errors
+are the same whether or not the declaration is empty, for example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">invariant in <span class="predefined-type">float</span> x; <span class="comment">// Error. An input cannot be invariant.</span>
+invariant in <span class="predefined-type">float</span>; <span class="comment">// Error even though no variable is declared.</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="operators-and-expressions">5. Operators and Expressions</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="operators">5.1. Operators</h3>
+<div class="paragraph">
+<p>The OpenGL ES Shading Language has the following operators.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Precedence</th>
+<th class="tableblock halign-left valign-top">Operator Class</th>
+<th class="tableblock halign-left valign-top">Operators</th>
+<th class="tableblock halign-left valign-top">Associativity</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1 (highest)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">parenthetical grouping</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">( )</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">NA</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">array subscript<br>
+ function call and constructor structure<br>
+ field or method selector, swizzle<br>
+ post fix increment and decrement</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">[ ]<br>
+ ( )<br>
+ .<br>
+ ++ --</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">prefix increment and decrement<br>
+ unary</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">++ --<br>
+ + - ~ !</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Right to Left</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">multiplicative</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">* / %</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">additive</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">+ -</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">6</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise shift</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><< >></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">7</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">relational</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">< > <= >=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">8</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">equality</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">== !=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">9</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise and</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">&</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">10</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise exclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">^</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">11</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise inclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">|</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">12</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical and</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">&&</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">13</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical exclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">^^</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">14</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical inclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">||</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">15</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">selection</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">? :</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Right to Left</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">16</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Assignment<br>
+ arithmetic assignments</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">=<br>
+ += -=<br>
+ *= /=<br>
+ %= <<= >>=<br>
+ &= ^= |=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Right to Left</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">17 (lowest)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">sequence</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">,</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>There is no address-of operator nor a dereference operator.
+There is no typecast operator; constructors are used instead.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="array-operations">5.2. Array Operations</h3>
+<div class="paragraph">
+<p>These are now described in “<a href="#structure-and-array-operations">Structure and
+Array Operations</a>”.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="function-calls">5.3. Function Calls</h3>
+<div class="paragraph">
+<p>If a function returns a value, then a call to that function may be used as
+an expression, whose type will be the type that was used to declare or
+define the function.</p>
+</div>
+<div class="paragraph">
+<p>Function definitions and calling conventions are discussed in
+“<a href="#function-definitions">Function Definitions</a>”.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="constructors">5.4. Constructors</h3>
+<div class="paragraph">
+<p>Constructors use the function call syntax, where the function name is a
+type, and the call makes an object of that type.
+Constructors are used the same way in both initializers and expressions.
+(See “<a href="#shading-language-grammar">Shading Language Grammar</a>” for details.)
+The parameters are used to initialize the constructed value.
+Constructors can be used to request a data type conversion to change from
+one scalar type to another scalar type, or to build larger types out of
+smaller types, or to reduce a larger type to a smaller type.</p>
+</div>
+<div class="paragraph">
+<p>In general, constructors are not built-in functions with predetermined
+prototypes.
+For arrays and structures, there must be exactly one argument in the
+constructor for each element or member.
+For the other types, the arguments must provide a sufficient number of
+components to perform the initialization, and it is an error to
+include so many arguments that they cannot all be used.
+Detailed rules follow.
+The prototypes actually listed below are merely a subset of examples.</p>
+</div>
+<div class="sect3">
+<h4 id="conversion-and-scalar-constructors">5.4.1. Conversion and Scalar Constructors</h4>
+<div class="paragraph">
+<p>Converting between scalar types is done as the following prototypes
+indicate:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span>(uint) <span class="comment">// converts an unsigned integer to a signed integer</span>
+<span class="predefined-type">int</span>(<span class="predefined-type">bool</span>) <span class="comment">// converts a Boolean value to an int</span>
+<span class="predefined-type">int</span>(<span class="predefined-type">float</span>) <span class="comment">// converts a float value to an int</span>
+uint(<span class="predefined-type">int</span>) <span class="comment">// converts a signed integer value to an unsigned integer</span>
+uint(<span class="predefined-type">bool</span>) <span class="comment">// converts a Boolean value to an unsigned integer</span>
+uint(<span class="predefined-type">float</span>) <span class="comment">// converts a float value to an unsigned integer</span>
+<span class="predefined-type">bool</span>(<span class="predefined-type">int</span>) <span class="comment">// converts a signed integer value to a Boolean</span>
+<span class="predefined-type">bool</span>(uint) <span class="comment">// converts an unsigned integer value to a Boolean value</span>
+<span class="predefined-type">bool</span>(<span class="predefined-type">float</span>) <span class="comment">// converts a float value to a Boolean</span>
+<span class="predefined-type">float</span>(<span class="predefined-type">int</span>) <span class="comment">// converts a signed integer value to a float</span>
+<span class="predefined-type">float</span>(uint) <span class="comment">// converts an unsigned integer value to a float value</span>
+<span class="predefined-type">float</span>(<span class="predefined-type">bool</span>) <span class="comment">// converts a Boolean value to a float</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When constructors are used to convert a floating-point type to an integer
+type, the fractional part of the floating-point value is dropped.
+It is undefined to convert a negative floating-point value to an <strong>uint</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Integer values having more bits of precision than a single-precision
+floating-point mantissa will lose precision when converted to <strong>float</strong>.</p>
+</div>
+<div class="paragraph">
+<p>When a constructor is used to convert any integer or floating-point type to
+a <strong>bool</strong>, 0 and 0.0 are converted to <strong>false</strong>, and non-zero values are
+converted to <strong>true</strong>.
+When a constructor is used to convert a <strong>bool</strong> to any integer or
+floating-point type, <strong>false</strong> is converted to 0 or 0.0, and <strong>true</strong> is
+converted to 1 or 1.0.</p>
+</div>
+<div class="paragraph">
+<p>The constructor <strong>int</strong>(<strong>uint</strong>) preserves the bit pattern in the argument,
+which will change the argument’s value if its sign bit is set.
+The constructor <strong>uint</strong>(<strong>int</strong>) preserves the bit pattern in the argument,
+which will change its value if it is negative.</p>
+</div>
+<div class="paragraph">
+<p>Identity constructors, like <strong>float</strong>(<strong>float</strong>) are also legal, but of little
+use.</p>
+</div>
+<div class="paragraph">
+<p>Scalar constructors with non-scalar parameters can be used to take the first
+element from a non-scalar.
+For example, the constructor <strong>float</strong>(<strong>vec3</strong>) will select the first component
+of the <strong>vec3</strong> parameter.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="vector-and-matrix-constructors">5.4.2. Vector and Matrix Constructors</h4>
+<div class="paragraph">
+<p>Constructors can be used to create vectors or matrices from a set of
+scalars, vectors, or matrices.
+This includes the ability to shorten vectors.</p>
+</div>
+<div class="paragraph">
+<p>If there is a single scalar parameter to a vector constructor, it is used to
+initialize all components of the constructed vector to that scalar’s value.
+If there is a single scalar parameter to a matrix constructor, it is used to
+initialize all the components on the matrix’s diagonal, with the remaining
+components initialized to 0.0.</p>
+</div>
+<div class="paragraph">
+<p>If a vector is constructed from multiple scalars, one or more vectors, or
+one or more matrices, or a mixture of these, the vector’s components will be
+constructed in order from the components of the arguments.
+The arguments will be consumed left to right, and each argument will have
+all its components consumed, in order, before any components from the next
+argument are consumed.
+Similarly for constructing a matrix from multiple scalars or vectors, or a
+mixture of these.
+Matrix components will be constructed and consumed in column major order.
+In these cases, there must be enough components provided in the arguments to
+provide an initializer for every component in the constructed value.
+It is an error to provide extra arguments beyond this last used
+argument.</p>
+</div>
+<div class="paragraph">
+<p>If a matrix is constructed from a matrix, then each component (column <em>i</em>,
+row <em>j</em>) in the result that has a corresponding component (column <em>i</em>, row
+<em>j</em>) in the argument will be initialized from there.
+All other components will be initialized to the identity matrix.
+If a matrix argument is given to a matrix constructor, it is
+an error to have any other arguments.</p>
+</div>
+<div class="paragraph">
+<p>If the basic type (<strong>bool</strong>, <strong>int</strong>,
+or <strong>float</strong>)
+of a parameter to a
+constructor does not match the basic type of the object being constructed,
+the scalar construction rules (above) are used to convert the parameters.</p>
+</div>
+<div class="paragraph">
+<p>Some useful vector constructors are as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3(<span class="predefined-type">float</span>) <span class="comment">// initializes each component of the vec3 with the float</span>
+vec4(ivec4) <span class="comment">// makes a vec4 with component-wise conversion</span>
+vec4(mat2) <span class="comment">// the vec4 is column 0 followed by column 1</span>
+vec2(<span class="predefined-type">float</span>, <span class="predefined-type">float</span>) <span class="comment">// initializes a vec2 with 2 floats</span>
+ivec3(<span class="predefined-type">int</span>, <span class="predefined-type">int</span>, <span class="predefined-type">int</span>) <span class="comment">// initializes an ivec3 with 3 ints</span>
+bvec4(<span class="predefined-type">int</span>, <span class="predefined-type">int</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>) <span class="comment">// uses 4 Boolean conversions</span>
+vec2(vec3) <span class="comment">// drops the third component of a vec3</span>
+vec3(vec4) <span class="comment">// drops the fourth component of a vec4</span>
+vec3(vec2, <span class="predefined-type">float</span>) <span class="comment">// vec3.x = vec2.x, vec3.y = vec2.y, vec3.z = float</span>
+vec3(<span class="predefined-type">float</span>, vec2) <span class="comment">// vec3.x = float, vec3.y = vec2.x, vec3.z = vec2.y</span>
+vec4(vec3, <span class="predefined-type">float</span>)
+vec4(<span class="predefined-type">float</span>, vec3)
+vec4(vec2, vec2)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Some examples of these are:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 color = vec4(<span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span>);
+vec4 rgba = vec4(<span class="float">1</span><span class="float">.0</span>); <span class="comment">// sets each component to 1.0</span>
+vec3 rgb = vec3(color); <span class="comment">// drop the 4th component</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>To initialize the diagonal of a matrix with all other elements set to zero:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat2(<span class="predefined-type">float</span>)
+mat3(<span class="predefined-type">float</span>)
+mat4(<span class="predefined-type">float</span>)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>That is, <em>result[i][j]</em> is set to the <em>float</em> argument for all \(i
+= j\) and set to 0 for all \(i \neq j\).</p>
+</div>
+<div class="paragraph">
+<p>To initialize a matrix by specifying vectors or scalars, the components are
+assigned to the matrix elements in column-major order.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat2(vec2, vec2); <span class="comment">// one column per argument</span>
+mat3(vec3, vec3, vec3); <span class="comment">// one column per argument</span>
+mat4(vec4, vec4, vec4, vec4); <span class="comment">// one column per argument</span>
+mat3x2(vec2, vec2, vec2); <span class="comment">// one column per argument</span>
+mat2(<span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// first column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>); <span class="comment">// second column</span>
+mat3(<span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// first column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// second column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>); <span class="comment">// third column</span>
+mat4(<span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// first column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// second column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// third column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>); <span class="comment">// fourth column</span>
+mat2x3(vec2, <span class="predefined-type">float</span>, <span class="comment">// first column</span>
+ vec2, <span class="predefined-type">float</span>); <span class="comment">// second column</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A wide range of other possibilities exist, to construct a matrix from
+vectors and scalars, as long as enough components are present to initialize
+the matrix.
+To construct a matrix from a matrix:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat3x3(mat4x4); <span class="comment">// takes the upper-left 3x3 of the mat4x4</span>
+mat2x3(mat4x2); <span class="comment">// takes the upper-left 2x2 of the mat4x4, last row is 0,0</span>
+mat4x4(mat3x3); <span class="comment">// puts the mat3x3 in the upper-left, sets the lower right</span>
+ <span class="comment">// component to 1, and the rest to 0</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="structure-constructors">5.4.3. Structure Constructors</h4>
+<div class="paragraph">
+<p>Once a structure is defined, and its type is given a name, a constructor is
+available with the same name to construct instances of that structure.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> light {
+ <span class="predefined-type">float</span> intensity;
+ vec3 position;
+};
+
+light lightVar = light(<span class="float">3</span><span class="float">.0</span>, vec3(<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>));</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The arguments to the constructor will be used to set the structure’s
+members, in order, using one argument per member.
+Each argument must be the same type as the member it
+sets.</p>
+</div>
+<div class="paragraph">
+<p>Structure constructors can be used as initializers or in expressions.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="array-constructors">5.4.4. Array Constructors</h4>
+<div class="paragraph">
+<p>Array types can also be used as constructor names, which can then be used in
+expressions or initializers.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">const</span> <span class="predefined-type">float</span> c[<span class="integer">3</span>] = <span class="predefined-type">float</span>[<span class="integer">3</span>](<span class="float">5</span><span class="float">.0</span>, <span class="float">7</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>);
+<span class="directive">const</span> <span class="predefined-type">float</span> d[<span class="integer">3</span>] = <span class="predefined-type">float</span>[](<span class="float">5</span><span class="float">.0</span>, <span class="float">7</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>);
+
+<span class="predefined-type">float</span> g;
+...
+<span class="predefined-type">float</span> a[<span class="integer">5</span>] = <span class="predefined-type">float</span>[<span class="integer">5</span>](g, <span class="integer">1</span>, g, <span class="float">2</span><span class="float">.3</span>, g);
+<span class="predefined-type">float</span> b[<span class="integer">3</span>];
+
+b = <span class="predefined-type">float</span>[<span class="integer">3</span>](g, g + <span class="float">1</span><span class="float">.0</span>, g + <span class="float">2</span><span class="float">.0</span>);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>There must be exactly the same number of arguments as the size of the array
+being constructed.
+If no size is present in the constructor, then the array is explicitly sized
+to the number of arguments provided.
+The arguments are assigned in order, starting at element 0, to the elements
+of the constructed array.
+Each argument must be the same type as the element type of the
+array.</p>
+</div>
+<div class="paragraph">
+<p>Arrays of arrays are similarly constructed, and the size for any dimension
+is <strong class="purple">optional</strong></p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 b[<span class="integer">2</span>] = ...;
+vec4[<span class="integer">3</span>][<span class="integer">2</span>](b, b, b); <span class="comment">// constructor</span>
+vec4[][<span class="integer">2</span>](b, b, b); <span class="comment">// constructor, valid, size deduced</span>
+vec4[<span class="integer">3</span>][](b, b, b); <span class="comment">// constructor, valid, size deduced</span>
+vec4[][](b, b, b); <span class="comment">// constructor, valid, both sizes deduced</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="vector-components">5.5. Vector Components</h3>
+<div class="paragraph">
+<p>The names of the components of a vector
+are denoted by a single letter.
+As a notational convenience, several letters are associated with each
+component based on common usage of position, color or texture coordinate
+vectors.
+The individual components can be selected by following the variable name
+with period (<strong>.</strong>) and then the component name.</p>
+</div>
+<div class="paragraph">
+<p>The component names supported are:</p>
+</div>
+<table class="tableblock frame-all grid-all fit-content">
+<colgroup>
+<col>
+<col>
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>{ x, y, z, w }</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Useful when accessing vectors that represent points or normals</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>{ r, g, b, a }</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Useful when accessing vectors that represent colors</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>{ s, t, p, q }</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Useful when accessing vectors that represent texture coordinates</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The component names <em>x</em>, <em>r</em>, and <em>s</em> are, for example, synonyms for the
+same (first) component in a vector.</p>
+</div>
+<div class="paragraph">
+<p>Note that the third component of the texture coordinate set, <em>r</em> in
+OpenGL ES, has been renamed <em>p</em> so as to avoid the confusion with <em>r</em> (for
+red) in a color.</p>
+</div>
+<div class="paragraph">
+<p>Accessing components beyond those declared for the type is
+an error so, for example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec2 pos;
+pos.x <span class="comment">// is legal</span>
+pos.z <span class="comment">// is illegal</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The component selection syntax allows multiple components to be selected by
+appending their names (from the same name set) after the period (<strong>.</strong>).</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 v4;
+v4.rgba; <span class="comment">// is a vec4 and the same as just using v4,</span>
+v4.rgb; <span class="comment">// is a vec3,</span>
+v4.b; <span class="comment">// is a float,</span>
+v4.xy; <span class="comment">// is a vec2,</span>
+v4.xgba; <span class="comment">// is illegal - the component names do not come from the same set</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>No more than 4 components can be selected.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 v4;
+v4.xyzwxy; <span class="comment">// is illegal since it has 6 components</span>
+(v4.xyzwxy).xy; <span class="comment">// is illegal since the intermediate value has 6</span>
+components</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The order of the components can be different to swizzle them, or replicated:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 pos = vec4(<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>, <span class="float">4</span><span class="float">.0</span>);
+vec4 swiz = pos.wzyx; <span class="comment">// swiz = (4.0, 3.0, 2.0, 1.0)</span>
+vec4 dup = pos.xxyy; <span class="comment">// dup = (1.0, 1.0, 2.0, 2.0)</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This notation is more concise than the constructor syntax.
+To form an r-value, it can be applied to any expression that results in a
+vector r-value.</p>
+</div>
+<div class="paragraph">
+<p>The component group notation can occur on the left hand side of an
+expression.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 pos = vec4(<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>, <span class="float">4</span><span class="float">.0</span>);
+pos.xw = vec2(<span class="float">5</span><span class="float">.0</span>, <span class="float">6</span><span class="float">.0</span>); <span class="comment">// pos = (5.0, 2.0, 3.0, 6.0)</span>
+pos.wx = vec2(<span class="float">7</span><span class="float">.0</span>, <span class="float">8</span><span class="float">.0</span>); <span class="comment">// pos = (8.0, 2.0, 3.0, 7.0)</span>
+pos.xx = vec2(<span class="float">3</span><span class="float">.0</span>, <span class="float">4</span><span class="float">.0</span>); <span class="comment">// illegal - 'x' used twice</span>
+pos.xy = vec3(<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>); <span class="comment">// illegal - mismatch between vec2 and vec3</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>To form an l-value, swizzling must further be applied to an l-value and
+contain no duplicate components. It results in an l-value of scalar or
+vector type, depending on number of components specified.</p>
+</div>
+<div class="paragraph">
+<p>Array subscripting syntax can also be applied to vectors (but not to
+scalars) to provide numeric indexing.
+So in</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 pos;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p><em>pos[2]</em> refers to the third element of <em>pos</em> and is equivalent to <em>pos.z</em>.
+This allows variable indexing into a vector, as well as a generic way of
+accessing components.
+Any integer expression can be used as the subscript.
+The first component is at index zero.
+Reading from or writing to a vector using a constant integral expression
+with a value that is negative or greater than or equal to the size of the
+vector results in an error.
+When indexing with non-constant expressions, behavior is undefined if the
+index is negative, or greater than or equal to the size of the vector.</p>
+</div>
+<div class="paragraph">
+<p>Note that scalars are not considered to be single-component vectors and
+therefore the use of component selection operators on scalars is illegal.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="matrix-components">5.6. Matrix Components</h3>
+<div class="paragraph">
+<p>The components of a matrix can be accessed using array subscripting syntax.
+Applying a single subscript to a matrix treats the matrix as an array of
+column vectors, and selects a single column, whose type is a vector of the
+same size as the (column size of the) matrix.
+The leftmost column is column 0.
+A second subscript would then operate on the resulting vector, as defined
+earlier for vectors.
+Hence, two subscripts select a column and then a row.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat4 m;
+m[<span class="integer">1</span>] = vec4(<span class="float">2</span><span class="float">.0</span>); <span class="comment">// sets the second column to all 2.0</span>
+m[<span class="integer">0</span>][<span class="integer">0</span>] = <span class="float">1</span><span class="float">.0</span>; <span class="comment">// sets the upper left element to 1.0</span>
+m[<span class="integer">2</span>][<span class="integer">3</span>] = <span class="float">2</span><span class="float">.0</span>; <span class="comment">// sets the 4th element of the third column to 2.0</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Behavior is undefined when accessing a component outside the bounds of a
+matrix with a non-constant expression.
+It is an error to access a matrix with a constant expression
+that is outside the bounds of the matrix.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="structure-and-array-operations">5.7. Structure and Array Operations</h3>
+<div class="paragraph">
+<p>The members of a structure and the <strong>length</strong>() method of an array are
+selected using the period (<strong>.</strong>).</p>
+</div>
+<div class="paragraph">
+<p>In total, only the following operators are allowed to operate on arrays and
+structures as whole entities:</p>
+</div>
+<table class="tableblock frame-all grid-all fit-content">
+<colgroup>
+<col>
+<col>
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">field selector</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>.</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">equality</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>==</strong> <strong>!=</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">assignment</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>=</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Ternary operator</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>?:</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Sequence operator</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>,</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">indexing (arrays only)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>[</strong> <strong>]</strong></p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The equality operators and assignment operator are only allowed if the two
+operands are same size and type.
+The operands cannot contain any opaque types.
+Structure types must be of the same declared structure.
+Both array operands must be
+compile-time
+sized.
+When using the equality operators, two structures are equal if and only if
+all the members are component-wise equal, and two arrays are equal if and
+only if all the elements are element-wise equal.</p>
+</div>
+<div class="paragraph">
+<p>Array elements are accessed using the array subscript operator (<strong>[ ]</strong>).
+An example of accessing an array element is</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">diffuseColor += lightIntensity[<span class="integer">3</span>] * NdotL;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Array indices start at zero.
+Array elements are accessed using an expression whose type is <strong>int</strong> or
+<strong>uint</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Behavior is undefined if a shader subscripts an array with an index less
+than 0 or greater than or equal to the size the array was declared with.</p>
+</div>
+<div class="paragraph">
+<p>Arrays can also be accessed with the method operator (<strong>.</strong>) and the <strong>length</strong>
+method to query the size of the array:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">lightIntensity.length() <span class="comment">// return the size of the array</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="assignments">5.8. Assignments</h3>
+<div class="paragraph">
+<p>Assignments of values to variable names are done with the assignment
+operator (<strong>=</strong>):</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><em>lvalue-expression</em> = <em>rvalue-expression</em></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The <em>lvalue-expression</em> evaluates to an l-value.
+The assignment operator stores the value of <em>rvalue-expression</em> into the
+l-value and returns an r-value with the type and precision of
+<em>lvalue-expression</em>.
+The <em>lvalue-expression</em> and <em>rvalue-expression</em> must have the same
+type.
+Any type-conversions must be specified explicitly via constructors.
+It is an error if the l-value is not writable.
+Variables that are built-in types, entire structures or arrays, structure
+members, l-values with the field selector (<strong>.</strong>) applied to select components
+or swizzles without repeated fields, l-values within parentheses, and
+l-values dereferenced with the array subscript operator (<strong>[ ]</strong>) are all
+l-values.
+Other binary or unary expressions, function names, swizzles with repeated
+fields, and constants cannot be l-values.
+The ternary operator (<strong>?:</strong>) is also not allowed as an l-value.
+Using an incorrect expression as an l-value results in an error.</p>
+</div>
+<div class="paragraph">
+<p>Expressions on the left of an assignment are evaluated before expressions on
+the right of the assignment.</p>
+</div>
+<div class="paragraph">
+<p>The other assignment operators are</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>add into (<strong>+=</strong>)</p>
+</li>
+<li>
+<p>subtract from (<strong>-=</strong>)</p>
+</li>
+<li>
+<p>multiply into (<strong>*=</strong>)</p>
+</li>
+<li>
+<p>divide into (<strong>/=</strong>)</p>
+</li>
+<li>
+<p>modulus into (<strong>%=</strong>)</p>
+</li>
+<li>
+<p>left shift by (<strong><<=</strong>)</p>
+</li>
+<li>
+<p>right shift by (<strong>>>=</strong>)</p>
+</li>
+<li>
+<p>and into (<strong>&=</strong>)</p>
+</li>
+<li>
+<p>inclusive-or into (<strong>|=</strong>)</p>
+</li>
+<li>
+<p>exclusive-or into (<strong>^=</strong>)</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>where the general expression</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><em>lvalue</em> <em>op</em>= <em>expression</em></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><em>lvalue</em> = <em>lvalue</em> <em>op</em> <em>expression</em></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>where <em>lvalue</em> is the value returned by <em>lvalue-expression</em>, <em>op</em> is as
+described below, and the <em>lvalue-expression</em> and <em>expression</em> must satisfy
+the semantic requirements of both <em>op</em> and equals (<strong>=</strong>).</p>
+</div>
+<div class="paragraph">
+<p>Reading a variable before writing (or initializing) it is legal, however the
+value is undefined.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="expressions">5.9. Expressions</h3>
+<div class="paragraph">
+<p>Expressions in the shading language are built from the following:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Constants of type <strong>bool</strong>, all integral types, all floating-point types,
+all vector types, and all matrix types.</p>
+</li>
+<li>
+<p>Constructors of all types.</p>
+</li>
+<li>
+<p>Variable names of all types.</p>
+</li>
+<li>
+<p>Arrays with the length method applied.</p>
+</li>
+<li>
+<p>Subscripted arrays.</p>
+</li>
+<li>
+<p>Function calls that return values.
+In some cases, function calls returning <strong>void</strong> are also allowed in
+expressions as specified below.</p>
+</li>
+<li>
+<p>Component field selectors and array subscript results.</p>
+</li>
+<li>
+<p>Parenthesized expressions.
+Any expression, including expressions with void type can be
+parenthesized.
+Parentheses can be used to group operations.
+Operations within parentheses are done before operations across
+parentheses.</p>
+</li>
+<li>
+<p>The arithmetic binary operators add (<strong>+</strong>), subtract (<strong>-</strong>), multiply
+(<strong>*</strong>), and divide (<strong>/</strong>) operate on integer and floating-point scalars,
+vectors, and matrices.
+If the operands are integral types, they must both be signed or both be
+unsigned.
+All arithmetic binary operators result in the same fundamental type
+(signed integer, unsigned integer, or floating-point) as the operands
+they operate on.
+The following cases are valid</p>
+<div class="ulist">
+<ul>
+<li>
+<p>The two operands are scalars.
+In this case the operation is applied, resulting in a scalar.</p>
+</li>
+<li>
+<p>One operand is a scalar, and the other is a vector or matrix.
+In this case, the scalar operation is applied independently to each
+component of the vector or matrix, resulting in the same size vector or
+matrix.</p>
+</li>
+<li>
+<p>The two operands are vectors of the same size.
+In this case, the operation is done component-wise resulting in the
+same size vector.</p>
+</li>
+<li>
+<p>The operator is add (<strong>+</strong>), subtract (<strong>-</strong>), or divide (<strong>/</strong>), and the
+operands are matrices with the same number of rows and the same number
+of columns.
+In this case, the operation is done component-wise resulting in the
+same size matrix.</p>
+</li>
+<li>
+<p>The operator is multiply (<strong>*</strong>), where both operands are matrices or one
+operand is a vector and the other a matrix.
+A right vector operand is treated as a column vector and a left vector
+operand as a row vector.
+In all these cases, it is required that the number of columns of the
+left operand is equal to the number of rows of the right operand.
+Then, the multiply (<strong>*</strong>) operation does a linear algebraic multiply,
+yielding an object that has the same number of rows as the left operand
+and the same number of columns as the right operand.
+“<a href="#vector-and-matrix-operations">Vector and Matrix Operations</a>”
+explains in more detail how vectors and matrices are operated on.</p>
+<div class="openblock">
+<div class="content">
+<div class="paragraph">
+<p>All other cases result in an error.</p>
+</div>
+<div class="paragraph">
+<p>Use the built-in functions <strong>dot</strong>, <strong>cross</strong>, <strong>matrixCompMult</strong>, and
+<strong>outerProduct</strong>, to get, respectively, vector dot product, vector cross
+product, matrix component-wise multiplication, and the matrix product of a
+column vector times a row vector.</p>
+</div>
+</div>
+</div>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>The operator modulus (<strong>%</strong>) operates on signed or unsigned integers or
+integer vectors.
+The operand types must both be signed or both be unsigned.
+The operands cannot be vectors of differing size; this is
+an error.
+If one operand is a scalar and the other vector, then the scalar is
+applied component-wise to the vector, resulting in the same type as the
+vector.
+If both are vectors of the same size, the result is computed
+component-wise.
+The resulting value is undefined for any component computed with a
+second operand that is zero, while results for other components with
+non-zero second operands remain defined.
+If both operands are non-negative, then the remainder is non-negative.
+Results are undefined if one or both operands are negative.
+The operator modulus (<strong>%</strong>) is not defined for any other data types
+(non-integer types).</p>
+</li>
+<li>
+<p>The arithmetic unary operators negate (<strong>-</strong>), post- and pre-increment and
+decrement (<strong>--</strong> and <strong>++</strong>) operate on integer or floating-point values
+(including vectors and matrices).
+All unary operators work component-wise on their operands.
+These result with the same type they operated on.
+For post- and pre-increment and decrement, the expression must be a writable
+l-value.
+Pre-increment and pre-decrement add or subtract 1 or 1.0 to the contents
+of the expression they operate on, and the value of the pre-increment or
+pre-decrement expression is the resulting value of that modification.
+Post-increment and post-decrement expressions add or subtract 1 or 1.0
+to the contents of the expression they operate on, but the resulting
+expression has the expression’s value before the post-increment or
+post-decrement was executed.</p>
+</li>
+<li>
+<p>The relational operators greater than (<strong>></strong>), less than (<strong><</strong>), greater
+than or equal (<strong>>=</strong>), and less than or equal (<strong><=</strong>) operate only on
+scalar integer and scalar floating-point expressions.
+The result is scalar Boolean.
+The types of the operands must match.
+To do component-wise relational comparisons on vectors, use the built-in
+functions <strong>lessThan</strong>, <strong>lessThanEqual</strong>, <strong>greaterThan</strong>, and
+<strong>greaterThanEqual.</strong></p>
+</li>
+<li>
+<p>The equality operators <strong>equal</strong> (<strong>==</strong>), and not equal (<strong>!=</strong>) operate on
+all types except opaque types.
+They result in a scalar Boolean.
+The types of the operands must match.
+For vectors, matrices, structures, and arrays, all components, members,
+or elements of one operand must equal the corresponding components,
+members, or elements in the other operand for the operands to be
+considered equal.
+To get a vector of component-wise equality results for vectors, use the
+built-in functions <strong>equal</strong> and <strong>notEqual</strong>.</p>
+</li>
+<li>
+<p>The logical binary operators and (<strong>&&</strong>), or (<strong>||</strong>), and exclusive or
+(<strong>^^</strong>) operate only on two Boolean expressions and result in a Boolean
+expression.
+And (<strong>&&</strong>) will only evaluate the right hand operand if the left hand
+operand evaluated to <strong>true</strong>.
+Or (<strong>||</strong>) will only evaluate the right hand operand if the left hand
+operand evaluated to <strong>false</strong>.
+Exclusive or (<strong>^^</strong>) will always evaluate both operands.</p>
+</li>
+<li>
+<p>The logical unary operator not (<strong>!</strong>).
+It operates only on a Boolean expression and results in a Boolean
+expression.
+To operate on a vector, use the built-in function <strong>not</strong>.</p>
+</li>
+<li>
+<p>The sequence (<strong>,</strong>) operator that operates on expressions by returning
+the type and value of the right-most expression in a comma separated
+list of expressions.
+All expressions are evaluated, in order, from left to right.
+The operands to the sequence operator may have <strong>void</strong> type.</p>
+</li>
+<li>
+<p>The ternary selection operator (<strong>?:</strong>).
+It operates on three expressions (<em>exp1</em> <strong>?</strong> <em>exp2</em> <strong>:</strong> <em>exp3</em>).
+This operator evaluates the first expression, which must result in a
+scalar Boolean.
+If the result is true, it selects to evaluate the second expression,
+otherwise it selects to evaluate the third expression.
+Only one of the second and third expressions is evaluated.
+The second and third expressions can be any type, including <strong>void</strong>, as
+long their types match.
+This resulting matching type is the type of the entire expression.</p>
+</li>
+<li>
+<p>The one’s complement operator (<strong>~</strong>).
+The operand must be of type signed or unsigned integer or integer
+vector, and the result is the one’s complement of its operand; each bit
+of each component is complemented, including any sign bits.</p>
+</li>
+<li>
+<p>The shift operators (<strong><<</strong>) and (<strong>>></strong>).
+For both operators, the operands must be signed or unsigned integers or
+integer vectors.
+One operand can be signed while the other is unsigned.
+In all cases, the resulting type will be the same type as the left
+operand.
+If the first operand is a scalar, the second operand has to be a scalar
+as well.
+If the first operand is a vector, the second operand must be a scalar or
+a vector with the same size as the first operand, and the result is
+computed component-wise.
+The result is undefined if the right operand is negative, or greater
+than or equal to the number of bits in the left expression’s base type.
+The value of E1 << E2 is E1 (interpreted as a bit pattern) left-shifted
+by E2 bits.
+The value of E1 >> E2 is E1 right-shifted by E2 bit positions.
+If E1 is a signed integer, the right-shift will extend the sign bit.
+If E1 is an unsigned integer, the right-shift will zero-extend.</p>
+</li>
+<li>
+<p>The bitwise operators and (<strong>&</strong>), exclusive-or (<strong>^</strong>), and inclusive-or
+(<strong>|</strong>).
+The operands must be of type signed or unsigned integers or integer
+vectors.
+The operands cannot be vectors of differing size; this is an error.
+If one operand is a scalar and the other a vector, the scalar is applied
+component-wise to the vector, resulting in the same type as the vector.
+The fundamental types of the operands (signed or unsigned) must match,
+and will be the resulting fundamental type.
+For and (<strong>&</strong>), the result is the bitwise-and function of the operands.
+For exclusive-or (<strong>^</strong>), the result is the bitwise exclusive-or function
+of the operands.
+For inclusive-or (<strong>|</strong>), the result is the bitwise inclusive-or function
+of the operands.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>For a complete specification of the syntax of expressions, see
+“<a href="#shading-language-grammar">Shading Language Grammar</a>”.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="vector-and-matrix-operations">5.10. Vector and Matrix Operations</h3>
+<div class="paragraph">
+<p>With a few exceptions, operations are component-wise.
+Usually, when an operator operates on a vector or matrix, it is operating
+independently on each component of the vector or matrix, in a component-wise
+fashion.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3 v, u;
+<span class="predefined-type">float</span> f;
+v = u + f;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will be equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">v.x = u.x + f;
+v.y = u.y + f;
+v.z = u.z + f;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>And</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3 v, u, w;
+w = v + u;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will be equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">w.x = v.x + u.x;
+w.y = v.y + u.y;
+w.z = v.z + u.z;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and likewise for most operators and all integer and floating-point vector
+and matrix types.
+The exceptions are matrix multiplied by vector, vector multiplied by matrix,
+and matrix multiplied by matrix.
+These do not operate component-wise, but rather perform the correct linear
+algebraic multiply.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3 v, u;
+mat3 m;
+u = v * m;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">u.x = dot(v, m[<span class="integer">0</span>]); <span class="comment">// m[0] is the left column of m</span>
+u.y = dot(v, m[<span class="integer">1</span>]); <span class="comment">// dot(a,b) is the inner (dot) product of a and b</span>
+u.z = dot(v, m[<span class="integer">2</span>]);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>And</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">u = m * v;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">u.x = m[<span class="integer">0</span>].x * v.x + m[<span class="integer">1</span>].x * v.y + m[<span class="integer">2</span>].x * v.z;
+u.y = m[<span class="integer">0</span>].y * v.x + m[<span class="integer">1</span>].y * v.y + m[<span class="integer">2</span>].y * v.z;
+u.z = m[<span class="integer">0</span>].z * v.x + m[<span class="integer">1</span>].z * v.y + m[<span class="integer">2</span>].z * v.z;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>And</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat3 m, n, r;
+r = m * n;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">r[<span class="integer">0</span>].x = m[<span class="integer">0</span>].x * n[<span class="integer">0</span>].x + m[<span class="integer">1</span>].x * n[<span class="integer">0</span>].y + m[<span class="integer">2</span>].x * n[<span class="integer">0</span>].z;
+r[<span class="integer">1</span>].x = m[<span class="integer">0</span>].x * n[<span class="integer">1</span>].x + m[<span class="integer">1</span>].x * n[<span class="integer">1</span>].y + m[<span class="integer">2</span>].x * n[<span class="integer">1</span>].z;
+r[<span class="integer">2</span>].x = m[<span class="integer">0</span>].x * n[<span class="integer">2</span>].x + m[<span class="integer">1</span>].x * n[<span class="integer">2</span>].y + m[<span class="integer">2</span>].x * n[<span class="integer">2</span>].z;
+r[<span class="integer">0</span>].y = m[<span class="integer">0</span>].y * n[<span class="integer">0</span>].x + m[<span class="integer">1</span>].y * n[<span class="integer">0</span>].y + m[<span class="integer">2</span>].y * n[<span class="integer">0</span>].z;
+r[<span class="integer">1</span>].y = m[<span class="integer">0</span>].y * n[<span class="integer">1</span>].x + m[<span class="integer">1</span>].y * n[<span class="integer">1</span>].y + m[<span class="integer">2</span>].y * n[<span class="integer">1</span>].z;
+r[<span class="integer">2</span>].y = m[<span class="integer">0</span>].y * n[<span class="integer">2</span>].x + m[<span class="integer">1</span>].y * n[<span class="integer">2</span>].y + m[<span class="integer">2</span>].y * n[<span class="integer">2</span>].z;
+r[<span class="integer">0</span>].z = m[<span class="integer">0</span>].z * n[<span class="integer">0</span>].x + m[<span class="integer">1</span>].z * n[<span class="integer">0</span>].y + m[<span class="integer">2</span>].z * n[<span class="integer">0</span>].z;
+r[<span class="integer">1</span>].z = m[<span class="integer">0</span>].z * n[<span class="integer">1</span>].x + m[<span class="integer">1</span>].z * n[<span class="integer">1</span>].y + m[<span class="integer">2</span>].z * n[<span class="integer">1</span>].z;
+r[<span class="integer">2</span>].z = m[<span class="integer">0</span>].z * n[<span class="integer">2</span>].x + m[<span class="integer">1</span>].z * n[<span class="integer">2</span>].y + m[<span class="integer">2</span>].z * n[<span class="integer">2</span>].z;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and similarly for other sizes of vectors and matrices.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="evaluation-of-expressions">5.11. Evaluation of Expressions</h3>
+<div class="paragraph">
+<p>In general expressions must be evaluated in the order specified by the
+precedence of operations and may only be regrouped if the result is the same or
+where the result is undefined.
+No other transforms may be applied that affect the result of an operation.
+GLSL ES relaxes these requirements for scalar operations in the following
+ways:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Addition and multiplication are assumed to be associative.</p>
+</li>
+<li>
+<p>Multiplication is assumed to be distributive over addition.
+Therefore expressions may be expanded and re-factored.</p>
+</li>
+<li>
+<p>Floating-point division may be replaced by reciprocal and
+multiplication.</p>
+</li>
+<li>
+<p>Multiplication may be replaced by repeated addition.</p>
+</li>
+<li>
+<p>Within the constraints of invariance (where applicable), the precision
+used may vary.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>These rules also apply to the built-in functions.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="statements-and-structure">6. Statements and Structure</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The fundamental building blocks of the OpenGL ES Shading Language are:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>statements and declarations</p>
+</li>
+<li>
+<p>function definitions</p>
+</li>
+<li>
+<p>selection (<strong>if</strong>-<strong>else</strong> and <strong>switch</strong>-<strong>case</strong>-<strong>default</strong>)</p>
+</li>
+<li>
+<p>iteration (<strong>for</strong>, <strong>while</strong>, and <strong>do</strong>-<strong>while</strong>)</p>
+</li>
+<li>
+<p>jumps (<strong>discard</strong>, <strong>return</strong>, <strong>break</strong>, and <strong>continue</strong>)</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>The overall structure of a shader is as follows</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>translation-unit</em> : </dt>
+<dd>
+<p><em>global-declaration</em><br>
+<em>translation-unit</em> <em>global-declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>global-declaration</em> : </dt>
+<dd>
+<p><em>function-definition</em><br>
+<em>declaration</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>That is, a shader is a sequence of declarations and function bodies.
+Function bodies are defined as</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>function-definition</em> : </dt>
+<dd>
+<p><em>function-prototype</em> <strong>{</strong> <em>statement-list</em> <strong>}</strong></p>
+</dd>
+<dt class="hdlist1"><em>statement-list</em> : </dt>
+<dd>
+<p><em>statement</em><br>
+<em>statement-list</em> <em>statement</em></p>
+</dd>
+<dt class="hdlist1"><em>statement</em> : </dt>
+<dd>
+<p><em>compound-statement</em><br>
+<em>simple-statement</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Curly braces are used to group sequences of statements into compound
+statements.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>compound-statement</em> : </dt>
+<dd>
+<p><strong>{</strong> <em>statement-list</em> <strong>}</strong></p>
+</dd>
+<dt class="hdlist1"><em>simple-statement</em> : </dt>
+<dd>
+<p><em>declaration-statement</em><br>
+<em>expression-statement</em><br>
+<em>selection-statement</em><br>
+<em>iteration-statement</em><br>
+<em>jump-statement</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Simple declaration, expression, and jump statements end in a semi-colon.</p>
+</div>
+<div class="paragraph">
+<p>This above is slightly simplified, and the complete grammar specified in
+“<a href="#shading-language-grammar">Shading Language Grammar</a>” should be used as
+the definitive specification.</p>
+</div>
+<div class="paragraph">
+<p>Declarations and expressions have already been discussed.</p>
+</div>
+<div class="sect2">
+<h3 id="function-definitions">6.1. Function Definitions</h3>
+<div class="paragraph">
+<p>As indicated by the grammar above, a valid shader is a sequence of global
+declarations and function definitions.
+A function is declared as the following example shows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// prototype</span>
+returnType functionName (type0 arg0, type1 arg1, ..., typen argn);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and a function is defined like</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// definition</span>
+returnType functionName (type0 arg0, type1 arg1, ..., typen argn)
+{
+ <span class="comment">// do some computation</span>
+ <span class="keyword">return</span> returnValue;
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>returnType</em> must be present and cannot be void, or:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> functionName (type0 arg0, type1 arg1, ..., typen argn)
+{
+ <span class="comment">// do some computation</span>
+ <span class="keyword">return</span>; <span class="comment">// optional</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Each of the <em>typeN</em> must include a type and can optionally include a
+parameter qualifier and/or <strong>const</strong>.</p>
+</div>
+<div class="paragraph">
+<p>A function is called by using its name followed by a list of arguments in
+parentheses.</p>
+</div>
+<div class="paragraph">
+<p>Arrays are allowed as arguments and as the return type.
+In both cases, the array must be
+compile-time
+sized.
+An array is passed or returned by using just its name, without brackets, and
+the size of the array must match the size specified in the function’s
+declaration.</p>
+</div>
+<div class="paragraph">
+<p>Structures are also allowed as argument types.
+The return type can also be a structure.</p>
+</div>
+<div class="paragraph">
+<p>See “<a href="#shading-language-grammar">Shading Language Grammar</a>” for the
+definitive reference on the syntax to declare and define functions.</p>
+</div>
+<div class="paragraph">
+<p>All functions must be either declared with a prototype or defined with a
+body before they are called.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> myfunc (<span class="predefined-type">float</span> f, <span class="comment">// f is an input parameter</span>
+ out <span class="predefined-type">float</span> g); <span class="comment">// g is an output parameter</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Functions that return no value must be declared as <strong>void</strong>.
+A <strong>void</strong> function can only use <strong>return</strong> without a return argument, even if
+the return argument has <strong>void</strong> type.
+Return statements only accept values:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> func1() { }
+<span class="directive">void</span> func2() { <span class="keyword">return</span> func1(); } <span class="comment">// illegal return statement</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Only a precision qualifier is allowed on the return type of a function.
+Formal parameters can have parameter, precision, and memory qualifiers, but
+no other qualifiers.</p>
+</div>
+<div class="paragraph">
+<p>Functions that accept no input arguments need not use <strong>void</strong> in the argument
+list because prototypes (or definitions) are required and therefore there is
+no ambiguity when an empty argument list “( )” is declared.
+The idiom “(<strong>void</strong>)” as a parameter list is provided for convenience.</p>
+</div>
+<div class="paragraph">
+<p>Function names can be overloaded.
+The same function name can be used for multiple functions, as long as the
+parameter types differ.
+If a function name is declared twice with the same parameter types, then the
+return types and all qualifiers must also match, and it is the same function
+being declared.
+When function calls are resolved, an exact type match for all the arguments
+is required.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 f(in vec4 x, out vec4 y);
+vec4 f(in vec4 x, out uvec4 y); <span class="comment">// allowed, different argument type</span>
+<span class="predefined-type">int</span> f(in vec4 x, out vec4 y); <span class="comment">// error, only return type differs</span>
+vec4 f(in vec4 x, in vec4 y); <span class="comment">// error, only qualifier differs</span>
+vec4 f(<span class="directive">const</span> in vec4 x, out vec4 y); <span class="comment">// error, only qualifier differs</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Calling the first two functions above with the following argument types
+yields</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">f(vec4, vec4) <span class="comment">// exact match of vec4 f(in vec4 x, out vec4 y)</span>
+f(vec4, uvec4) <span class="comment">// exact match of vec4 f(in vec4 x, out uvec4 y)</span>
+f(ivec4, vec4) <span class="comment">// error, no exact match.</span>
+f(ivec4, uvec4) <span class="comment">// error, no exact match.</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>User-defined functions can have multiple declarations, but only one
+definition.</p>
+</div>
+<div class="paragraph">
+<p>A shader cannot redefine or overload built-in functions.</p>
+</div>
+<div class="paragraph">
+<p>The function <em>main</em> is used as the entry point to a shader executable.
+All shaders must define a function named <em>main</em>.
+This function takes no arguments, returns no value, and must be declared as
+type <strong>void</strong>:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> main()
+{
+ ...
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The function <em>main</em> can contain uses of <strong>return</strong>.
+See “<a href="#jumps">Jumps</a>” for more details.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time or link-time error to declare or define a function
+<strong>main</strong> with any other parameters or return type.</p>
+</div>
+<div class="sect3">
+<h4 id="function-calling-conventions">6.1.1. Function Calling Conventions</h4>
+<div class="paragraph">
+<p>Functions are called by value-return.
+This means input arguments are copied into the function at call time, and
+output arguments are copied back to the caller before function exit.
+Because the function works with local copies of parameters, there are no
+issues regarding aliasing of variables within a function.
+To control what parameters are copied in and/or out through a function
+definition or declaration:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The keyword <strong>in</strong> is used as a qualifier to denote a parameter is to be
+copied in, but not copied out.</p>
+</li>
+<li>
+<p>The keyword <strong>out</strong> is used as a qualifier to denote a parameter is to be
+copied out, but not copied in.
+This should be used whenever possible to avoid unnecessarily copying
+parameters in.</p>
+</li>
+<li>
+<p>The keyword <strong>inout</strong> is used as a qualifier to denote the parameter is to
+be both copied in and copied out.
+It means the same thing as specifying both <strong>in</strong> and <strong>out</strong>.</p>
+</li>
+<li>
+<p>A function parameter declared with no such qualifier means the same
+thing as specifying <strong>in</strong>.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>All arguments are evaluated at call time, exactly once, in order, from left
+to right.
+Evaluation of an <strong>in</strong> parameter results in a value that is copied to the
+formal parameter.
+Evaluation of an <strong>out</strong> parameter results in an l-value that is used to copy
+out a value when the function returns.
+Evaluation of an <strong>inout</strong> parameter results in both a value and an l-value;
+the value is copied to the formal parameter at call time and the l-value is
+used to copy out a value when the function returns.</p>
+</div>
+<div class="paragraph">
+<p>The order in which output parameters are copied back to the caller is
+undefined.</p>
+</div>
+<div class="paragraph">
+<p>In a function, writing to an input-only parameter is allowed.
+Only the function’s copy is modified.
+This can be prevented by declaring a parameter with the <strong>const</strong> qualifier.</p>
+</div>
+<div class="paragraph">
+<p>When calling a function, expressions that do not evaluate to l-values cannot
+be passed to parameters declared as <strong>out</strong> or <strong>inout</strong>, or an error
+results.</p>
+</div>
+<div class="paragraph">
+<p>Only precision qualifiers are allowed on the return type of a function.</p>
+</div>
+<div class="paragraph">
+<p>The syntax for function prototypes can be informally expressed as:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>function_prototype</em> : </dt>
+<dd>
+<p>[ <em>type_qualifier</em> ] <em>type_specifier</em> <em>IDENTIFIER</em> <em>LEFT_PAREN</em>
+<em>parameter_declaration</em> , <em>parameter_declaration</em> , …​
+, <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>parameter_declaration</em> : </dt>
+<dd>
+<p>[ <em>type_qualifier</em> ] <em>type_specifier</em> [ <em>IDENTIFIER</em> [ <em>array_specifier</em>
+] ]</p>
+</dd>
+<dt class="hdlist1"><em>type_qualifier</em> : </dt>
+<dd>
+<p><em>single_type_qualifier</em> , <em>single_type_qualifier</em> , …​</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The qualifiers allowed on formal parameters are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><em>empty</em><br>
+<strong>const</strong><br>
+<strong>in</strong><br>
+<strong>out</strong><br>
+<strong>inout</strong><br>
+<strong>precise</strong><br>
+<em>memory-qualifier</em><br>
+<em>precision-qualifier</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>const</strong> qualifier cannot be used with <strong>out</strong> or <strong>inout</strong>, or
+an error results.
+The above is used both for function declarations (i.e., prototypes) and for
+function definitions.
+Hence, function definitions can have unnamed arguments.</p>
+</div>
+<div class="paragraph">
+<p>Static, and hence dynamic recursion, are not allowed.
+Static recursion is present if the static function-call graph of a program
+contains cycles.
+Dynamic recursion occurs if at any time control flow has entered but not
+exited a single function more than once.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="selection">6.2. Selection</h3>
+<div class="paragraph">
+<p>Conditional control flow in the shading language is done by either <strong>if</strong>,
+<strong>if</strong>-<strong>else</strong>, or <strong>switch</strong> statements:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>selection-statement</em> : </dt>
+<dd>
+<p><strong>if</strong> <strong>(</strong> <em>bool-expression</em> <strong>)</strong> <em>statement</em><br>
+<strong>if</strong> <strong>(</strong> <em>bool-expression</em> <strong>)</strong> <em>statement</em> <strong>else</strong> <em>statement</em><br>
+<strong>switch</strong> <strong>(</strong> <em>init-expression</em> <strong>)</strong> <strong>{</strong> <em>switch-statement-list<sub>opt</sub></em> <strong>}</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Where <em>switch-statement-list</em> is a nested scope containing a list of zero or
+more <em>switch-statement</em> and other statements defined by the language, where
+<em>switch-statement</em> adds some forms of labels.
+That is</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>switch-statement-list</em> : </dt>
+<dd>
+<p><em>switch-statement</em><br>
+<em>switch-statement-list</em> <em>switch-statement</em></p>
+</dd>
+<dt class="hdlist1"><em>switch-statement</em> : </dt>
+<dd>
+<p><strong>case</strong> <em>constant-expression</em> <strong>:</strong><br>
+<strong>default</strong> <strong>:</strong> <em>statement</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Note the above grammar’s purpose is to aid discussion in this section; the
+normative grammar is in “<a href="#shading-language-grammar">Shading Language
+Grammar</a>”.</p>
+</div>
+<div class="paragraph">
+<p>If an <strong>if</strong>-expression evaluates to <strong>true</strong>, then the first <em>statement</em> is
+executed.
+If it evaluates to <strong>false</strong> and there is an <strong>else</strong> part then the second
+<em>statement</em> is executed.</p>
+</div>
+<div class="paragraph">
+<p>Any expression whose type evaluates to a Boolean can be used as the
+conditional expression <em>bool-expression</em>.
+Vector types are not accepted as the expression to <strong>if</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Conditionals can be nested.</p>
+</div>
+<div class="paragraph">
+<p>The type of <em>init-expression</em> in a <strong>switch</strong> statement must be a scalar
+integer.
+The type of <em>init-expression</em> must match the type of the <strong>case</strong> labels
+within each <strong>switch</strong> statement.
+Either signed integers or unsigned integers are allowed but there is no
+implicit type conversion between the two.
+If a <strong>case</strong> label has a <em>constant-expression</em> of equal value to
+<em>init-expression</em>, execution will continue after that label.
+Otherwise, if there is a <strong>default</strong> label, execution will continue after that
+label.
+Otherwise, execution skips the rest of the switch statement.
+It is an error to have more than one <strong>default</strong> or a replicated
+<em>constant-expression</em>.
+A <strong>break</strong> statement not nested in a loop or other switch statement (either
+not nested or nested only in <strong>if</strong> or <strong>if</strong>-<strong>else</strong> statements) will also skip
+the rest of the switch statement.
+Fall through labels are allowed, but it is an error to have no
+statement between a label and the end of the switch statement.
+No statements are allowed in a switch statement before the first <strong>case</strong>
+statement.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>case</strong> and <strong>default</strong> labels can only appear within a <strong>switch</strong> statement.
+No <strong>case</strong> or <strong>default</strong> labels can be nested inside other statements or
+compound statements within their corresponding <strong>switch</strong>.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="iteration">6.3. Iteration</h3>
+<div class="paragraph">
+<p>For, while, and do loops are allowed as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">for</span> (init-expression; condition-expression; loop-expression)
+ sub-statement
+<span class="keyword">while</span> (condition-expression)
+ sub-statement
+<span class="keyword">do</span>
+ statement
+<span class="keyword">while</span> (condition-expression)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>See “<a href="#shading-language-grammar">Shading Language Grammar</a>” for the
+definitive specification of loops.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>for</strong> loop first evaluates the <em>init-expression</em>, then the
+<em>condition-expression</em>.
+If the <em>condition-expression</em> evaluates to <strong>true</strong>, then the body of the loop
+is executed.
+After the body is executed, a <strong>for</strong> loop will then evaluate the
+<em>loop-expression</em>, and then loop back to evaluate the
+<em>condition-expression</em>, repeating until the <em>condition-expression</em> evaluates
+to <strong>false</strong>.
+The loop is then exited, skipping its body and skipping its
+<em>loop-expression</em>.
+Variables modified by the <em>loop-expression</em> maintain their value after the
+loop is exited, provided they are still in scope.
+Variables declared in <em>init-expression</em> or <em>condition-expression</em> are only
+in scope until the end of the sub-statement of the <strong>for</strong> loop.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>while</strong> loop first evaluates the <em>condition-expression</em>.
+If <strong>true</strong>, then the body is executed.
+This is then repeated, until the <em>condition-expression</em> evaluates to
+<strong>false</strong>, exiting the loop and skipping its body.
+Variables declared in the <em>condition-expression</em> are only in scope until the
+end of the sub-statement of the <strong>while</strong> loop.</p>
+</div>
+<div class="paragraph">
+<p>For both <strong>for</strong> and <strong>while</strong> loops, the sub-statement does not introduce a new
+scope for variable names, so the following has a redeclaration error:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">for</span> (<span class="predefined-type">int</span> i = <span class="integer">0</span>; i < <span class="integer">10</span>; i++) +
+{
+ <span class="predefined-type">int</span> i; <span class="comment">// redeclaration error +</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>do</strong>-<strong>while</strong> loop first executes the body, then executes the
+<em>condition-expression</em>.
+This is repeated until <em>condition-expression</em> evaluates to <strong>false</strong>, and then
+the loop is exited.</p>
+</div>
+<div class="paragraph">
+<p>Expressions for <em>condition-expression</em> must evaluate to a Boolean.</p>
+</div>
+<div class="paragraph">
+<p>Both the <em>condition-expression</em> and the <em>init-expression</em> can declare and
+initialize a variable, except in the <strong>do</strong>-<strong>while</strong> loop, which cannot declare
+a variable in its <em>condition-expression</em>.
+The variable’s scope lasts only until the end of the sub-statement that
+forms the body of the loop.</p>
+</div>
+<div class="paragraph">
+<p>Loops can be nested.</p>
+</div>
+<div class="paragraph">
+<p>Non-terminating loops are allowed.
+The consequences of very long or non-terminating loops are platform
+dependent.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="jumps">6.4. Jumps</h3>
+<div class="paragraph">
+<p>These are the jumps:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>jump_statement</em> : </dt>
+<dd>
+<p><strong>continue</strong> <strong>;</strong><br>
+<strong>break</strong> <strong>;</strong><br>
+<strong>return</strong> <strong>;</strong><br>
+<strong>return</strong> <em>expression</em> <strong>;</strong><br>
+<strong>discard</strong> <strong>;</strong> // in the fragment shader language only</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>There is no “goto” or other non-structured flow of control.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>continue</strong> jump is used only in loops.
+It skips the remainder of the body of the inner-most loop of which it is
+inside.
+For <strong>while</strong> and <strong>do</strong>-<strong>while</strong> loops, this jump is to the next evaluation of
+the loop <em>condition-expression</em> from which the loop continues as previously
+defined.
+For <strong>for</strong> loops, the jump is to the <em>loop-expression</em>, followed by the
+<em>condition-expression</em>.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>break</strong> jump can also be used only in loops and <strong>switch</strong> statements.
+It is simply an immediate exit of the inner-most loop or <strong>switch</strong> statements
+containing the <strong>break</strong>.
+No further execution of <em>condition-expression</em>, <em>loop-expression</em>, or
+<em>switch-statement</em> is done.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>discard</strong> keyword is only allowed within fragment shaders.
+It can be used within a fragment shader to abandon the operation on the
+current fragment.
+This keyword causes the fragment to be discarded and no updates to any
+buffers will occur.
+Any prior writes to other buffers such as shader storage buffers are
+unaffected.
+Control flow exits the shader, and subsequent implicit or explicit
+derivatives are undefined when this control flow is non-uniform (meaning
+different fragments within the primitive take different control paths).
+It would typically be used within a conditional statement, for example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">if</span> (intensity < <span class="float">0</span><span class="float">.0</span>)
+ discard;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A fragment shader may test a fragment’s alpha value and discard the fragment
+based on that test.
+However, it should be noted that coverage testing occurs after the fragment
+shader runs, and the coverage test can change the alpha value.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>return</strong> jump causes immediate exit of the current function.
+If it has <em>expression</em> then that is the return value for the function.</p>
+</div>
+<div class="paragraph">
+<p>The function <em>main</em> can use <strong>return</strong>.
+This simply causes <em>main</em> to exit in the same way as when the end of the
+function had been reached.
+It does not imply a use of <strong>discard</strong> in a fragment shader.
+Using <strong>return</strong> in <em>main</em> before defining outputs will have the same behavior
+as reaching the end of <em>main</em> before defining outputs.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="built-in-variables">7. Built-In Variables</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="built-in-language-variables">7.1. Built-In Language Variables</h3>
+<div class="paragraph">
+<p>Some OpenGL ES operations occur in fixed functionality and need to provide
+values to or receive values from shader executables.
+Shaders communicate with fixed-function OpenGL ES pipeline stages, and
+optionally with other shader executables, through the use of built-in input
+and output variables.</p>
+</div>
+<div class="sect3">
+<h4 id="vertex-shader-special-variables">7.1.1. Vertex Shader Special Variables</h4>
+<div class="paragraph">
+<p>The built-in vertex shader variables are intrinsically declared as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in highp <span class="predefined-type">int</span> gl_VertexID;
+in highp <span class="predefined-type">int</span> gl_InstanceID;
+
+out gl_PerVertex {
+ out highp vec4 gl_Position;
+ out highp <span class="predefined-type">float</span> gl_PointSize;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_Position</em> is intended for writing the homogeneous vertex
+position.
+It can be written at any time during shader execution.
+This value will be used by primitive assembly, clipping, culling, and other
+fixed functionality operations, if present, that operate on primitives after
+vertex processing has occurred.
+Its value is undefined after the vertex processing stage if the vertex
+shader executable does not write <em>gl_Position</em>.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_PointSize</em> is intended for a shader to write the size of
+the point to be rasterized.
+It is measured in pixels.
+If <em>gl_PointSize</em> is not written to, its value is undefined in subsequent
+pipe stages.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_VertexID</em> is a vertex shader input variable that holds an
+integer index for the vertex, as defined under “Shader Inputs” in section
+11.1.3.9 “Shader Inputs” of the <a href="#references">OpenGL ES Specification</a>.
+While the variable <em>gl_VertexID</em> is always present, its value is not always
+defined.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_InstanceID</em> is a vertex shader input variable that holds
+the instance number of the current primitive in an instanced draw call (see
+“Shader Inputs” in section 11.1.3.9 “Shader Inputs” of the
+<a href="#references">OpenGL ES Specification</a>).
+If the current primitive does not come from an instanced draw call, the
+value of <em>gl_InstanceID</em> is zero.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="tessellation-control-shader-special-variables">7.1.2. Tessellation Control Shader Special Variables</h4>
+<div class="paragraph">
+<p>In the tessellation control shader, built-in variables are intrinsically
+declared as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerVertex {
+ highp vec4 gl_Position;
+} gl_in[gl_MaxPatchVertices];
+
+in highp <span class="predefined-type">int</span> gl_PatchVerticesIn;
+in highp <span class="predefined-type">int</span> gl_PrimitiveID;
+in highp <span class="predefined-type">int</span> gl_InvocationID;
+
+out gl_PerVertex {
+ highp vec4 gl_Position;
+} gl_out[];
+
+patch out highp <span class="predefined-type">float</span> gl_TessLevelOuter[<span class="integer">4</span>];
+patch out highp <span class="predefined-type">float</span> gl_TessLevelInner[<span class="integer">2</span>];
+patch out highp vec4 gl_BoundingBox[<span class="integer">2</span>];</code></pre>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-control-input-variables">Tessellation Control Input Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em> contains the output written in the previous shader stage to
+<em>gl_Position</em>.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PatchVerticesIn</em> contains the number of vertices in the input patch
+being processed by the shader.
+A single shader can read patches of differing sizes, so the value of
+<em>gl_PatchVerticesIn</em> may differ between patches.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PrimitiveID</em> contains the number of primitives processed by the shader
+since the current set of rendering primitives was started.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_InvocationID</em> contains the number of the output patch vertex assigned to
+the tessellation control shader invocation.
+It is assigned integer values in the range [0, N-1], where N is the number
+of output patch vertices per primitive.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-control-output-variables">Tessellation Control Output Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em> is used in the same fashion as the corresponding output
+variable in the vertex shader.</p>
+</div>
+<div class="paragraph">
+<p>The values written to <em>gl_TessLevelOuter</em> and <em>gl_TessLevelInner</em> are
+assigned to the corresponding outer and inner tessellation levels of the
+output patch.
+They are used by the tessellation primitive generator to control primitive
+tessellation and may be read by tessellation evaluation shaders.</p>
+</div>
+<div class="paragraph">
+<p>The values written to <em>gl_BoundingBox</em> specify the minimum and maximum
+clip-space extents of a bounding box containing all primitives generated
+from the patch by the primitive generator, geometry shader, and clipping.
+Fragments may or may not be generated for portions of these primitives
+that extend outside the window-coordinate projection of this bounding
+box.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="tessellation-evaluation-shader-special-variables">7.1.3. Tessellation Evaluation Shader Special Variables</h4>
+<div class="paragraph">
+<p>In the tessellation evaluation shader, built-in variables are intrinsically
+declared as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerVertex {
+ highp vec4 gl_Position;
+} gl_in[gl_MaxPatchVertices];
+
+in highp <span class="predefined-type">int</span> gl_PatchVerticesIn;
+in highp <span class="predefined-type">int</span> gl_PrimitiveID;
+in highp vec3 gl_TessCoord;
+patch in highp <span class="predefined-type">float</span> gl_TessLevelOuter[<span class="integer">4</span>];
+patch in highp <span class="predefined-type">float</span> gl_TessLevelInner[<span class="integer">2</span>];
+
+out gl_PerVertex {
+ highp vec4 gl_Position;
+};</code></pre>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-evaluation-input-variables">Tessellation Evaluation Input Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em> contains the output written in the previous shader stage to
+<em>gl_Position</em>.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PatchVerticesIn</em> and <em>gl_PrimitiveID</em> are defined in the same fashion as
+the corresponding input variables in the tessellation control shader.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_TessCoord</em> specifies a three-component <em>(u,v,w)</em> vector identifying the
+position of the vertex being processed by the shader relative to the
+primitive being tessellated.
+Its values will obey the properties</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_TessCoord.x == <span class="float">1</span><span class="float">.0</span> - (<span class="float">1</span><span class="float">.0</span> - gl_TessCoord.x) <span class="comment">// two operations performed</span>
+gl_TessCoord.y == <span class="float">1</span><span class="float">.0</span> - (<span class="float">1</span><span class="float">.0</span> - gl_TessCoord.y) <span class="comment">// two operations performed</span>
+gl_TessCoord.z == <span class="float">1</span><span class="float">.0</span> - (<span class="float">1</span><span class="float">.0</span> - gl_TessCoord.z) <span class="comment">// two operations performed</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>to aid in replicating subdivision computations.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_TessLevelOuter</em> and <em>gl_TessLevelInner</em> are filled with the
+corresponding outputs written by the active tessellation control shader.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-evaluation-output-variables">Tessellation Evaluation Output Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em> is used in the same fashion as the corresponding output
+variable in the vertex shader.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="geometry-shader-special-variables">7.1.4. Geometry Shader Special Variables</h4>
+<div class="paragraph">
+<p>In the geometry shader, built-in variables are intrinsically declared as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerVertex {
+ highp vec4 gl_Position;
+} gl_in[];
+
+in highp <span class="predefined-type">int</span> gl_PrimitiveIDIn;
+in highp <span class="predefined-type">int</span> gl_InvocationID;
+
+out gl_PerVertex {
+ highp vec4 gl_Position;
+};
+
+out highp <span class="predefined-type">int</span> gl_PrimitiveID;
+out highp <span class="predefined-type">int</span> gl_Layer;</code></pre>
+</div>
+</div>
+<div class="sect4">
+<h5 id="geometry-shader-input-variables">Geometry Shader Input Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em> contains the output written in the previous shader stage to
+<em>gl_Position</em>.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PrimitiveIDIn</em> contains the number of primitives processed by the shader
+since the current set of rendering primitives was started.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_InvocationID</em> contains the invocation number assigned to the geometry
+shader invocation.
+It is assigned integer values in the range [0, N-1], where N is the number
+of geometry shader invocations per primitive.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="geometry-shader-output-variables">Geometry Shader Output Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em> is used in the same fashion as the corresponding output
+variable in the vertex shader.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PrimitiveID</em> is filled with a single integer that serves as a primitive
+identifier to the fragment shader.
+This is then available to fragment shaders, which will select the written
+primitive ID from the provoking vertex of the primitive being shaded.
+If a fragment shader using <em>gl_PrimitiveID</em> is active and a geometry shader
+is also active, the geometry shader must write to <em>gl_PrimitiveID</em> or the
+fragment shader input <em>gl_PrimitiveID</em> is undefined.
+See section 11.3.4.4 “Geometry Shader Outputs” of the
+<a href="#references">OpenGL ES Specification</a> for more information.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_Layer</em> is used to select a specific layer (or face and layer of a cube
+map) of a multi-layer framebuffer attachment.
+The actual layer used will come from one of the vertices in the primitive
+being shaded.
+Which vertex the layer comes from is determined as discussed in section
+11.3.4.4 of the <a href="#references">OpenGL ES Specification</a>
+but may be undefined, so it is best to write the same layer value for all
+vertices of a primitive.
+If a shader statically assigns a value to <em>gl_Layer</em>, layered rendering mode
+is enabled.
+See section 11.3.4.4 “Geometry Shader Outputs” and section 9.8 “Layered
+Framebuffers” of the <a href="#references">OpenGL ES Specification</a> for more information.
+If a shader statically assigns a value to <em>gl_Layer</em>, and there is an
+execution path through the shader that does not set <em>gl_Layer</em>, then the
+value of <em>gl_Layer</em> is undefined for executions of the shader that take that
+path.</p>
+</div>
+<div class="paragraph">
+<p>The output variable <em>gl_Layer</em> takes on a special value when used with an
+array of cube map textures.
+Instead of only referring to the layer, it is used to select a cube map face
+and a layer.
+Setting <em>gl_Layer</em> to the value <em>layer*6+face</em> will render to face <em>face</em> of
+the cube defined in layer <em>layer</em>.
+The face values are defined in table 8.25 of the <a href="#references">OpenGL ES Specification</a>,
+but repeated below for clarity.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Face Value</th>
+<th class="tableblock halign-left valign-top">Resulting Target</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">0</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_POSITIVE_X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_NEGATIVE_X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_POSITIVE_Y</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_NEGATIVE_Y</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_POSITIVE_Z</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_NEGATIVE_Z</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>For example, to render to the positive <em>y</em> cube map face located in the 5th
+layer of the cube map array, <em>gl_Layer</em> should be set to <em>5 * 6 + 2</em>.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="fragment-shader-special-variables">7.1.5. Fragment Shader Special Variables</h4>
+<div class="paragraph">
+<p>The built-in special variables that are accessible from a fragment shader
+are intrinsically declared as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in highp vec4 gl_FragCoord;
+in <span class="predefined-type">bool</span> gl_FrontFacing;
+out highp <span class="predefined-type">float</span> gl_FragDepth;
+in mediump vec2 gl_PointCoord;
+in <span class="predefined-type">bool</span> gl_HelperInvocation;
+in highp <span class="predefined-type">int</span> gl_PrimitiveID;
+in highp <span class="predefined-type">int</span> gl_Layer;
+in lowp <span class="predefined-type">int</span> gl_SampleID;
+in mediump vec2 gl_SamplePosition;
+in highp <span class="predefined-type">int</span> gl_SampleMaskIn[(gl_MaxSamples+<span class="integer">31</span>)/<span class="integer">32</span>];
+out highp <span class="predefined-type">int</span> gl_SampleMask[(gl_MaxSamples+<span class="integer">31</span>)/<span class="integer">32</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The output of the fragment shader executable is processed by the fixed
+function operations at the back end of the OpenGL ES pipeline.</p>
+</div>
+<div class="paragraph">
+<p>The fixed functionality computed depth for a fragment may be obtained by
+reading <em>gl_FragCoord.z</em>, described below.</p>
+</div>
+<div class="paragraph">
+<p>Writing to <em>gl_FragDepth</em> will establish the depth value for the fragment
+being processed.
+If depth buffering is enabled, and no shader writes <em>gl_FragDepth</em>, then the
+fixed function value for depth will be used as the fragment’s depth value.
+If a shader statically assigns a value to <em>gl_FragDepth</em>, and there is an
+execution path through the shader that does not set <em>gl_FragDepth</em>, then the
+value of the fragment’s depth may be undefined for executions of the shader
+that take that path.
+That is, if the set of linked fragment shaders statically contain a write to
+<em>gl_FragDepth</em>, then it is responsible for always writing it.</p>
+</div>
+<div class="paragraph">
+<p>If a shader executes the <strong>discard</strong> keyword, the fragment is discarded, and
+the values of any user-defined fragment outputs, <em>gl_FragDepth</em>, and
+<em>gl_SampleMask</em> become irrelevant.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_FragCoord</em> is available as an input variable from within
+fragment shaders and it holds the window relative coordinates (<em>x</em>, <em>y</em>,
+<em>z</em>, <em>1/w</em>) values for the fragment.
+If multi-sampling, this value can be for any location within the pixel, or
+one of the fragment samples.
+The use of <strong>centroid</strong> does not further restrict this value to be inside the
+current primitive.
+This value is the result of the fixed functionality that interpolates
+primitives after vertex processing to generate fragments.
+The <em>z</em> component is the depth value that would be used for the fragment’s
+depth if no shader contained any writes to <em>gl_FragDepth</em>.
+This is useful for invariance if a shader conditionally computes
+<em>gl_FragDepth</em> but otherwise wants the fixed functionality fragment depth.</p>
+</div>
+<div class="paragraph">
+<p>Fragment shaders have access to the input built-in variable
+<em>gl_FrontFacing</em>, whose value is <strong>true</strong> if the fragment belongs to a
+front-facing primitive.
+One use of this is to emulate two-sided lighting by selecting one of two
+colors calculated by a vertex or geometry shader.</p>
+</div>
+<div class="paragraph">
+<p>The values in <em>gl_PointCoord</em> are two-dimensional coordinates indicating
+where within a point primitive the current fragment is located, when point
+sprites are enabled.
+They range from 0.0 to 1.0 across the point.
+If the current primitive is not a point, or if point sprites are not
+enabled, then the values read from <em>gl_PointCoord</em> are undefined.</p>
+</div>
+<div class="paragraph">
+<p>For both the input array <em>gl_SampleMaskIn[]</em> and the output array
+<em>gl_SampleMask[]</em>, bit <em>B</em> of mask <em>M</em> (<em>gl_SampleMaskIn[M]</em> or
+<em>gl_SampleMask[M]</em>) corresponds to sample <em>32*M+B</em>.
+These arrays have <strong>ceil</strong>(<em>s</em>/32) elements, where <em>s</em> is the maximum number
+of color samples supported by the implementation.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_SampleMaskIn</em> indicates the set of samples covered by
+the primitive generating the fragment during multisample rasterization.
+It has a sample bit set if and only if the sample is considered covered for
+this fragment shader invocation.</p>
+</div>
+<div class="paragraph">
+<p>The output array <em>gl_SampleMask[]</em> sets the sample mask for the fragment
+being processed.
+Coverage for the current fragment will become the logical AND of the
+coverage mask and the output <em>gl_SampleMask</em>.
+This array must be sized in the fragment shader either implicitly or
+explicitly, to be no larger than the implementation-dependent maximum
+sample-mask (as an array of 32bit elements), determined by the maximum
+number of samples..
+If the fragment shader statically assigns a value to <em>gl_SampleMask</em>, the
+sample mask will be undefined for any array elements of any fragment shader
+invocations that fail to assign a value.
+If a shader does not statically assign a value to <em>gl_SampleMask</em>, the
+sample mask has no effect on the processing of a fragment.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_SampleID</em> is filled with the sample number of the
+sample currently being processed.
+This variable is in the range <em>0</em> to <em>gl_NumSamples-1</em>, where
+<em>gl_NumSamples</em> is the total number of samples in the framebuffer, or 1 if
+rendering to a non-multisample framebuffer.
+Any static use of this variable in a fragment shader causes the entire
+shader to be evaluated per-sample.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_SamplePosition</em> contains the position of the current
+sample within the multisample draw buffer.
+The <em>x</em> and <em>y</em> components of <em>gl_SamplePosition</em> contain the sub-pixel
+coordinate of the current sample and will have values in the range 0.0 to
+1.0.
+Any static use of this variable in a fragment shader causes the entire
+shader to be evaluated per sample.</p>
+</div>
+<div class="paragraph">
+<p>The value <em>gl_HelperInvocation</em> is <strong>true</strong> if the fragment shader invocation
+is considered a <em>helper invocation</em> and is <strong>false</strong> otherwise.
+A helper invocation is a fragment shader invocation that is created solely
+for the purposes of evaluating derivatives for use in non-helper fragment
+shader invocations.
+Such derivatives are computed implicitly in the built-in function
+<strong>texture</strong>() (see “<a href="#texture-functions">Texture Functions</a>”), and
+explicitly in the derivative functions in
+“<a href="#derivative-functions">Derivative Functions</a>”, for example <strong>dFdx</strong>() and
+<strong>dFdy</strong>().</p>
+</div>
+<div class="paragraph">
+<p>Fragment shader helper invocations execute the same shader code as
+non-helper invocations, but will not have side effects that modify the
+framebuffer or other shader-accessible memory.
+In particular:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Fragments corresponding to helper invocations are discarded when shader
+execution is complete, without updating the framebuffer.</p>
+</li>
+<li>
+<p>Stores to image and buffer variables performed by helper invocations
+have no effect on the underlying image or buffer memory.</p>
+</li>
+<li>
+<p>Atomic operations to image, buffer, or atomic counter variables
+performed by helper invocations have no effect on the underlying image
+or buffer memory.
+The values returned by such atomic operations are undefined.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Helper invocations may be generated for pixels not covered by a primitive
+being rendered.
+While fragment shader inputs qualified with <strong>centroid</strong> are normally required
+to be sampled in the intersection of the pixel and the primitive, the
+requirement is ignored for such pixels since there is no intersection
+between the pixel and primitive.</p>
+</div>
+<div class="paragraph">
+<p>Helper invocations may also be generated for fragments that are covered by a
+primitive being rendered when the fragment is killed by early fragment tests
+(using the <strong>early_fragment_tests</strong> qualifier) or where the implementation is
+able to determine that executing the fragment shader would have no effect
+other than assisting in computing derivatives for other fragment shader
+invocations.</p>
+</div>
+<div class="paragraph">
+<p>The set of helper invocations generated when processing any set of
+primitives is implementation-dependent.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_PrimitiveID</em> is filled with the value written to the
+<em>gl_PrimitiveID</em> geometry shader output, if a geometry shader is present.
+Otherwise, it is filled with the number of primitives processed by the
+shader since the current set of rendering primitives was started.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_Layer</em> is filled with the value written to the
+<em>gl_Layer</em> geometry shader output, if a geometry shader is present.
+If the geometry stage does not dynamically assign a value to <em>gl_Layer</em>, the
+value of <em>gl_Layer</em> in the fragment stage will be undefined.
+If the geometry stage makes no static assignment to <em>gl_Layer</em>, the input
+value in the fragment stage will be zero.
+Otherwise, the fragment stage will read the same value written by the
+geometry stage, even if that value is out of range.
+If a fragment shader contains a static access to <em>gl_Layer</em>, it will count
+against the implementation defined limit for the maximum number of inputs to
+the fragment stage.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="compute-shader-special-variables">7.1.6. Compute Shader Special Variables</h4>
+<div class="paragraph">
+<p>In the compute shader, built-in variables are declared as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// workgroup dimensions</span>
+in uvec3 gl_NumWorkGroups;
+<span class="directive">const</span> uvec3 gl_WorkGroupSize;
+
+<span class="comment">// workgroup and invocation IDs</span>
+in uvec3 gl_WorkGroupID;
+in uvec3 gl_LocalInvocationID;
+
+<span class="comment">// derived variables</span>
+in uvec3 gl_GlobalInvocationID;
+in uint gl_LocalInvocationIndex;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_NumWorkGroups</em> is a compute-shader input variable
+containing the number of workgroups in each dimension of the dispatch that
+will execute the compute shader.
+Its content is equal to the values specified in the <em>num_groups_x</em>,
+<em>num_groups_y</em>, and <em>num_groups_z</em> parameters passed to the
+<em>DispatchCompute</em> API entry point.</p>
+</div>
+<div class="paragraph">
+<p>The built-in constant <em>gl_WorkGroupSize</em> is a compute-shader constant
+containing the workgroup size of the shader.
+The size of the workgroup in the <em>X</em>, <em>Y</em>, and <em>Z</em> dimensions is stored in
+the <em>x</em>, <em>y</em>, and <em>z</em> components.
+The constants values in <em>gl_WorkGroupSize</em> will match those specified in the
+required <strong>local_size_x</strong>, <strong>local_size_y</strong>, and <strong>local_size_z</strong> layout
+qualifiers for the current shader.
+This is a constant so that it can be used to size arrays of memory that can
+be shared within the workgroup.
+It is a compile-time error to use <em>gl_WorkGroupSize</em> in a shader that does
+not declare a fixed workgroup size, or before that shader has declared a
+fixed workgroup size, using <strong>local_size_x</strong>, <strong>local_size_y</strong>, and
+<strong>local_size_z</strong>.</p>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_WorkGroupID</em> is a compute-shader input variable
+containing the three-dimensional index of the workgroup that the
+current invocation is executing in.
+The possible values range across the parameters passed into
+<em>DispatchCompute</em>, i.e., from (0, 0, 0) to (<em>gl_NumWorkGroups.x</em> - 1,
+<em>gl_NumWorkGroups.y</em> - 1, <em>gl_NumWorkGroups.z</em> -1).</p>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_LocalInvocationID</em> is a compute-shader input
+variable containing the three-dimensional index of the current work item
+within the workgroup.
+The possible values for this variable range across the workgroup
+size, i.e., (0,0,0) to (<em>gl_WorkGroupSize.x</em> - 1, <em>gl_WorkGroupSize.y</em> - 1,
+<em>gl_WorkGroupSize.z</em> - 1).</p>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_GlobalInvocationID</em> is a compute shader input
+variable containing the global index of the current work item.
+This value uniquely identifies this invocation from all other invocations
+across all workgroups initiated by the current <em>DispatchCompute</em> call.
+This is computed as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_GlobalInvocationID =
+ gl_WorkGroupID * gl_WorkGroupSize + gl_LocalInvocationID;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_LocalInvocationIndex</em> is a compute shader input
+variable that contains the one-dimensional representation of the
+<em>gl_LocalInvocationID</em>.
+This is computed as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_LocalInvocationIndex =
+ gl_LocalInvocationID.z * gl_WorkGroupSize.x * gl_WorkGroupSize.y +
+ gl_LocalInvocationID.y * gl_WorkGroupSize.x +
+ gl_LocalInvocationID.x;</code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="built-in-constants">7.2. Built-In Constants</h3>
+<div class="paragraph">
+<p>The following built-in constants are provided to all shaders.
+The actual values used are implementation-dependent, but must be at least
+the value shown.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">//</span>
+<span class="comment">// Implementation-dependent constants. The example values below</span>
+<span class="comment">// are the minimum values allowed for these maximums.</span>
+<span class="comment">//</span></code></pre>
+</div>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxVertexAttribs = <span class="integer">16</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxVertexUniformVectors = <span class="integer">256</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxVertexOutputVectors = <span class="integer">16</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxVertexTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxVertexImageUniforms = <span class="integer">0</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxVertexAtomicCounters = <span class="integer">0</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxVertexAtomicCounterBuffers = <span class="integer">0</span>;
+
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessControlInputComponents = <span class="integer">64</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessControlOutputComponents = <span class="integer">64</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessControlTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessControlUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessControlTotalOutputComponents = <span class="integer">2048</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessControlImageUniforms = <span class="integer">0</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessControlAtomicCounters = <span class="integer">0</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessControlAtomicCounterBuffers = <span class="integer">0</span>;
+
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessPatchComponents = <span class="integer">120</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxPatchVertices = <span class="integer">32</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessGenLevel = <span class="integer">64</span>;
+
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessEvaluationInputComponents = <span class="integer">64</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessEvaluationOutputComponents = <span class="integer">64</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessEvaluationTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessEvaluationUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessEvaluationImageUniforms = <span class="integer">0</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessEvaluationAtomicCounters = <span class="integer">0</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTessEvaluationAtomicCounterBuffers = <span class="integer">0</span>;
+
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryInputComponents = <span class="integer">64</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryOutputComponents = <span class="integer">64</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryImageUniforms = <span class="integer">0</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryOutputVertices = <span class="integer">256</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryTotalOutputComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryAtomicCounters = <span class="integer">0</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxGeometryAtomicCounterBuffers = <span class="integer">0</span>;
+
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxFragmentInputVectors = <span class="integer">15</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxFragmentImageUniforms = <span class="integer">4</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxFragmentUniformVectors = <span class="integer">256</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxFragmentAtomicCounters = <span class="integer">8</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxFragmentAtomicCounterBuffers = <span class="integer">1</span>;
+
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxDrawBuffers = <span class="integer">4</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MinProgramTexelOffset = -<span class="integer">8</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxProgramTexelOffset = <span class="integer">7</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxImageUnits = <span class="integer">4</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxSamples = <span class="integer">4</span>;
+
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxComputeImageUniforms = <span class="integer">4</span>;
+<span class="directive">const</span> highp ivec3 gl_MaxComputeWorkGroupCount = ivec3(<span class="integer">65535</span>, <span class="integer">65535</span>, <span class="integer">65535</span>);
+<span class="directive">const</span> highp ivec3 gl_MaxComputeWorkGroupSize = ivec3(<span class="integer">128</span>, <span class="integer">128</span>, <span class="integer">64</span>);
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxComputeUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxComputeTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxComputeAtomicCounters = <span class="integer">8</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxComputeAtomicCounterBuffers = <span class="integer">1</span>;
+
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxCombinedTextureImageUnits = <span class="integer">96</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxCombinedImageUniforms = <span class="integer">4</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxCombinedShaderOutputResources = <span class="integer">4</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxCombinedAtomicCounters = <span class="integer">8</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxCombinedAtomicCounterBuffers = <span class="integer">1</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxAtomicCounterBindings = <span class="integer">1</span>;
+<span class="directive">const</span> mediump <span class="predefined-type">int</span> gl_MaxAtomicCounterBufferSize = <span class="integer">32</span>;</code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="built-in-uniform-state">7.3. Built-In Uniform State</h3>
+<div class="paragraph">
+<p>As an aid to accessing OpenGL ES processing state, the following uniform
+variables are built into the OpenGL ES Shading Language.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">//</span>
+<span class="comment">// Depth range in window coordinates,</span>
+<span class="comment">// section 12.5.1 "Controlling the Viewport" in the</span>
+<span class="comment">// OpenGL ES Specification.</span>
+<span class="comment">//</span>
+<span class="keyword">struct</span> gl_DepthRangeParameters {
+ highp <span class="predefined-type">float</span> near; <span class="comment">// n</span>
+ highp <span class="predefined-type">float</span> far; <span class="comment">// f</span>
+ highp <span class="predefined-type">float</span> diff; <span class="comment">// f - n</span>
+};
+uniform gl_DepthRangeParameters gl_DepthRange;
+uniform lowp <span class="predefined-type">int</span> gl_NumSamples;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>These variables are only guaranteed to be available in the fragment stage.
+In other stages, their presence and function is implementation-defined.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="redeclaring-built-in-blocks">7.4. Redeclaring Built-In Blocks</h3>
+<div class="paragraph">
+<p>The <em>gl_PerVertex</em> block can be redeclared in a shader to explicitly
+indicate what subset of the fixed pipeline interface will be used.
+This is necessary to establish the interface between multiple programs.</p>
+</div>
+<div class="paragraph">
+<p>If the <em>gl_PerVertex</em> block is not redefined in a given program, the
+intrinsically declared definition of that block is used for the program
+interface.</p>
+</div>
+<div class="paragraph">
+<p>For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out gl_PerVertex {
+ highp vec4 gl_Position; <span class="comment">// will use gl_Position</span>
+ highp vec4 t; <span class="comment">// error, only gl_PerVertex members allowed</span>
+}; <span class="comment">// no other members of gl_PerVertex will be used</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This establishes the output interface the shader will use with the
+subsequent pipeline stage.
+It must be a subset of the built-in members of <em>gl_PerVertex</em>.
+Such a redeclaration can also add the <strong>invariant</strong> qualifier and
+interpolation qualifiers.</p>
+</div>
+<div class="paragraph">
+<p>Other layout qualifiers, like <strong>location</strong>, cannot be added to such a
+redeclaration, unless specifically stated.</p>
+</div>
+<div class="paragraph">
+<p>If a built-in interface block is redeclared, it must appear in the shader
+before any use of any member included in the built-in declaration, or a
+compile-time error will result.
+It is also a compile-time error to redeclare the block more than once or to
+redeclare a built-in block and then use a member from that built-in block
+that was not included in the redeclaration.
+Also, if a built-in interface block is redeclared, no member of the built-in
+declaration can be redeclared outside the block redeclaration.
+If multiple shaders using members of a built-in block belonging to the same
+interface are linked together in the same program, they must all redeclare
+the built-in block in the same way, as described in
+“<a href="#interface-blocks">Interface Blocks</a>” for interface block matching, or a
+link-time error will result.
+It will also be a link-time error if some shaders in a program redeclare a
+specific built-in interface block while another shader in that program does
+not redeclare that interface block yet still uses a member of that interface
+block.
+If a built-in block interface is formed across shaders in different
+programs, the shaders must all redeclare the built-in block in the same way
+(as described for a single program), or the values passed along the
+interface are undefined.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="built-in-functions">8. Built-In Functions</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The OpenGL ES Shading Language defines an assortment of built-in convenience functions for
+scalar and vector operations.
+Many of these built-in functions can be used in more than one type of
+shader, but some are intended to provide a direct mapping to hardware and so
+are available only for a specific type of shader.</p>
+</div>
+<div class="paragraph">
+<p>The built-in functions basically fall into three categories:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>They expose some necessary hardware functionality in a convenient way
+such as accessing a texture map.
+There is no way in the language for these functions to be emulated by a
+shader.</p>
+</li>
+<li>
+<p>They represent a trivial operation (clamp, mix, etc.) that is very
+simple for the user to write, but they are very common and may have
+direct hardware support.
+It is a very hard problem for the compiler to map expressions to complex
+assembler instructions.</p>
+</li>
+<li>
+<p>They represent an operation graphics hardware is likely to accelerate at
+some point.
+The trigonometry functions fall into this category.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Many of the functions are similar to the same named ones in common C
+libraries, but they support vector input as well as the more traditional
+scalar input.</p>
+</div>
+<div class="paragraph">
+<p>Applications should be encouraged to use the built-in functions rather than
+do the equivalent computations in their own shader code since the built-in
+functions are assumed to be optimal (e.g. perhaps supported directly in
+hardware).</p>
+</div>
+<div class="paragraph">
+<p>When the built-in functions are specified below, where the input arguments
+(and corresponding output) can be <strong>float</strong>, <strong>vec2</strong>, <strong>vec3</strong>, or <strong>vec4</strong>,
+<em>genFType</em> is used as the argument.
+Where the input arguments (and corresponding output) can be <strong>int</strong>, <strong>ivec2</strong>,
+<strong>ivec3</strong>, or <strong>ivec4</strong>, <em>genIType</em> is used as the argument.
+Where the input arguments (and corresponding output) can be <strong>uint</strong>, <strong>uvec2</strong>,
+<strong>uvec3</strong>, or <strong>uvec4</strong>, <em>genUType</em> is used as the argument.
+Where the input arguments (or corresponding output) can be <strong>bool</strong>, <strong>bvec2</strong>,
+<strong>bvec3</strong>, or <strong>bvec4</strong>, <em>genBType</em> is used as the argument.
+For any specific use of a function, the actual types substituted for
+<em>genFType</em>, <em>genIType</em>, <em>genUType</em>, or <em>genBType</em> have to have the same
+number of components for all arguments and for the return type.
+Similarly, <em>mat</em> is used for any matrix basic
+type.</p>
+</div>
+<div class="paragraph">
+<p>Built-in functions have an effective precision qualification.
+This qualification cannot be set explicitly and may be different from the
+precision qualification of the result.</p>
+</div>
+<div class="paragraph">
+<p>The precision qualification of the operation of a built-in function is based
+on the precision qualification of its formal parameters and actual
+parameters (input arguments): When a formal parameter specifies a precision
+qualifier, that is used, otherwise, the precision qualification of the
+actual (calling) argument is used.
+The highest precision of these will be the precision of the operation of the
+built-in function.
+Generally, this is applied across all arguments to a built-in function, with
+the exceptions being:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>bitfieldExtract</strong> and <strong>bitfieldInsert</strong> ignore the <em>offset</em> and <em>bits</em>
+arguments.</p>
+</li>
+<li>
+<p><strong>interpolateAt</strong> functions only look at the <em>interpolant</em> argument.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>The precision qualification of the result of a built-in function is
+determined in one of the following ways:</p>
+</div>
+<div class="paragraph">
+<p>For the texture sampling, image load and image store functions, the
+precision of the return type matches the precision of the sampler type:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">uniform lowp sampler2D sampler;
+highp vec2 coord;
+...
+lowp vec4 col = texture (sampler, coord); <span class="comment">// texture() returns lowp</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Otherwise:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>For prototypes that do not specify a resulting precision qualifier, the
+precision will be the same as the precision of the operation (as defined
+earlier).</p>
+</li>
+<li>
+<p>For prototypes that do specify a resulting precision qualifier, the
+specified precision qualifier is the precision qualification of the
+result.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Where the built-in functions in the following sections specify an equation,
+the entire equation will be evaluated at the operation’s precision.
+This may lead to underflow or overflow in the result, even when the correct
+result could be represented in the operation precision.</p>
+</div>
+<div class="sect2">
+<h3 id="angle-and-trigonometry-functions">8.1. Angle and Trigonometry Functions</h3>
+<div class="paragraph">
+<p>Function parameters specified as <em>angle</em> are assumed to be in units of
+radians.
+In no case will any of these functions result in a divide by zero error.
+If the divisor of a ratio is 0, then results will be undefined.</p>
+</div>
+<div class="paragraph">
+<p>These all operate component-wise.
+The description is per component.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>radians</strong>(genFType <em>degrees</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Converts <em>degrees</em> to radians, i.e.,
+ <span class="eq">(π / 180) · degrees</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>degrees</strong>(genFType <em>radians</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Converts <em>radians</em> to degrees, i.e.,
+ <span class="eq">(180 / π) · radians</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>sin</strong>(genFType <em>angle</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The standard trigonometric sine function.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>cos</strong>(genFType <em>angle</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The standard trigonometric cosine function.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>tan</strong>(genFType <em>angle</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The standard trigonometric tangent.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>asin</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc sine.
+ Returns an angle whose sine is <em>x</em>.
+ The range of values returned by this function is
+ <span class="eq">[-π / 2, π / 2]</span>.
+ Results are undefined if <span class="eq">|x| > 1</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>acos</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc cosine.
+ Returns an angle whose cosine is <em>x</em>.
+ The range of values returned by this function is <span class="eq">[0,π]</span>.
+ Results are undefined if <span class="eq">|x| > 1</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>atan</strong>(genFType <em>y</em>, genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc tangent.
+ Returns an angle whose tangent is <span class="eq">y / x</span>.
+ The signs of <em>x</em> and <em>y</em> are used to determine what quadrant the angle
+ is in.
+ The range of values returned by this function is <span class="eq">[-π, π</span>.
+ Results are undefined if <em>x</em> and <em>y</em> are both 0.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>atan</strong>(genFType <em>y_over_x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc tangent.
+ Returns an angle whose tangent is <em>y_over_x</em>.
+ The range of values returned by this function is
+ <span class="eq">[-π / 2, π / 2]</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>sinh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the hyperbolic sine function <span class="eq">(e<sup>x</sup> - e<sup>-x</sup>) / 2</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>cosh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the hyperbolic cosine function <span class="eq">(e<sup>x</sup> + e<sup>-x</sup>) / 2</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>tanh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the hyperbolic tangent function <span class="eq">sinh(x) / cosh(x)</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>asinh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc hyperbolic sine; returns the inverse of <strong>sinh</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>acosh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc hyperbolic cosine; returns the non-negative inverse of <strong>cosh</strong>.
+ Results are undefined if <span class="eq">x < 1</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>atanh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc hyperbolic tangent; returns the inverse of <strong>tanh</strong>.
+ Results are undefined if <span class="eq">x ≥ 1</span>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="exponential-functions">8.2. Exponential Functions</h3>
+<div class="paragraph">
+<p>These all operate component-wise.
+The description is per component.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>pow</strong>(genFType <em>x</em>, genFType <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> raised to the <em>y</em> power, i.e., <span class="eq">x<sup>y</sup></span>.
+ Results are undefined if <span class="eq">x < 0</span>.
+ Results are undefined if <span class="eq">x = 0</span> and <span class="eq">y ≤ 0</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>exp</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the natural exponentiation of <em>x</em>, i.e., <span class="eq">e<sup>x</sup></span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>log</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the natural logarithm of <em>x</em>, i.e., returns the value <em>y</em>
+ which satisfies the equation <span class="eq">x = e<sup>y</sup></span>.
+ Results are undefined if <span class="eq">x ≤ 0</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>exp2</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns 2 raised to the <em>x</em> power, i.e., <span class="eq">2<sup>x</sup></span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>log2</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the base 2 logarithm of <em>x</em>, i.e., returns the value <em>y</em> which
+ satisfies the equation <span class="eq">x = 2<sup>y</sup></span>.
+ Results are undefined if <span class="eq">x ≤ 0</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>sqrt</strong>(genFType <em>x</em>)<br>
+ genDType <strong>sqrt</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <span class="eq">sqrt(x)</span>.
+ Results are undefined if <span class="eq">x < 0</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>inversesqrt</strong>(genFType <em>x</em>)<br>
+ genDType <strong>inversesqrt</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <span class="eq">1 / sqrt(x)</span>.
+ Results are undefined if <span class="eq">x ≤ 0</span>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="common-functions">8.3. Common Functions</h3>
+<div class="paragraph">
+<p>These all operate component-wise.
+The description is per component.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>abs</strong>(genFType <em>x</em>)<br>
+ genIType <strong>abs</strong>(genIType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> if <span class="eq">x ≥ 0</span>; otherwise it returns -<em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>sign</strong>(genFType <em>x</em>)<br>
+ genIType <strong>sign</strong>(genIType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns 1.0 if <em>x</em> > 0, 0.0 if <em>x</em> = 0, or -1.0 if <em>x</em> < 0.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>floor</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer that is less than or
+ equal to <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>trunc</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer to <em>x</em> whose absolute
+ value is not larger than the absolute value of <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>round</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer to <em>x</em>.
+ The fraction 0.5 will round in a direction chosen by the
+ implementation, presumably the direction that is fastest.
+ This includes the possibility that <strong>round</strong>(<em>x</em>) returns the same value
+ as <strong>roundEven</strong>(<em>x</em>) for all values of <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>roundEven</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer to <em>x</em>.
+ A fractional part of 0.5 will round toward the nearest even integer.
+ (Both 3.5 and 4.5 for x will return 4.0.)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>ceil</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer that is greater than or
+ equal to <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>fract</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> - <strong>floor</strong>(<em>x</em>).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>mod</strong>(genFType <em>x</em>, float <em>y</em>)<br>
+ genFType <strong>mod</strong>(genFType <em>x</em>, genFType <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Modulus.
+ Returns <span class="eq">x - y · <strong>floor</strong>(x / y)</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>modf</strong>(genFType <em>x</em>, out genFType <em>i</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the fractional part of <em>x</em> and sets <em>i</em> to the integer part (as
+ a whole number floating-point value).
+ Both the return value and the output parameter will have the same sign
+ as <em>x</em>.
+ If <em>x</em> has the value +/- Inf, the return value should be NaN and must
+ be either NaN or 0.0.
+ For <strong>highp</strong> non-constant expressions, the value returned must be
+ consistent.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>min</strong>(genFType <em>x</em>, genFType <em>y</em>)<br>
+ genFType <strong>min</strong>(genFType <em>x</em>, float <em>y</em>)<br>
+ genIType <strong>min</strong>(genIType <em>x</em>, genIType <em>y</em>)<br>
+ genIType <strong>min</strong>(genIType <em>x</em>, int <em>y</em>)<br>
+ genUType <strong>min</strong>(genUType <em>x</em>, genUType <em>y</em>)<br>
+ genUType <strong>min</strong>(genUType <em>x</em>, uint <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>y</em> if <em>y</em> < <em>x;</em> otherwise it returns <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>max</strong>(genFType <em>x</em>, genFType <em>y</em>)<br>
+ genFType <strong>max</strong>(genFType <em>x</em>, float <em>y</em>)<br>
+ genIType <strong>max</strong>(genIType <em>x</em>, genIType <em>y</em>)<br>
+ genIType <strong>max</strong>(genIType <em>x</em>, int <em>y</em>)<br>
+ genUType <strong>max</strong>(genUType <em>x</em>, genUType <em>y</em>)<br>
+ genUType <strong>max</strong>(genUType <em>x</em>, uint <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>y</em> if <em>x</em> < <em>y;</em> otherwise it returns <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>clamp</strong>(genFType <em>x</em>, genFType <em>minVal</em>, genFType <em>maxVal</em>)<br>
+ genFType <strong>clamp</strong>(genFType <em>x</em>, float <em>minVal</em>, float <em>maxVal</em>)<br>
+ genIType <strong>clamp</strong>(genIType <em>x</em>, genIType <em>minVal</em>, genIType <em>maxVal</em>)<br>
+ genIType <strong>clamp</strong>(genIType <em>x</em>, int <em>minVal</em>, int <em>maxVal</em>)<br>
+ genUType <strong>clamp</strong>(genUType <em>x</em>, genUType <em>minVal</em>, genUType <em>maxVal</em>)<br>
+ genUType <strong>clamp</strong>(genUType <em>x</em>, uint <em>minVal</em>, uint <em>maxVal</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>min</strong>(<strong>max</strong>(<em>x</em>, <em>minVal</em>), <em>maxVal</em>).
+ Results are undefined if <em>minVal</em> > <em>maxVal</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>mix</strong>(genFType <em>x</em>, genFType <em>y</em>, genFType <em>a</em>)<br>
+ genFType <strong>mix</strong>(genFType <em>x</em>, genFType <em>y</em>, float <em>a</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the linear blend of <em>x</em> and <em>y</em>, i.e.,
+ <span class="eq">x · (1 - a) + y · a</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>mix</strong>(genFType <em>x</em>, genFType <em>y</em>, genBType <em>a</em>)<br>
+ genIType <strong>mix</strong>(genIType <em>x</em>, genIType <em>y</em>, genBType <em>a</em>)<br>
+ genUType <strong>mix</strong>(genUType <em>x</em>, genUType <em>y</em>, genBType <em>a</em>)<br>
+ genBType <strong>mix</strong>(genBType <em>x</em>, genBType <em>y</em>, genBType <em>a</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Selects which vector each returned component comes from.
+ For a component of <em>a</em> that is <strong>false</strong>, the corresponding component of
+ <em>x</em> is returned.
+ For a component of <em>a</em> that is <strong>true</strong>, the corresponding component of
+ <em>y</em> is returned.
+ Components of <em>x</em> and <em>y</em> that are not selected are allowed to be
+ invalid floating-point values and will have no effect on the results.
+ Thus, this provides different functionality than, for example,<br>
+ genFType <strong>mix</strong>(genFType <em>x</em>, genFType <em>y</em>, genFType(<em>a</em>))<br>
+ where <em>a</em> is a Boolean vector.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>step</strong>(genFType <em>edge</em>, genFType <em>x</em>)<br>
+ genFType <strong>step</strong>(float <em>edge</em>, genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns 0.0 if <em>x</em> < <em>edge;</em> otherwise it returns 1.0.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>smoothstep</strong>(genFType <em>edge0</em>, genFType <em>edge1</em>, genFType <em>x</em>)<br>
+ genFType <strong>smoothstep</strong>(float <em>edge0</em>, float <em>edge1</em>, genFType <em>x</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Returns 0.0 if <span class="eq">x ≤ edge0</span> and 1.0 if <span class="eq">x ≥ edge1</span>, and
+ performs smooth Hermite interpolation between 0 and 1 when <span class="eq">edge0
+ < x < edge1</span>.
+ This is useful in cases where you would want a threshold function with
+ a smooth transition.
+ This is equivalent to:</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">genFType t;
+t = clamp ((x - edge0) / (edge1 - edge0), <span class="integer">0</span>, <span class="integer">1</span>);
+<span class="keyword">return</span> t * t * (<span class="integer">3</span> - <span class="integer">2</span> * t);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>(And similarly for doubles.) Results are undefined if <span class="eq">edge0 ≥
+edge1</span>.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genBType <strong>isnan</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if <em>x</em> holds a NaN.
+ Returns <strong>false</strong> otherwise.
+ Always returns <strong>false</strong> if NaNs are not implemented.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genBType <strong>isinf</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if <em>x</em> holds a positive infinity or negative infinity.
+ Returns <strong>false</strong> otherwise.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genIType <strong>floatBitsToInt</strong>(highp genFType <em>value</em>)<br>
+ genUType <strong>floatBitsToUint</strong>(highp genFType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a signed or unsigned integer value representing the encoding
+ of a floating-point value.
+ The <strong>float</strong> value’s bit-level representation is preserved.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>intBitsToFloat</strong>(highp genIType <em>value</em>)<br>
+ genFType <strong>uintBitsToFloat</strong>(highp genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a floating-point value corresponding to a signed or unsigned
+ integer encoding of a floating-point value.
+ If an Inf or NaN is passed in, it will not signal, and the resulting
+ floating-point value is unspecified.
+ Otherwise, the bit-level representation is preserved.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>fma</strong>(genFType <em>a</em>, genFType <em>b</em>, genFType <em>c</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Computes and returns <span class="eq">a * b + c</span>.
+ In uses where the return value is eventually consumed by a variable
+ declared as <strong>precise</strong>:</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="ulist">
+<ul>
+<li>
+<p><strong>fma</strong>() is considered a single operation, whereas the expression “a * b
++ c” consumed by a variable declared <strong>precise</strong> is considered two
+operations.</p>
+</li>
+<li>
+<p>The precision of <strong>fma</strong>() can differ from the precision of the expression
+“a * b + c”.</p>
+</li>
+<li>
+<p><strong>fma</strong>() will be computed with the same precision as any other <strong>fma</strong>()
+consumed by a precise variable, giving invariant results for the same
+input values of <em>a</em>, <em>b</em>, and <em>c</em>.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Otherwise, in the absence of <strong>precise</strong> consumption, there are no special
+constraints on the number of operations or difference in precision between
+<strong>fma</strong>() and the expression “_a_ * <em>b</em> + <em>c</em>”.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>frexp</strong>(highp genFType <em>x</em>, out highp genIType <em>exp</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Splits <em>x</em> into a floating-point significand in the range
+ <span class="eq">[0.5,1.0]</span>, and an integral exponent of two, such that</p>
+<p class="tableblock"> <span class="eq">x = significant · 2<sup>exponent</sup></span></p>
+<p class="tableblock"> The significand is returned by the function and the exponent is
+ returned in the parameter <em>exp</em>.
+ For a floating-point value of zero, the significand and exponent are
+ both zero.</p>
+<p class="tableblock"> If an implementation supports signed zero, an input value of minus
+ zero should return a significand of minus zero.
+ For a floating-point value that is an infinity or is not a number, the
+ results are undefined.</p>
+<p class="tableblock"> If the input <em>x</em> is a vector, this operation is performed in a
+ component-wise manner; the value returned by the function and the
+ value written to <em>exp</em> are vectors with the same number of components
+ as <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>ldexp</strong>(highp genFType <em>x</em>, highp genIType <em>exp</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Builds a floating-point number from <em>x</em> and the corresponding integral
+ exponent of two in <em>exp</em>, returning:</p>
+<p class="tableblock"> <span class="eq">significand · 2<sup>exponent</sup></span></p>
+<p class="tableblock"> If this product is too large to be represented in the floating-point
+ type, the result is undefined.</p>
+<p class="tableblock"> If <em>exp</em> is greater than +128, the value returned is undefined.
+ If <em>exp</em> is less than -126, the value returned may be flushed to zero.
+ Additionally, splitting the value into a significand and exponent
+ using <strong>frexp</strong>() and then reconstructing a floating-point value using
+ <strong>ldexp</strong>() should yield the original input for zero and all finite
+ non-denormalized values.<br>
+ If the input <em>x</em> is a vector, this operation is performed in a
+ component-wise manner; the value passed in <em>exp</em> and returned by the
+ function are vectors with the same number of components as <em>x</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="floating-point-pack-and-unpack-functions">8.4. Floating-Point Pack and Unpack Functions</h3>
+<div class="paragraph">
+<p>These functions do not operate component-wise, rather, as described in each
+case.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>packUnorm2x16</strong>(vec2 <em>v</em>)<br>
+ highp uint <strong>packSnorm2x16</strong>(vec2 <em>v</em>)<br>
+ highp uint <strong>packUnorm4x8</strong>(vec4 <em>v</em>)<br>
+ highp uint <strong>packSnorm4x8</strong>(vec4 <em>v</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">First, converts each component of the normalized floating-point value
+ <em>v</em> into 16-bit (<strong>2x16</strong>) or 8-bit (<strong>4x8</strong>) integer values.
+ Then, the results are packed into the returned 32-bit unsigned
+ integer.</p>
+<p class="tableblock"> The conversion for component <em>c</em> of <em>v</em> to fixed point is done as
+ follows:</p>
+<p class="tableblock"> <strong>packUnorm2x16</strong>: <strong>round</strong>(<strong>clamp</strong>(<em>c</em>, 0, +1) * 65535.0)<br>
+ <strong>packSnorm2x16:</strong> <strong>round</strong>(<strong>clamp</strong>(<em>c</em>, -1, +1) * 32767.0)<br>
+ <strong>packUnorm4x8</strong>: <strong>round</strong>(<strong>clamp</strong>(<em>c</em>, 0, +1) * 255.0)<br>
+ <strong>packSnorm4x8</strong>: <strong>round</strong>(<strong>clamp</strong>(<em>c</em>, -1, +1) * 127.0)</p>
+<p class="tableblock"> The first component of the vector will be written to the least
+ significant bits of the output; the last component will be written to
+ the most significant bits.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp vec2 <strong>unpackUnorm2x16</strong>(highp uint <em>p</em>)<br>
+ highp vec2 <strong>unpackSnorm2x16</strong>(highp uint <em>p</em>)<br>
+ mediump vec4 <strong>unpackUnorm4x8</strong>(highp uint <em>p</em>)<br>
+ mediump vec4 <strong>unpackSnorm4x8</strong>(highp uint <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">First, unpacks a single 32-bit unsigned integer <em>p</em> into a pair of
+ 16-bit unsigned integers, a pair of 16-bit signed integers, four 8-bit
+ unsigned integers, or four 8-bit signed integers, respectively.
+ Then, each component is converted to a normalized floating-point value
+ to generate the returned two- or four-component vector.</p>
+<p class="tableblock"> The conversion for unpacked fixed-point value <em>f</em> to floating-point is
+ done as follows:</p>
+<p class="tableblock"> <strong>unpackUnorm2x16</strong>: <em>f</em> / 65535.0<br>
+ <strong>unpackSnorm2x16</strong>: <strong>clamp</strong>(<em>f</em> / 32767.0, -1, +1)<br>
+ <strong>unpackUnorm4x8</strong>: <em>f</em> / 255.0<br>
+ <strong>unpackSnorm4x8</strong>: <strong>clamp</strong>(<em>f</em> / 127.0, -1, +1)</p>
+<p class="tableblock"> The first component of the returned vector will be extracted from the
+ least significant bits of the input; the last component will be
+ extracted from the most significant bits.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>packHalf2x16</strong>(vec2 <em>v</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns an unsigned integer obtained by converting the components of a
+ two-component floating-point vector to the 16-bit floating-point
+ representation found in the <a href="#references">OpenGL ES Specification</a>, and
+ then packing these two 16-bit integers into a 32-bit unsigned integer.</p>
+<p class="tableblock"> The first vector component specifies the 16 least-significant bits of
+ the result; the second component specifies the 16 most-significant
+ bits.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mediump vec2 <strong>unpackHalf2x16</strong>(highp uint <em>v</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a two-component floating-point vector with components obtained
+ by unpacking a 32-bit unsigned integer into a pair of 16-bit values,
+ interpreting those values as 16-bit floating-point numbers according
+ to the <a href="#references">OpenGL ES Specification</a>, and converting them to
+ 32-bit floating-point values.</p>
+<p class="tableblock"> The first component of the vector is obtained from the 16
+ least-significant bits of <em>v</em>; the second component is obtained from
+ the 16 most-significant bits of <em>v</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="geometric-functions">8.5. Geometric Functions</h3>
+<div class="paragraph">
+<p>These operate on vectors as vectors, not component-wise.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>length</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Returns the length of vector <em>x</em>, i.e.,
+ <span class="eq">sqrt( x<sub>0</sub><sup>2</sup> + x<sub>1</sub><sup>2</sup> + …​ )</span>.</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>distance</strong>(genFType <em>p0</em>, genFType <em>p1</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the distance between <em>p0</em> and <em>p1</em>, i.e.,
+ <strong>length</strong>(<em>p0</em> - <em>p1</em>)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>dot</strong>(genFType <em>x</em>, genFType <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the dot product of <em>x</em> and <em>y</em>, i.e.,
+ <span class="eq">x<sub>0</sub> · y<sub>0</sub> + x<sub>1</sub> · y<sub>1</sub> + …​</span></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec3 <strong>cross</strong>(vec3 <em>x</em>, vec3 <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Returns the cross product of <em>x</em> and <em>y</em>, i.e.,
+ <span class="eq">(x<sub>1</sub> · y<sub>2</sub> - y<sub>1</sub> · x<sub>2</sub>,
+ x<sub>2</sub> · y<sub>0</sub> - y<sub>2</sub> · x<sub>0</sub>,
+ x<sub>0</sub> · y<sub>1</sub> - y<sub>0</sub> · x<sub>1</sub>)</span>.</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>normalize</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a vector in the same direction as <em>x</em> but with a length of 1,
+ i.e. <em>x</em> / <strong>length</strong>(x).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>faceforward</strong>(genFType <em>N</em>, genFType <em>I</em>, genFType <em>Nref</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">If <strong>dot</strong>(<em>Nref</em>, <em>I</em>) < 0 return <em>N</em>, otherwise return -<em>N</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>reflect</strong>(genFType <em>I</em>, genFType <em>N</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">For the incident vector <em>I</em> and surface orientation <em>N</em>, returns the
+ reflection direction: <span class="eq">I - 2 · <strong>dot</strong>(N, I) · N</span>.
+ <em>N</em> must already be normalized in order to achieve the desired result.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>refract</strong>(genFType <em>I</em>, genFType <em>N</em>, float <em>eta</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>For the incident vector <em>I</em> and surface normal <em>N</em>, and the ratio of
+ indices of refraction <em>eta</em>, return the refraction vector.
+ The result is computed by the <a href="#refraction-equation">refraction
+ equation</a> shown below.</p>
+</div>
+<div class="literalblock">
+<div class="content">
+<pre>The input parameters for the incident vector _I_ and the surface
+normal _N_ must already be normalized to get the desired results.</pre>
+</div>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+<div class="sect3">
+<h4 id="refraction-equation">8.5.1. Refraction Equation</h4>
+<div class="stemblock">
+<div class="content">
+\[k = 1.0 - eta * eta * (1.0 - \textbf{dot}(N,I) \cdot \textbf{dot}(N,I))\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+result &=
+ \begin{cases}
+ genFType(0.0), & k < 0.0 \\
+ eta * I - (eta * \textbf{dot}(N,I) + \sqrt { k }) * N, & \textbf{otherwise}
+ \end{cases}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="matrix-functions">8.6. Matrix Functions</h3>
+<div class="paragraph">
+<p>For each of the following built-in matrix functions, there is both a
+single-precision floating-point version, where all arguments and return
+values are single precision, and a double-precision floating-point version,
+where all arguments and return values are double precision.
+Only the single-precision floating-point version is shown.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mat <strong>matrixCompMult</strong>(mat <em>x</em>, mat <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Multiply matrix <em>x</em> by matrix <em>y</em> component-wise, i.e., result[i][j]
+ is the scalar product of <em>x</em>[i][j] and <em>y</em>[i][j].<br></p>
+<p class="tableblock"> Note: to get linear algebraic matrix multiplication, use the multiply
+ operator (<strong>*</strong>).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mat2 <strong>outerProduct</strong>(vec2 <em>c</em>, vec2 <em>r</em>)<br>
+ mat3 <strong>outerProduct</strong>(vec3 <em>c</em>, vec3 <em>r</em>)<br>
+ mat4 <strong>outerProduct</strong>(vec4 <em>c</em>, vec4 <em>r</em>)<br>
+ mat2x3 <strong>outerProduct</strong>(vec3 <em>c</em>, vec2 <em>r</em>)<br>
+ mat3x2 <strong>outerProduct</strong>(vec2 <em>c</em>, vec3 <em>r</em>)<br>
+ mat2x4 <strong>outerProduct</strong>(vec4 <em>c</em>, vec2 <em>r</em>)<br>
+ mat4x2 <strong>outerProduct</strong>(vec2 <em>c</em>, vec4 <em>r</em>)<br>
+ mat3x4 <strong>outerProduct</strong>(vec4 <em>c</em>, vec3 <em>r</em>)<br>
+ mat4x3 <strong>outerProduct</strong>(vec3 <em>c</em>, vec4 <em>r</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Treats the first parameter <em>c</em> as a column vector (matrix with one
+ column) and the second parameter <em>r</em> as a row vector (matrix with one
+ row) and does a linear algebraic matrix multiply <em>c</em> * <em>r</em>, yielding a
+ matrix whose number of rows is the number of components in <em>c</em> and
+ whose number of columns is the number of components in <em>r</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mat2 <strong>transpose</strong>(mat2 <em>m</em>)<br>
+ mat3 <strong>transpose</strong>(mat3 <em>m</em>)<br>
+ mat4 <strong>transpose</strong>(mat4 <em>m</em>)<br>
+ mat2x3 <strong>transpose</strong>(mat3x2 <em>m</em>)<br>
+ mat3x2 <strong>transpose</strong>(mat2x3 <em>m</em>)<br>
+ mat2x4 <strong>transpose</strong>(mat4x2 <em>m</em>)<br>
+ mat4x2 <strong>transpose</strong>(mat2x4 <em>m</em>)<br>
+ mat3x4 <strong>transpose</strong>(mat4x3 <em>m</em>)<br>
+ mat4x3 <strong>transpose</strong>(mat3x4 <em>m</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a matrix that is the transpose of <em>m</em>.
+ The input matrix <em>m</em> is not modified.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>determinant</strong>(mat2 <em>m</em>)<br>
+ float <strong>determinant</strong>(mat3 <em>m</em>)<br>
+ float <strong>determinant</strong>(mat4 <em>m</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the determinant of <em>m</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mat2 <strong>inverse</strong>(mat2 <em>m</em>)<br>
+ mat3 <strong>inverse</strong>(mat3 <em>m</em>)<br>
+ mat4 <strong>inverse</strong>(mat4 <em>m</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a matrix that is the inverse of <em>m</em>.
+ The input matrix <em>m</em> is not modified.
+ The values in the returned matrix are undefined if <em>m</em> is singular or
+ poorly-conditioned (nearly singular).</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="vector-relational-functions">8.7. Vector Relational Functions</h3>
+<div class="paragraph">
+<p>Relational and equality operators (<strong><</strong>, <strong><=</strong>, <strong>></strong>, <strong>>=</strong>, <strong>==</strong>, <strong>!=</strong>) are
+defined to operate on scalars and produce scalar Boolean results.
+For vector results, use the following built-in functions.
+Below, the following placeholders are used for the listed specific types:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Placeholder</th>
+<th class="tableblock halign-left valign-top">Specific Types Allowed</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec2, bvec3, bvec4</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">ivec</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">ivec2, ivec3, ivec4</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uvec</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uvec2, uvec3, uvec4</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec2, vec3, vec4</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>In all cases, the sizes of all the input and return vectors for any
+particular call must match.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>lessThan</strong>(vec x, vec y)<br>
+ bvec <strong>lessThan</strong>(ivec x, ivec y)<br>
+ bvec <strong>lessThan</strong>(uvec x, uvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x < y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>lessThanEqual</strong>(vec x, vec y)<br>
+ bvec <strong>lessThanEqual</strong>(ivec x, ivec y)<br>
+ bvec <strong>lessThanEqual</strong>(uvec x, uvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x ≤ y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>greaterThan</strong>(vec x, vec y)<br>
+ bvec <strong>greaterThan</strong>(ivec x, ivec y)<br>
+ bvec <strong>greaterThan</strong>(uvec x, uvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x > y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>greaterThanEqual</strong>(vec x, vec y)<br>
+ bvec <strong>greaterThanEqual</strong>(ivec x, ivec y)<br>
+ bvec <strong>greaterThanEqual</strong>(uvec x, uvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x ≥ y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>equal</strong>(vec x, vec y)<br>
+ bvec <strong>equal</strong>(ivec x, ivec y)<br>
+ bvec <strong>equal</strong>(uvec x, uvec y)<br>
+ bvec <strong>equal</strong>(bvec x, bvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x == y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>notEqual</strong>(vec x, vec y)<br>
+ bvec <strong>notEqual</strong>(ivec x, ivec y)<br>
+ bvec <strong>notEqual</strong>(uvec x, uvec y)<br>
+ bvec <strong>notEqual</strong>(bvec x, bvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x ≠ y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bool <strong>any</strong>(bvec x)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if any component of <em>x</em> is <strong>true</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bool <strong>all</strong>(bvec x)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> only if all components of <em>x</em> are <strong>true</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>not</strong>(bvec x)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise logical complement of <em>x</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="integer-functions">8.8. Integer Functions</h3>
+<div class="paragraph">
+<p>These all operate component-wise.
+The description is per component.
+The notation [<em>a</em>, <em>b</em>] means the set of bits from bit-number <em>a</em> through
+bit-number <em>b</em>, inclusive.
+The lowest-order bit is bit 0.
+“Bit number” will always refer to counting up from the lowest-order bit as
+bit 0.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genUType <strong>uaddCarry</strong>(highp genUType <em>x</em>, highp genUType <em>y</em>, out lowp genUType <em>carry</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Adds 32-bit unsigned integers <em>x</em> and <em>y</em>, returning the sum modulo
+ 2<sup>32</sup>.
+ The value <em>carry</em> is set to zero if the sum was less than 2<sup>32</sup>, or
+ one otherwise.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genUType <strong>usubBorrow</strong>(highp genUType <em>x</em>, highp genUType <em>y</em>, out lowp genUType <em>borrow</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Subtracts the 32-bit unsigned integer <em>y</em> from <em>x</em>, returning the
+ difference if non-negative, or 2<sup>32</sup> plus the difference otherwise.
+ The value <em>borrow</em> is set to zero if <span class="eq">x ≥ y</span>, or one
+ otherwise.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>umulExtended</strong>(highp genUType <em>x</em>, highp genUType <em>y</em>, out highp genUType <em>msb</em>, out highp genUType <em>lsb</em>)<br>
+ void <strong>imulExtended</strong>(highp genIType <em>x</em>, highp genIType <em>y</em>, out highp genIType <em>msb</em>, out highp genIType <em>lsb</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Multiplies 32-bit unsigned or signed integers <em>x</em> and <em>y</em>, producing a
+ 64-bit result.
+ The 32 least-significant bits are returned in <em>lsb</em>.
+ The 32 most-significant bits are returned in <em>msb</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genIType <strong>bitfieldExtract</strong>(genIType <em>value</em>, int <em>offset</em>, int <em>bits</em>)<br>
+ genUType <strong>bitfieldExtract</strong>(genUType <em>value</em>, int <em>offset</em>, int <em>bits</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Extracts bits <span class="eq">[offset, offset + bits - 1]</span> from <em>value</em>,
+ returning them in the least significant bits of the result.<br></p>
+<p class="tableblock"> For unsigned data types, the most significant bits of the result will
+ be set to zero.
+ For signed data types, the most significant bits will be set to the
+ value of bit <span class="eq">offset + bits - 1</span>.<br></p>
+<p class="tableblock"> If <em>bits</em> is zero, the result will be zero.
+ The result will be undefined if <em>offset</em> or <em>bits</em> is negative, or if
+ the sum of <em>offset</em> and <em>bits</em> is greater than the number of bits used
+ to store the operand.
+ Note that for vector versions of <strong>bitfieldExtract</strong>(), a single pair of
+ <em>offset</em> and <em>bits</em> values is shared for all components.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genIType <strong>bitfieldInsert</strong>(genIType <em>base</em>, genIType <em>insert</em>, int <em>offset</em>, int <em>bits</em>)<br>
+ genUType <strong>bitfieldInsert</strong>(genUType <em>base</em>, genUType <em>insert</em>, int <em>offset</em>, int <em>bits</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inserts the <em>bits</em> least significant bits of <em>insert</em> into <em>base</em>.</p>
+<p class="tableblock"> The result will have bits <span class="eq">[offset, offset + bits - 1]</span> taken from
+ bits <span class="eq">[0, bits - 1]</span> of <em>insert</em>, and all other bits taken
+ directly from the corresponding bits of <em>base</em>.
+ If <em>bits</em> is zero, the result will simply be <em>base</em>.
+ The result will be undefined if <em>offset</em> or <em>bits</em> is negative, or if
+ the sum of <em>offset</em> and <em>bits</em> is greater than the number of bits used
+ to store the operand.<br>
+ Note that for vector versions of <strong>bitfieldInsert</strong>(), a single pair of
+ <em>offset</em> and <em>bits</em> values is shared for all components.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genIType <strong>bitfieldReverse</strong>(highp genIType <em>value</em>)<br>
+ genUType <strong>bitfieldReverse</strong>(highp genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Reverses the bits of <em>value</em>.
+ The bit numbered <em>n</em> of the result will be taken from bit <span class="eq">(bits -
+ 1) - n</span> of <em>value</em>, where <em>bits</em> is the total number of bits used to
+ represent <em>value</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">lowp genIType <strong>bitCount</strong>(genIType <em>value</em>)<br>
+ lowp genIType <strong>bitCount</strong>(genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the number of one bits in the binary representation of
+ <em>value</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">lowp genIType <strong>findLSB</strong>(genIType <em>value</em>)<br>
+ lowp genIType <strong>findLSB</strong>(genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the bit number of the least significant one bit in the binary
+ representation of <em>value</em>.
+ If <em>value</em> is zero, -1 will be returned.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">lowp genIType <strong>findMSB</strong>(highp genIType <em>value</em>)<br>
+ lowp genIType <strong>findMSB</strong>(highp genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the bit number of the most significant bit in the binary
+ representation of <em>value</em>.</p>
+<p class="tableblock"> For positive integers, the result will be the bit number of the most
+ significant one bit.
+ For negative integers, the result will be the bit number of the most
+ significant zero bit.
+ For a <em>value</em> of zero or negative one, -1 will be returned.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="texture-functions">8.9. Texture Functions</h3>
+<div class="paragraph">
+<p>Texture lookup functions are available in all shading stages.
+However, level-of-detail is implicitly computed only for fragment shaders.
+Other shaders operate as though the base level-of-detail were computed as
+zero.
+The functions in the table below provide access to textures through
+samplers, as set up through the OpenGL ES API.
+Texture properties such as size, pixel format, number of dimensions,
+filtering method, number of mipmap levels, depth comparison, and so on are
+also defined by OpenGL ES API calls.
+Such properties are taken into account as the texture is accessed via the
+built-in functions defined below.</p>
+</div>
+<div class="paragraph">
+<p>Texture data can be stored by the GL as single-precision floating-point,
+unsigned normalized integer, unsigned integer or signed integer data.
+This is determined by the type of the internal format of the texture.</p>
+</div>
+<div class="paragraph">
+<p>Texture lookup functions are provided that can return their result as
+floating-point, unsigned integer or signed integer, depending on the sampler
+type passed to the lookup function.
+Care must be taken to use the right sampler type for texture access.
+The following table lists the supported combinations of sampler types and
+texture internal formats.
+Blank entries are unsupported.
+Doing a texture lookup will return undefined values for unsupported
+combinations.</p>
+</div>
+<div class="paragraph">
+<p>For depth/stencil textures, the internal texture format is determined by the
+component being accessed as set through the OpenGL ES API.
+When the depth/stencil texture mode is set to DEPTH_COMPONENT, the internal
+format of the depth component should be used.
+When the depth/stencil texture mode is set to STENCIL_INDEX, the internal format
+of the stencil component should be used.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Internal Texture Format</th>
+<th class="tableblock halign-left valign-top">Floating-Point Sampler Types</th>
+<th class="tableblock halign-left valign-top">Signed Integer Sampler Types</th>
+<th class="tableblock halign-left valign-top">Unsigned Integer Sampler Types</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Floating-point</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Supported</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Normalized Integer</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Supported</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Signed Integer</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Supported</p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Unsigned Integer</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Supported</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>If an integer sampler type is used, the result of a texture lookup is an
+<strong>ivec4</strong>.
+If an unsigned integer sampler type is used, the result of a texture lookup
+is a <strong>uvec4</strong>.
+If a floating-point sampler type is used, the result of a texture lookup is
+a <strong>vec4</strong>.</p>
+</div>
+<div class="paragraph">
+<p>In the prototypes below, the “_g_” in the return type “_gvec4_” is used
+as a placeholder for nothing, “_i_”, or “_u_” making a return type of
+<strong>vec4</strong>, <strong>ivec4</strong>, or <strong>uvec4</strong>.
+In these cases, the sampler argument type also starts with “_g_”,
+indicating the same substitution done on the return type; it is either a
+floating-point, signed integer, or unsigned integer sampler, matching the
+basic type of the return type, as described above.</p>
+</div>
+<div class="paragraph">
+<p>For shadow forms (the sampler parameter is a shadow-type), a depth
+comparison lookup on the depth texture bound to <em>sampler</em> is done as
+described in section
+8.20
+“Texture Comparison Modes” of the
+<a href="#references">OpenGL ES Specification</a>.
+See the table below for which component specifies <em>D<sub>ref</sub></em>.
+The texture bound to <em>sampler</em> must be a depth texture, or results are
+undefined.
+If a non-shadow texture call is made to a sampler that represents a depth
+texture with depth comparisons turned on, then results are undefined.
+If a shadow texture call is made to a sampler that represents a depth
+texture with depth comparisons turned off, then results are undefined.
+If a shadow texture call is made to a sampler that does not represent a
+depth texture, then results are undefined.</p>
+</div>
+<div class="paragraph">
+<p>In all functions below, the <em>bias</em> parameter is optional for fragment
+shaders.
+The <em>bias</em> parameter is not accepted in any other shader stage.
+For a fragment shader, if <em>bias</em> is present, it is added to the implicit
+level-of-detail prior to performing the texture access operation.
+No <em>bias</em> or <em>lod</em> parameters for
+multisample textures, or texture buffers
+are supported because mipmaps are not allowed for these types of textures.</p>
+</div>
+<div class="paragraph">
+<p>The implicit level-of-detail is selected as follows: For a texture that is
+not mipmapped, the texture is used directly.
+If it is mipmapped and running in a fragment shader, the level-of-detail
+computed by the implementation is used to do the texture lookup.
+If it is mipmapped and running in a non-fragment shader, then the base
+texture is used.</p>
+</div>
+<div class="paragraph">
+<p>Some texture functions (non-“<strong>Lod</strong>” and non-“<strong>Grad</strong>” versions) may
+require implicit derivatives.
+Implicit derivatives are undefined within non-uniform control flow and for
+non-fragment shader texture fetches.</p>
+</div>
+<div class="paragraph">
+<p>For <strong>Cube</strong> forms, the direction of <em>P</em> is used to select which face to do a
+2-dimensional texture lookup in, as described in section 8.13 “Cube Map
+Texture Selection” of the <a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>For <strong>Array</strong> forms, the array layer used will be</p>
+</div>
+<div class="paragraph">
+<p>\(\max(0,\min(d-1,\left\lfloor layer + 0.5\right\rfloor))\)</p>
+</div>
+<div class="paragraph">
+<p>where <em>d</em> is the depth of the texture array and <em>layer</em> comes from the
+component indicated in the tables below.</p>
+</div>
+<div class="sect3">
+<h4 id="texture-query-functions">8.9.1. Texture Query Functions</h4>
+<div class="paragraph">
+<p>The <strong>textureSize</strong> functions query the dimensions of a specific texture level
+for a sampler.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp ivec2 <strong>textureSize</strong>(gsampler2D <em>sampler</em>, int <em>lod</em>)<br>
+ highp ivec3 <strong>textureSize</strong>(gsampler3D <em>sampler</em>, int <em>lod</em>)<br>
+ highp ivec2 <strong>textureSize</strong>(gsamplerCube <em>sampler</em>, int <em>lod</em>)<br>
+ highp ivec2 <strong>textureSize</strong>(sampler2DShadow <em>sampler</em>, int <em>lod</em>)<br>
+ highp ivec2 <strong>textureSize</strong>(samplerCubeShadow <em>sampler</em>, int <em>lod</em>)
+ highp ivec3 <strong>textureSize</strong>(gsamplerCubeArray <em>sampler</em>, int <em>lod</em>)<br>
+ highp ivec3 <strong>textureSize</strong>(samplerCubeArrayShadow <em>sampler</em>, int <em>lod</em>)<br>
+ highp ivec3 <strong>textureSize</strong>(gsampler2DArray <em>sampler</em>, int <em>lod</em>)<br>
+ highp ivec3 <strong>textureSize</strong>(sampler2DArrayShadow <em>sampler</em>, int <em>lod</em>)<br>
+ highp int <strong>textureSize</strong>(gsamplerBuffer <em>sampler</em>)<br>
+ highp ivec2 <strong>textureSize</strong>(gsampler2DMS <em>sampler</em>)<br>
+ highp ivec3 <strong>textureSize</strong>(gsampler2DMSArray <em>sampler</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the dimensions of level <em>lod</em> (if present) for the texture
+ bound to <em>sampler</em>, as described in section
+ 8.11 “Texture Queries” of the <a href="#references">OpenGL ES Specification</a>.<br>
+ The components in the return value are filled in, in order, with the
+ width, height, and depth of the texture.</p>
+<p class="tableblock"> For the array forms, the last component of the return value is the
+ number of layers in the texture array, or the number of cubes in the
+ texture cube map array.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="texel-lookup-functions">8.9.2. Texel Lookup Functions</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>texture</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em>[, float <em>bias</em>] )<br>
+ float <strong>texture</strong>(sampler2DShadow <em>sampler</em>, <em>vec3 _P</em> [, float <em>bias</em>])<br>
+ float <strong>texture</strong>(samplerCubeShadow <em>sampler</em>, <em>vec4 _P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsamplerCubeArray <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ float <strong>texture</strong>(sampler2DArrayShadow <em>sampler</em>, vec4 <em>P</em>)<br>
+ float <strong>texture</strong>(samplerCubeArrayShadow <em>sampler</em>, vec4 <em>P</em>, float <em>compare</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Use the texture coordinate <em>P</em> to do a texture lookup in the texture
+ currently bound to <em>sampler</em>.</p>
+<p class="tableblock"> For shadow forms: When <em>compare</em> is present, it is used as <em>D<sub>ref</sub></em>
+ and the array layer comes from the last component of <em>P</em>.
+ When <em>compare</em> is not present, the last component of <em>P</em> is used as
+ <em>D<sub>ref</sub></em> and the array layer comes from the second to last component
+ of <em>P</em>.</p>
+<p class="tableblock"> For non-shadow forms: the array layer comes from the last component of
+ <em>P</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProj</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProj</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProj</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ float <strong>textureProj</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup with projection.
+ The texture coordinates consumed from <em>P</em>, not including the last
+ component of <em>P</em>, are divided by the last component of <em>P</em> to
+ form projected coordinates <em>P'</em>.
+ The resulting third component of <em>P</em> in the shadow forms is used as
+ <em>D<sub>ref</sub></em>.
+ The third component of <em>P</em> is ignored when <em>sampler</em> has type
+ <strong>gsampler2D</strong> and <em>P</em> has type <strong>vec4</strong>.
+ After these values are computed, texture lookup proceeds as in
+ <strong>texture</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureLod</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ float <strong>textureLod</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsamplerCubeArray <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Do a texture lookup as in <strong>texture</strong> but with explicit level-of-detail;
+ <em>lod</em> specifies <span class="eq">λ<sub>base</sub>]</span> and sets the partial derivatives
+ as follows:<br>
+ (See section 8.14 “Texture Minification” and equations 8.4-8.6 of
+ the <a href="#references">OpenGL ES Specification</a>.)<br>
+ <br>
+ <span class="eq">∂u / ∂x =
+ ∂v / ∂x =
+ ∂w / ∂x = 0</span>
+ <br>
+ <span class="eq">∂u / ∂y =
+ ∂v / ∂y =
+ ∂w / ∂y = 0</span></p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureOffset</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureOffset</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>, ivec3 <em>offset</em> [, float <em>bias</em>] )<br>
+ float <strong>textureOffset</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureOffset</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup as in <strong>texture</strong> but with <em>offset</em> added to the
+ <span class="eq">(u,v,w)</span> texel coordinates before looking up each texel.
+ The offset value must be a constant expression.
+ A limited range of offset values are supported; the minimum and
+ maximum offset values are implementation-dependent and given by
+ <em>gl_MinProgramTexelOffset</em> and <em>gl_MaxProgramTexelOffset</em>,
+ respectively.</p>
+<p class="tableblock"> Note that <em>offset</em> does not apply to the layer coordinate for texture
+ arrays.
+ This is explained in detail in section 8.14.2 “Coordinate Wrapping
+ and Texel Selection” of the <a href="#references">OpenGL ES Specification</a>, where <em>offset</em>
+ is <span class="eq">(δ<sub>u</sub>, δ<sub>v</sub>, δ<sub>w</sub>)</span>.<br>
+ Note that texel offsets are also not supported for cube maps.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>texelFetch</strong>(gsampler2D <em>sampler</em>, ivec2 <em>P</em>, int <em>lod</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler3D <em>sampler</em>, ivec3 <em>P</em>, int <em>lod</em>)
+ gvec4 <strong>texelFetch</strong>(gsampler2DArray <em>sampler</em>, ivec3 <em>P</em>, int <em>lod</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsamplerBuffer <em>sampler</em>, int <em>P</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler2DMS <em>sampler</em>, ivec2 <em>P</em>, int <em>sample</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler2DMSArray <em>sampler</em>, ivec3 <em>P</em>, int <em>sample</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Use integer texture coordinate <em>P</em> to lookup a single texel from
+ <em>sampler</em>.
+ The array layer comes from the last component of <em>P</em> for the array
+ forms.
+ The level-of-detail <em>lod</em> (if present) is as described in sections
+ 11.1.3.2 “Texel Fetches” and 8.14.1 “Scale Factor and Level of
+ Detail” of the <a href="#references">OpenGL ES Specification</a>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>texelFetchOffset</strong>(gsampler2D <em>sampler</em>, ivec2 <em>P</em>, int <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>texelFetchOffset</strong>(gsampler3D <em>sampler</em>, ivec3 <em>P</em>, int <em>lod</em>, ivec3 <em>offset</em>)<br>
+ gvec4 <strong>texelFetchOffset</strong>(gsampler2DArray <em>sampler</em>, ivec3 <em>P</em>, int <em>lod</em>, ivec2 <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Fetch a single texel as in <strong>texelFetch</strong>, offset by <em>offset</em> as
+ described in <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjOffset</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProjOffset</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProjOffset</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, ivec3 <em>offset</em> [, float <em>bias</em>] )<br>
+ float <strong>textureProjOffset</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a projective texture lookup as described in <strong>textureProj</strong>, offset
+ by <em>offset</em> as described in <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureLodOffset</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureLodOffset</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>, ivec3 <em>offset</em>)<br>
+ float <strong>textureLodOffset</strong>(sampler2DShadow <em>sampler</em>, <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureLodOffset</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do an offset texture lookup with explicit level-of-detail.
+ See <strong>textureLod</strong> and <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjLod</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureProjLod</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureProjLod</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)<br>
+ float <strong>textureProjLod</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a projective texture lookup with explicit level-of-detail.
+ See <strong>textureProj</strong> and <strong>textureLod</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjLodOffset</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjLodOffset</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjLodOffset</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>, ivec3 <em>offset</em>)<br>
+ float <strong>textureProjLodOffset</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do an offset projective texture lookup with explicit level-of-detail.
+ See <strong>textureProj</strong>, <strong>textureLod</strong>, and <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGrad</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsampler3D <em>sampler</em>, <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(samplerCubeShadow <em>sampler</em>, vec4 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(sampler2DArrayShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsamplerCubeArray <em>sampler</em>, vec4 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Do a texture lookup as in <strong>texture</strong> but with <a href="#explicit-gradients">explicit gradients</a> as shown below.
+ The partial derivatives of <em>P</em> are with respect to window <em>x</em> and
+ window <em>y</em>.
+ For the cube version, the partial derivatives of <em>P</em> are assumed to be
+ in the coordinate system used before texture coordinates are projected
+ onto the appropriate cube face.</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGradOffset</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureGradOffset</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>, ivec3 <em>offset</em>)<br>
+ float <strong>textureGradOffset</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureGradOffset</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureGradOffset</strong>(sampler2DArrayShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup with both explicit gradient and offset, as
+ described in <strong>textureGrad</strong> and <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjGrad</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureProjGrad</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureProjGrad</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br>
+ float <strong>textureProjGrad</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup both projectively, as described in <strong>textureProj</strong>,
+ and with explicit gradient as described in <strong>textureGrad</strong>.
+ The partial derivatives <em>dPdx</em> and <em>dPdy</em> are assumed to be already
+ projected.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjGradOffset</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjGradOffset</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjGradOffset</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>, ivec3 <em>offset</em>)<br>
+ float <strong>textureProjGradOffset</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup projectively and with explicit gradient as
+ described in <strong>textureProjGrad</strong>, as well as with offset, as described in
+ <strong>textureOffset</strong>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="explicit-gradients">8.9.3. Explicit Gradients</h4>
+<div class="paragraph">
+<p>In the <strong>textureGrad</strong> functions described above, explicit gradients control
+texture lookups as follows:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+ \frac{\partial{s}}{\partial{x}} & = \frac{\partial{P.s}}{\partial{x}} \\[0.8em]
+ \frac{\partial{s}}{\partial{y}} & = \frac{\partial{P.s}}{\partial{y}} \\[0.8em]
+ \frac{\partial{t}}{\partial{x}} & = \frac{\partial{P.t}}{\partial{x}} \\[0.8em]
+ \frac{\partial{t}}{\partial{y}} & = \frac{\partial{P.t}}{\partial{y}} \\[0.8em]
+ \frac{\partial{r}}{\partial{x}} & =
+ \begin{cases}
+ 0.0, & \text{for 2D} \\[0.8em]
+ \frac{\partial{P.p}}{\partial{x}}, & \text{cube, other}
+ \end{cases} \\[2.5em]
+ \frac{\partial{r}}{\partial{y}} & =
+ \begin{cases}
+ 0.0, & \text{for 2D} \\[0.8em]
+ \frac{\partial{P.p}}{\partial{y}}, & \text{cube, other}
+ \end{cases}
+\end{aligned}\]
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="texture-gather-functions">8.9.4. Texture Gather Functions</h4>
+<div class="paragraph">
+<p>The texture gather functions take components of a single floating-point
+vector operand as a texture coordinate, determine a set of four texels to
+sample from the base level-of-detail of the specified texture image, and
+return one component from each texel in a four-component result vector.</p>
+</div>
+<div class="paragraph">
+<p>When performing a texture gather operation, the minification and
+magnification filters are ignored, and the rules for LINEAR filtering in the
+<a href="#references">OpenGL ES Specification</a> are applied to the base level of the texture image
+to identify the four texels <em>i<sub>0</sub> j<sub>1</sub></em>, <em>i<sub>1</sub> j<sub>1</sub></em>, <em>i<sub>1</sub> j<sub>0</sub></em>, and <em>i<sub>0</sub>
+j<sub>0</sub></em>.
+The texels are then converted to texture base colors (<em>R<sub>s</sub></em>, <em>G<sub>s</sub></em>,
+<em>B<sub>s</sub></em>, <em>A<sub>s</sub></em>) according to table 15.1, followed by application of the
+texture swizzle as described in section 15.2.1 “Texture Access” of the
+<a href="#references">OpenGL ES Specification</a>.
+A four-component vector is assembled by taking the selected component from
+each of the post-swizzled texture source colors in the order (<em>i<sub>0</sub> j<sub>1</sub></em>,
+<em>i<sub>1</sub> j<sub>1</sub></em>, <em>i<sub>1</sub> j<sub>0</sub></em>, <em>i<sub>0</sub> j<sub>0</sub></em>).</p>
+</div>
+<div class="paragraph">
+<p>The selected component is identified by the optional <em>comp</em> argument, where
+the values zero, one, two, and three identify the <em>R<sub>s</sub></em>, <em>G<sub>s</sub></em>, <em>B<sub>s</sub></em>, or
+<em>A<sub>s</sub></em> component, respectively.
+If <em>comp</em> is omitted, it is treated as identifying the <em>R<sub>s</sub></em> component.</p>
+</div>
+<div class="paragraph">
+<p>Incomplete textures (see section 8.16 “Texture Completeness” of the
+<a href="#references">OpenGL ES Specification</a>) return a texture source color of (0,0,0,1) for all
+four source texels.</p>
+</div>
+<div class="paragraph">
+<p>For texture gather functions using a shadow sampler type, each of the four
+texel lookups perform a depth comparison against the depth reference value
+passed in (<em>refZ</em>), and returns the result of that comparison in the
+appropriate component of the result vector.</p>
+</div>
+<div class="paragraph">
+<p>As with other texture lookup functions, the results of a texture gather are
+undefined for shadow samplers if the texture referenced is not a depth
+texture or has depth comparisons disabled; or for non-shadow samplers if the
+texture referenced is a depth texture with depth comparisons enabled.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>textureGatherOffset</strong> built-in functions from the OpenGL ES Shading Language return a vector
+derived from sampling four texels in the image array of level <em>level<sub>base</sub></em>.
+For each of the four texel offsets specified by the <em>offsets</em> argument, the
+rules for the LINEAR minification filter are applied to identify a 2 ×
+2 texel footprint, from which the single texel T<sub>i0j0</sub> is selected.
+A four-component vector is then assembled by taking a single component from
+each of the four T<sub>i0j0</sub> texels in the same manner as for the
+<strong>textureGather</strong> function.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGather</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em> [, int <em>comp</em>])<br>
+ gvec4 <strong>textureGather</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em> [, int <em>comp</em>])<br>
+ gvec4 <strong>textureGather</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em> [, int <em>comp</em>])<br>
+ gvec4 <strong>textureGather</strong>(gsamplerCubeArray <em>sampler</em>, vec4 <em>P</em>[, int <em>comp</em>])<br>
+ vec4 <strong>textureGather</strong>(sampler2DShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>)<br>
+ vec4 <strong>textureGather</strong>(sampler2DArrayShadow <em>sampler</em>, vec3 <em>P</em>, float <em>refZ</em>)<br>
+ vec4 <strong>textureGather</strong>(samplerCubeShadow <em>sampler</em>, vec3 <em>P</em>, float <em>refZ</em>)<br>
+ vec4 <strong>textureGather</strong>(samplerCubeArrayShadow <em>sampler</em>, vec4 <em>P</em>, float <em>refZ</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Returns the value<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4(Sample_i0_j1(P, base).comp,
+ Sample_i1_j1(P, base).comp,
+ Sample_i1_j0(P, base).comp,
+ Sample_i0_j0(P, base).comp)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>If specified, the value of <em>comp</em> must be a constant integer expression with
+a value of 0, 1, 2, or 3, identifying the <em>x</em>, <em>y</em>, <em>z</em>, or <em>w</em>
+post-swizzled component of the four-component vector lookup result for each
+texel, respectively.
+If <em>comp</em> is not specified, it is treated as 0, selecting the <em>x</em> component
+of each texel to generate the result.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGatherOffset</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offset</em>, [ int <em>comp</em>])<br>
+ gvec4 <strong>textureGatherOffset</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em> [ int <em>comp</em>])<br>
+ vec4 <strong>textureGatherOffset</strong>(sampler2DShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>, ivec2 <em>offset</em>)<br>
+ vec4 <strong>textureGatherOffset</strong>(sampler2DArrayShadow <em>sampler</em>, vec3 <em>P</em>, float <em>refZ</em>, ivec2 <em>offset</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Perform a texture gather operation as in <strong>textureGather</strong> by <em>offset</em>
+ as described in <strong>textureOffset</strong> except that the <em>offset</em> can be
+ variable (non constant) and the implementation-dependent minimum and
+ maximum offset values are given by MIN_PROGRAM_TEXTURE_GATHER_OFFSET
+ and MAX_PROGRAM_TEXTURE_GATHER_OFFSET, respectively.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGatherOffsets</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offsets</em>[4] [, int <em>comp</em>])<br>
+ gvec4 <strong>textureGatherOffsets</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offsets</em>[4] [, int <em>comp</em>])<br>
+ vec4 <strong>textureGatherOffsets</strong>(sampler2DShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>, ivec2 <em>offsets</em>[4])<br>
+ vec4 <strong>textureGatherOffsets</strong>(sampler2DArrayShadow <em>sampler</em>, vec3 <em>P</em>, float <em>refZ</em>, ivec2 <em>offsets</em>[4])<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Operate identically to <strong>textureGatherOffset</strong> except that <em>offsets</em> is
+ used to determine the location of the four texels to sample.
+ Each of the four texels is obtained by applying the corresponding
+ offset in <em>offsets</em> as a (<em>u</em>, <em>v</em>) coordinate offset to <em>P</em>,
+ identifying the four-texel LINEAR footprint, and then selecting the
+ texel <em>i<sub>0</sub> j<sub>0</sub></em> of that footprint.
+ The specified values in <em>offsets</em> must be constant integral
+ expressions.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect2">
+<h3 id="atomic-counter-functions">8.10. Atomic Counter Functions</h3>
+<div class="paragraph">
+<p>The atomic-counter operations in this section operate atomically with
+respect to each other.
+They are atomic for any single counter, meaning any of these operations on a
+specific counter in one shader instantiation will be indivisible by any of
+these operations on the same counter from another shader instantiation.
+There is no guarantee that these operations are atomic with respect to other
+forms of access to the counter or that they are serialized when applied to
+separate counters.
+Such cases would require additional use of fences, barriers, or other forms
+of synchronization, if atomicity or serialization is desired.</p>
+</div>
+<div class="paragraph">
+<p>The underlying counter is a 32-bit unsigned integer.
+The result of operations will wrap to [0, 2<sup>32</sup>-1].</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterIncrement</strong>(atomic_uint <em>c</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>increments the counter for <em>c</em>, and</p>
+</li>
+<li>
+<p>returns its value prior to the increment operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterDecrement</strong>(atomic_uint <em>c</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>decrements the counter for <em>c</em>, and</p>
+</li>
+<li>
+<p>returns the value resulting from the decrement operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounter</strong>(atomic_uint <em>c</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the counter value for <em>c</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="atomic-memory-functions">8.11. Atomic Memory Functions</h3>
+<div class="paragraph">
+<p>Atomic memory functions perform atomic operations on an individual signed or
+unsigned integer stored in buffer object or shared variable storage.
+All of the atomic memory operations read a value from memory, compute a new
+value using one of the operations described below, write the new value to
+memory, and return the original value read.
+The contents of the memory being updated by the atomic operation are
+guaranteed not to be modified by any other assignment or atomic memory
+function in any shader invocation between the time the original value is
+read and the time the new value is written.</p>
+</div>
+<div class="paragraph">
+<p>Atomic memory functions are supported only for a limited set of variables.
+A shader will fail to compile if the value passed to the <em>mem</em> argument of
+an atomic memory function does not correspond to a buffer or shared
+variable.
+It is acceptable to pass an element of an array or a single component of a
+vector to the <em>mem</em> argument of an atomic memory function, as long as the
+underlying array or vector is a buffer or shared variable.</p>
+</div>
+<div class="paragraph">
+<p>All the built-in functions in this section accept arguments with
+combinations of <strong>restrict</strong>, <strong>coherent</strong>, and <strong>volatile</strong> memory qualification,
+despite not having them listed in the prototypes.
+The atomic operation will operate as required by the calling argument’s
+memory qualification, not by the built-in function’s formal parameter memory
+qualification.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicAdd</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicAdd</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by adding the value of <em>data</em> to the contents
+ <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicMin</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicMin</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by taking the minimum of the value of <em>data</em> and
+ the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicMax</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicMax</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by taking the maximum of the value of <em>data</em> and
+ the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicAnd</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicAnd</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise AND of the value of
+ <em>data</em> and the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicOr</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicOr</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise OR of the value of
+ <em>data</em> and the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicXor</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicXor</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise EXCLUSIVE OR of the
+ value of <em>data</em> and the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicExchange</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicExchange</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by simply copying the value of <em>data</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCompSwap</strong>(inout uint <em>mem</em>, uint <em>compare</em>, uint <em>data</em>)<br>
+ int <strong>atomicCompSwap</strong>(inout int <em>mem</em>, int <em>compare</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Compares the value of <em>compare</em> and the contents of <em>mem</em>.
+ If the values are equal, the new value is given by <em>data</em>; otherwise,
+ it is taken from the original contents of <em>mem</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="image-functions">8.12. Image Functions</h3>
+<div class="paragraph">
+<p>Variables using one of the image basic types may be used by the built-in
+shader image memory functions defined in this section to read and write
+individual texels of a texture.
+Each image variable references an image unit, which has a texture image
+attached.</p>
+</div>
+<div class="paragraph">
+<p>When image memory functions below access memory, an individual texel in the
+image is identified using an (<em>i</em>), (<em>i, j</em>), or (<em>i, j, k</em>) coordinate
+corresponding to the values of <em>P</em>.
+The coordinates
+are used to select an individual texel in the manner described in section
+8.22
+“Texture Image Loads and Stores” of the <a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>Loads and stores support float, integer, and unsigned integer types.
+The data types below starting “_gimage_” serve as placeholders meaning
+types starting either “<strong>image</strong>”, “<strong>iimage</strong>”, or “<strong>uimage</strong>” in the same
+way as <em>gvec</em> or <em>gsampler</em> in earlier sections.</p>
+</div>
+<div class="paragraph">
+<p>The <em>IMAGE_PARAMS</em> in the prototypes below is a placeholder representing
+18
+separate functions, each for a different type of image variable.
+The <em>IMAGE_PARAMS</em> placeholder is replaced by one of the following parameter
+lists:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage2D <em>image</em>, ivec2 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage3D <em>image</em>, ivec3 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimageCube <em>image</em>, ivec3 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimageBuffer <em>image</em>, int <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage2DArray <em>image</em>, ivec3 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimageCubeArray <em>image</em>, ivec3 <em>P</em></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>where each of the lines represents one of three different image variable
+types, and <em>image</em>,
+<em>P</em>
+specify the individual texel to operate on.
+The method for identifying the individual texel operated on from <em>image</em>,
+<em>P</em>,
+and the method for reading and writing the texel are specified in section
+8.22
+“Texture Image Loads and Stores” of the <a href="#references">OpenGL ES Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>The atomic functions perform operations on individual texels or samples of
+an image variable.
+Atomic memory operations read a value from the selected texel, compute a new
+value using one of the operations described below, write the new value to
+the selected texel, and return the original value read.
+The contents of the texel being updated by the atomic operation are
+guaranteed not to be modified by any other image store or atomic function
+between the time the original value is read and the time the new value is
+written.</p>
+</div>
+<div class="paragraph">
+<p>Atomic memory operations are supported on only a subset of all image
+variable types; <em>image</em> must be either:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>a signed integer image variable (type starts “<strong>iimage</strong>”) and a format
+qualifier of <strong>r32i</strong>, used with a <em>data</em> argument of type <strong>int</strong>, or</p>
+</li>
+<li>
+<p>an unsigned integer image variable (type starts “<strong>uimage</strong>”) and a
+format qualifier of <strong>r32ui</strong>, used with a <em>data</em> argument of type <strong>uint</strong>,
+or</p>
+</li>
+<li>
+<p>a float image variable (type starts “<strong>image</strong>”) and a format qualifier
+of <strong>r32f</strong>, used with a <em>data</em> argument of type <strong>float</strong>
+(<strong>imageAtomicExchange</strong> only).</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>All the built-in functions in this section accept arguments with
+combinations of <strong>restrict</strong>, <strong>coherent</strong>, and <strong>volatile</strong> memory qualification,
+despite not having them listed in the prototypes.
+The image operation will operate as required by the calling argument’s
+memory qualification, not by the built-in function’s formal parameter memory
+qualification.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp ivec2 <strong>imageSize</strong>(readonly writeonly gimage2D <em>image</em>)<br>
+ highp ivec3 <strong>imageSize</strong>(readonly writeonly gimage3D <em>image</em>)<br>
+ highp ivec2 <strong>imageSize</strong>(readonly writeonly gimageCube <em>image</em>)<br>
+ highp ivec3 <strong>imageSize</strong>(readonly writeonly gimageCubeArray <em>image</em>)<br>
+ highp ivec3 <strong>imageSize</strong>(readonly writeonly gimage2DArray <em>image</em>)<br>
+ highp int <strong>imageSize</strong>(readonly writeonly gimageBuffer <em>image</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the dimensions of the image or images bound to <em>image</em>.
+ For arrayed images, the last component of the return value will hold
+ the size of the array.
+ Cube images only return the dimensions of one face, and the number of
+ cubes in the cube map array, if arrayed.<br>
+ Note: The qualification <strong>readonly writeonly</strong> accepts a variable
+ qualified with <strong>readonly</strong>, <strong>writeonly</strong>, both, or neither.
+ It means the formal argument will be used for neither reading nor
+ writing to the underlying memory.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp gvec4 <strong>imageLoad</strong>(readonly <em>IMAGE_PARAMS</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Loads the texel at the coordinate <em>P</em> from the image unit <em>image</em> (in
+ <em>IMAGE_PARAMS</em>).
+ When <em>image</em> and <em>P</em>
+ identify a valid texel, the bits used to represent the selected texel in
+ memory are converted to a <strong>vec4</strong>, <strong>ivec4</strong>, or <strong>uvec4</strong> in the manner
+ described in section
+8.23
+ “Texture Image Loads and Stores” of the
+ <a href="#references">OpenGL ES Specification</a> and returned.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>imageStore</strong>(writeonly <em>IMAGE_PARAMS</em>, gvec4 <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Stores <em>data</em> into the texel at the coordinate <em>P</em> from the image
+ specified by <em>image</em>.
+ When <em>image</em> and <em>P</em>
+ identify a valid texel, the bits used to represent <em>data</em> are converted
+ to the format of the image unit in the manner described in section
+8.23
+ “Texture Image Loads and Stores” of the <a href="#references">OpenGL ES Specification</a>
+ and stored to the specified texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>imageAtomicAdd</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ highp int <strong>imageAtomicAdd</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by adding the value of <em>data</em> to the contents of
+ the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>imageAtomicMin</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ highp int <strong>imageAtomicMin</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by taking the minimum of the value of <em>data</em> and
+ the contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>imageAtomicMax</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ highp int <strong>imageAtomicMax</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by taking the maximum of the value <em>data</em> and the
+ contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>imageAtomicAnd</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ highp int <strong>imageAtomicAnd</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise AND of the value of
+ <em>data</em> and the contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>imageAtomicOr</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ highp int <strong>imageAtomicOr</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise OR of the value of
+ <em>data</em> and the contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>imageAtomicXor</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ highp int <strong>imageAtomicXor</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise EXCLUSIVE OR of the
+ value of <em>data</em> and the contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>imageAtomicExchange</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ highp int <strong>imageAtomicExchange</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)<br>
+ highp float <strong>imageAtomicExchange</strong>(<em>IMAGE_PARAMS</em>, float <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by simply copying the value of <em>data</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">highp uint <strong>imageAtomicCompSwap</strong>(<em>IMAGE_PARAMS</em>, uint <em>compare</em>, uint <em>data</em>)<br>
+ highp int <strong>imageAtomicCompSwap</strong>(<em>IMAGE_PARAMS</em>, int <em>compare</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Compares the value of <em>compare</em> and the contents of the selected
+ texel.
+ If the values are equal, the new value is given by <em>data</em>; otherwise,
+ it is taken from the original value loaded from the texel.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="geometry-shader-functions">8.13. Geometry Shader Functions</h3>
+<div class="paragraph">
+<p>These functions are only available in geometry shaders.
+They are described in more depth following the table.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>EmitVertex</strong>()<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Emits the current values of output variables to the current output
+ primitive.
+ On return from this call, the values of output variables are
+ undefined.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>EndPrimitive</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Completes the current output primitive and starts a new one.
+ No vertex is emitted.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The function <strong>EmitVertex</strong>() specifies that a vertex is completed.
+A vertex is added to the current output primitive using the current values
+of all built-in and user-defined output variables.
+The values of all output variables are undefined after a call to
+<strong>EmitVertex</strong>().
+If a geometry shader invocation has emitted more vertices than permitted by
+the output layout qualifier <strong>max_vertices</strong>, the results of calling
+<strong>EmitVertex</strong>() are undefined.</p>
+</div>
+<div class="paragraph">
+<p>The function <strong>EndPrimitive</strong>() specifies that the current output primitive is
+completed and a new output primitive (of the same type) will be started by
+any subsequent <strong>EmitVertex</strong>().
+This function does not emit a vertex.
+If the output layout is declared to be <strong>points</strong>, calling <strong>EndPrimitive</strong>() is
+optional.</p>
+</div>
+<div class="paragraph">
+<p>A geometry shader starts with an output primitive containing no vertices.
+When a geometry shader terminates, the current output primitive is
+automatically completed.
+It is not necessary to call <strong>EndPrimitive</strong>() if the geometry shader writes
+only a single primitive.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="fragment-processing-functions">8.14. Fragment Processing Functions</h3>
+<div class="paragraph">
+<p>Fragment processing functions are only available in fragment shaders.</p>
+</div>
+<div class="sect3">
+<h4 id="derivative-functions">8.14.1. Derivative Functions</h4>
+<div class="paragraph">
+<p>Derivatives may be computationally expensive and/or numerically unstable.
+Therefore, an OpenGL ES implementation may approximate the true derivatives
+by using a fast but not entirely accurate derivative computation.
+Derivatives are undefined within non-uniform control flow.</p>
+</div>
+<div class="paragraph">
+<p>The expected behavior of a derivative is specified using forward/backward
+differencing.</p>
+</div>
+<div class="paragraph">
+<p>Forward differencing:</p>
+</div>
+<div class="paragraph">
+<p>\(F(x+dx) - F(x) \sim dFdx(x) \cdot dx (1a)\)</p>
+</div>
+<div class="paragraph">
+<p>\(dFdx(x) \sim \frac{F(x+dx) - F(x)}{dx} (1b)\)</p>
+</div>
+<div class="paragraph">
+<p>Backward differencing:</p>
+</div>
+<div class="paragraph">
+<p>\(F(x-dx) - F(x) \sim -dFdx(x) \cdot dx (2a)\)</p>
+</div>
+<div class="paragraph">
+<p>\(dFdx(x) \sim \frac{F(x) - F(x-dx)}{dx} (2b)\)</p>
+</div>
+<div class="paragraph">
+<p>With single-sample rasterization, \(dx \leq 1.0\) in equations 1b
+and 2b.
+For multisample rasterization, \(dx < 2.0\) in equations 1b and 2b.</p>
+</div>
+<div class="paragraph">
+<p>\(dFdy\) is approximated similarly, with <em>y</em> replacing <em>x</em>.</p>
+</div>
+<div class="paragraph">
+<p>An OpenGL ES implementation may use the above or other methods to perform
+the calculation, subject to the following conditions:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>The method may use piecewise linear approximations.
+Such linear approximations imply that higher order derivatives,
+<strong>dFdx</strong>(<strong>dFdx</strong>(<em>x</em>)) and above, are undefined.</p>
+</li>
+<li>
+<p>The method may assume that the function evaluated is continuous.
+Therefore derivatives within the body of a non-uniform conditional are
+undefined.</p>
+</li>
+<li>
+<p>The method may differ per fragment, subject to the constraint that the
+method may vary by window coordinates, not screen coordinates.
+The invariance requirement described in section 13.2 “Invariance” of
+the <a href="#references">OpenGL ES Specification</a>, is relaxed for derivative calculations,
+because the method may be a function of fragment location.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Other properties that are desirable, but not required, are:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Functions should be evaluated within the interior of a primitive
+(interpolated, not extrapolated).</p>
+</li>
+<li>
+<p>Functions for <strong>dFdx</strong> should be evaluated while holding <em>y</em> constant.
+Functions for <strong>dFdy</strong> should be evaluated while holding <em>x</em> constant.
+However, mixed higher order derivatives, like <strong>dFdx</strong>(<strong>dFdy</strong>(<em>y</em>)) and
+<strong>dFdy</strong>(<strong>dFdx</strong>(<em>x</em>)) are undefined.</p>
+</li>
+<li>
+<p>Derivatives of constant arguments should be 0.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>In some implementations, varying degrees of derivative accuracy may be
+obtained by providing GL hints (see section 19.1 “Hints” of the
+<a href="#references">OpenGL ES Specification</a>), allowing a user to make an image quality versus
+speed trade off.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>dFdx</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the derivative in x using local differencing for the input
+ argument <em>p</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>dFdy</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the derivative in y using local differencing for the input
+ argument <em>p</em>.<br>
+ <br>
+ These two functions are commonly used to estimate the filter width used
+ to anti-alias procedural textures. We are assuming that the expression
+ is being evaluated in parallel on a SIMD array so that at any given
+ point in time the value of the function is known at the grid points
+ represented by the SIMD array. Local differencing between SIMD array
+ elements can therefore be used to derive <strong>dFdx</strong>, <strong>dFdy</strong>, etc.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>fwidth</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the sum of the absolute derivative in x and y using local
+ differencing for the input argument <em>p</em>, i.e., <strong>abs</strong>(<strong>dFdx</strong>(<em>p</em>))
+ + <strong>abs</strong>(<strong>dFdy</strong>(<em>p</em>));</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="interpolation-functions">8.14.2. Interpolation Functions</h4>
+<div class="paragraph">
+<p>Built-in interpolation functions are available to compute an interpolated
+value of a fragment shader input variable at a shader-specified (<em>x</em>, <em>y</em>)
+location.
+A separate (<em>x</em>, <em>y</em>) location may be used for each invocation of the
+built-in function, and those locations may differ from the default (<em>x</em>,
+<em>y</em>) location used to produce the default value of the input.
+For the <strong>interpolateAt*</strong> functions, the call will return a precision
+qualification matching the precision of the <em>interpolant</em> argument to the
+function call.</p>
+</div>
+<div class="paragraph">
+<p>For all of the interpolation functions, <em>interpolant</em> must be an l-value
+from an <strong>in</strong> declaration;
+this can be either a variable, or an array element.
+Component selection operators (e.g. <strong>.xy</strong>) may not be used when specifying <em>interpolant</em>.</p>
+</div>
+<div class="paragraph">
+<p>If <em>interpolant</em> is declared with the <strong>flat</strong> qualifier, the interpolated
+value will have the same value everywhere for a single primitive, so the
+location used for interpolation has no effect and the functions just return
+that same value.
+If <em>interpolant</em> is declared with the <strong>centroid</strong> qualifier, the value
+returned by <strong>interpolateAtSample</strong>() and <strong>interpolateAtOffset</strong>() will be
+evaluated at the specified location, ignoring the location normally used
+with the <strong>centroid</strong> qualifier.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>interpolateAtCentroid</strong>(float <em>interpolant</em>)<br>
+ vec2 <strong>interpolateAtCentroid</strong>(vec2 <em>interpolant</em>)<br>
+ vec3 <strong>interpolateAtCentroid</strong>(vec3 <em>interpolant</em>)<br>
+ vec4 <strong>interpolateAtCentroid</strong>(vec4 <em>interpolant</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the value of the input <em>interpolant</em> sampled at a location
+ inside both the pixel and the primitive being processed.
+ The value obtained would be the same value assigned to the input
+ variable if declared with the <strong>centroid</strong> qualifier.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>interpolateAtSample</strong>(float <em>interpolant</em>, int <em>sample</em>)<br>
+ vec2 <strong>interpolateAtSample</strong>(vec2 <em>interpolant</em>, int <em>sample</em>)<br>
+ vec3 <strong>interpolateAtSample</strong>(vec3 <em>interpolant</em>, int <em>sample</em>)<br>
+ vec4 <strong>interpolateAtSample</strong>(vec4 <em>interpolant</em>, int <em>sample</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the value of the input <em>interpolant</em> variable at the location
+ of sample number <em>sample</em>.
+ If multisample buffers are not available, the input variable will be
+ evaluated at the center of the pixel.
+ If sample <em>sample</em> does not exist, the position used to interpolate
+ the input variable is undefined.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>interpolateAtOffset</strong>(float <em>interpolant</em>, vec2 offset)<br>
+ vec2 <strong>interpolateAtOffset</strong>(vec2 <em>interpolant</em>, vec2 offset)<br>
+ vec3 <strong>interpolateAtOffset</strong>(vec3 <em>interpolant</em>, vec2 offset)<br>
+ vec4 <strong>interpolateAtOffset</strong>(vec4 <em>interpolant</em>, vec2 offset)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the value of the input <em>interpolant</em> variable sampled at an
+ offset from the center of the pixel specified by <em>offset</em>.
+ The two floating-point components of <em>offset</em>, give the offset in
+ pixels in the <em>x</em> and <em>y</em> directions, respectively.<br>
+ An offset of (0, 0) identifies the center of the pixel.
+ The range and granularity of offsets supported by this function is
+ implementation-dependent.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect2">
+<h3 id="shader-invocation-control-functions">8.15. Shader Invocation Control Functions</h3>
+<div class="paragraph">
+<p>The shader invocation control function is only available in tessellation
+control and compute shaders.
+It is used to control the relative execution order of multiple shader
+invocations used to process a patch (in the case of tessellation control
+shaders) or a workgroup (in the case of compute shaders), which are
+otherwise executed with an undefined relative order.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>barrier</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">For any given static instance of <strong>barrier</strong>(), all tessellation control
+ shader invocations for a single input patch must enter it before any
+ will be allowed to continue beyond it, or all compute shader
+ invocations for a single workgroup must enter it before any will
+ continue beyond it.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The function <strong>barrier</strong>() provides a partially defined order of execution
+between shader invocations.
+The ensures that, for some types of memory accesses, values written by one
+invocation prior to a given static instance of <strong>barrier</strong>() can be safely read
+by other invocations after their call to the same static instance <strong>barrier</strong>().
+Because invocations may execute in an undefined order between these barrier
+calls, the values of a per-vertex or per-patch output variable for tessellation
+control shaders, or the values of <strong>shared</strong> variables for compute shaders will be
+undefined in a number of cases enumerated in
+“<a href="#output-variables">Output Variables</a>” (for tessellation control shaders)
+and “<a href="#shared-variables">Shared Variables</a>” (for compute shaders).</p>
+</div>
+<div class="paragraph">
+<p>For tessellation control shaders, the <strong>barrier</strong>() function may only be
+placed inside the function <strong>main</strong>() of the shader and may not be called
+within any control flow.
+Barriers are also disallowed after a return statement in the function
+<strong>main</strong>().
+Any such misplaced barriers result in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>A <strong>barrier</strong>() affects control flow but only synchronizes memory accesses
+to <strong>shared</strong> variables and tessellation control output variables.
+For other memory accesses, it does not ensure that values written by one invocation
+prior to a given static instance of <strong>barrier</strong>() can be safely read by other
+invocations after their call to the same static instance of <strong>barrier</strong>().
+To achieve this requires the use of both <strong>barrier</strong>() and a memory barrier.</p>
+</div>
+<div class="paragraph">
+<p>For compute shaders, the <strong>barrier</strong>() function may be placed within control
+flow, but that control flow must be uniform control flow.
+That is, all the controlling expressions that lead to execution of the
+barrier must be dynamically uniform expressions.
+This ensures that if any shader invocation enters a conditional statement,
+then all invocations will enter it.
+While compilers are encouraged to give warnings if they can detect this
+might not happen, compilers cannot completely determine this.
+Hence, it is the author’s responsibility to ensure <strong>barrier</strong>() only exists
+inside uniform control flow.
+Otherwise, some shader invocations will stall indefinitely, waiting for a
+barrier that is never reached by other invocations.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="shader-memory-control-functions">8.16. Shader Memory Control Functions</h3>
+<div class="paragraph">
+<p>Within a single shader invocation, the visibility and order of writes made
+by that invocation are well-defined.
+However, the relative order of reads and writes to a single shared memory
+address from multiple separate shader invocations is largely undefined.
+Additionally, the order of accesses to multiple memory addresses performed
+by a single shader invocation, as observed by other shader invocations, is
+also undefined.</p>
+</div>
+<div class="paragraph">
+<p>The following built-in functions can be used to control the ordering of
+reads and writes:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrier</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of memory transactions issued by a single shader
+ invocation.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrierAtomicCounter</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of accesses to atomic-counter variables issued by
+ a single shader invocation.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrierBuffer</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of memory transactions to buffer variables issued
+ within a single shader invocation.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrierShared</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of memory transactions to shared variables issued
+ within a single shader invocation, as viewed by other invocations in
+ the same workgroup.<br>
+ Only available in compute shaders.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrierImage</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of memory transactions to images issued within a
+ single shader invocation.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>groupMemoryBarrier</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of all memory transactions issued within a single
+ shader invocation, as viewed by other invocations in the same workgroup.<br>
+ Only available in compute shaders.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The memory barrier built-in functions can be used to order reads and writes
+to variables stored in memory accessible to other shader invocations.
+When called, these functions will wait for the completion of all reads and
+writes previously performed by the caller that access selected variable
+types, and then return with no other effect.
+The built-in functions <strong>memoryBarrierAtomicCounter</strong>(),
+<strong>memoryBarrierBuffer</strong>(), <strong>memoryBarrierImage</strong>(), and <strong>memoryBarrierShared</strong>()
+wait for the completion of accesses to atomic counter, buffer, image, and
+shared variables, respectively.
+The built-in functions <strong>memoryBarrier</strong>() and <strong>groupMemoryBarrier</strong>() wait for
+the completion of accesses to all of the above variable types.
+The functions <strong>memoryBarrierShared</strong>() and <strong>groupMemoryBarrier</strong>() are
+available only in compute shaders; the other functions are available in all
+shader types.</p>
+</div>
+<div class="paragraph">
+<p>When these functions return, the effects of any memory stores performed
+using coherent variables prior to the call will be visible to any
+future<sup>1</sup> coherent access to the same memory performed by any other shader
+invocation.
+In particular, the values written this way in one shader stage are
+guaranteed to be visible to coherent memory accesses performed by shader
+invocations in subsequent stages when those invocations were triggered by
+the execution of the original shader invocation (e.g. fragment shader
+invocations for a primitive resulting from a particular geometry
+shader invocation).</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">1</dt>
+<dd>
+<p>An access is only a <em>future</em> access if a <em>happens-before</em> relation can
+be established between the store and the load.</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>Additionally, memory barrier functions order stores performed by the calling
+invocation, as observed by other shader invocations.
+Without memory barriers, if one shader invocation performs two stores to
+coherent variables, a second shader invocation might see the values written
+by the second store prior to seeing those written by the first.
+However, if the first shader invocation calls a memory barrier function
+between the two stores, selected other shader invocations will never see the
+results of the second store before seeing those of the first.
+When using the functions <strong>groupMemoryBarrier</strong>() or <strong>memoryBarrierShared</strong>(),
+this ordering guarantee applies only to other shader invocations in the same
+compute shader workgroup; all other memory barrier functions provide the
+guarantee to all other shader invocations.
+No memory barrier is required to guarantee the order of memory stores as
+observed by the invocation performing the stores; an invocation reading from
+a variable that it previously wrote will always see the most recently
+written value unless another shader invocation also wrote to the same
+memory.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="shader-interface-matching">9. Shader Interface Matching</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>As described in chapter 7 of the <a href="#references">OpenGL ES Specification</a>, shaders may be
+linked together to form a <em>program object</em> before being bound to the
+pipeline or may be linked and bound individually as <em>separable program
+objects</em>.</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="title">Note</div>
+<div class="paragraph">
+<p>These were previously known as separate shader objects (SSOs) but the
+mechanism has been extended to support future versions of the specification
+that have more than two shader stages.
+It allows a subset of the shaders to be linked together.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="paragraph">
+<p>Within a <em>program object</em> or a <em>separable program object</em>, qualifiers for
+matching variables must themselves match according to the rules specified in
+this section.
+There are also matching rules for qualifiers of matching variables between
+separable program objects but only for variables across an input/output
+boundary between shader stages.
+For other shader interface variables such as uniforms, each program object
+or separable program object has its own name space and so the same name can
+refer to multiple independent variables.
+Consequently, there are no matching rules for qualifiers in these cases.</p>
+</div>
+<div class="sect2">
+<h3 id="input-output-matching-by-name-in-linked-programs">9.1. Input Output Matching by Name in Linked Programs</h3>
+<div class="paragraph">
+<p>When linking shaders, the type of declared vertex outputs and fragment
+inputs with the same name must match, otherwise the link command will fail.
+Only those fragment inputs statically used (i.e. read) in the fragment
+shader must be declared as outputs in the vertex shader; declaring
+superfluous vertex shader outputs is permissible.</p>
+</div>
+<div class="paragraph">
+<p>The following table summarizes the rules for matching shader outputs to
+shader inputs in consecutive stages when shaders are linked together.</p>
+</div>
+<table class="tableblock frame-all grid-all fit-content">
+<colgroup>
+<col>
+<col>
+<col>
+<col>
+<col>
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top" colspan="2" rowspan="2"><p class="tableblock">Treatment of Mismatched Input Variables</p></td>
+<td class="tableblock halign-left valign-top" colspan="3"><p class="tableblock">Consuming Shader (input variables)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No Declaration</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Declared but no Static Use</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Declared and Static Use</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top" rowspan="3"><p class="tableblock">Generating Shader (output variables)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No Declaration</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">error</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Declares; no static Use</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed (values are undefined)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Declares and static Use</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed (values are potentially undefined)</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>See “<a href="#static-use">Static Use</a>” for the definition of <em>static use</em>.</p>
+</div>
+<div class="paragraph">
+<p>The precision of a vertex output does not need to match the precision of the
+corresponding fragment input.
+The minimum precision at which vertex outputs are interpolated is the
+minimum of the vertex output precision and the fragment input precision,
+with the exception that for <strong>highp</strong>, implementations do not have to support
+full IEEE 754 precision.</p>
+</div>
+<div class="paragraph">
+<p>The precision of values exported to a transform feedback buffer is the
+precision of the outputs of the vertex shader.
+However, they are converted to <strong>highp</strong> format before being written.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="matching-of-qualifiers">9.2. Matching of Qualifiers</h3>
+<div class="paragraph">
+<p>The following tables summarize the requirements for matching of qualifiers.
+It applies whenever there are two or more matching variables in a shader
+interface.</p>
+</div>
+<div class="paragraph">
+<p>Notes:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p><em>Yes</em> means the qualifiers must match.</p>
+</li>
+<li>
+<p><em>No</em> means the qualifiers do not need to match.</p>
+</li>
+<li>
+<p><em>Consistent</em> means qualifiers may be missing from a subset of
+declarations but they cannot conflict</p>
+</li>
+<li>
+<p>If there are conflicting qualifiers, only the last of these is
+significant.</p>
+</li>
+<li>
+<p>Matching is based only on the resulting qualification, not on the
+presence or otherwise of qualifiers.</p>
+</li>
+<li>
+<p>The rules apply to all declared variables, irrespective of whether they
+are statically used, with the exception of inputs and outputs when
+shaders are linked (see
+“<a href="#input-output-matching-by-name-in-linked-programs">Input Output
+Matching by Name in Linked Programs</a>”).</p>
+</li>
+<li>
+<p>Errors are generated for any conflicts.</p>
+</li>
+</ol>
+</div>
+<div class="sect3">
+<h4 id="linked-shaders">9.2.1. Linked Shaders</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier Class</th>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top"><strong>in</strong>/<strong>out</strong></th>
+<th class="tableblock halign-left valign-top">Default Uniforms</th>
+<th class="tableblock halign-left valign-top"><strong>uniform</strong> Block</th>
+<th class="tableblock halign-left valign-top"><strong>buffer</strong> Block</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Storage<sup>1</sup></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>in</strong><br>
+ <strong>out</strong><br>
+ <strong>uniform</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top" rowspan="2"><p class="tableblock">Auxiliary</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>centroid</strong><br>
+ <strong>sample</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>patch</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top" rowspan="5"><p class="tableblock">Layout</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>location</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes<sup>2</sup></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Consistent</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Block layout<sup>3,4</sup></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>binding</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Consistent</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>offset</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">format</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Interpolation</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>smooth</strong><br>
+ <strong>flat</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Precision</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lowp</strong><br>
+ <strong>mediump</strong><br>
+ <strong>highp</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Variance</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>invariant</strong><br>
+ <strong>precise</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Memory</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">all</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+</tr>
+</tbody>
+</table>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">1</dt>
+<dd>
+<p>Storage qualifiers determine <em>when</em> variables match rather than being
+<em>required</em> to match for matching variables.
+Note also that each shader interface has a separate name space so for
+example, it is possible to use the same name for a vertex output and
+fragment uniform.</p>
+</dd>
+<dt class="hdlist1">2</dt>
+<dd>
+<p>If present, the <strong>location</strong> qualifier determines the matching of inputs
+and outputs.
+See section 7.4.1 “Shader interface Matching” of the
+<a href="#references">OpenGL ES Specification</a> for details.]</p>
+</dd>
+<dt class="hdlist1">3</dt>
+<dd>
+<p>The <strong>row_major</strong> and <strong>column_major</strong> layout qualifiers do not need to
+match when applied to non-matrix types.</p>
+</dd>
+<dt class="hdlist1">4</dt>
+<dd>
+<p>In cases where a layout qualifier overrides a previous layout qualifier
+or a default, only the resulting qualification must match.</p>
+</dd>
+</dl>
+</div>
+</div>
+<div class="sect3">
+<h4 id="separable-programs">9.2.2. Separable Programs</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 33.3333%;">
+<col style="width: 33.3333%;">
+<col style="width: 33.3334%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier Class</th>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top"><strong>in</strong>/<strong>out</strong></th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Storage</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>in</strong><br>
+ <strong>out</strong><br>
+ <strong>uniform</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top" rowspan="2"><p class="tableblock">Auxiliary</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>centroid</strong><br>
+ <strong>sample</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>patch</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top" rowspan="5"><p class="tableblock">Layout</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>location</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Block layout</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>binding</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>offset</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">format</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Interpolation</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>smooth</strong><br>
+ <strong>flat</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Precision</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lowp</strong><br>
+ <strong>mediump</strong><br>
+ <strong>highp</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Yes</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Variance</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>invariant</strong><br>
+ <strong>precise</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Memory</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">all</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">N/A</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="shading-language-grammar">10. Shading Language Grammar</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The grammar is fed from the output of lexical analysis.
+The tokens returned from lexical analysis are</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">CONST BOOL FLOAT INT UINT
+
+BVEC2 BVEC3 BVEC4 IVEC2 IVEC3 IVEC4 UVEC2 UVEC3 UVEC4 VEC2 VEC3 VEC4
+
+MAT2 MAT3 MAT4 CENTROID IN OUT INOUT
+
+UNIFORM PATCH SAMPLE BUFFER SHARED
+
+COHERENT VOLATILE RESTRICT READONLY WRITEONLY
+
+FLAT SMOOTH LAYOUT
+
+MAT2X2 MAT2X3 MAT2X4
+
+MAT3X2 MAT3X3 MAT3X4
+
+MAT4X2 MAT4X3 MAT4X4
+
+
+
+
+
+ATOMIC_UINT
+
+SAMPLER2D SAMPLER3D SAMPLERCUBE SAMPLER2DSHADOW
+
+SAMPLERCUBESHADOW SAMPLER2DARRAY SAMPLER2DARRAYSHADOW
+
+ISAMPLER2D ISAMPLER3D ISAMPLERCUBE ISAMPLER2DARRAY
+
+USAMPLER2D USAMPLER3D USAMPLERCUBE USAMPLER2DARRAY
+
+
+
+
+
+
+SAMPLERBUFFER ISAMPLERBUFFER USAMPLERBUFFER
+
+SAMPLERCUBEARRAY SAMPLERCUBEARRAYSHADOW
+
+ISAMPLERCUBEARRAY USAMPLERCUBEARRAY
+
+SAMPLER2DMS ISAMPLER2DMS USAMPLER2DMS
+
+SAMPLER2DMSARRAY ISAMPLER2DMSARRAY USAMPLER2DMSARRAY
+
+IMAGE2D IIMAGE2D UIMAGE2D
+
+IMAGE3D IIMAGE3D UIMAGE3D
+
+IMAGECUBE IIMAGECUBE UIMAGECUBE
+
+IMAGEBUFFER IIMAGEBUFFER UIMAGEBUFFER
+
+IMAGE2DARRAY IIMAGE2DARRAY UIMAGE2DARRAY
+
+IMAGECUBEARRAY IIMAGECUBEARRAY UIMAGECUBEARRAY
+
+
+
+
+
+
+STRUCT VOID
+
+WHILE BREAK CONTINUE DO ELSE FOR IF DISCARD RETURN SWITCH CASE DEFAULT
+
+IDENTIFIER TYPE_NAME
+
+FLOATCONSTANT INTCONSTANT UINTCONSTANT BOOLCONSTANT
+
+FIELD_SELECTION
+
+LEFT_OP RIGHT_OP
+
+INC_OP DEC_OP LE_OP GE_OP EQ_OP NE_OP
+
+AND_OP OR_OP XOR_OP MUL_ASSIGN DIV_ASSIGN ADD_ASSIGN
+
+MOD_ASSIGN LEFT_ASSIGN RIGHT_ASSIGN AND_ASSIGN XOR_ASSIGN OR_ASSIGN
+
+SUB_ASSIGN
+
+LEFT_PAREN RIGHT_PAREN LEFT_BRACKET RIGHT_BRACKET LEFT_BRACE RIGHT_BRACE DOT
+
+COMMA COLON EQUAL SEMICOLON BANG DASH TILDE PLUS STAR SLASH PERCENT
+
+LEFT_ANGLE RIGHT_ANGLE VERTICAL_BAR CARET AMPERSAND QUESTION
+
+INVARIANT PRECISE
+
+HIGH_PRECISION MEDIUM_PRECISION LOW_PRECISION PRECISION</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The following describes the grammar for the OpenGL ES Shading Language in terms of the above
+tokens.
+The starting rule is <em>translation_unit</em>.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>variable_identifier</em> : </dt>
+<dd>
+<p><em>IDENTIFIER</em></p>
+</dd>
+<dt class="hdlist1"><em>primary_expression</em> : </dt>
+<dd>
+<p><em>variable_identifier</em><br>
+<em>INTCONSTANT</em><br>
+<em>UINTCONSTANT</em><br>
+<em>FLOATCONSTANT</em><br>
+<em>BOOLCONSTANT</em><br>
+<em>LEFT_PAREN</em> <em>expression</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>postfix_expression</em> : </dt>
+<dd>
+<p><em>primary_expression</em><br>
+<em>postfix_expression</em> <em>LEFT_BRACKET</em> <em>integer_expression</em> <em>RIGHT_BRACKET</em><br>
+<em>function_call</em><br>
+<em>postfix_expression</em> <em>DOT</em> <em>FIELD_SELECTION</em><br>
+<em>postfix_expression</em> <em>INC_OP</em><br>
+<em>postfix_expression</em> <em>DEC_OP</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>FIELD_SELECTION includes members in structures, component selection for
+vectors and the 'length' identifier for the length() method</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>integer_expression</em> : </dt>
+<dd>
+<p><em>expression</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call</em> : </dt>
+<dd>
+<p><em>function_call_or_method</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_or_method</em> : </dt>
+<dd>
+<p><em>function_call_generic</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_generic</em> : </dt>
+<dd>
+<p><em>function_call_header_with_parameters</em> <em>RIGHT_PAREN</em><br>
+<em>function_call_header_no_parameters</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_header_no_parameters</em> : </dt>
+<dd>
+<p><em>function_call_header</em> <em>VOID</em><br>
+<em>function_call_header</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_header_with_parameters</em> : </dt>
+<dd>
+<p><em>function_call_header</em> <em>assignment_expression</em><br>
+<em>function_call_header_with_parameters</em> <em>COMMA</em> <em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_header</em> : </dt>
+<dd>
+<p><em>function_identifier</em> <em>LEFT_PAREN</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: Constructors look like functions, but lexical analysis
+recognized most of them as keywords.
+They are now recognized through <em>type_specifier</em>.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Methods (<strong>.length</strong>) and identifiers are recognized through
+<em>postfix_expression</em>.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>function_identifier</em> : </dt>
+<dd>
+<p><em>type_specifier</em><br>
+<em>postfix_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>unary_expression</em> : </dt>
+<dd>
+<p><em>postfix_expression</em><br>
+<em>INC_OP</em> <em>unary_expression</em><br>
+<em>DEC_OP</em> <em>unary_expression</em><br>
+<em>unary_operator</em> <em>unary_expression</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: No traditional style type casts.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>unary_operator</em> : </dt>
+<dd>
+<p><em>PLUS</em><br>
+<em>DASH</em><br>
+<em>BANG</em><br>
+<em>TILDE</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: No '*' or '&' unary ops.
+Pointers are not supported.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>multiplicative_expression</em> : </dt>
+<dd>
+<p><em>unary_expression</em><br>
+<em>multiplicative_expression</em> <em>STAR</em> <em>unary_expression</em><br>
+<em>multiplicative_expression</em> <em>SLASH</em> <em>unary_expression</em><br>
+<em>multiplicative_expression</em> <em>PERCENT</em> <em>unary_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>additive_expression</em> : </dt>
+<dd>
+<p><em>multiplicative_expression</em><br>
+<em>additive_expression</em> <em>PLUS</em> <em>multiplicative_expression</em><br>
+<em>additive_expression</em> <em>DASH</em> <em>multiplicative_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>shift_expression</em> : </dt>
+<dd>
+<p><em>additive_expression</em><br>
+<em>shift_expression</em> <em>LEFT_OP</em> <em>additive_expression</em><br>
+<em>shift_expression</em> <em>RIGHT_OP</em> <em>additive_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>relational_expression</em> : </dt>
+<dd>
+<p><em>shift_expression</em><br>
+<em>relational_expression</em> <em>LEFT_ANGLE</em> <em>shift_expression</em><br>
+<em>relational_expression</em> <em>RIGHT_ANGLE</em> <em>shift_expression</em><br>
+<em>relational_expression</em> <em>LE_OP</em> <em>shift_expression</em><br>
+<em>relational_expression</em> <em>GE_OP</em> <em>shift_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>equality_expression</em> : </dt>
+<dd>
+<p><em>relational_expression</em><br>
+<em>equality_expression</em> <em>EQ_OP</em> <em>relational_expression</em><br>
+<em>equality_expression</em> <em>NE_OP</em> <em>relational_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>and_expression</em> : </dt>
+<dd>
+<p><em>equality_expression</em><br>
+<em>and_expression</em> <em>AMPERSAND</em> <em>equality_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>exclusive_or_expression</em> : </dt>
+<dd>
+<p><em>and_expression</em><br>
+<em>exclusive_or_expression</em> <em>CARET</em> <em>and_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>inclusive_or_expression</em> : </dt>
+<dd>
+<p><em>exclusive_or_expression</em><br>
+<em>inclusive_or_expression</em> <em>VERTICAL_BAR</em> <em>exclusive_or_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>logical_and_expression</em> : </dt>
+<dd>
+<p><em>inclusive_or_expression</em><br>
+<em>logical_and_expression</em> <em>AND_OP</em> <em>inclusive_or_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>logical_xor_expression</em> : </dt>
+<dd>
+<p><em>logical_and_expression</em><br>
+<em>logical_xor_expression</em> <em>XOR_OP</em> <em>logical_and_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>logical_or_expression</em> : </dt>
+<dd>
+<p><em>logical_xor_expression</em><br>
+<em>logical_or_expression</em> <em>OR_OP</em> <em>logical_xor_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>conditional_expression</em> : </dt>
+<dd>
+<p><em>logical_or_expression</em><br>
+<em>logical_or_expression</em> <em>QUESTION</em> <em>expression</em> <em>COLON</em>
+<em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>assignment_expression</em> : </dt>
+<dd>
+<p><em>conditional_expression</em><br>
+<em>unary_expression</em> <em>assignment_operator</em> <em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>assignment_operator</em> : </dt>
+<dd>
+<p><em>EQUAL</em><br>
+<em>MUL_ASSIGN</em><br>
+<em>DIV_ASSIGN</em><br>
+<em>MOD_ASSIGN</em><br>
+<em>ADD_ASSIGN</em><br>
+<em>SUB_ASSIGN</em><br>
+<em>LEFT_ASSIGN</em><br>
+<em>RIGHT_ASSIGN</em><br>
+<em>AND_ASSIGN</em><br>
+<em>XOR_ASSIGN</em><br>
+<em>OR_ASSIGN</em></p>
+</dd>
+<dt class="hdlist1"><em>expression</em> : </dt>
+<dd>
+<p><em>assignment_expression</em><br>
+<em>expression</em> <em>COMMA</em> <em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>constant_expression</em> : </dt>
+<dd>
+<p><em>conditional_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>declaration</em> : </dt>
+<dd>
+<p><em>function_prototype</em> <em>SEMICOLON</em><br>
+<em>init_declarator_list</em> <em>SEMICOLON</em><br>
+<em>PRECISION</em> <em>precision_qualifier</em> <em>type_specifier</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em>
+<em>RIGHT_BRACE</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em>
+<em>RIGHT_BRACE</em> <em>IDENTIFIER</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em>
+<em>RIGHT_BRACE</em> <em>IDENTIFIER</em> <em>array_specifier</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>identifier_list</em> <em>SEMICOLON</em></p>
+</dd>
+<dt class="hdlist1"><em>identifier_list</em> : </dt>
+<dd>
+<p><em>COMMA</em> <em>IDENTIFIER</em><br>
+<em>identifier_list</em> <em>COMMA</em> <em>IDENTIFIER</em></p>
+</dd>
+<dt class="hdlist1"><em>function_prototype</em> : </dt>
+<dd>
+<p><em>function_declarator</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>function_declarator</em> : </dt>
+<dd>
+<p><em>function_header</em><br>
+<em>function_header_with_parameters</em></p>
+</dd>
+<dt class="hdlist1"><em>function_header_with_parameters</em> : </dt>
+<dd>
+<p><em>function_header</em> <em>parameter_declaration</em><br>
+<em>function_header_with_parameters</em> <em>COMMA</em> <em>parameter_declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>function_header</em> : </dt>
+<dd>
+<p><em>fully_specified_type</em> <em>IDENTIFIER</em> <em>LEFT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>parameter_declarator</em> : </dt>
+<dd>
+<p><em>type_specifier</em> <em>IDENTIFIER</em><br>
+<em>type_specifier</em> <em>IDENTIFIER</em> <em>array_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>parameter_declaration</em> : </dt>
+<dd>
+<p><em>type_qualifier</em> <em>parameter_declarator</em><br>
+<em>parameter_declarator</em><br>
+<em>type_qualifier</em> <em>parameter_type_specifier</em><br>
+<em>parameter_type_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>parameter_type_specifier</em> : </dt>
+<dd>
+<p><em>type_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>init_declarator_list</em> : </dt>
+<dd>
+<p><em>single_declaration</em><br>
+<em>init_declarator_list</em> <em>COMMA</em> <em>IDENTIFIER</em><br>
+<em>init_declarator_list</em> <em>COMMA</em> <em>IDENTIFIER</em> <em>array_specifier</em><br>
+<em>init_declarator_list</em> <em>COMMA</em> <em>IDENTIFIER</em> <em>array_specifier</em> <em>EQUAL</em>
+<em>initializer</em><br>
+<em>init_declarator_list</em> <em>COMMA</em> <em>IDENTIFIER</em> <em>EQUAL</em> <em>initializer</em></p>
+</dd>
+<dt class="hdlist1"><em>single_declaration</em> : </dt>
+<dd>
+<p><em>fully_specified_type</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em> <em>array_specifier</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em> <em>array_specifier</em> <em>EQUAL</em>
+<em>initializer</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em> <em>EQUAL</em> <em>initializer</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: No 'enum', or 'typedef'.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>fully_specified_type</em> : </dt>
+<dd>
+<p><em>type_specifier</em><br>
+<em>type_qualifier</em> <em>type_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>invariant_qualifier</em> : </dt>
+<dd>
+<p><em>INVARIANT</em></p>
+</dd>
+<dt class="hdlist1"><em>interpolation_qualifier</em> : </dt>
+<dd>
+<p><em>SMOOTH</em><br>
+<em>FLAT</em></p>
+</dd>
+<dt class="hdlist1"><em>layout_qualifier</em> : </dt>
+<dd>
+<p><em>LAYOUT</em> <em>LEFT_PAREN</em> <em>layout_qualifier_id_list</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>layout_qualifier_id_list</em> : </dt>
+<dd>
+<p><em>layout_qualifier_id</em><br>
+<em>layout_qualifier_id_list</em> <em>COMMA</em> <em>layout_qualifier_id</em></p>
+</dd>
+<dt class="hdlist1"><em>layout_qualifier_id</em> : </dt>
+<dd>
+<p><em>IDENTIFIER</em><br>
+<em>IDENTIFIER</em> <em>EQUAL</em> <em>INTCONSTANT</em><br>
+<em>IDENTIFIER</em> <em>EQUAL</em> <em>UINTCONSTANT</em><br>
+<em>SHARED</em></p>
+</dd>
+<dt class="hdlist1"><em>precise_qualifier</em> : </dt>
+<dd>
+<p><em>PRECISE</em></p>
+</dd>
+<dt class="hdlist1"><em>type_qualifier</em> : </dt>
+<dd>
+<p><em>single_type_qualifier</em><br>
+<em>type_qualifier</em> <em>single_type_qualifier</em></p>
+</dd>
+<dt class="hdlist1"><em>single_type_qualifier</em> : </dt>
+<dd>
+<p><em>storage_qualifier</em><br>
+<em>layout_qualifier</em><br>
+<em>precision_qualifier</em><br>
+<em>interpolation_qualifier</em><br>
+<em>invariant_qualifier</em><br>
+<em>precise_qualifier</em></p>
+</dd>
+<dt class="hdlist1"><em>storage_qualifier</em> : </dt>
+<dd>
+<p><em>CONST</em><br>
+<em>IN</em><br>
+<em>OUT</em><br>
+<em>INOUT</em><br>
+<em>CENTROID</em><br>
+<em>PATCH</em><br>
+<em>SAMPLE</em><br>
+<em>UNIFORM</em><br>
+<em>BUFFER</em><br>
+<em>SHARED</em><br>
+<em>COHERENT</em><br>
+<em>VOLATILE</em><br>
+<em>RESTRICT</em><br>
+<em>READONLY</em><br>
+<em>WRITEONLY</em></p>
+</dd>
+<dt class="hdlist1"><em>type_specifier</em> : </dt>
+<dd>
+<p><em>type_specifier_nonarray</em><br>
+<em>type_specifier_nonarray</em> <em>array_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>array_specifier</em> : </dt>
+<dd>
+<p><em>LEFT_BRACKET</em> <em>RIGHT_BRACKET</em><br>
+<em>LEFT_BRACKET</em> <em>constant_expression</em> <em>RIGHT_BRACKET</em><br>
+<em>array_specifier</em> <em>LEFT_BRACKET</em> <em>RIGHT_BRACKET</em><br>
+<em>array_specifier</em> <em>LEFT_BRACKET</em> <em>constant_expression</em> <em>RIGHT_BRACKET</em></p>
+</dd>
+<dt class="hdlist1"><em>type_specifier_nonarray</em> : </dt>
+<dd>
+<p><em>VOID</em><br>
+<em>FLOAT</em><br>
+<em>INT</em><br>
+<em>UINT</em><br>
+<em>BOOL</em><br>
+<em>VEC2</em><br>
+<em>VEC3</em><br>
+<em>VEC4</em><br>
+<em>BVEC2</em><br>
+<em>BVEC3</em><br>
+<em>BVEC4</em><br>
+<em>IVEC2</em><br>
+<em>IVEC3</em><br>
+<em>IVEC4</em><br>
+<em>UVEC2</em><br>
+<em>UVEC3</em><br>
+<em>UVEC4</em><br>
+<em>MAT2</em><br>
+<em>MAT3</em><br>
+<em>MAT4</em><br>
+<em>MAT2X2</em><br>
+<em>MAT2X3</em><br>
+<em>MAT2X4</em><br>
+<em>MAT3X2</em><br>
+<em>MAT3X3</em><br>
+<em>MAT3X4</em><br>
+<em>MAT4X2</em><br>
+<em>MAT4X3</em><br>
+<em>MAT4X4</em><br>
+<em>ATOMIC_UINT</em><br>
+<em>SAMPLER2D</em><br>
+<em>SAMPLER3D</em><br>
+<em>SAMPLERCUBE</em><br>
+<em>SAMPLER2DSHADOW</em><br>
+<em>SAMPLERCUBESHADOW</em><br>
+<em>SAMPLER2DARRAY</em><br>
+<em>SAMPLER2DARRAYSHADOW</em><br>
+<em>SAMPLERCUBEARRAY</em><br>
+<em>SAMPLERCUBEARRAYSHADOW</em><br>
+<em>ISAMPLER2D</em><br>
+<em>ISAMPLER3D</em><br>
+<em>ISAMPLERCUBE</em><br>
+<em>ISAMPLER2DARRAY</em><br>
+<em>ISAMPLERCUBEARRAY</em><br>
+<em>USAMPLER2D</em><br>
+<em>USAMPLER3D</em><br>
+<em>USAMPLERCUBE</em><br>
+<em>USAMPLER2DARRAY</em><br>
+<em>USAMPLERCUBEARRAY</em><br>
+<em>SAMPLERBUFFER</em><br>
+<em>ISAMPLERBUFFER</em><br>
+<em>USAMPLERBUFFER</em><br>
+<em>SAMPLER2DMS</em><br>
+<em>ISAMPLER2DMS</em><br>
+<em>USAMPLER2DMS</em><br>
+<em>SAMPLER2DMSARRAY</em><br>
+<em>ISAMPLER2DMSARRAY</em><br>
+<em>USAMPLER2DMSARRAY</em><br>
+<em>IMAGE2D</em><br>
+<em>IIMAGE2D</em><br>
+<em>UIMAGE2D</em><br>
+<em>IMAGE3D</em><br>
+<em>IIMAGE3D</em><br>
+<em>UIMAGE3D</em><br>
+<em>IMAGECUBE</em><br>
+<em>IIMAGECUBE</em><br>
+<em>UIMAGECUBE</em><br>
+<em>IMAGEBUFFER</em><br>
+<em>IIMAGEBUFFER</em><br>
+<em>UIMAGEBUFFER</em><br>
+<em>IMAGE2DARRAY</em><br>
+<em>IIMAGE2DARRAY</em><br>
+<em>UIMAGE2DARRAY</em><br>
+<em>IMAGECUBEARRAY</em><br>
+<em>IIMAGECUBEARRAY</em><br>
+<em>UIMAGECUBEARRAY</em><br>
+<em>struct_specifier</em><br>
+<em>TYPE_NAME</em></p>
+</dd>
+<dt class="hdlist1"><em>precision_qualifier</em> : </dt>
+<dd>
+<p><em>HIGH_PRECISION</em><br>
+<em>MEDIUM_PRECISION</em><br>
+<em>LOW_PRECISION</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_specifier</em> : </dt>
+<dd>
+<p><em>STRUCT</em> <em>IDENTIFIER</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em>
+<em>RIGHT_BRACE</em><br>
+<em>STRUCT</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em> <em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_declaration_list</em> : </dt>
+<dd>
+<p><em>struct_declaration</em><br>
+<em>struct_declaration_list</em> <em>struct_declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_declaration</em> : </dt>
+<dd>
+<p><em>type_specifier</em> <em>struct_declarator_list</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>type_specifier</em> <em>struct_declarator_list</em> <em>SEMICOLON</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_declarator_list</em> : </dt>
+<dd>
+<p><em>struct_declarator</em><br>
+<em>struct_declarator_list</em> <em>COMMA</em> <em>struct_declarator</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_declarator</em> : </dt>
+<dd>
+<p><em>IDENTIFIER</em><br>
+<em>IDENTIFIER</em> <em>array_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>initializer</em> : </dt>
+<dd>
+<p><em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>declaration_statement</em> : </dt>
+<dd>
+<p><em>declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>statement</em> : </dt>
+<dd>
+<p><em>compound_statement</em><br>
+<em>simple_statement</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: labeled statements for SWITCH only; 'goto' is not supported.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>simple_statement</em> : </dt>
+<dd>
+<p><em>declaration_statement</em><br>
+<em>expression_statement</em><br>
+<em>selection_statement</em><br>
+<em>switch_statement</em><br>
+<em>case_label</em><br>
+<em>iteration_statement</em><br>
+<em>jump_statement</em></p>
+</dd>
+<dt class="hdlist1"><em>compound_statement</em> : </dt>
+<dd>
+<p><em>LEFT_BRACE</em> <em>RIGHT_BRACE</em><br>
+<em>LEFT_BRACE</em> <em>statement_list</em> <em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>statement_no_new_scope</em> : </dt>
+<dd>
+<p><em>compound_statement_no_new_scope</em><br>
+<em>simple_statement</em></p>
+</dd>
+<dt class="hdlist1"><em>compound_statement_no_new_scope</em> : </dt>
+<dd>
+<p><em>LEFT_BRACE</em> <em>RIGHT_BRACE</em><br>
+<em>LEFT_BRACE</em> <em>statement_list</em> <em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>statement_list</em> : </dt>
+<dd>
+<p><em>statement</em><br>
+<em>statement_list</em> <em>statement</em></p>
+</dd>
+<dt class="hdlist1"><em>expression_statement</em> : </dt>
+<dd>
+<p><em>SEMICOLON</em><br>
+<em>expression</em> <em>SEMICOLON</em></p>
+</dd>
+<dt class="hdlist1"><em>selection_statement</em> : </dt>
+<dd>
+<p><em>IF</em> <em>LEFT_PAREN</em> <em>expression</em> <em>RIGHT_PAREN</em> <em>selection_rest_statement</em></p>
+</dd>
+<dt class="hdlist1"><em>selection_rest_statement</em> : </dt>
+<dd>
+<p><em>statement</em> <em>ELSE</em> <em>statement</em><br>
+<em>statement</em></p>
+</dd>
+<dt class="hdlist1"><em>condition</em> : </dt>
+<dd>
+<p><em>expression</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em> <em>EQUAL</em> <em>initializer</em></p>
+</dd>
+<dt class="hdlist1"><em>switch_statement</em> : </dt>
+<dd>
+<p><em>SWITCH</em> <em>LEFT_PAREN</em> <em>expression</em> <em>RIGHT_PAREN</em> <em>LEFT_BRACE</em>
+<em>switch_statement_list</em><br>
+<em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>switch_statement_list</em> : </dt>
+<dd>
+<p>/* <em>nothing</em> */<br>
+<em>statement_list</em></p>
+</dd>
+<dt class="hdlist1"><em>case_label</em> : </dt>
+<dd>
+<p><em>CASE</em> <em>expression</em> <em>COLON</em><br>
+<em>DEFAULT</em> <em>COLON</em></p>
+</dd>
+<dt class="hdlist1"><em>iteration_statement</em> : </dt>
+<dd>
+<p><em>WHILE</em> <em>LEFT_PAREN</em> <em>condition</em> <em>RIGHT_PAREN</em> <em>statement_no_new_scope</em><br>
+<em>DO</em> <em>statement</em> <em>WHILE</em> <em>LEFT_PAREN</em> <em>expression</em> <em>RIGHT_PAREN</em>
+<em>SEMICOLON</em><br>
+<em>FOR</em> <em>LEFT_PAREN</em> <em>for_init_statement</em> <em>for_rest_statement</em>
+<em>RIGHT_PAREN</em> <em>statement_no_new_scope</em></p>
+</dd>
+<dt class="hdlist1"><em>for_init_statement</em> : </dt>
+<dd>
+<p><em>expression_statement</em><br>
+<em>declaration_statement</em></p>
+</dd>
+<dt class="hdlist1"><em>conditionopt</em> : </dt>
+<dd>
+<p><em>condition</em><br>
+/* <em>empty</em> */</p>
+</dd>
+<dt class="hdlist1"><em>for_rest_statement</em> : </dt>
+<dd>
+<p><em>conditionopt</em> <em>SEMICOLON</em><br>
+<em>conditionopt</em> <em>SEMICOLON</em> <em>expression</em></p>
+</dd>
+<dt class="hdlist1"><em>jump_statement</em> : </dt>
+<dd>
+<p><em>CONTINUE</em> <em>SEMICOLON</em><br>
+<em>BREAK</em> <em>SEMICOLON</em><br>
+<em>RETURN</em> <em>SEMICOLON</em><br>
+<em>RETURN</em> <em>expression</em> <em>SEMICOLON</em><br>
+<em>DISCARD</em> <em>SEMICOLON</em> // Fragment shader only.</p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: No 'goto'.
+Gotos are not supported.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>translation_unit</em> : </dt>
+<dd>
+<p><em>external_declaration</em><br>
+<em>translation_unit</em> <em>external_declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>external_declaration</em> : </dt>
+<dd>
+<p><em>function_definition</em><br>
+<em>declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>function_definition</em> : </dt>
+<dd>
+<p><em>function_prototype</em> <em>compound_statement_no_new_scope</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>In general the above grammar describes a super set of the OpenGL ES Shading Language.
+Certain constructs that are valid purely in terms of the grammar are
+disallowed by statements elsewhere in this specification.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="counting-of-inputs-and-outputs">11. Counting of Inputs and Outputs</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>This section applies to outputs from the last active vertex processing stage
+and inputs to the fragment stage.
+GLSL ES 3.20 specifies the storage available for such variables in terms of
+an array of 4-vectors.
+The assumption is that variables will be packed into these arrays without
+wasting space.
+This places significant burden on implementations since optimal packing is
+computationally intensive.
+Implementations may have more internal resources than exposed to the
+application and so avoid the need to perform packing but this is also
+considered an expensive solution.</p>
+</div>
+<div class="paragraph">
+<p>GLSL ES 3.20 therefore relaxes the requirements for packing by
+specifying a simpler algorithm that may be used.
+This algorithm specifies a minimum requirement for when a set of variables
+must be supported by an implementation.
+The implementation is allowed to support more than the minimum and so may
+use a more efficient algorithm and/or may support more registers than the
+virtual target machine.</p>
+</div>
+<div class="paragraph">
+<p>Outputs from the last active vertex stage and inputs to the fragment stage are
+counted separately.
+If statically used in the fragment shader, the built-in special variables
+(<em>gl_FragCoord</em>, <em>gl_FrontFacing</em> and <em>gl_PointCoord</em>) are included when
+calculating the storage requirements of fragment inputs.</p>
+</div>
+<div class="paragraph">
+<p>If the last active vertex-pipeline shader and fragment shader are linked
+together, inputs and outputs are only counted if they are statically used
+within the shader.
+If the shaders are each compiled into a separable program, all declared inputs
+and outputs are counted.</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="title">Note</div>
+<div class="paragraph">
+<p>GLSL ES 3.20 does not require the implementation to remove
+outputs which are not statically used in the fragment shader.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="paragraph">
+<p>For the algorithm used, failing resource allocation for a variable must
+result in an error.</p>
+</div>
+<div class="paragraph">
+<p>The resource allocation of variables must succeed for all cases where the
+following packing algorithm succeeds:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The target architecture consists of a grid of registers, 16 rows by 4
+columns.
+Each register can contain a scalar value, i.e. a float, int or uint.</p>
+</li>
+<li>
+<p>Variables with an explicit location are allocated first.
+When attempting to allocate a location for other variables, if there is
+a conflict, the search moves to the next available free location.</p>
+</li>
+<li>
+<p>Structures are assumed to be flattened.
+Each data member is treated as if it were at global scope.</p>
+</li>
+<li>
+<p>Variables are packed into the registers one at a time so that they each
+occupy a contiguous sub-rectangle.
+No splitting of variables is permitted.</p>
+</li>
+<li>
+<p>The orientation of variables is fixed.
+Vectors always occupy registers in a single row.
+Elements of an array must be in different rows.
+E.g. <strong>vec4</strong> will always occupy one row; float[16] will occupy one
+column.
+Since it is not permitted to split a variable, large arrays e.g.
+float[32] will always fail with this algorithm.</p>
+</li>
+<li>
+<p>Non-square matrices of type <strong>matCxR</strong> consume the same space as a square
+matrix of type <strong>matN</strong> where N is the greater of C and R.
+Variables of type <strong>mat2</strong> occupies 2 complete rows.
+These rules allow implementations more flexibility in how variables are
+stored.
++ Other variables consume only the minimum space required.</p>
+</li>
+<li>
+<p>Arrays of size N are assumed to take N times the size of the base type.</p>
+</li>
+<li>
+<p>Variables are packed in the following order:</p>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p><strong>mat4</strong> and arrays of <strong>mat4</strong>.</p>
+</li>
+<li>
+<p><strong>mat2</strong> and arrays of <strong>mat2</strong> (since they occupy full rows)</p>
+</li>
+<li>
+<p><strong>vec4</strong> and arrays of <strong>vec4</strong></p>
+</li>
+<li>
+<p><strong>mat3</strong> and arrays of <strong>mat3</strong></p>
+</li>
+<li>
+<p><strong>vec3</strong> and arrays of <strong>vec3</strong></p>
+</li>
+<li>
+<p><strong>vec2</strong> and arrays of <strong>vec2</strong></p>
+</li>
+<li>
+<p>Scalar types and arrays of scalar types</p>
+</li>
+</ol>
+</div>
+</li>
+<li>
+<p>For each of the above types, the arrays are processed in order of size,
+largest first.
+Arrays of size 1 and the base type are considered equivalent.
+The first type to be packed will be mat4[4], mat4[3], mat[2] followed by
+mat4, mat2[4]…​mat2[2], mat2, vec4[8], vec4[7],…​vec4[1], vec4,
+mat3[2], mat3 and so on.
+The last variables to be packed will be float (and float[1]).</p>
+</li>
+<li>
+<p>For 2,3 and 4 component variables packing is started using the 1<sup>st</sup>
+column of the 1<sup>st</sup> row.
+Variables are then allocated to successive rows, aligning them to the
+1<sup>st</sup> column.</p>
+</li>
+<li>
+<p>For 2 component variables, when there are no spare rows, the strategy is
+switched to using the highest numbered row and the lowest numbered
+column where the variable will fit.
+(In practice, this means they will be aligned to the x or z component.)
+Packing of any further 3 or 4 component variables will fail at this
+point.</p>
+</li>
+<li>
+<p>1 component variables (e.g. floats and arrays of floats) have their own
+packing rule.
+They are packed in order of size, largest first.
+Each variable is placed in the column that leaves the least amount of
+space in the column and aligned to the lowest available rows within that
+column.
+During this phase of packing, space will be available in up to 4
+columns.
+The space within each column is always contiguous in the case where no
+variables have explicit locations.</p>
+</li>
+<li>
+<p>For each type, variables with the 'smooth' property are packed first,
+followed by variables with the 'flat' property.</p>
+</li>
+<li>
+<p>Each row can contain either values with the 'smooth' property or the
+'flat' property but not both.
+If this situation is encountered during allocation, the algorithm skips
+the component location and continues with the next available location.
+These skipped locations may be used for other values later in the
+allocation process.</p>
+</li>
+<li>
+<p>There is no backtracking.
+Once a value is assigned a location, it cannot be changed, even if such
+a change is required for a successful allocation.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Example: pack the following types:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out vec4 a; <span class="comment">// top left</span>
+out mat3 b; <span class="comment">// align to left, lowest numbered rows</span>
+out mat2x3 c; <span class="comment">// same size as mat3, align to left</span>
+out vec2 d[<span class="integer">6</span>]; <span class="comment">// align to left, lowest numbered rows</span>
+out vec2 e[<span class="integer">4</span>]; <span class="comment">// Cannot align to left so align to z column, highest</span>
+ <span class="comment">// numbered rows</span>
+out vec2 f; <span class="comment">// Align to left, lowest numbered rows.</span>
+out <span class="predefined-type">float</span> g[<span class="integer">3</span>] <span class="comment">// Column with minimum space</span>
+out <span class="predefined-type">float</span> h[<span class="integer">2</span>]; <span class="comment">// Column with minimum space (choice of 3, any</span>
+ <span class="comment">// can be used)</span>
+out <span class="predefined-type">float</span> i; <span class="comment">// Column with minimum space</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>In this example, the variables happen to be listed in the order in which
+they are packed.
+Packing is independent of the order of declaration.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 20%;">
+<col style="width: 20%;">
+<col style="width: 20%;">
+<col style="width: 20%;">
+<col style="width: 20%;">
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>x</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>y</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>z</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>w</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">0</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>a</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>a</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>a</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>a</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>b</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">6</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>c</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">7</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>g</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">8</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>g</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">9</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>g</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">10</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">11</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">12</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>d</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>e</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>e</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">13</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>f</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>f</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>e</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>e</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">14</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>h</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>i</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>e</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>e</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">15</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>h</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>e</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>e</strong></p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Some types e.g. mat4[8] will be too large to fit.
+These always fail with this algorithm.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="acknowledgments">12. Acknowledgments</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>This specification is based on the work of those who contributed to past
+versions of the Open GL and Open GL ES Language Specifications and the
+following contributors to this version:</p>
+</div>
+<div class="paragraph">
+<p>Acorn Pooley, NVIDIA<br>
+Alberto Moreira, Qualcomm<br>
+Aleksandra Krstic, Qualcomm<br>
+Alon Or-bach, Nokia & Samsung<br>
+Andrzej Kacprowski, Intel<br>
+Arzhange Safdarzadeh, Intel<br>
+Aske Simon Christensen, ARM<br>
+Avi Shapira, Graphic Remedy<br>
+Barthold Lichtenbelt, NVIDIA<br>
+Ben Bowman, Imagination Technologies<br>
+Ben Brierton, Broadcom<br>
+Benj Lipchak, Apple<br>
+Benson Tao, Vivante<br>
+Bill Licea-Kane, Qualcomm<br>
+Brent Insko, Intel<br>
+Brian Murray, Freescale<br>
+Bruce Merry, ARM<br>
+Carlos Santa, TI<br>
+Cass Everitt, Epic Games & NVIDIA<br>
+Cemil Azizoglu, TI<br>
+Chang-Hyo Yu, Samsung<br>
+Chris Dodd, NVIDIA<br>
+Chris Knox, NVIDIA<br>
+Chris Tserng, TI<br>
+Clay Montgomery, TI<br>
+Daniel Kartch, NVIDIA<br>
+Daniel Koch, Transgaming& NVIDIA<br>
+Daoxiang Gong, Imagination Technologies<br>
+Dave Shreiner, ARM<br>
+David Garcia, AMD<br>
+David Jarmon, Vivante<br>
+Derek Cornish, Epic Games<br>
+Dominick Witczak, Mobica<br>
+Eben Upton, Broadcom<br>
+Ed Plowman, ARM<br>
+Eisaku Ohbuchi, DMP<br>
+Elan Lennard, ARM<br>
+Erik Faye-Lund, ARM<br>
+Graham Connor, Imagination Technologies<br>
+Graham Sellers, AMD<br>
+Greg Roth, NVIDIA<br>
+Guillaume Portier, Hi Corporation<br>
+Guofang Jiao, Qualcomm<br>
+Hans-Martin Will, Vincent<br>
+Hwanyong Lee, Huone<br>
+I-Gene Leong, NVIDIA<br>
+Ian Romanick, Intel<br>
+Ian South-Dickinson, NVIDIA<br>
+Ilan Aelion-Exch, Samsung<br>
+Inkyun Lee, Huone<br>
+Jacob Strm, Ericsson<br>
+James Adams, Broadcom<br>
+James Jones, Imagination Technologies<br>
+James McCombe, Imagination Technologies<br>
+Jamie Gennis, Google<br>
+Jan-Harald Fredriksen, ARM<br>
+Jani Vaisanen, Nokia<br>
+Jarkko Kemppainen, Symbio<br>
+Jeff Bolz, NVIDIA<br>
+Jeff Leger, Qualcomm<br>
+Jeff Vigil, Qualcomm<br>
+Jeremy Sandmel, Apple<br>
+Jeremy Thorne, Broadcom<br>
+Jim Hauxwell, Broadcom<br>
+Jinsung Kim, Huone<br>
+Jiyoung Yoon, Huone<br>
+John Kessenich, Google<br>
+Jon Kennedy, 3DLabs<br>
+Jon Leech, Khronos<br>
+Jonathan Putsman, Imagination Technologies<br>
+Joohoon Lee, Samsung<br>
+JoukoKylmäoja, Symbio<br>
+Jrn Nystad, ARM<br>
+Jussi Rasanen, NVIDIA<br>
+Kalle Raita, drawElements<br>
+Kari Pulli, Nokia<br>
+Keith Whitwell, VMware<br>
+Kent Miller, Netlogic Microsystems<br>
+Kimmo Nikkanen, Nokia<br>
+Konsta Karsisto, Nokia<br>
+Krzysztof Kaminski, Intel<br>
+Larry Seiler, Intel<br>
+Lars Remes, Symbio<br>
+Lee Thomason, Adobe<br>
+Lefan Zhong, Vivante<br>
+Marcus Lorentzon, Ericsson<br>
+Mark Butler, Imagination Technologies<br>
+Mark Callow, Hi Corporation<br>
+Mark Cresswell, Broadcom<br>
+Mark Snyder, Alt Software<br>
+Mark Young, AMD<br>
+Mathieu Robart, STM<br>
+Matt Netsch, Qualcomm<br>
+Matt Russo, Matrox<br>
+Maurice Ribble, Qualcomm<br>
+Max Kazakov, DMP<br>
+Mika Pesonen, Nokia<br>
+Mike Cai, Vivante<br>
+Mike Weiblen, Zebra Imaging & Qualcomm<br>
+Mila Smith, AMD<br>
+Nakhoon Baek, Kyungpook Univeristy<br>
+Nate Huang, NVIDIA<br>
+Neil Trevett, NVIDIA<br>
+Nelson Kidd, Intel<br>
+Nick Haemel, NVIDIA<br>
+Nick Penwarden, Epic Games<br>
+Niklas Smedberg, Epic Games<br>
+Nizar Romdan, ARM<br>
+Oliver Wohlmuth , Fujitsu<br>
+Pat Brown, NVIDIA<br>
+Paul Ruggieri, Qualcomm<br>
+Per Wennersten, Ericsson<br>
+Petri Talala, Symbio<br>
+Phil Huxley, ZiiLabs<br>
+Philip Hatcher, Freescale & Intel<br>
+Piers Daniell, NVIDIA<br>
+Pyry Haulos, drawElements<br>
+Piotr Tomaszewski, Ericsson<br>
+Piotr Uminski, Intel<br>
+Rami Mayer, Samsung<br>
+Rauli Laatikainen, RightWare<br>
+Rob Barris, NVIDIA<br>
+Rob Simpson, Qualcomm<br>
+Roj Langhi, Vivante<br>
+Rune Holm, ARM<br>
+Sami Kyostila, Nokia<br>
+Sean Ellis, ARM<br>
+Shereef Shehata, TI<br>
+Sila Kayo, Nokia<br>
+Slawomir Cygan, Intel<br>
+Slawomir Grajewski, Intel<br>
+Steve Hill, STM & Broadcom<br>
+Steven Olney, DMP<br>
+Suman Sharma, Intel<br>
+Tapani Palli, Nokia<br>
+Teemu Laakso, Symbio<br>
+Tero Karras, NVIDIA<br>
+Timo Suoranta, Imagination Technologies & Broadcom<br>
+Tom Cooksey, ARM<br>
+Tom McReynolds, NVIDIA<br>
+Tom Olson, TI & ARM<br>
+Tomi Aarnio, Nokia<br>
+Tommy Asano, Takumi<br>
+Wes Bang, Nokia<br>
+Yanjun Zhang, Vivante</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="references">13. Normative References</h2>
+<div class="sectionbody">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>International Standard ISO/IEC 14882:1998(E).
+Programming Languages - C++. Referenced for preprocessor only</p>
+</li>
+<li>
+<p>“OpenGL<sup>R</sup> ES, Version 3.2”,
+<a href="https://www.khronos.org/registry/OpenGL/index_es.php" class="bare">https://www.khronos.org/registry/OpenGL/index_es.php</a>, November 3, 2016.</p>
+</li>
+<li>
+<p>“The OpenGL<sup>R</sup> Graphics System: A Specification, Version 4.6 (Core
+Profile)”, <a href="https://www.khronos.org/registry/OpenGL/index_gl.php" class="bare">https://www.khronos.org/registry/OpenGL/index_gl.php</a>, June
+1, 2016.</p>
+</li>
+<li>
+<p>International Standard ISO/IEC 646:1991.
+Information technology - ISO 7-bit coded character set for information
+interchange</p>
+</li>
+<li>
+<p>The Unicode Standard Version 6.0 - Core Specification.</p>
+</li>
+<li>
+<p>IEEE 754-2008.
+<em>IEEE Standard for Floating-Point Arithmetic</em></p>
+</li>
+</ol>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Version 3.20.5<br>
+Last updated 2018-12-12 16:37:44 MST
+</div>
+</div>
+
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
diff --git a/specs/es/3.2/GLSL_ES_Specification_3.20.pdf b/specs/es/3.2/GLSL_ES_Specification_3.20.pdf
index 70434ce..c436090 100644
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+++ b/specs/es/3.2/GLSL_ES_Specification_3.20.pdf
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diff --git a/specs/es/3.2/es_spec_3.2.pdf b/specs/es/3.2/es_spec_3.2.pdf
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--- a/specs/es/3.2/es_spec_3.2.pdf
+++ b/specs/es/3.2/es_spec_3.2.pdf
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diff --git a/specs/es/3.2/es_spec_3.2.withchanges.pdf b/specs/es/3.2/es_spec_3.2.withchanges.pdf
index 2d0ead6..04d5acc 100644
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+++ b/specs/es/3.2/es_spec_3.2.withchanges.pdf
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diff --git a/specs/es/3.2/katex/README.md b/specs/es/3.2/katex/README.md
new file mode 100644
index 0000000..5f8caa8
--- /dev/null
+++ b/specs/es/3.2/katex/README.md
@@ -0,0 +1,68 @@
+# [<img src="https://khan.github.io/KaTeX/katex-logo.svg" width="130" alt="KaTeX">](https://khan.github.io/KaTeX/) [](https://travis-ci.org/Khan/KaTeX)
+
+[](https://gitter.im/Khan/KaTeX?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge)
+
+KaTeX is a fast, easy-to-use JavaScript library for TeX math rendering on the web.
+
+ * **Fast:** KaTeX renders its math synchronously and doesn't need to reflow the page. See how it compares to a competitor in [this speed test](http://jsperf.com/katex-vs-mathjax/).
+ * **Print quality:** KaTeX’s layout is based on Donald Knuth’s TeX, the gold standard for math typesetting.
+ * **Self contained:** KaTeX has no dependencies and can easily be bundled with your website resources.
+ * **Server side rendering:** KaTeX produces the same output regardless of browser or environment, so you can pre-render expressions using Node.js and send them as plain HTML.
+
+KaTeX supports all major browsers, including Chrome, Safari, Firefox, Opera, and IE 8 - IE 11. A list of supported commands can be on the [wiki](https://github.com/Khan/KaTeX/wiki/Function-Support-in-KaTeX).
+
+## Usage
+
+You can [download KaTeX](https://github.com/khan/katex/releases) and host it on your server or include the `katex.min.js` and `katex.min.css` files on your page directly from a CDN:
+
+```html
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.6.0/katex.min.css">
+<script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.6.0/katex.min.js"></script>
+```
+
+#### In-browser rendering
+
+Call `katex.render` with a TeX expression and a DOM element to render into:
+
+```js
+katex.render("c = \\pm\\sqrt{a^2 + b^2}", element);
+```
+
+If KaTeX can't parse the expression, it throws a `katex.ParseError` error.
+
+#### Server side rendering or rendering to a string
+
+To generate HTML on the server or to generate an HTML string of the rendered math, you can use `katex.renderToString`:
+
+```js
+var html = katex.renderToString("c = \\pm\\sqrt{a^2 + b^2}");
+// '<span class="katex">...</span>'
+```
+
+Make sure to include the CSS and font files, but there is no need to include the JavaScript. Like `render`, `renderToString` throws if it can't parse the expression.
+
+#### Rendering options
+
+You can provide an object of options as the last argument to `katex.render` and `katex.renderToString`. Available options are:
+
+- `displayMode`: `boolean`. If `true` the math will be rendered in display mode, which will put the math in display style (so `\int` and `\sum` are large, for example), and will center the math on the page on its own line. If `false` the math will be rendered in inline mode. (default: `false`)
+- `throwOnError`: `boolean`. If `true`, KaTeX will throw a `ParseError` when it encounters an unsupported command. If `false`, KaTeX will render the unsupported command as text in the color given by `errorColor`. (default: `true`)
+- `errorColor`: `string`. A color string given in the format `"#XXX"` or `"#XXXXXX"`. This option determines the color which unsupported commands are rendered in. (default: `#cc0000`)
+
+For example:
+
+```js
+katex.render("c = \\pm\\sqrt{a^2 + b^2}", element, { displayMode: true });
+```
+
+#### Automatic rendering of math on a page
+
+Math on the page can be automatically rendered using the auto-render extension. See [the Auto-render README](contrib/auto-render/README.md) for more information.
+
+## Contributing
+
+See [CONTRIBUTING.md](CONTRIBUTING.md)
+
+## License
+
+KaTeX is licensed under the [MIT License](http://opensource.org/licenses/MIT).
diff --git a/specs/es/3.2/katex/contrib/auto-render.min.js b/specs/es/3.2/katex/contrib/auto-render.min.js
new file mode 100644
index 0000000..30cc312
--- /dev/null
+++ b/specs/es/3.2/katex/contrib/auto-render.min.js
@@ -0,0 +1 @@
+(function(e){if(typeof exports==="object"&&typeof module!=="undefined"){module.exports=e()}else if(typeof define==="function"&&define.amd){define([],e)}else{var t;if(typeof window!=="undefined"){t=window}else if(typeof global!=="undefined"){t=global}else if(typeof self!=="undefined"){t=self}else{t=this}t.renderMathInElement=e()}})(function(){var e,t,r;return function n(e,t,r){function a(o,l){if(!t[o]){if(!e[o]){var f=typeof require=="function"&&require;if(!l&&f)return f(o,!0);if(i)return i(o,!0);var d=new Error("Cannot find module '"+o+"'");throw d.code="MODULE_NOT_FOUND",d}var s=t[o]={exports:{}};e[o][0].call(s.exports,function(t){var r=e[o][1][t];return a(r?r:t)},s,s.exports,n,e,t,r)}return t[o].exports}var i=typeof require=="function"&&require;for(var o=0;o<r.length;o++)a(r[o]);return a}({1:[function(e,t,r){var n=e("./splitAtDelimiters");var a=function(e,t){var r=[{type:"text",data:e}];for(var a=0;a<t.length;a++){var i=t[a];r=n(r,i.left,i.right,i.display||false)}return r};var i=function(e,t){var r=a(e,t);var n=document.createDocumentFragment();for(var i=0;i<r.length;i++){if(r[i].type==="text"){n.appendChild(document.createTextNode(r[i].data))}else{var o=document.createElement("span");var l=r[i].data;try{katex.render(l,o,{displayMode:r[i].display})}catch(f){if(!(f instanceof katex.ParseError)){throw f}console.error("KaTeX auto-render: Failed to parse `"+r[i].data+"` with ",f);n.appendChild(document.createTextNode(r[i].rawData));continue}n.appendChild(o)}}return n};var o=function(e,t,r){for(var n=0;n<e.childNodes.length;n++){var a=e.childNodes[n];if(a.nodeType===3){var l=i(a.textContent,t);n+=l.childNodes.length-1;e.replaceChild(l,a)}else if(a.nodeType===1){var f=r.indexOf(a.nodeName.toLowerCase())===-1;if(f){o(a,t,r)}}}};var l={delimiters:[{left:"$$",right:"$$",display:true},{left:"\\[",right:"\\]",display:true},{left:"\\(",right:"\\)",display:false}],ignoredTags:["script","noscript","style","textarea","pre","code"]};var f=function(e){var t;var r;for(var n=1,a=arguments.length;n<a;n++){t=arguments[n];for(r in t){if(Object.prototype.hasOwnProperty.call(t,r)){e[r]=t[r]}}}return e};var d=function(e,t){if(!e){throw new Error("No element provided to render")}t=f({},l,t);o(e,t.delimiters,t.ignoredTags)};t.exports=d},{"./splitAtDelimiters":2}],2:[function(e,t,r){var n=function(e,t,r){var n=r;var a=0;var i=e.length;while(n<t.length){var o=t[n];if(a<=0&&t.slice(n,n+i)===e){return n}else if(o==="\\"){n++}else if(o==="{"){a++}else if(o==="}"){a--}n++}return-1};var a=function(e,t,r,a){var i=[];for(var o=0;o<e.length;o++){if(e[o].type==="text"){var l=e[o].data;var f=true;var d=0;var s;s=l.indexOf(t);if(s!==-1){d=s;i.push({type:"text",data:l.slice(0,d)});f=false}while(true){if(f){s=l.indexOf(t,d);if(s===-1){break}i.push({type:"text",data:l.slice(d,s)});d=s}else{s=n(r,l,d+t.length);if(s===-1){break}i.push({type:"math",data:l.slice(d+t.length,s),rawData:l.slice(d,s+r.length),display:a});d=s+r.length}f=!f}i.push({type:"text",data:l.slice(d)})}else{i.push(e[o])}}return i};t.exports=a},{}]},{},[1])(1)});
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+++ b/specs/es/3.2/katex/katex.css
@@ -0,0 +1,976 @@
+@font-face {
+ font-family: 'KaTeX_AMS';
+ src: url('fonts/KaTeX_AMS-Regular.eot');
+ src: url('fonts/KaTeX_AMS-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_AMS-Regular.woff2') format('woff2'), url('fonts/KaTeX_AMS-Regular.woff') format('woff'), url('fonts/KaTeX_AMS-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Caligraphic';
+ src: url('fonts/KaTeX_Caligraphic-Bold.eot');
+ src: url('fonts/KaTeX_Caligraphic-Bold.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Caligraphic-Bold.woff2') format('woff2'), url('fonts/KaTeX_Caligraphic-Bold.woff') format('woff'), url('fonts/KaTeX_Caligraphic-Bold.ttf') format('truetype');
+ font-weight: bold;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Caligraphic';
+ src: url('fonts/KaTeX_Caligraphic-Regular.eot');
+ src: url('fonts/KaTeX_Caligraphic-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Caligraphic-Regular.woff2') format('woff2'), url('fonts/KaTeX_Caligraphic-Regular.woff') format('woff'), url('fonts/KaTeX_Caligraphic-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Fraktur';
+ src: url('fonts/KaTeX_Fraktur-Bold.eot');
+ src: url('fonts/KaTeX_Fraktur-Bold.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Fraktur-Bold.woff2') format('woff2'), url('fonts/KaTeX_Fraktur-Bold.woff') format('woff'), url('fonts/KaTeX_Fraktur-Bold.ttf') format('truetype');
+ font-weight: bold;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Fraktur';
+ src: url('fonts/KaTeX_Fraktur-Regular.eot');
+ src: url('fonts/KaTeX_Fraktur-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Fraktur-Regular.woff2') format('woff2'), url('fonts/KaTeX_Fraktur-Regular.woff') format('woff'), url('fonts/KaTeX_Fraktur-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Main';
+ src: url('fonts/KaTeX_Main-Bold.eot');
+ src: url('fonts/KaTeX_Main-Bold.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Main-Bold.woff2') format('woff2'), url('fonts/KaTeX_Main-Bold.woff') format('woff'), url('fonts/KaTeX_Main-Bold.ttf') format('truetype');
+ font-weight: bold;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Main';
+ src: url('fonts/KaTeX_Main-Italic.eot');
+ src: url('fonts/KaTeX_Main-Italic.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Main-Italic.woff2') format('woff2'), url('fonts/KaTeX_Main-Italic.woff') format('woff'), url('fonts/KaTeX_Main-Italic.ttf') format('truetype');
+ font-weight: normal;
+ font-style: italic;
+}
+@font-face {
+ font-family: 'KaTeX_Main';
+ src: url('fonts/KaTeX_Main-Regular.eot');
+ src: url('fonts/KaTeX_Main-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Main-Regular.woff2') format('woff2'), url('fonts/KaTeX_Main-Regular.woff') format('woff'), url('fonts/KaTeX_Main-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Math';
+ src: url('fonts/KaTeX_Math-Italic.eot');
+ src: url('fonts/KaTeX_Math-Italic.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Math-Italic.woff2') format('woff2'), url('fonts/KaTeX_Math-Italic.woff') format('woff'), url('fonts/KaTeX_Math-Italic.ttf') format('truetype');
+ font-weight: normal;
+ font-style: italic;
+}
+@font-face {
+ font-family: 'KaTeX_SansSerif';
+ src: url('fonts/KaTeX_SansSerif-Regular.eot');
+ src: url('fonts/KaTeX_SansSerif-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_SansSerif-Regular.woff2') format('woff2'), url('fonts/KaTeX_SansSerif-Regular.woff') format('woff'), url('fonts/KaTeX_SansSerif-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Script';
+ src: url('fonts/KaTeX_Script-Regular.eot');
+ src: url('fonts/KaTeX_Script-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Script-Regular.woff2') format('woff2'), url('fonts/KaTeX_Script-Regular.woff') format('woff'), url('fonts/KaTeX_Script-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Size1';
+ src: url('fonts/KaTeX_Size1-Regular.eot');
+ src: url('fonts/KaTeX_Size1-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Size1-Regular.woff2') format('woff2'), url('fonts/KaTeX_Size1-Regular.woff') format('woff'), url('fonts/KaTeX_Size1-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Size2';
+ src: url('fonts/KaTeX_Size2-Regular.eot');
+ src: url('fonts/KaTeX_Size2-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Size2-Regular.woff2') format('woff2'), url('fonts/KaTeX_Size2-Regular.woff') format('woff'), url('fonts/KaTeX_Size2-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Size3';
+ src: url('fonts/KaTeX_Size3-Regular.eot');
+ src: url('fonts/KaTeX_Size3-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Size3-Regular.woff2') format('woff2'), url('fonts/KaTeX_Size3-Regular.woff') format('woff'), url('fonts/KaTeX_Size3-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Size4';
+ src: url('fonts/KaTeX_Size4-Regular.eot');
+ src: url('fonts/KaTeX_Size4-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Size4-Regular.woff2') format('woff2'), url('fonts/KaTeX_Size4-Regular.woff') format('woff'), url('fonts/KaTeX_Size4-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Typewriter';
+ src: url('fonts/KaTeX_Typewriter-Regular.eot');
+ src: url('fonts/KaTeX_Typewriter-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Typewriter-Regular.woff2') format('woff2'), url('fonts/KaTeX_Typewriter-Regular.woff') format('woff'), url('fonts/KaTeX_Typewriter-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+.katex-display {
+ display: block;
+ margin: 1em 0;
+ text-align: center;
+}
+.katex-display > .katex {
+ display: inline-block;
+ text-align: initial;
+}
+.katex {
+ font: normal 1.21em KaTeX_Main, Times New Roman, serif;
+ line-height: 1.2;
+ white-space: nowrap;
+ text-indent: 0;
+}
+.katex .katex-html {
+ display: inline-block;
+}
+.katex .katex-mathml {
+ position: absolute;
+ clip: rect(1px, 1px, 1px, 1px);
+ padding: 0;
+ border: 0;
+ height: 1px;
+ width: 1px;
+ overflow: hidden;
+}
+.katex .base {
+ display: inline-block;
+}
+.katex .strut {
+ display: inline-block;
+}
+.katex .mathrm {
+ font-style: normal;
+}
+.katex .textit {
+ font-style: italic;
+}
+.katex .mathit {
+ font-family: KaTeX_Math;
+ font-style: italic;
+}
+.katex .mathbf {
+ font-family: KaTeX_Main;
+ font-weight: bold;
+}
+.katex .amsrm {
+ font-family: KaTeX_AMS;
+}
+.katex .mathbb {
+ font-family: KaTeX_AMS;
+}
+.katex .mathcal {
+ font-family: KaTeX_Caligraphic;
+}
+.katex .mathfrak {
+ font-family: KaTeX_Fraktur;
+}
+.katex .mathtt {
+ font-family: KaTeX_Typewriter;
+}
+.katex .mathscr {
+ font-family: KaTeX_Script;
+}
+.katex .mathsf {
+ font-family: KaTeX_SansSerif;
+}
+.katex .mainit {
+ font-family: KaTeX_Main;
+ font-style: italic;
+}
+.katex .mord + .mop {
+ margin-left: 0.16667em;
+}
+.katex .mord + .mbin {
+ margin-left: 0.22222em;
+}
+.katex .mord + .mrel {
+ margin-left: 0.27778em;
+}
+.katex .mord + .minner {
+ margin-left: 0.16667em;
+}
+.katex .mop + .mord {
+ margin-left: 0.16667em;
+}
+.katex .mop + .mop {
+ margin-left: 0.16667em;
+}
+.katex .mop + .mrel {
+ margin-left: 0.27778em;
+}
+.katex .mop + .minner {
+ margin-left: 0.16667em;
+}
+.katex .mbin + .mord {
+ margin-left: 0.22222em;
+}
+.katex .mbin + .mop {
+ margin-left: 0.22222em;
+}
+.katex .mbin + .mopen {
+ margin-left: 0.22222em;
+}
+.katex .mbin + .minner {
+ margin-left: 0.22222em;
+}
+.katex .mrel + .mord {
+ margin-left: 0.27778em;
+}
+.katex .mrel + .mop {
+ margin-left: 0.27778em;
+}
+.katex .mrel + .mopen {
+ margin-left: 0.27778em;
+}
+.katex .mrel + .minner {
+ margin-left: 0.27778em;
+}
+.katex .mclose + .mop {
+ margin-left: 0.16667em;
+}
+.katex .mclose + .mbin {
+ margin-left: 0.22222em;
+}
+.katex .mclose + .mrel {
+ margin-left: 0.27778em;
+}
+.katex .mclose + .minner {
+ margin-left: 0.16667em;
+}
+.katex .mpunct + .mord {
+ margin-left: 0.16667em;
+}
+.katex .mpunct + .mop {
+ margin-left: 0.16667em;
+}
+.katex .mpunct + .mrel {
+ margin-left: 0.16667em;
+}
+.katex .mpunct + .mopen {
+ margin-left: 0.16667em;
+}
+.katex .mpunct + .mclose {
+ margin-left: 0.16667em;
+}
+.katex .mpunct + .mpunct {
+ margin-left: 0.16667em;
+}
+.katex .mpunct + .minner {
+ margin-left: 0.16667em;
+}
+.katex .minner + .mord {
+ margin-left: 0.16667em;
+}
+.katex .minner + .mop {
+ margin-left: 0.16667em;
+}
+.katex .minner + .mbin {
+ margin-left: 0.22222em;
+}
+.katex .minner + .mrel {
+ margin-left: 0.27778em;
+}
+.katex .minner + .mopen {
+ margin-left: 0.16667em;
+}
+.katex .minner + .mpunct {
+ margin-left: 0.16667em;
+}
+.katex .minner + .minner {
+ margin-left: 0.16667em;
+}
+.katex .mord.mtight {
+ margin-left: 0;
+}
+.katex .mop.mtight {
+ margin-left: 0;
+}
+.katex .mbin.mtight {
+ margin-left: 0;
+}
+.katex .mrel.mtight {
+ margin-left: 0;
+}
+.katex .mopen.mtight {
+ margin-left: 0;
+}
+.katex .mclose.mtight {
+ margin-left: 0;
+}
+.katex .mpunct.mtight {
+ margin-left: 0;
+}
+.katex .minner.mtight {
+ margin-left: 0;
+}
+.katex .mord + .mop.mtight {
+ margin-left: 0.16667em;
+}
+.katex .mop + .mord.mtight {
+ margin-left: 0.16667em;
+}
+.katex .mop + .mop.mtight {
+ margin-left: 0.16667em;
+}
+.katex .mclose + .mop.mtight {
+ margin-left: 0.16667em;
+}
+.katex .minner + .mop.mtight {
+ margin-left: 0.16667em;
+}
+.katex .reset-textstyle.textstyle {
+ font-size: 1em;
+}
+.katex .reset-textstyle.scriptstyle {
+ font-size: 0.7em;
+}
+.katex .reset-textstyle.scriptscriptstyle {
+ font-size: 0.5em;
+}
+.katex .reset-scriptstyle.textstyle {
+ font-size: 1.42857em;
+}
+.katex .reset-scriptstyle.scriptstyle {
+ font-size: 1em;
+}
+.katex .reset-scriptstyle.scriptscriptstyle {
+ font-size: 0.71429em;
+}
+.katex .reset-scriptscriptstyle.textstyle {
+ font-size: 2em;
+}
+.katex .reset-scriptscriptstyle.scriptstyle {
+ font-size: 1.4em;
+}
+.katex .reset-scriptscriptstyle.scriptscriptstyle {
+ font-size: 1em;
+}
+.katex .style-wrap {
+ position: relative;
+}
+.katex .vlist {
+ display: inline-block;
+}
+.katex .vlist > span {
+ display: block;
+ height: 0;
+ position: relative;
+}
+.katex .vlist > span > span {
+ display: inline-block;
+}
+.katex .vlist .baseline-fix {
+ display: inline-table;
+ table-layout: fixed;
+}
+.katex .msupsub {
+ text-align: left;
+}
+.katex .mfrac > span > span {
+ text-align: center;
+}
+.katex .mfrac .frac-line {
+ width: 100%;
+}
+.katex .mfrac .frac-line:before {
+ border-bottom-style: solid;
+ border-bottom-width: 1px;
+ content: "";
+ display: block;
+}
+.katex .mfrac .frac-line:after {
+ border-bottom-style: solid;
+ border-bottom-width: 0.04em;
+ content: "";
+ display: block;
+ margin-top: -1px;
+}
+.katex .mspace {
+ display: inline-block;
+}
+.katex .mspace.negativethinspace {
+ margin-left: -0.16667em;
+}
+.katex .mspace.thinspace {
+ width: 0.16667em;
+}
+.katex .mspace.negativemediumspace {
+ margin-left: -0.22222em;
+}
+.katex .mspace.mediumspace {
+ width: 0.22222em;
+}
+.katex .mspace.thickspace {
+ width: 0.27778em;
+}
+.katex .mspace.sixmuspace {
+ width: 0.333333em;
+}
+.katex .mspace.eightmuspace {
+ width: 0.444444em;
+}
+.katex .mspace.enspace {
+ width: 0.5em;
+}
+.katex .mspace.twelvemuspace {
+ width: 0.666667em;
+}
+.katex .mspace.quad {
+ width: 1em;
+}
+.katex .mspace.qquad {
+ width: 2em;
+}
+.katex .llap,
+.katex .rlap {
+ width: 0;
+ position: relative;
+}
+.katex .llap > .inner,
+.katex .rlap > .inner {
+ position: absolute;
+}
+.katex .llap > .fix,
+.katex .rlap > .fix {
+ display: inline-block;
+}
+.katex .llap > .inner {
+ right: 0;
+}
+.katex .rlap > .inner {
+ left: 0;
+}
+.katex .katex-logo .a {
+ font-size: 0.75em;
+ margin-left: -0.32em;
+ position: relative;
+ top: -0.2em;
+}
+.katex .katex-logo .t {
+ margin-left: -0.23em;
+}
+.katex .katex-logo .e {
+ margin-left: -0.1667em;
+ position: relative;
+ top: 0.2155em;
+}
+.katex .katex-logo .x {
+ margin-left: -0.125em;
+}
+.katex .rule {
+ display: inline-block;
+ border: solid 0;
+ position: relative;
+}
+.katex .overline .overline-line,
+.katex .underline .underline-line {
+ width: 100%;
+}
+.katex .overline .overline-line:before,
+.katex .underline .underline-line:before {
+ border-bottom-style: solid;
+ border-bottom-width: 1px;
+ content: "";
+ display: block;
+}
+.katex .overline .overline-line:after,
+.katex .underline .underline-line:after {
+ border-bottom-style: solid;
+ border-bottom-width: 0.04em;
+ content: "";
+ display: block;
+ margin-top: -1px;
+}
+.katex .sqrt > .sqrt-sign {
+ position: relative;
+}
+.katex .sqrt .sqrt-line {
+ width: 100%;
+}
+.katex .sqrt .sqrt-line:before {
+ border-bottom-style: solid;
+ border-bottom-width: 1px;
+ content: "";
+ display: block;
+}
+.katex .sqrt .sqrt-line:after {
+ border-bottom-style: solid;
+ border-bottom-width: 0.04em;
+ content: "";
+ display: block;
+ margin-top: -1px;
+}
+.katex .sqrt > .root {
+ margin-left: 0.27777778em;
+ margin-right: -0.55555556em;
+}
+.katex .sizing,
+.katex .fontsize-ensurer {
+ display: inline-block;
+}
+.katex .sizing.reset-size1.size1,
+.katex .fontsize-ensurer.reset-size1.size1 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size1.size2,
+.katex .fontsize-ensurer.reset-size1.size2 {
+ font-size: 1.4em;
+}
+.katex .sizing.reset-size1.size3,
+.katex .fontsize-ensurer.reset-size1.size3 {
+ font-size: 1.6em;
+}
+.katex .sizing.reset-size1.size4,
+.katex .fontsize-ensurer.reset-size1.size4 {
+ font-size: 1.8em;
+}
+.katex .sizing.reset-size1.size5,
+.katex .fontsize-ensurer.reset-size1.size5 {
+ font-size: 2em;
+}
+.katex .sizing.reset-size1.size6,
+.katex .fontsize-ensurer.reset-size1.size6 {
+ font-size: 2.4em;
+}
+.katex .sizing.reset-size1.size7,
+.katex .fontsize-ensurer.reset-size1.size7 {
+ font-size: 2.88em;
+}
+.katex .sizing.reset-size1.size8,
+.katex .fontsize-ensurer.reset-size1.size8 {
+ font-size: 3.46em;
+}
+.katex .sizing.reset-size1.size9,
+.katex .fontsize-ensurer.reset-size1.size9 {
+ font-size: 4.14em;
+}
+.katex .sizing.reset-size1.size10,
+.katex .fontsize-ensurer.reset-size1.size10 {
+ font-size: 4.98em;
+}
+.katex .sizing.reset-size2.size1,
+.katex .fontsize-ensurer.reset-size2.size1 {
+ font-size: 0.71428571em;
+}
+.katex .sizing.reset-size2.size2,
+.katex .fontsize-ensurer.reset-size2.size2 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size2.size3,
+.katex .fontsize-ensurer.reset-size2.size3 {
+ font-size: 1.14285714em;
+}
+.katex .sizing.reset-size2.size4,
+.katex .fontsize-ensurer.reset-size2.size4 {
+ font-size: 1.28571429em;
+}
+.katex .sizing.reset-size2.size5,
+.katex .fontsize-ensurer.reset-size2.size5 {
+ font-size: 1.42857143em;
+}
+.katex .sizing.reset-size2.size6,
+.katex .fontsize-ensurer.reset-size2.size6 {
+ font-size: 1.71428571em;
+}
+.katex .sizing.reset-size2.size7,
+.katex .fontsize-ensurer.reset-size2.size7 {
+ font-size: 2.05714286em;
+}
+.katex .sizing.reset-size2.size8,
+.katex .fontsize-ensurer.reset-size2.size8 {
+ font-size: 2.47142857em;
+}
+.katex .sizing.reset-size2.size9,
+.katex .fontsize-ensurer.reset-size2.size9 {
+ font-size: 2.95714286em;
+}
+.katex .sizing.reset-size2.size10,
+.katex .fontsize-ensurer.reset-size2.size10 {
+ font-size: 3.55714286em;
+}
+.katex .sizing.reset-size3.size1,
+.katex .fontsize-ensurer.reset-size3.size1 {
+ font-size: 0.625em;
+}
+.katex .sizing.reset-size3.size2,
+.katex .fontsize-ensurer.reset-size3.size2 {
+ font-size: 0.875em;
+}
+.katex .sizing.reset-size3.size3,
+.katex .fontsize-ensurer.reset-size3.size3 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size3.size4,
+.katex .fontsize-ensurer.reset-size3.size4 {
+ font-size: 1.125em;
+}
+.katex .sizing.reset-size3.size5,
+.katex .fontsize-ensurer.reset-size3.size5 {
+ font-size: 1.25em;
+}
+.katex .sizing.reset-size3.size6,
+.katex .fontsize-ensurer.reset-size3.size6 {
+ font-size: 1.5em;
+}
+.katex .sizing.reset-size3.size7,
+.katex .fontsize-ensurer.reset-size3.size7 {
+ font-size: 1.8em;
+}
+.katex .sizing.reset-size3.size8,
+.katex .fontsize-ensurer.reset-size3.size8 {
+ font-size: 2.1625em;
+}
+.katex .sizing.reset-size3.size9,
+.katex .fontsize-ensurer.reset-size3.size9 {
+ font-size: 2.5875em;
+}
+.katex .sizing.reset-size3.size10,
+.katex .fontsize-ensurer.reset-size3.size10 {
+ font-size: 3.1125em;
+}
+.katex .sizing.reset-size4.size1,
+.katex .fontsize-ensurer.reset-size4.size1 {
+ font-size: 0.55555556em;
+}
+.katex .sizing.reset-size4.size2,
+.katex .fontsize-ensurer.reset-size4.size2 {
+ font-size: 0.77777778em;
+}
+.katex .sizing.reset-size4.size3,
+.katex .fontsize-ensurer.reset-size4.size3 {
+ font-size: 0.88888889em;
+}
+.katex .sizing.reset-size4.size4,
+.katex .fontsize-ensurer.reset-size4.size4 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size4.size5,
+.katex .fontsize-ensurer.reset-size4.size5 {
+ font-size: 1.11111111em;
+}
+.katex .sizing.reset-size4.size6,
+.katex .fontsize-ensurer.reset-size4.size6 {
+ font-size: 1.33333333em;
+}
+.katex .sizing.reset-size4.size7,
+.katex .fontsize-ensurer.reset-size4.size7 {
+ font-size: 1.6em;
+}
+.katex .sizing.reset-size4.size8,
+.katex .fontsize-ensurer.reset-size4.size8 {
+ font-size: 1.92222222em;
+}
+.katex .sizing.reset-size4.size9,
+.katex .fontsize-ensurer.reset-size4.size9 {
+ font-size: 2.3em;
+}
+.katex .sizing.reset-size4.size10,
+.katex .fontsize-ensurer.reset-size4.size10 {
+ font-size: 2.76666667em;
+}
+.katex .sizing.reset-size5.size1,
+.katex .fontsize-ensurer.reset-size5.size1 {
+ font-size: 0.5em;
+}
+.katex .sizing.reset-size5.size2,
+.katex .fontsize-ensurer.reset-size5.size2 {
+ font-size: 0.7em;
+}
+.katex .sizing.reset-size5.size3,
+.katex .fontsize-ensurer.reset-size5.size3 {
+ font-size: 0.8em;
+}
+.katex .sizing.reset-size5.size4,
+.katex .fontsize-ensurer.reset-size5.size4 {
+ font-size: 0.9em;
+}
+.katex .sizing.reset-size5.size5,
+.katex .fontsize-ensurer.reset-size5.size5 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size5.size6,
+.katex .fontsize-ensurer.reset-size5.size6 {
+ font-size: 1.2em;
+}
+.katex .sizing.reset-size5.size7,
+.katex .fontsize-ensurer.reset-size5.size7 {
+ font-size: 1.44em;
+}
+.katex .sizing.reset-size5.size8,
+.katex .fontsize-ensurer.reset-size5.size8 {
+ font-size: 1.73em;
+}
+.katex .sizing.reset-size5.size9,
+.katex .fontsize-ensurer.reset-size5.size9 {
+ font-size: 2.07em;
+}
+.katex .sizing.reset-size5.size10,
+.katex .fontsize-ensurer.reset-size5.size10 {
+ font-size: 2.49em;
+}
+.katex .sizing.reset-size6.size1,
+.katex .fontsize-ensurer.reset-size6.size1 {
+ font-size: 0.41666667em;
+}
+.katex .sizing.reset-size6.size2,
+.katex .fontsize-ensurer.reset-size6.size2 {
+ font-size: 0.58333333em;
+}
+.katex .sizing.reset-size6.size3,
+.katex .fontsize-ensurer.reset-size6.size3 {
+ font-size: 0.66666667em;
+}
+.katex .sizing.reset-size6.size4,
+.katex .fontsize-ensurer.reset-size6.size4 {
+ font-size: 0.75em;
+}
+.katex .sizing.reset-size6.size5,
+.katex .fontsize-ensurer.reset-size6.size5 {
+ font-size: 0.83333333em;
+}
+.katex .sizing.reset-size6.size6,
+.katex .fontsize-ensurer.reset-size6.size6 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size6.size7,
+.katex .fontsize-ensurer.reset-size6.size7 {
+ font-size: 1.2em;
+}
+.katex .sizing.reset-size6.size8,
+.katex .fontsize-ensurer.reset-size6.size8 {
+ font-size: 1.44166667em;
+}
+.katex .sizing.reset-size6.size9,
+.katex .fontsize-ensurer.reset-size6.size9 {
+ font-size: 1.725em;
+}
+.katex .sizing.reset-size6.size10,
+.katex .fontsize-ensurer.reset-size6.size10 {
+ font-size: 2.075em;
+}
+.katex .sizing.reset-size7.size1,
+.katex .fontsize-ensurer.reset-size7.size1 {
+ font-size: 0.34722222em;
+}
+.katex .sizing.reset-size7.size2,
+.katex .fontsize-ensurer.reset-size7.size2 {
+ font-size: 0.48611111em;
+}
+.katex .sizing.reset-size7.size3,
+.katex .fontsize-ensurer.reset-size7.size3 {
+ font-size: 0.55555556em;
+}
+.katex .sizing.reset-size7.size4,
+.katex .fontsize-ensurer.reset-size7.size4 {
+ font-size: 0.625em;
+}
+.katex .sizing.reset-size7.size5,
+.katex .fontsize-ensurer.reset-size7.size5 {
+ font-size: 0.69444444em;
+}
+.katex .sizing.reset-size7.size6,
+.katex .fontsize-ensurer.reset-size7.size6 {
+ font-size: 0.83333333em;
+}
+.katex .sizing.reset-size7.size7,
+.katex .fontsize-ensurer.reset-size7.size7 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size7.size8,
+.katex .fontsize-ensurer.reset-size7.size8 {
+ font-size: 1.20138889em;
+}
+.katex .sizing.reset-size7.size9,
+.katex .fontsize-ensurer.reset-size7.size9 {
+ font-size: 1.4375em;
+}
+.katex .sizing.reset-size7.size10,
+.katex .fontsize-ensurer.reset-size7.size10 {
+ font-size: 1.72916667em;
+}
+.katex .sizing.reset-size8.size1,
+.katex .fontsize-ensurer.reset-size8.size1 {
+ font-size: 0.28901734em;
+}
+.katex .sizing.reset-size8.size2,
+.katex .fontsize-ensurer.reset-size8.size2 {
+ font-size: 0.40462428em;
+}
+.katex .sizing.reset-size8.size3,
+.katex .fontsize-ensurer.reset-size8.size3 {
+ font-size: 0.46242775em;
+}
+.katex .sizing.reset-size8.size4,
+.katex .fontsize-ensurer.reset-size8.size4 {
+ font-size: 0.52023121em;
+}
+.katex .sizing.reset-size8.size5,
+.katex .fontsize-ensurer.reset-size8.size5 {
+ font-size: 0.57803468em;
+}
+.katex .sizing.reset-size8.size6,
+.katex .fontsize-ensurer.reset-size8.size6 {
+ font-size: 0.69364162em;
+}
+.katex .sizing.reset-size8.size7,
+.katex .fontsize-ensurer.reset-size8.size7 {
+ font-size: 0.83236994em;
+}
+.katex .sizing.reset-size8.size8,
+.katex .fontsize-ensurer.reset-size8.size8 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size8.size9,
+.katex .fontsize-ensurer.reset-size8.size9 {
+ font-size: 1.19653179em;
+}
+.katex .sizing.reset-size8.size10,
+.katex .fontsize-ensurer.reset-size8.size10 {
+ font-size: 1.43930636em;
+}
+.katex .sizing.reset-size9.size1,
+.katex .fontsize-ensurer.reset-size9.size1 {
+ font-size: 0.24154589em;
+}
+.katex .sizing.reset-size9.size2,
+.katex .fontsize-ensurer.reset-size9.size2 {
+ font-size: 0.33816425em;
+}
+.katex .sizing.reset-size9.size3,
+.katex .fontsize-ensurer.reset-size9.size3 {
+ font-size: 0.38647343em;
+}
+.katex .sizing.reset-size9.size4,
+.katex .fontsize-ensurer.reset-size9.size4 {
+ font-size: 0.43478261em;
+}
+.katex .sizing.reset-size9.size5,
+.katex .fontsize-ensurer.reset-size9.size5 {
+ font-size: 0.48309179em;
+}
+.katex .sizing.reset-size9.size6,
+.katex .fontsize-ensurer.reset-size9.size6 {
+ font-size: 0.57971014em;
+}
+.katex .sizing.reset-size9.size7,
+.katex .fontsize-ensurer.reset-size9.size7 {
+ font-size: 0.69565217em;
+}
+.katex .sizing.reset-size9.size8,
+.katex .fontsize-ensurer.reset-size9.size8 {
+ font-size: 0.83574879em;
+}
+.katex .sizing.reset-size9.size9,
+.katex .fontsize-ensurer.reset-size9.size9 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size9.size10,
+.katex .fontsize-ensurer.reset-size9.size10 {
+ font-size: 1.20289855em;
+}
+.katex .sizing.reset-size10.size1,
+.katex .fontsize-ensurer.reset-size10.size1 {
+ font-size: 0.20080321em;
+}
+.katex .sizing.reset-size10.size2,
+.katex .fontsize-ensurer.reset-size10.size2 {
+ font-size: 0.2811245em;
+}
+.katex .sizing.reset-size10.size3,
+.katex .fontsize-ensurer.reset-size10.size3 {
+ font-size: 0.32128514em;
+}
+.katex .sizing.reset-size10.size4,
+.katex .fontsize-ensurer.reset-size10.size4 {
+ font-size: 0.36144578em;
+}
+.katex .sizing.reset-size10.size5,
+.katex .fontsize-ensurer.reset-size10.size5 {
+ font-size: 0.40160643em;
+}
+.katex .sizing.reset-size10.size6,
+.katex .fontsize-ensurer.reset-size10.size6 {
+ font-size: 0.48192771em;
+}
+.katex .sizing.reset-size10.size7,
+.katex .fontsize-ensurer.reset-size10.size7 {
+ font-size: 0.57831325em;
+}
+.katex .sizing.reset-size10.size8,
+.katex .fontsize-ensurer.reset-size10.size8 {
+ font-size: 0.69477912em;
+}
+.katex .sizing.reset-size10.size9,
+.katex .fontsize-ensurer.reset-size10.size9 {
+ font-size: 0.8313253em;
+}
+.katex .sizing.reset-size10.size10,
+.katex .fontsize-ensurer.reset-size10.size10 {
+ font-size: 1em;
+}
+.katex .delimsizing.size1 {
+ font-family: KaTeX_Size1;
+}
+.katex .delimsizing.size2 {
+ font-family: KaTeX_Size2;
+}
+.katex .delimsizing.size3 {
+ font-family: KaTeX_Size3;
+}
+.katex .delimsizing.size4 {
+ font-family: KaTeX_Size4;
+}
+.katex .delimsizing.mult .delim-size1 > span {
+ font-family: KaTeX_Size1;
+}
+.katex .delimsizing.mult .delim-size4 > span {
+ font-family: KaTeX_Size4;
+}
+.katex .nulldelimiter {
+ display: inline-block;
+ width: 0.12em;
+}
+.katex .op-symbol {
+ position: relative;
+}
+.katex .op-symbol.small-op {
+ font-family: KaTeX_Size1;
+}
+.katex .op-symbol.large-op {
+ font-family: KaTeX_Size2;
+}
+.katex .op-limits > .vlist > span {
+ text-align: center;
+}
+.katex .accent > .vlist > span {
+ text-align: center;
+}
+.katex .accent .accent-body > span {
+ width: 0;
+}
+.katex .accent .accent-body.accent-vec > span {
+ position: relative;
+ left: 0.326em;
+}
+.katex .mtable .vertical-separator {
+ display: inline-block;
+ margin: 0 -0.025em;
+ border-right: 0.05em solid black;
+}
+.katex .mtable .arraycolsep {
+ display: inline-block;
+}
+.katex .mtable .col-align-c > .vlist {
+ text-align: center;
+}
+.katex .mtable .col-align-l > .vlist {
+ text-align: left;
+}
+.katex .mtable .col-align-r > .vlist {
+ text-align: right;
+}
diff --git a/specs/es/3.2/katex/katex.js b/specs/es/3.2/katex/katex.js
new file mode 100644
index 0000000..e104be9
--- /dev/null
+++ b/specs/es/3.2/katex/katex.js
@@ -0,0 +1,9075 @@
+(function(f){if(typeof exports==="object"&&typeof module!=="undefined"){module.exports=f()}else if(typeof define==="function"&&define.amd){define([],f)}else{var g;if(typeof window!=="undefined"){g=window}else if(typeof global!=="undefined"){g=global}else if(typeof self!=="undefined"){g=self}else{g=this}g.katex = f()}})(function(){var define,module,exports;return (function e(t,n,r){function s(o,u){if(!n[o]){if(!t[o]){var a=typeof require=="function"&&require;if(!u&&a)return a(o,!0);if(i)return i(o,!0);var f=new Error("Cannot find module '"+o+"'");throw f.code="MODULE_NOT_FOUND",f}var l=n[o]={exports:{}};t[o][0].call(l.exports,function(e){var n=t[o][1][e];return s(n?n:e)},l,l.exports,e,t,n,r)}return n[o].exports}var i=typeof require=="function"&&require;for(var o=0;o<r.length;o++)s(r[o]);return s})({1:[function(require,module,exports){
+/* eslint no-console:0 */
+/**
+ * This is the main entry point for KaTeX. Here, we expose functions for
+ * rendering expressions either to DOM nodes or to markup strings.
+ *
+ * We also expose the ParseError class to check if errors thrown from KaTeX are
+ * errors in the expression, or errors in javascript handling.
+ */
+
+var ParseError = require("./src/ParseError");
+var Settings = require("./src/Settings");
+
+var buildTree = require("./src/buildTree");
+var parseTree = require("./src/parseTree");
+var utils = require("./src/utils");
+
+/**
+ * Parse and build an expression, and place that expression in the DOM node
+ * given.
+ */
+var render = function(expression, baseNode, options) {
+ utils.clearNode(baseNode);
+
+ var settings = new Settings(options);
+
+ var tree = parseTree(expression, settings);
+ var node = buildTree(tree, expression, settings).toNode();
+
+ baseNode.appendChild(node);
+};
+
+// KaTeX's styles don't work properly in quirks mode. Print out an error, and
+// disable rendering.
+if (typeof document !== "undefined") {
+ if (document.compatMode !== "CSS1Compat") {
+ typeof console !== "undefined" && console.warn(
+ "Warning: KaTeX doesn't work in quirks mode. Make sure your " +
+ "website has a suitable doctype.");
+
+ render = function() {
+ throw new ParseError("KaTeX doesn't work in quirks mode.");
+ };
+ }
+}
+
+/**
+ * Parse and build an expression, and return the markup for that.
+ */
+var renderToString = function(expression, options) {
+ var settings = new Settings(options);
+
+ var tree = parseTree(expression, settings);
+ return buildTree(tree, expression, settings).toMarkup();
+};
+
+/**
+ * Parse an expression and return the parse tree.
+ */
+var generateParseTree = function(expression, options) {
+ var settings = new Settings(options);
+ return parseTree(expression, settings);
+};
+
+module.exports = {
+ render: render,
+ renderToString: renderToString,
+ /**
+ * NOTE: This method is not currently recommended for public use.
+ * The internal tree representation is unstable and is very likely
+ * to change. Use at your own risk.
+ */
+ __parse: generateParseTree,
+ ParseError: ParseError
+};
+
+},{"./src/ParseError":6,"./src/Settings":8,"./src/buildTree":13,"./src/parseTree":22,"./src/utils":25}],2:[function(require,module,exports){
+/** @flow */
+
+"use strict";
+
+function getRelocatable(re) {
+ // In the future, this could use a WeakMap instead of an expando.
+ if (!re.__matchAtRelocatable) {
+ // Disjunctions are the lowest-precedence operator, so we can make any
+ // pattern match the empty string by appending `|()` to it:
+ // https://people.mozilla.org/~jorendorff/es6-draft.html#sec-patterns
+ var source = re.source + "|()";
+
+ // We always make the new regex global.
+ var flags = "g" + (re.ignoreCase ? "i" : "") + (re.multiline ? "m" : "") + (re.unicode ? "u" : "")
+ // sticky (/.../y) doesn't make sense in conjunction with our relocation
+ // logic, so we ignore it here.
+ ;
+
+ re.__matchAtRelocatable = new RegExp(source, flags);
+ }
+ return re.__matchAtRelocatable;
+}
+
+function matchAt(re, str, pos) {
+ if (re.global || re.sticky) {
+ throw new Error("matchAt(...): Only non-global regexes are supported");
+ }
+ var reloc = getRelocatable(re);
+ reloc.lastIndex = pos;
+ var match = reloc.exec(str);
+ // Last capturing group is our sentinel that indicates whether the regex
+ // matched at the given location.
+ if (match[match.length - 1] == null) {
+ // Original regex matched.
+ match.length = match.length - 1;
+ return match;
+ } else {
+ return null;
+ }
+}
+
+module.exports = matchAt;
+},{}],3:[function(require,module,exports){
+/**
+ * The Lexer class handles tokenizing the input in various ways. Since our
+ * parser expects us to be able to backtrack, the lexer allows lexing from any
+ * given starting point.
+ *
+ * Its main exposed function is the `lex` function, which takes a position to
+ * lex from and a type of token to lex. It defers to the appropriate `_innerLex`
+ * function.
+ *
+ * The various `_innerLex` functions perform the actual lexing of different
+ * kinds.
+ */
+
+var matchAt = require("match-at");
+
+var ParseError = require("./ParseError");
+
+// The main lexer class
+function Lexer(input) {
+ this.input = input;
+ this.pos = 0;
+}
+
+/**
+ * The resulting token returned from `lex`.
+ *
+ * It consists of the token text plus some position information.
+ * The position information is essentially a range in an input string,
+ * but instead of referencing the bare input string, we refer to the lexer.
+ * That way it is possible to attach extra metadata to the input string,
+ * like for example a file name or similar.
+ *
+ * The position information (all three parameters) is optional,
+ * so it is OK to construct synthetic tokens if appropriate.
+ * Not providing available position information may lead to
+ * degraded error reporting, though.
+ *
+ * @param {string} text the text of this token
+ * @param {number=} start the start offset, zero-based inclusive
+ * @param {number=} end the end offset, zero-based exclusive
+ * @param {Lexer=} lexer the lexer which in turn holds the input string
+ */
+function Token(text, start, end, lexer) {
+ this.text = text;
+ this.start = start;
+ this.end = end;
+ this.lexer = lexer;
+}
+
+/**
+ * Given a pair of tokens (this and endToken), compute a “Token” encompassing
+ * the whole input range enclosed by these two.
+ *
+ * @param {Token} endToken last token of the range, inclusive
+ * @param {string} text the text of the newly constructed token
+ */
+Token.prototype.range = function(endToken, text) {
+ if (endToken.lexer !== this.lexer) {
+ return new Token(text); // sorry, no position information available
+ }
+ return new Token(text, this.start, endToken.end, this.lexer);
+};
+
+/* The following tokenRegex
+ * - matches typical whitespace (but not NBSP etc.) using its first group
+ * - does not match any control character \x00-\x1f except whitespace
+ * - does not match a bare backslash
+ * - matches any ASCII character except those just mentioned
+ * - does not match the BMP private use area \uE000-\uF8FF
+ * - does not match bare surrogate code units
+ * - matches any BMP character except for those just described
+ * - matches any valid Unicode surrogate pair
+ * - matches a backslash followed by one or more letters
+ * - matches a backslash followed by any BMP character, including newline
+ * Just because the Lexer matches something doesn't mean it's valid input:
+ * If there is no matching function or symbol definition, the Parser will
+ * still reject the input.
+ */
+var tokenRegex = new RegExp(
+ "([ \r\n\t]+)|" + // whitespace
+ "([!-\\[\\]-\u2027\u202A-\uD7FF\uF900-\uFFFF]" + // single codepoint
+ "|[\uD800-\uDBFF][\uDC00-\uDFFF]" + // surrogate pair
+ "|\\\\(?:[a-zA-Z]+|[^\uD800-\uDFFF])" + // function name
+ ")"
+);
+
+/**
+ * This function lexes a single token.
+ */
+Lexer.prototype.lex = function() {
+ var input = this.input;
+ var pos = this.pos;
+ if (pos === input.length) {
+ return new Token("EOF", pos, pos, this);
+ }
+ var match = matchAt(tokenRegex, input, pos);
+ if (match === null) {
+ throw new ParseError(
+ "Unexpected character: '" + input[pos] + "'",
+ new Token(input[pos], pos, pos + 1, this));
+ }
+ var text = match[2] || " ";
+ var start = this.pos;
+ this.pos += match[0].length;
+ var end = this.pos;
+ return new Token(text, start, end, this);
+};
+
+module.exports = Lexer;
+
+},{"./ParseError":6,"match-at":2}],4:[function(require,module,exports){
+/**
+ * This file contains the “gullet” where macros are expanded
+ * until only non-macro tokens remain.
+ */
+
+var Lexer = require("./Lexer");
+
+function MacroExpander(input, macros) {
+ this.lexer = new Lexer(input);
+ this.macros = macros;
+ this.stack = []; // contains tokens in REVERSE order
+ this.discardedWhiteSpace = [];
+}
+
+/**
+ * Recursively expand first token, then return first non-expandable token.
+ */
+MacroExpander.prototype.nextToken = function() {
+ for (;;) {
+ if (this.stack.length === 0) {
+ this.stack.push(this.lexer.lex());
+ }
+ var topToken = this.stack.pop();
+ var name = topToken.text;
+ if (!(name.charAt(0) === "\\" && this.macros.hasOwnProperty(name))) {
+ return topToken;
+ }
+ var expansion = this.macros[name];
+ if (typeof expansion === "string") {
+ var bodyLexer = new Lexer(expansion);
+ expansion = [];
+ var tok = bodyLexer.lex();
+ while (tok.text !== "EOF") {
+ expansion.push(tok);
+ tok = bodyLexer.lex();
+ }
+ expansion.reverse(); // to fit in with stack using push and pop
+ this.macros[name] = expansion;
+ }
+ this.stack = this.stack.concat(expansion);
+ }
+};
+
+MacroExpander.prototype.get = function(ignoreSpace) {
+ this.discardedWhiteSpace = [];
+ var token = this.nextToken();
+ if (ignoreSpace) {
+ while (token.text === " ") {
+ this.discardedWhiteSpace.push(token);
+ token = this.nextToken();
+ }
+ }
+ return token;
+};
+
+/**
+ * Undo the effect of the preceding call to the get method.
+ * A call to this method MUST be immediately preceded and immediately followed
+ * by a call to get. Only used during mode switching, i.e. after one token
+ * was got in the old mode but should get got again in a new mode
+ * with possibly different whitespace handling.
+ */
+MacroExpander.prototype.unget = function(token) {
+ this.stack.push(token);
+ while (this.discardedWhiteSpace.length !== 0) {
+ this.stack.push(this.discardedWhiteSpace.pop());
+ }
+};
+
+module.exports = MacroExpander;
+
+},{"./Lexer":3}],5:[function(require,module,exports){
+/**
+ * This file contains information about the options that the Parser carries
+ * around with it while parsing. Data is held in an `Options` object, and when
+ * recursing, a new `Options` object can be created with the `.with*` and
+ * `.reset` functions.
+ */
+
+/**
+ * This is the main options class. It contains the style, size, color, and font
+ * of the current parse level. It also contains the style and size of the parent
+ * parse level, so size changes can be handled efficiently.
+ *
+ * Each of the `.with*` and `.reset` functions passes its current style and size
+ * as the parentStyle and parentSize of the new options class, so parent
+ * handling is taken care of automatically.
+ */
+function Options(data) {
+ this.style = data.style;
+ this.color = data.color;
+ this.size = data.size;
+ this.phantom = data.phantom;
+ this.font = data.font;
+
+ if (data.parentStyle === undefined) {
+ this.parentStyle = data.style;
+ } else {
+ this.parentStyle = data.parentStyle;
+ }
+
+ if (data.parentSize === undefined) {
+ this.parentSize = data.size;
+ } else {
+ this.parentSize = data.parentSize;
+ }
+}
+
+/**
+ * Returns a new options object with the same properties as "this". Properties
+ * from "extension" will be copied to the new options object.
+ */
+Options.prototype.extend = function(extension) {
+ var data = {
+ style: this.style,
+ size: this.size,
+ color: this.color,
+ parentStyle: this.style,
+ parentSize: this.size,
+ phantom: this.phantom,
+ font: this.font
+ };
+
+ for (var key in extension) {
+ if (extension.hasOwnProperty(key)) {
+ data[key] = extension[key];
+ }
+ }
+
+ return new Options(data);
+};
+
+/**
+ * Create a new options object with the given style.
+ */
+Options.prototype.withStyle = function(style) {
+ return this.extend({
+ style: style
+ });
+};
+
+/**
+ * Create a new options object with the given size.
+ */
+Options.prototype.withSize = function(size) {
+ return this.extend({
+ size: size
+ });
+};
+
+/**
+ * Create a new options object with the given color.
+ */
+Options.prototype.withColor = function(color) {
+ return this.extend({
+ color: color
+ });
+};
+
+/**
+ * Create a new options object with "phantom" set to true.
+ */
+Options.prototype.withPhantom = function() {
+ return this.extend({
+ phantom: true
+ });
+};
+
+/**
+ * Create a new options objects with the give font.
+ */
+Options.prototype.withFont = function(font) {
+ return this.extend({
+ font: font || this.font
+ });
+};
+
+/**
+ * Create a new options object with the same style, size, and color. This is
+ * used so that parent style and size changes are handled correctly.
+ */
+Options.prototype.reset = function() {
+ return this.extend({});
+};
+
+/**
+ * A map of color names to CSS colors.
+ * TODO(emily): Remove this when we have real macros
+ */
+var colorMap = {
+ "katex-blue": "#6495ed",
+ "katex-orange": "#ffa500",
+ "katex-pink": "#ff00af",
+ "katex-red": "#df0030",
+ "katex-green": "#28ae7b",
+ "katex-gray": "gray",
+ "katex-purple": "#9d38bd",
+ "katex-blueA": "#ccfaff",
+ "katex-blueB": "#80f6ff",
+ "katex-blueC": "#63d9ea",
+ "katex-blueD": "#11accd",
+ "katex-blueE": "#0c7f99",
+ "katex-tealA": "#94fff5",
+ "katex-tealB": "#26edd5",
+ "katex-tealC": "#01d1c1",
+ "katex-tealD": "#01a995",
+ "katex-tealE": "#208170",
+ "katex-greenA": "#b6ffb0",
+ "katex-greenB": "#8af281",
+ "katex-greenC": "#74cf70",
+ "katex-greenD": "#1fab54",
+ "katex-greenE": "#0d923f",
+ "katex-goldA": "#ffd0a9",
+ "katex-goldB": "#ffbb71",
+ "katex-goldC": "#ff9c39",
+ "katex-goldD": "#e07d10",
+ "katex-goldE": "#a75a05",
+ "katex-redA": "#fca9a9",
+ "katex-redB": "#ff8482",
+ "katex-redC": "#f9685d",
+ "katex-redD": "#e84d39",
+ "katex-redE": "#bc2612",
+ "katex-maroonA": "#ffbde0",
+ "katex-maroonB": "#ff92c6",
+ "katex-maroonC": "#ed5fa6",
+ "katex-maroonD": "#ca337c",
+ "katex-maroonE": "#9e034e",
+ "katex-purpleA": "#ddd7ff",
+ "katex-purpleB": "#c6b9fc",
+ "katex-purpleC": "#aa87ff",
+ "katex-purpleD": "#7854ab",
+ "katex-purpleE": "#543b78",
+ "katex-mintA": "#f5f9e8",
+ "katex-mintB": "#edf2df",
+ "katex-mintC": "#e0e5cc",
+ "katex-grayA": "#f6f7f7",
+ "katex-grayB": "#f0f1f2",
+ "katex-grayC": "#e3e5e6",
+ "katex-grayD": "#d6d8da",
+ "katex-grayE": "#babec2",
+ "katex-grayF": "#888d93",
+ "katex-grayG": "#626569",
+ "katex-grayH": "#3b3e40",
+ "katex-grayI": "#21242c",
+ "katex-kaBlue": "#314453",
+ "katex-kaGreen": "#71B307"
+};
+
+/**
+ * Gets the CSS color of the current options object, accounting for the
+ * `colorMap`.
+ */
+Options.prototype.getColor = function() {
+ if (this.phantom) {
+ return "transparent";
+ } else {
+ return colorMap[this.color] || this.color;
+ }
+};
+
+module.exports = Options;
+
+},{}],6:[function(require,module,exports){
+/**
+ * This is the ParseError class, which is the main error thrown by KaTeX
+ * functions when something has gone wrong. This is used to distinguish internal
+ * errors from errors in the expression that the user provided.
+ *
+ * If possible, a caller should provide a Token or ParseNode with information
+ * about where in the source string the problem occurred.
+ *
+ * @param {string} message The error message
+ * @param {(Token|ParseNode)=} token An object providing position information
+ */
+function ParseError(message, token) {
+ var error = "KaTeX parse error: " + message;
+ var start;
+ var end;
+
+ if (token && token.lexer && token.start <= token.end) {
+ // If we have the input and a position, make the error a bit fancier
+
+ // Get the input
+ var input = token.lexer.input;
+
+ // Prepend some information
+ start = token.start;
+ end = token.end;
+ if (start === input.length) {
+ error += " at end of input: ";
+ } else {
+ error += " at position " + (start + 1) + ": ";
+ }
+
+ // Underline token in question using combining underscores
+ var underlined = input.slice(start, end).replace(/[^]/g, "$&\u0332");
+
+ // Extract some context from the input and add it to the error
+ var left;
+ if (start > 15) {
+ left = "…" + input.slice(start - 15, start);
+ } else {
+ left = input.slice(0, start);
+ }
+ var right;
+ if (end + 15 < input.length) {
+ right = input.slice(end, end + 15) + "…";
+ } else {
+ right = input.slice(end);
+ }
+ error += left + underlined + right;
+ }
+
+ // Some hackery to make ParseError a prototype of Error
+ // See http://stackoverflow.com/a/8460753
+ var self = new Error(error);
+ self.name = "ParseError";
+ self.__proto__ = ParseError.prototype;
+
+ self.position = start;
+ return self;
+}
+
+// More hackery
+ParseError.prototype.__proto__ = Error.prototype;
+
+module.exports = ParseError;
+
+},{}],7:[function(require,module,exports){
+/* eslint no-constant-condition:0 */
+var functions = require("./functions");
+var environments = require("./environments");
+var MacroExpander = require("./MacroExpander");
+var symbols = require("./symbols");
+var utils = require("./utils");
+var cjkRegex = require("./unicodeRegexes").cjkRegex;
+
+var parseData = require("./parseData");
+var ParseError = require("./ParseError");
+
+/**
+ * This file contains the parser used to parse out a TeX expression from the
+ * input. Since TeX isn't context-free, standard parsers don't work particularly
+ * well.
+ *
+ * The strategy of this parser is as such:
+ *
+ * The main functions (the `.parse...` ones) take a position in the current
+ * parse string to parse tokens from. The lexer (found in Lexer.js, stored at
+ * this.lexer) also supports pulling out tokens at arbitrary places. When
+ * individual tokens are needed at a position, the lexer is called to pull out a
+ * token, which is then used.
+ *
+ * The parser has a property called "mode" indicating the mode that
+ * the parser is currently in. Currently it has to be one of "math" or
+ * "text", which denotes whether the current environment is a math-y
+ * one or a text-y one (e.g. inside \text). Currently, this serves to
+ * limit the functions which can be used in text mode.
+ *
+ * The main functions then return an object which contains the useful data that
+ * was parsed at its given point, and a new position at the end of the parsed
+ * data. The main functions can call each other and continue the parsing by
+ * using the returned position as a new starting point.
+ *
+ * There are also extra `.handle...` functions, which pull out some reused
+ * functionality into self-contained functions.
+ *
+ * The earlier functions return ParseNodes.
+ * The later functions (which are called deeper in the parse) sometimes return
+ * ParseFuncOrArgument, which contain a ParseNode as well as some data about
+ * whether the parsed object is a function which is missing some arguments, or a
+ * standalone object which can be used as an argument to another function.
+ */
+
+/**
+ * Main Parser class
+ */
+function Parser(input, settings) {
+ // Create a new macro expander (gullet) and (indirectly via that) also a
+ // new lexer (mouth) for this parser (stomach, in the language of TeX)
+ this.gullet = new MacroExpander(input, settings.macros);
+ // Store the settings for use in parsing
+ this.settings = settings;
+ // Count leftright depth (for \middle errors)
+ this.leftrightDepth = 0;
+}
+
+var ParseNode = parseData.ParseNode;
+
+/**
+ * An initial function (without its arguments), or an argument to a function.
+ * The `result` argument should be a ParseNode.
+ */
+function ParseFuncOrArgument(result, isFunction, token) {
+ this.result = result;
+ // Is this a function (i.e. is it something defined in functions.js)?
+ this.isFunction = isFunction;
+ this.token = token;
+}
+
+/**
+ * Checks a result to make sure it has the right type, and throws an
+ * appropriate error otherwise.
+ *
+ * @param {boolean=} consume whether to consume the expected token,
+ * defaults to true
+ */
+Parser.prototype.expect = function(text, consume) {
+ if (this.nextToken.text !== text) {
+ throw new ParseError(
+ "Expected '" + text + "', got '" + this.nextToken.text + "'",
+ this.nextToken
+ );
+ }
+ if (consume !== false) {
+ this.consume();
+ }
+};
+
+/**
+ * Considers the current look ahead token as consumed,
+ * and fetches the one after that as the new look ahead.
+ */
+Parser.prototype.consume = function() {
+ this.nextToken = this.gullet.get(this.mode === "math");
+};
+
+Parser.prototype.switchMode = function(newMode) {
+ this.gullet.unget(this.nextToken);
+ this.mode = newMode;
+ this.consume();
+};
+
+/**
+ * Main parsing function, which parses an entire input.
+ *
+ * @return {?Array.<ParseNode>}
+ */
+Parser.prototype.parse = function() {
+ // Try to parse the input
+ this.mode = "math";
+ this.consume();
+ var parse = this.parseInput();
+ return parse;
+};
+
+/**
+ * Parses an entire input tree.
+ */
+Parser.prototype.parseInput = function() {
+ // Parse an expression
+ var expression = this.parseExpression(false);
+ // If we succeeded, make sure there's an EOF at the end
+ this.expect("EOF", false);
+ return expression;
+};
+
+var endOfExpression = ["}", "\\end", "\\right", "&", "\\\\", "\\cr"];
+
+/**
+ * Parses an "expression", which is a list of atoms.
+ *
+ * @param {boolean} breakOnInfix Should the parsing stop when we hit infix
+ * nodes? This happens when functions have higher precendence
+ * than infix nodes in implicit parses.
+ *
+ * @param {?string} breakOnTokenText The text of the token that the expression
+ * should end with, or `null` if something else should end the
+ * expression.
+ *
+ * @return {ParseNode}
+ */
+Parser.prototype.parseExpression = function(breakOnInfix, breakOnTokenText) {
+ var body = [];
+ // Keep adding atoms to the body until we can't parse any more atoms (either
+ // we reached the end, a }, or a \right)
+ while (true) {
+ var lex = this.nextToken;
+ if (endOfExpression.indexOf(lex.text) !== -1) {
+ break;
+ }
+ if (breakOnTokenText && lex.text === breakOnTokenText) {
+ break;
+ }
+ if (breakOnInfix && functions[lex.text] && functions[lex.text].infix) {
+ break;
+ }
+ var atom = this.parseAtom();
+ if (!atom) {
+ if (!this.settings.throwOnError && lex.text[0] === "\\") {
+ var errorNode = this.handleUnsupportedCmd();
+ body.push(errorNode);
+ continue;
+ }
+
+ break;
+ }
+ body.push(atom);
+ }
+ return this.handleInfixNodes(body);
+};
+
+/**
+ * Rewrites infix operators such as \over with corresponding commands such
+ * as \frac.
+ *
+ * There can only be one infix operator per group. If there's more than one
+ * then the expression is ambiguous. This can be resolved by adding {}.
+ *
+ * @returns {Array}
+ */
+Parser.prototype.handleInfixNodes = function(body) {
+ var overIndex = -1;
+ var funcName;
+
+ for (var i = 0; i < body.length; i++) {
+ var node = body[i];
+ if (node.type === "infix") {
+ if (overIndex !== -1) {
+ throw new ParseError(
+ "only one infix operator per group",
+ node.value.token);
+ }
+ overIndex = i;
+ funcName = node.value.replaceWith;
+ }
+ }
+
+ if (overIndex !== -1) {
+ var numerNode;
+ var denomNode;
+
+ var numerBody = body.slice(0, overIndex);
+ var denomBody = body.slice(overIndex + 1);
+
+ if (numerBody.length === 1 && numerBody[0].type === "ordgroup") {
+ numerNode = numerBody[0];
+ } else {
+ numerNode = new ParseNode("ordgroup", numerBody, this.mode);
+ }
+
+ if (denomBody.length === 1 && denomBody[0].type === "ordgroup") {
+ denomNode = denomBody[0];
+ } else {
+ denomNode = new ParseNode("ordgroup", denomBody, this.mode);
+ }
+
+ var value = this.callFunction(
+ funcName, [numerNode, denomNode], null);
+ return [new ParseNode(value.type, value, this.mode)];
+ } else {
+ return body;
+ }
+};
+
+// The greediness of a superscript or subscript
+var SUPSUB_GREEDINESS = 1;
+
+/**
+ * Handle a subscript or superscript with nice errors.
+ */
+Parser.prototype.handleSupSubscript = function(name) {
+ var symbolToken = this.nextToken;
+ var symbol = symbolToken.text;
+ this.consume();
+ var group = this.parseGroup();
+
+ if (!group) {
+ if (!this.settings.throwOnError && this.nextToken.text[0] === "\\") {
+ return this.handleUnsupportedCmd();
+ } else {
+ throw new ParseError(
+ "Expected group after '" + symbol + "'",
+ symbolToken
+ );
+ }
+ } else if (group.isFunction) {
+ // ^ and _ have a greediness, so handle interactions with functions'
+ // greediness
+ var funcGreediness = functions[group.result].greediness;
+ if (funcGreediness > SUPSUB_GREEDINESS) {
+ return this.parseFunction(group);
+ } else {
+ throw new ParseError(
+ "Got function '" + group.result + "' with no arguments " +
+ "as " + name, symbolToken);
+ }
+ } else {
+ return group.result;
+ }
+};
+
+/**
+ * Converts the textual input of an unsupported command into a text node
+ * contained within a color node whose color is determined by errorColor
+ */
+Parser.prototype.handleUnsupportedCmd = function() {
+ var text = this.nextToken.text;
+ var textordArray = [];
+
+ for (var i = 0; i < text.length; i++) {
+ textordArray.push(new ParseNode("textord", text[i], "text"));
+ }
+
+ var textNode = new ParseNode(
+ "text",
+ {
+ body: textordArray,
+ type: "text"
+ },
+ this.mode);
+
+ var colorNode = new ParseNode(
+ "color",
+ {
+ color: this.settings.errorColor,
+ value: [textNode],
+ type: "color"
+ },
+ this.mode);
+
+ this.consume();
+ return colorNode;
+};
+
+/**
+ * Parses a group with optional super/subscripts.
+ *
+ * @return {?ParseNode}
+ */
+Parser.prototype.parseAtom = function() {
+ // The body of an atom is an implicit group, so that things like
+ // \left(x\right)^2 work correctly.
+ var base = this.parseImplicitGroup();
+
+ // In text mode, we don't have superscripts or subscripts
+ if (this.mode === "text") {
+ return base;
+ }
+
+ // Note that base may be empty (i.e. null) at this point.
+
+ var superscript;
+ var subscript;
+ while (true) {
+ // Lex the first token
+ var lex = this.nextToken;
+
+ if (lex.text === "\\limits" || lex.text === "\\nolimits") {
+ // We got a limit control
+ if (!base || base.type !== "op") {
+ throw new ParseError(
+ "Limit controls must follow a math operator",
+ lex);
+ } else {
+ var limits = lex.text === "\\limits";
+ base.value.limits = limits;
+ base.value.alwaysHandleSupSub = true;
+ }
+ this.consume();
+ } else if (lex.text === "^") {
+ // We got a superscript start
+ if (superscript) {
+ throw new ParseError("Double superscript", lex);
+ }
+ superscript = this.handleSupSubscript("superscript");
+ } else if (lex.text === "_") {
+ // We got a subscript start
+ if (subscript) {
+ throw new ParseError("Double subscript", lex);
+ }
+ subscript = this.handleSupSubscript("subscript");
+ } else if (lex.text === "'") {
+ // We got a prime
+ var prime = new ParseNode("textord", "\\prime", this.mode);
+
+ // Many primes can be grouped together, so we handle this here
+ var primes = [prime];
+ this.consume();
+ // Keep lexing tokens until we get something that's not a prime
+ while (this.nextToken.text === "'") {
+ // For each one, add another prime to the list
+ primes.push(prime);
+ this.consume();
+ }
+ // Put them into an ordgroup as the superscript
+ superscript = new ParseNode("ordgroup", primes, this.mode);
+ } else {
+ // If it wasn't ^, _, or ', stop parsing super/subscripts
+ break;
+ }
+ }
+
+ if (superscript || subscript) {
+ // If we got either a superscript or subscript, create a supsub
+ return new ParseNode("supsub", {
+ base: base,
+ sup: superscript,
+ sub: subscript
+ }, this.mode);
+ } else {
+ // Otherwise return the original body
+ return base;
+ }
+};
+
+// A list of the size-changing functions, for use in parseImplicitGroup
+var sizeFuncs = [
+ "\\tiny", "\\scriptsize", "\\footnotesize", "\\small", "\\normalsize",
+ "\\large", "\\Large", "\\LARGE", "\\huge", "\\Huge"
+];
+
+// A list of the style-changing functions, for use in parseImplicitGroup
+var styleFuncs = [
+ "\\displaystyle", "\\textstyle", "\\scriptstyle", "\\scriptscriptstyle"
+];
+
+/**
+ * Parses an implicit group, which is a group that starts at the end of a
+ * specified, and ends right before a higher explicit group ends, or at EOL. It
+ * is used for functions that appear to affect the current style, like \Large or
+ * \textrm, where instead of keeping a style we just pretend that there is an
+ * implicit grouping after it until the end of the group. E.g.
+ * small text {\Large large text} small text again
+ * It is also used for \left and \right to get the correct grouping.
+ *
+ * @return {?ParseNode}
+ */
+Parser.prototype.parseImplicitGroup = function() {
+ var start = this.parseSymbol();
+
+ if (start == null) {
+ // If we didn't get anything we handle, fall back to parseFunction
+ return this.parseFunction();
+ }
+
+ var func = start.result;
+ var body;
+
+ if (func === "\\left") {
+ // If we see a left:
+ // Parse the entire left function (including the delimiter)
+ var left = this.parseFunction(start);
+ // Parse out the implicit body
+ ++this.leftrightDepth;
+ body = this.parseExpression(false);
+ --this.leftrightDepth;
+ // Check the next token
+ this.expect("\\right", false);
+ var right = this.parseFunction();
+ return new ParseNode("leftright", {
+ body: body,
+ left: left.value.value,
+ right: right.value.value
+ }, this.mode);
+ } else if (func === "\\begin") {
+ // begin...end is similar to left...right
+ var begin = this.parseFunction(start);
+ var envName = begin.value.name;
+ if (!environments.hasOwnProperty(envName)) {
+ throw new ParseError(
+ "No such environment: " + envName, begin.value.nameGroup);
+ }
+ // Build the environment object. Arguments and other information will
+ // be made available to the begin and end methods using properties.
+ var env = environments[envName];
+ var args = this.parseArguments("\\begin{" + envName + "}", env);
+ var context = {
+ mode: this.mode,
+ envName: envName,
+ parser: this,
+ positions: args.pop()
+ };
+ var result = env.handler(context, args);
+ this.expect("\\end", false);
+ var endNameToken = this.nextToken;
+ var end = this.parseFunction();
+ if (end.value.name !== envName) {
+ throw new ParseError(
+ "Mismatch: \\begin{" + envName + "} matched " +
+ "by \\end{" + end.value.name + "}",
+ endNameToken);
+ }
+ result.position = end.position;
+ return result;
+ } else if (utils.contains(sizeFuncs, func)) {
+ // If we see a sizing function, parse out the implict body
+ body = this.parseExpression(false);
+ return new ParseNode("sizing", {
+ // Figure out what size to use based on the list of functions above
+ size: "size" + (utils.indexOf(sizeFuncs, func) + 1),
+ value: body
+ }, this.mode);
+ } else if (utils.contains(styleFuncs, func)) {
+ // If we see a styling function, parse out the implict body
+ body = this.parseExpression(true);
+ return new ParseNode("styling", {
+ // Figure out what style to use by pulling out the style from
+ // the function name
+ style: func.slice(1, func.length - 5),
+ value: body
+ }, this.mode);
+ } else {
+ // Defer to parseFunction if it's not a function we handle
+ return this.parseFunction(start);
+ }
+};
+
+/**
+ * Parses an entire function, including its base and all of its arguments.
+ * The base might either have been parsed already, in which case
+ * it is provided as an argument, or it's the next group in the input.
+ *
+ * @param {ParseFuncOrArgument=} baseGroup optional as described above
+ * @return {?ParseNode}
+ */
+Parser.prototype.parseFunction = function(baseGroup) {
+ if (!baseGroup) {
+ baseGroup = this.parseGroup();
+ }
+
+ if (baseGroup) {
+ if (baseGroup.isFunction) {
+ var func = baseGroup.result;
+ var funcData = functions[func];
+ if (this.mode === "text" && !funcData.allowedInText) {
+ throw new ParseError(
+ "Can't use function '" + func + "' in text mode",
+ baseGroup.token);
+ }
+
+ var args = this.parseArguments(func, funcData);
+ var token = baseGroup.token;
+ var result = this.callFunction(func, args, args.pop(), token);
+ return new ParseNode(result.type, result, this.mode);
+ } else {
+ return baseGroup.result;
+ }
+ } else {
+ return null;
+ }
+};
+
+/**
+ * Call a function handler with a suitable context and arguments.
+ */
+Parser.prototype.callFunction = function(name, args, positions, token) {
+ var context = {
+ funcName: name,
+ parser: this,
+ positions: positions,
+ token: token
+ };
+ return functions[name].handler(context, args);
+};
+
+/**
+ * Parses the arguments of a function or environment
+ *
+ * @param {string} func "\name" or "\begin{name}"
+ * @param {{numArgs:number,numOptionalArgs:number|undefined}} funcData
+ * @return the array of arguments, with the list of positions as last element
+ */
+Parser.prototype.parseArguments = function(func, funcData) {
+ var totalArgs = funcData.numArgs + funcData.numOptionalArgs;
+ if (totalArgs === 0) {
+ return [[this.pos]];
+ }
+
+ var baseGreediness = funcData.greediness;
+ var positions = [this.pos];
+ var args = [];
+
+ for (var i = 0; i < totalArgs; i++) {
+ var nextToken = this.nextToken;
+ var argType = funcData.argTypes && funcData.argTypes[i];
+ var arg;
+ if (i < funcData.numOptionalArgs) {
+ if (argType) {
+ arg = this.parseGroupOfType(argType, true);
+ } else {
+ arg = this.parseGroup(true);
+ }
+ if (!arg) {
+ args.push(null);
+ positions.push(this.pos);
+ continue;
+ }
+ } else {
+ if (argType) {
+ arg = this.parseGroupOfType(argType);
+ } else {
+ arg = this.parseGroup();
+ }
+ if (!arg) {
+ if (!this.settings.throwOnError &&
+ this.nextToken.text[0] === "\\") {
+ arg = new ParseFuncOrArgument(
+ this.handleUnsupportedCmd(this.nextToken.text),
+ false);
+ } else {
+ throw new ParseError(
+ "Expected group after '" + func + "'", nextToken);
+ }
+ }
+ }
+ var argNode;
+ if (arg.isFunction) {
+ var argGreediness =
+ functions[arg.result].greediness;
+ if (argGreediness > baseGreediness) {
+ argNode = this.parseFunction(arg);
+ } else {
+ throw new ParseError(
+ "Got function '" + arg.result + "' as " +
+ "argument to '" + func + "'", nextToken);
+ }
+ } else {
+ argNode = arg.result;
+ }
+ args.push(argNode);
+ positions.push(this.pos);
+ }
+
+ args.push(positions);
+
+ return args;
+};
+
+
+/**
+ * Parses a group when the mode is changing.
+ *
+ * @return {?ParseFuncOrArgument}
+ */
+Parser.prototype.parseGroupOfType = function(innerMode, optional) {
+ var outerMode = this.mode;
+ // Handle `original` argTypes
+ if (innerMode === "original") {
+ innerMode = outerMode;
+ }
+
+ if (innerMode === "color") {
+ return this.parseColorGroup(optional);
+ }
+ if (innerMode === "size") {
+ return this.parseSizeGroup(optional);
+ }
+
+ this.switchMode(innerMode);
+ if (innerMode === "text") {
+ // text mode is special because it should ignore the whitespace before
+ // it
+ while (this.nextToken.text === " ") {
+ this.consume();
+ }
+ }
+ // By the time we get here, innerMode is one of "text" or "math".
+ // We switch the mode of the parser, recurse, then restore the old mode.
+ var res = this.parseGroup(optional);
+ this.switchMode(outerMode);
+ return res;
+};
+
+/**
+ * Parses a group, essentially returning the string formed by the
+ * brace-enclosed tokens plus some position information.
+ *
+ * @param {string} modeName Used to describe the mode in error messages
+ * @param {boolean=} optional Whether the group is optional or required
+ */
+Parser.prototype.parseStringGroup = function(modeName, optional) {
+ if (optional && this.nextToken.text !== "[") {
+ return null;
+ }
+ var outerMode = this.mode;
+ this.mode = "text";
+ this.expect(optional ? "[" : "{");
+ var str = "";
+ var firstToken = this.nextToken;
+ var lastToken = firstToken;
+ while (this.nextToken.text !== (optional ? "]" : "}")) {
+ if (this.nextToken.text === "EOF") {
+ throw new ParseError(
+ "Unexpected end of input in " + modeName,
+ firstToken.range(this.nextToken, str));
+ }
+ lastToken = this.nextToken;
+ str += lastToken.text;
+ this.consume();
+ }
+ this.mode = outerMode;
+ this.expect(optional ? "]" : "}");
+ return firstToken.range(lastToken, str);
+};
+
+/**
+ * Parses a regex-delimited group: the largest sequence of tokens
+ * whose concatenated strings match `regex`. Returns the string
+ * formed by the tokens plus some position information.
+ *
+ * @param {RegExp} regex
+ * @param {string} modeName Used to describe the mode in error messages
+ */
+Parser.prototype.parseRegexGroup = function(regex, modeName) {
+ var outerMode = this.mode;
+ this.mode = "text";
+ var firstToken = this.nextToken;
+ var lastToken = firstToken;
+ var str = "";
+ while (this.nextToken.text !== "EOF"
+ && regex.test(str + this.nextToken.text)) {
+ lastToken = this.nextToken;
+ str += lastToken.text;
+ this.consume();
+ }
+ if (str === "") {
+ throw new ParseError(
+ "Invalid " + modeName + ": '" + firstToken.text + "'",
+ firstToken);
+ }
+ this.mode = outerMode;
+ return firstToken.range(lastToken, str);
+};
+
+/**
+ * Parses a color description.
+ */
+Parser.prototype.parseColorGroup = function(optional) {
+ var res = this.parseStringGroup("color", optional);
+ if (!res) {
+ return null;
+ }
+ var match = (/^(#[a-z0-9]+|[a-z]+)$/i).exec(res.text);
+ if (!match) {
+ throw new ParseError("Invalid color: '" + res.text + "'", res);
+ }
+ return new ParseFuncOrArgument(
+ new ParseNode("color", match[0], this.mode),
+ false);
+};
+
+/**
+ * Parses a size specification, consisting of magnitude and unit.
+ */
+Parser.prototype.parseSizeGroup = function(optional) {
+ var res;
+ if (!optional && this.nextToken.text !== "{") {
+ res = this.parseRegexGroup(
+ /^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2}$/, "size");
+ } else {
+ res = this.parseStringGroup("size", optional);
+ }
+ if (!res) {
+ return null;
+ }
+ var match = (/([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/).exec(res.text);
+ if (!match) {
+ throw new ParseError("Invalid size: '" + res.text + "'", res);
+ }
+ var data = {
+ number: +(match[1] + match[2]), // sign + magnitude, cast to number
+ unit: match[3]
+ };
+ if (data.unit !== "em" && data.unit !== "ex" && data.unit !== "mu") {
+ throw new ParseError("Invalid unit: '" + data.unit + "'", res);
+ }
+ return new ParseFuncOrArgument(
+ new ParseNode("color", data, this.mode),
+ false);
+};
+
+/**
+ * If the argument is false or absent, this parses an ordinary group,
+ * which is either a single nucleus (like "x") or an expression
+ * in braces (like "{x+y}").
+ * If the argument is true, it parses either a bracket-delimited expression
+ * (like "[x+y]") or returns null to indicate the absence of a
+ * bracket-enclosed group.
+ *
+ * @param {boolean=} optional Whether the group is optional or required
+ * @return {?ParseFuncOrArgument}
+ */
+Parser.prototype.parseGroup = function(optional) {
+ var firstToken = this.nextToken;
+ // Try to parse an open brace
+ if (this.nextToken.text === (optional ? "[" : "{")) {
+ // If we get a brace, parse an expression
+ this.consume();
+ var expression = this.parseExpression(false, optional ? "]" : null);
+ var lastToken = this.nextToken;
+ // Make sure we get a close brace
+ this.expect(optional ? "]" : "}");
+ if (this.mode === "text") {
+ this.formLigatures(expression);
+ }
+ return new ParseFuncOrArgument(
+ new ParseNode("ordgroup", expression, this.mode,
+ firstToken, lastToken),
+ false);
+ } else {
+ // Otherwise, just return a nucleus, or nothing for an optional group
+ return optional ? null : this.parseSymbol();
+ }
+};
+
+/**
+ * Form ligature-like combinations of characters for text mode.
+ * This includes inputs like "--", "---", "``" and "''".
+ * The result will simply replace multiple textord nodes with a single
+ * character in each value by a single textord node having multiple
+ * characters in its value. The representation is still ASCII source.
+ *
+ * @param {Array.<ParseNode>} group the nodes of this group,
+ * list will be moified in place
+ */
+Parser.prototype.formLigatures = function(group) {
+ var i;
+ var n = group.length - 1;
+ for (i = 0; i < n; ++i) {
+ var a = group[i];
+ var v = a.value;
+ if (v === "-" && group[i + 1].value === "-") {
+ if (i + 1 < n && group[i + 2].value === "-") {
+ group.splice(i, 3, new ParseNode(
+ "textord", "---", "text", a, group[i + 2]));
+ n -= 2;
+ } else {
+ group.splice(i, 2, new ParseNode(
+ "textord", "--", "text", a, group[i + 1]));
+ n -= 1;
+ }
+ }
+ if ((v === "'" || v === "`") && group[i + 1].value === v) {
+ group.splice(i, 2, new ParseNode(
+ "textord", v + v, "text", a, group[i + 1]));
+ n -= 1;
+ }
+ }
+};
+
+/**
+ * Parse a single symbol out of the string. Here, we handle both the functions
+ * we have defined, as well as the single character symbols
+ *
+ * @return {?ParseFuncOrArgument}
+ */
+Parser.prototype.parseSymbol = function() {
+ var nucleus = this.nextToken;
+
+ if (functions[nucleus.text]) {
+ this.consume();
+ // If there exists a function with this name, we return the function and
+ // say that it is a function.
+ return new ParseFuncOrArgument(
+ nucleus.text,
+ true, nucleus);
+ } else if (symbols[this.mode][nucleus.text]) {
+ this.consume();
+ // Otherwise if this is a no-argument function, find the type it
+ // corresponds to in the symbols map
+ return new ParseFuncOrArgument(
+ new ParseNode(symbols[this.mode][nucleus.text].group,
+ nucleus.text, this.mode, nucleus),
+ false, nucleus);
+ } else if (this.mode === "text" && cjkRegex.test(nucleus.text)) {
+ this.consume();
+ return new ParseFuncOrArgument(
+ new ParseNode("textord", nucleus.text, this.mode, nucleus),
+ false, nucleus);
+ } else {
+ return null;
+ }
+};
+
+Parser.prototype.ParseNode = ParseNode;
+
+module.exports = Parser;
+
+},{"./MacroExpander":4,"./ParseError":6,"./environments":16,"./functions":19,"./parseData":21,"./symbols":23,"./unicodeRegexes":24,"./utils":25}],8:[function(require,module,exports){
+/**
+ * This is a module for storing settings passed into KaTeX. It correctly handles
+ * default settings.
+ */
+
+/**
+ * Helper function for getting a default value if the value is undefined
+ */
+function get(option, defaultValue) {
+ return option === undefined ? defaultValue : option;
+}
+
+/**
+ * The main Settings object
+ *
+ * The current options stored are:
+ * - displayMode: Whether the expression should be typeset by default in
+ * textstyle or displaystyle (default false)
+ */
+function Settings(options) {
+ // allow null options
+ options = options || {};
+ this.displayMode = get(options.displayMode, false);
+ this.throwOnError = get(options.throwOnError, true);
+ this.errorColor = get(options.errorColor, "#cc0000");
+ this.macros = options.macros || {};
+}
+
+module.exports = Settings;
+
+},{}],9:[function(require,module,exports){
+/**
+ * This file contains information and classes for the various kinds of styles
+ * used in TeX. It provides a generic `Style` class, which holds information
+ * about a specific style. It then provides instances of all the different kinds
+ * of styles possible, and provides functions to move between them and get
+ * information about them.
+ */
+
+var sigmas = require("./fontMetrics.js").sigmas;
+
+var metrics = [{}, {}, {}];
+var i;
+for (var key in sigmas) {
+ if (sigmas.hasOwnProperty(key)) {
+ for (i = 0; i < 3; i++) {
+ metrics[i][key] = sigmas[key][i];
+ }
+ }
+}
+for (i = 0; i < 3; i++) {
+ metrics[i].emPerEx = sigmas.xHeight[i] / sigmas.quad[i];
+}
+
+/**
+ * The main style class. Contains a unique id for the style, a size (which is
+ * the same for cramped and uncramped version of a style), a cramped flag, and a
+ * size multiplier, which gives the size difference between a style and
+ * textstyle.
+ */
+function Style(id, size, multiplier, cramped) {
+ this.id = id;
+ this.size = size;
+ this.cramped = cramped;
+ this.sizeMultiplier = multiplier;
+ this.metrics = metrics[size > 0 ? size - 1 : 0];
+}
+
+/**
+ * Get the style of a superscript given a base in the current style.
+ */
+Style.prototype.sup = function() {
+ return styles[sup[this.id]];
+};
+
+/**
+ * Get the style of a subscript given a base in the current style.
+ */
+Style.prototype.sub = function() {
+ return styles[sub[this.id]];
+};
+
+/**
+ * Get the style of a fraction numerator given the fraction in the current
+ * style.
+ */
+Style.prototype.fracNum = function() {
+ return styles[fracNum[this.id]];
+};
+
+/**
+ * Get the style of a fraction denominator given the fraction in the current
+ * style.
+ */
+Style.prototype.fracDen = function() {
+ return styles[fracDen[this.id]];
+};
+
+/**
+ * Get the cramped version of a style (in particular, cramping a cramped style
+ * doesn't change the style).
+ */
+Style.prototype.cramp = function() {
+ return styles[cramp[this.id]];
+};
+
+/**
+ * HTML class name, like "displaystyle cramped"
+ */
+Style.prototype.cls = function() {
+ return sizeNames[this.size] + (this.cramped ? " cramped" : " uncramped");
+};
+
+/**
+ * HTML Reset class name, like "reset-textstyle"
+ */
+Style.prototype.reset = function() {
+ return resetNames[this.size];
+};
+
+/**
+ * Return if this style is tightly spaced (scriptstyle/scriptscriptstyle)
+ */
+Style.prototype.isTight = function() {
+ return this.size >= 2;
+};
+
+// IDs of the different styles
+var D = 0;
+var Dc = 1;
+var T = 2;
+var Tc = 3;
+var S = 4;
+var Sc = 5;
+var SS = 6;
+var SSc = 7;
+
+// String names for the different sizes
+var sizeNames = [
+ "displaystyle textstyle",
+ "textstyle",
+ "scriptstyle",
+ "scriptscriptstyle"
+];
+
+// Reset names for the different sizes
+var resetNames = [
+ "reset-textstyle",
+ "reset-textstyle",
+ "reset-scriptstyle",
+ "reset-scriptscriptstyle"
+];
+
+// Instances of the different styles
+var styles = [
+ new Style(D, 0, 1.0, false),
+ new Style(Dc, 0, 1.0, true),
+ new Style(T, 1, 1.0, false),
+ new Style(Tc, 1, 1.0, true),
+ new Style(S, 2, 0.7, false),
+ new Style(Sc, 2, 0.7, true),
+ new Style(SS, 3, 0.5, false),
+ new Style(SSc, 3, 0.5, true)
+];
+
+// Lookup tables for switching from one style to another
+var sup = [S, Sc, S, Sc, SS, SSc, SS, SSc];
+var sub = [Sc, Sc, Sc, Sc, SSc, SSc, SSc, SSc];
+var fracNum = [T, Tc, S, Sc, SS, SSc, SS, SSc];
+var fracDen = [Tc, Tc, Sc, Sc, SSc, SSc, SSc, SSc];
+var cramp = [Dc, Dc, Tc, Tc, Sc, Sc, SSc, SSc];
+
+// We only export some of the styles. Also, we don't export the `Style` class so
+// no more styles can be generated.
+module.exports = {
+ DISPLAY: styles[D],
+ TEXT: styles[T],
+ SCRIPT: styles[S],
+ SCRIPTSCRIPT: styles[SS]
+};
+
+},{"./fontMetrics.js":17}],10:[function(require,module,exports){
+/* eslint no-console:0 */
+/**
+ * This module contains general functions that can be used for building
+ * different kinds of domTree nodes in a consistent manner.
+ */
+
+var domTree = require("./domTree");
+var fontMetrics = require("./fontMetrics");
+var symbols = require("./symbols");
+var utils = require("./utils");
+
+var greekCapitals = [
+ "\\Gamma",
+ "\\Delta",
+ "\\Theta",
+ "\\Lambda",
+ "\\Xi",
+ "\\Pi",
+ "\\Sigma",
+ "\\Upsilon",
+ "\\Phi",
+ "\\Psi",
+ "\\Omega"
+];
+
+// The following have to be loaded from Main-Italic font, using class mainit
+var mainitLetters = [
+ "\u0131", // dotless i, \imath
+ "\u0237", // dotless j, \jmath
+ "\u00a3" // \pounds
+];
+
+/**
+ * Makes a symbolNode after translation via the list of symbols in symbols.js.
+ * Correctly pulls out metrics for the character, and optionally takes a list of
+ * classes to be attached to the node.
+ *
+ * TODO: make argument order closer to makeSpan
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ */
+var makeSymbol = function(value, fontFamily, mode, options, classes) {
+ // Replace the value with its replaced value from symbol.js
+ if (symbols[mode][value] && symbols[mode][value].replace) {
+ value = symbols[mode][value].replace;
+ }
+
+ var metrics = fontMetrics.getCharacterMetrics(value, fontFamily);
+
+ var symbolNode;
+ if (metrics) {
+ var italic = metrics.italic;
+ if (mode === "text") {
+ italic = 0;
+ }
+ symbolNode = new domTree.symbolNode(
+ value, metrics.height, metrics.depth, italic, metrics.skew,
+ classes);
+ } else {
+ // TODO(emily): Figure out a good way to only print this in development
+ typeof console !== "undefined" && console.warn(
+ "No character metrics for '" + value + "' in style '" +
+ fontFamily + "'");
+ symbolNode = new domTree.symbolNode(value, 0, 0, 0, 0, classes);
+ }
+
+ if (options) {
+ if (options.style.isTight()) {
+ symbolNode.classes.push("mtight");
+ }
+ if (options.getColor()) {
+ symbolNode.style.color = options.getColor();
+ }
+ }
+
+ return symbolNode;
+};
+
+/**
+ * Makes a symbol in Main-Regular or AMS-Regular.
+ * Used for rel, bin, open, close, inner, and punct.
+ */
+var mathsym = function(value, mode, options, classes) {
+ // Decide what font to render the symbol in by its entry in the symbols
+ // table.
+ // Have a special case for when the value = \ because the \ is used as a
+ // textord in unsupported command errors but cannot be parsed as a regular
+ // text ordinal and is therefore not present as a symbol in the symbols
+ // table for text
+ if (value === "\\" || symbols[mode][value].font === "main") {
+ return makeSymbol(value, "Main-Regular", mode, options, classes);
+ } else {
+ return makeSymbol(
+ value, "AMS-Regular", mode, options, classes.concat(["amsrm"]));
+ }
+};
+
+/**
+ * Makes a symbol in the default font for mathords and textords.
+ */
+var mathDefault = function(value, mode, options, classes, type) {
+ if (type === "mathord") {
+ return mathit(value, mode, options, classes);
+ } else if (type === "textord") {
+ return makeSymbol(
+ value, "Main-Regular", mode, options, classes.concat(["mathrm"]));
+ } else {
+ throw new Error("unexpected type: " + type + " in mathDefault");
+ }
+};
+
+/**
+ * Makes a symbol in the italic math font.
+ */
+var mathit = function(value, mode, options, classes) {
+ if (/[0-9]/.test(value.charAt(0)) ||
+ // glyphs for \imath and \jmath do not exist in Math-Italic so we
+ // need to use Main-Italic instead
+ utils.contains(mainitLetters, value) ||
+ utils.contains(greekCapitals, value)) {
+ return makeSymbol(
+ value, "Main-Italic", mode, options, classes.concat(["mainit"]));
+ } else {
+ return makeSymbol(
+ value, "Math-Italic", mode, options, classes.concat(["mathit"]));
+ }
+};
+
+/**
+ * Makes either a mathord or textord in the correct font and color.
+ */
+var makeOrd = function(group, options, type) {
+ var mode = group.mode;
+ var value = group.value;
+ if (symbols[mode][value] && symbols[mode][value].replace) {
+ value = symbols[mode][value].replace;
+ }
+
+ var classes = ["mord"];
+
+ var font = options.font;
+ if (font) {
+ if (font === "mathit" || utils.contains(mainitLetters, value)) {
+ return mathit(value, mode, options, classes);
+ } else {
+ var fontName = fontMap[font].fontName;
+ if (fontMetrics.getCharacterMetrics(value, fontName)) {
+ return makeSymbol(
+ value, fontName, mode, options, classes.concat([font]));
+ } else {
+ return mathDefault(value, mode, options, classes, type);
+ }
+ }
+ } else {
+ return mathDefault(value, mode, options, classes, type);
+ }
+};
+
+/**
+ * Calculate the height, depth, and maxFontSize of an element based on its
+ * children.
+ */
+var sizeElementFromChildren = function(elem) {
+ var height = 0;
+ var depth = 0;
+ var maxFontSize = 0;
+
+ if (elem.children) {
+ for (var i = 0; i < elem.children.length; i++) {
+ if (elem.children[i].height > height) {
+ height = elem.children[i].height;
+ }
+ if (elem.children[i].depth > depth) {
+ depth = elem.children[i].depth;
+ }
+ if (elem.children[i].maxFontSize > maxFontSize) {
+ maxFontSize = elem.children[i].maxFontSize;
+ }
+ }
+ }
+
+ elem.height = height;
+ elem.depth = depth;
+ elem.maxFontSize = maxFontSize;
+};
+
+/**
+ * Makes a span with the given list of classes, list of children, and options.
+ *
+ * TODO: Ensure that `options` is always provided (currently some call sites
+ * don't pass it).
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ */
+var makeSpan = function(classes, children, options) {
+ var span = new domTree.span(classes, children, options);
+
+ sizeElementFromChildren(span);
+
+ return span;
+};
+
+/**
+ * Prepends the given children to the given span, updating height, depth, and
+ * maxFontSize.
+ */
+var prependChildren = function(span, children) {
+ span.children = children.concat(span.children);
+
+ sizeElementFromChildren(span);
+};
+
+/**
+ * Makes a document fragment with the given list of children.
+ */
+var makeFragment = function(children) {
+ var fragment = new domTree.documentFragment(children);
+
+ sizeElementFromChildren(fragment);
+
+ return fragment;
+};
+
+/**
+ * Makes an element placed in each of the vlist elements to ensure that each
+ * element has the same max font size. To do this, we create a zero-width space
+ * with the correct font size.
+ */
+var makeFontSizer = function(options, fontSize) {
+ var fontSizeInner = makeSpan([], [new domTree.symbolNode("\u200b")]);
+ fontSizeInner.style.fontSize =
+ (fontSize / options.style.sizeMultiplier) + "em";
+
+ var fontSizer = makeSpan(
+ ["fontsize-ensurer", "reset-" + options.size, "size5"],
+ [fontSizeInner]);
+
+ return fontSizer;
+};
+
+/**
+ * Makes a vertical list by stacking elements and kerns on top of each other.
+ * Allows for many different ways of specifying the positioning method.
+ *
+ * Arguments:
+ * - children: A list of child or kern nodes to be stacked on top of each other
+ * (i.e. the first element will be at the bottom, and the last at
+ * the top). Element nodes are specified as
+ * {type: "elem", elem: node}
+ * while kern nodes are specified as
+ * {type: "kern", size: size}
+ * - positionType: The method by which the vlist should be positioned. Valid
+ * values are:
+ * - "individualShift": The children list only contains elem
+ * nodes, and each node contains an extra
+ * "shift" value of how much it should be
+ * shifted (note that shifting is always
+ * moving downwards). positionData is
+ * ignored.
+ * - "top": The positionData specifies the topmost point of
+ * the vlist (note this is expected to be a height,
+ * so positive values move up)
+ * - "bottom": The positionData specifies the bottommost point
+ * of the vlist (note this is expected to be a
+ * depth, so positive values move down
+ * - "shift": The vlist will be positioned such that its
+ * baseline is positionData away from the baseline
+ * of the first child. Positive values move
+ * downwards.
+ * - "firstBaseline": The vlist will be positioned such that
+ * its baseline is aligned with the
+ * baseline of the first child.
+ * positionData is ignored. (this is
+ * equivalent to "shift" with
+ * positionData=0)
+ * - positionData: Data used in different ways depending on positionType
+ * - options: An Options object
+ *
+ */
+var makeVList = function(children, positionType, positionData, options) {
+ var depth;
+ var currPos;
+ var i;
+ if (positionType === "individualShift") {
+ var oldChildren = children;
+ children = [oldChildren[0]];
+
+ // Add in kerns to the list of children to get each element to be
+ // shifted to the correct specified shift
+ depth = -oldChildren[0].shift - oldChildren[0].elem.depth;
+ currPos = depth;
+ for (i = 1; i < oldChildren.length; i++) {
+ var diff = -oldChildren[i].shift - currPos -
+ oldChildren[i].elem.depth;
+ var size = diff -
+ (oldChildren[i - 1].elem.height +
+ oldChildren[i - 1].elem.depth);
+
+ currPos = currPos + diff;
+
+ children.push({type: "kern", size: size});
+ children.push(oldChildren[i]);
+ }
+ } else if (positionType === "top") {
+ // We always start at the bottom, so calculate the bottom by adding up
+ // all the sizes
+ var bottom = positionData;
+ for (i = 0; i < children.length; i++) {
+ if (children[i].type === "kern") {
+ bottom -= children[i].size;
+ } else {
+ bottom -= children[i].elem.height + children[i].elem.depth;
+ }
+ }
+ depth = bottom;
+ } else if (positionType === "bottom") {
+ depth = -positionData;
+ } else if (positionType === "shift") {
+ depth = -children[0].elem.depth - positionData;
+ } else if (positionType === "firstBaseline") {
+ depth = -children[0].elem.depth;
+ } else {
+ depth = 0;
+ }
+
+ // Make the fontSizer
+ var maxFontSize = 0;
+ for (i = 0; i < children.length; i++) {
+ if (children[i].type === "elem") {
+ maxFontSize = Math.max(maxFontSize, children[i].elem.maxFontSize);
+ }
+ }
+ var fontSizer = makeFontSizer(options, maxFontSize);
+
+ // Create a new list of actual children at the correct offsets
+ var realChildren = [];
+ currPos = depth;
+ for (i = 0; i < children.length; i++) {
+ if (children[i].type === "kern") {
+ currPos += children[i].size;
+ } else {
+ var child = children[i].elem;
+
+ var shift = -child.depth - currPos;
+ currPos += child.height + child.depth;
+
+ var childWrap = makeSpan([], [fontSizer, child]);
+ childWrap.height -= shift;
+ childWrap.depth += shift;
+ childWrap.style.top = shift + "em";
+
+ realChildren.push(childWrap);
+ }
+ }
+
+ // Add in an element at the end with no offset to fix the calculation of
+ // baselines in some browsers (namely IE, sometimes safari)
+ var baselineFix = makeSpan(
+ ["baseline-fix"], [fontSizer, new domTree.symbolNode("\u200b")]);
+ realChildren.push(baselineFix);
+
+ var vlist = makeSpan(["vlist"], realChildren);
+ // Fix the final height and depth, in case there were kerns at the ends
+ // since the makeSpan calculation won't take that in to account.
+ vlist.height = Math.max(currPos, vlist.height);
+ vlist.depth = Math.max(-depth, vlist.depth);
+ return vlist;
+};
+
+// A table of size -> font size for the different sizing functions
+var sizingMultiplier = {
+ size1: 0.5,
+ size2: 0.7,
+ size3: 0.8,
+ size4: 0.9,
+ size5: 1.0,
+ size6: 1.2,
+ size7: 1.44,
+ size8: 1.73,
+ size9: 2.07,
+ size10: 2.49
+};
+
+// A map of spacing functions to their attributes, like size and corresponding
+// CSS class
+var spacingFunctions = {
+ "\\qquad": {
+ size: "2em",
+ className: "qquad"
+ },
+ "\\quad": {
+ size: "1em",
+ className: "quad"
+ },
+ "\\enspace": {
+ size: "0.5em",
+ className: "enspace"
+ },
+ "\\;": {
+ size: "0.277778em",
+ className: "thickspace"
+ },
+ "\\:": {
+ size: "0.22222em",
+ className: "mediumspace"
+ },
+ "\\,": {
+ size: "0.16667em",
+ className: "thinspace"
+ },
+ "\\!": {
+ size: "-0.16667em",
+ className: "negativethinspace"
+ }
+};
+
+/**
+ * Maps TeX font commands to objects containing:
+ * - variant: string used for "mathvariant" attribute in buildMathML.js
+ * - fontName: the "style" parameter to fontMetrics.getCharacterMetrics
+ */
+// A map between tex font commands an MathML mathvariant attribute values
+var fontMap = {
+ // styles
+ "mathbf": {
+ variant: "bold",
+ fontName: "Main-Bold"
+ },
+ "mathrm": {
+ variant: "normal",
+ fontName: "Main-Regular"
+ },
+ "textit": {
+ variant: "italic",
+ fontName: "Main-Italic"
+ },
+
+ // "mathit" is missing because it requires the use of two fonts: Main-Italic
+ // and Math-Italic. This is handled by a special case in makeOrd which ends
+ // up calling mathit.
+
+ // families
+ "mathbb": {
+ variant: "double-struck",
+ fontName: "AMS-Regular"
+ },
+ "mathcal": {
+ variant: "script",
+ fontName: "Caligraphic-Regular"
+ },
+ "mathfrak": {
+ variant: "fraktur",
+ fontName: "Fraktur-Regular"
+ },
+ "mathscr": {
+ variant: "script",
+ fontName: "Script-Regular"
+ },
+ "mathsf": {
+ variant: "sans-serif",
+ fontName: "SansSerif-Regular"
+ },
+ "mathtt": {
+ variant: "monospace",
+ fontName: "Typewriter-Regular"
+ }
+};
+
+module.exports = {
+ fontMap: fontMap,
+ makeSymbol: makeSymbol,
+ mathsym: mathsym,
+ makeSpan: makeSpan,
+ makeFragment: makeFragment,
+ makeVList: makeVList,
+ makeOrd: makeOrd,
+ prependChildren: prependChildren,
+ sizingMultiplier: sizingMultiplier,
+ spacingFunctions: spacingFunctions
+};
+
+},{"./domTree":15,"./fontMetrics":17,"./symbols":23,"./utils":25}],11:[function(require,module,exports){
+/* eslint no-console:0 */
+/**
+ * This file does the main work of building a domTree structure from a parse
+ * tree. The entry point is the `buildHTML` function, which takes a parse tree.
+ * Then, the buildExpression, buildGroup, and various groupTypes functions are
+ * called, to produce a final HTML tree.
+ */
+
+var ParseError = require("./ParseError");
+var Style = require("./Style");
+
+var buildCommon = require("./buildCommon");
+var delimiter = require("./delimiter");
+var domTree = require("./domTree");
+var fontMetrics = require("./fontMetrics");
+var utils = require("./utils");
+
+var makeSpan = buildCommon.makeSpan;
+
+var isSpace = function(node) {
+ return node instanceof domTree.span && node.classes[0] === "mspace";
+};
+
+// Binary atoms (first class `mbin`) change into ordinary atoms (`mord`)
+// depending on their surroundings. See TeXbook pg. 442-446, Rules 5 and 6,
+// and the text before Rule 19.
+
+var isBin = function(node) {
+ return node && node.classes[0] === "mbin";
+};
+
+var isBinLeftCanceller = function(node, isRealGroup) {
+ // TODO: This code assumes that a node's math class is the first element
+ // of its `classes` array. A later cleanup should ensure this, for
+ // instance by changing the signature of `makeSpan`.
+ if (node) {
+ return utils.contains(["mbin", "mopen", "mrel", "mop", "mpunct"],
+ node.classes[0]);
+ } else {
+ return isRealGroup;
+ }
+};
+
+var isBinRightCanceller = function(node, isRealGroup) {
+ if (node) {
+ return utils.contains(["mrel", "mclose", "mpunct"], node.classes[0]);
+ } else {
+ return isRealGroup;
+ }
+};
+
+/**
+ * Take a list of nodes, build them in order, and return a list of the built
+ * nodes. documentFragments are flattened into their contents, so the
+ * returned list contains no fragments. `isRealGroup` is true if `expression`
+ * is a real group (no atoms will be added on either side), as opposed to
+ * a partial group (e.g. one created by \color).
+ */
+var buildExpression = function(expression, options, isRealGroup) {
+ // Parse expressions into `groups`.
+ var groups = [];
+ for (var i = 0; i < expression.length; i++) {
+ var group = expression[i];
+ var output = buildGroup(group, options);
+ if (output instanceof domTree.documentFragment) {
+ Array.prototype.push.apply(groups, output.children);
+ } else {
+ groups.push(output);
+ }
+ }
+ // At this point `groups` consists entirely of `symbolNode`s and `span`s.
+
+ // Explicit spaces (e.g., \;, \,) should be ignored with respect to atom
+ // spacing (e.g., "add thick space between mord and mrel"). Since CSS
+ // adjacency rules implement atom spacing, spaces should be invisible to
+ // CSS. So we splice them out of `groups` and into the atoms themselves.
+ var spaces = null;
+ for (i = 0; i < groups.length; i++) {
+ if (isSpace(groups[i])) {
+ spaces = spaces || [];
+ spaces.push(groups[i]);
+ groups.splice(i, 1);
+ i--;
+ } else if (spaces) {
+ if (groups[i] instanceof domTree.symbolNode) {
+ groups[i] = makeSpan([].concat(groups[i].classes), [groups[i]]);
+ }
+ buildCommon.prependChildren(groups[i], spaces);
+ spaces = null;
+ }
+ }
+ if (spaces) {
+ Array.prototype.push.apply(groups, spaces);
+ }
+
+ // Binary operators change to ordinary symbols in some contexts.
+ for (i = 0; i < groups.length; i++) {
+ if (isBin(groups[i])
+ && (isBinLeftCanceller(groups[i - 1], isRealGroup)
+ || isBinRightCanceller(groups[i + 1], isRealGroup))) {
+ groups[i].classes[0] = "mord";
+ }
+ }
+
+ return groups;
+};
+
+// Return math atom class (mclass) of a domTree.
+var getTypeOfDomTree = function(node) {
+ if (node instanceof domTree.documentFragment) {
+ if (node.children.length) {
+ return getTypeOfDomTree(
+ node.children[node.children.length - 1]);
+ }
+ } else {
+ if (utils.contains(["mord", "mop", "mbin", "mrel", "mopen", "mclose",
+ "mpunct", "minner"], node.classes[0])) {
+ return node.classes[0];
+ }
+ }
+ return null;
+};
+
+/**
+ * Sometimes, groups perform special rules when they have superscripts or
+ * subscripts attached to them. This function lets the `supsub` group know that
+ * its inner element should handle the superscripts and subscripts instead of
+ * handling them itself.
+ */
+var shouldHandleSupSub = function(group, options) {
+ if (!group) {
+ return false;
+ } else if (group.type === "op") {
+ // Operators handle supsubs differently when they have limits
+ // (e.g. `\displaystyle\sum_2^3`)
+ return group.value.limits &&
+ (options.style.size === Style.DISPLAY.size ||
+ group.value.alwaysHandleSupSub);
+ } else if (group.type === "accent") {
+ return isCharacterBox(group.value.base);
+ } else {
+ return null;
+ }
+};
+
+/**
+ * Sometimes we want to pull out the innermost element of a group. In most
+ * cases, this will just be the group itself, but when ordgroups and colors have
+ * a single element, we want to pull that out.
+ */
+var getBaseElem = function(group) {
+ if (!group) {
+ return false;
+ } else if (group.type === "ordgroup") {
+ if (group.value.length === 1) {
+ return getBaseElem(group.value[0]);
+ } else {
+ return group;
+ }
+ } else if (group.type === "color") {
+ if (group.value.value.length === 1) {
+ return getBaseElem(group.value.value[0]);
+ } else {
+ return group;
+ }
+ } else if (group.type === "font") {
+ return getBaseElem(group.value.body);
+ } else {
+ return group;
+ }
+};
+
+/**
+ * TeXbook algorithms often reference "character boxes", which are simply groups
+ * with a single character in them. To decide if something is a character box,
+ * we find its innermost group, and see if it is a single character.
+ */
+var isCharacterBox = function(group) {
+ var baseElem = getBaseElem(group);
+
+ // These are all they types of groups which hold single characters
+ return baseElem.type === "mathord" ||
+ baseElem.type === "textord" ||
+ baseElem.type === "bin" ||
+ baseElem.type === "rel" ||
+ baseElem.type === "inner" ||
+ baseElem.type === "open" ||
+ baseElem.type === "close" ||
+ baseElem.type === "punct";
+};
+
+var makeNullDelimiter = function(options, classes) {
+ return makeSpan(classes.concat([
+ "sizing", "reset-" + options.size, "size5",
+ options.style.reset(), Style.TEXT.cls(),
+ "nulldelimiter"]));
+};
+
+/**
+ * This is a map of group types to the function used to handle that type.
+ * Simpler types come at the beginning, while complicated types come afterwards.
+ */
+var groupTypes = {};
+
+groupTypes.mathord = function(group, options) {
+ return buildCommon.makeOrd(group, options, "mathord");
+};
+
+groupTypes.textord = function(group, options) {
+ return buildCommon.makeOrd(group, options, "textord");
+};
+
+groupTypes.bin = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mbin"]);
+};
+
+groupTypes.rel = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mrel"]);
+};
+
+groupTypes.open = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mopen"]);
+};
+
+groupTypes.close = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mclose"]);
+};
+
+groupTypes.inner = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["minner"]);
+};
+
+groupTypes.punct = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mpunct"]);
+};
+
+groupTypes.ordgroup = function(group, options) {
+ return makeSpan(
+ ["mord", options.style.cls()],
+ buildExpression(group.value, options.reset(), true),
+ options
+ );
+};
+
+groupTypes.text = function(group, options) {
+ var newOptions = options.withFont(group.value.style);
+ var inner = buildExpression(group.value.body, newOptions, true);
+ for (var i = 0; i < inner.length - 1; i++) {
+ if (inner[i].tryCombine(inner[i + 1])) {
+ inner.splice(i + 1, 1);
+ i--;
+ }
+ }
+ return makeSpan(["mord", "text", newOptions.style.cls()],
+ inner, newOptions);
+};
+
+groupTypes.color = function(group, options) {
+ var elements = buildExpression(
+ group.value.value,
+ options.withColor(group.value.color),
+ false
+ );
+
+ // \color isn't supposed to affect the type of the elements it contains.
+ // To accomplish this, we wrap the results in a fragment, so the inner
+ // elements will be able to directly interact with their neighbors. For
+ // example, `\color{red}{2 +} 3` has the same spacing as `2 + 3`
+ return new buildCommon.makeFragment(elements);
+};
+
+groupTypes.supsub = function(group, options) {
+ // Superscript and subscripts are handled in the TeXbook on page
+ // 445-446, rules 18(a-f).
+
+ // Here is where we defer to the inner group if it should handle
+ // superscripts and subscripts itself.
+ if (shouldHandleSupSub(group.value.base, options)) {
+ return groupTypes[group.value.base.type](group, options);
+ }
+
+ var base = buildGroup(group.value.base, options.reset());
+ var supmid;
+ var submid;
+ var sup;
+ var sub;
+
+ var style = options.style;
+ var newOptions;
+
+ if (group.value.sup) {
+ newOptions = options.withStyle(style.sup());
+ sup = buildGroup(group.value.sup, newOptions);
+ supmid = makeSpan([style.reset(), style.sup().cls()],
+ [sup], newOptions);
+ }
+
+ if (group.value.sub) {
+ newOptions = options.withStyle(style.sub());
+ sub = buildGroup(group.value.sub, newOptions);
+ submid = makeSpan([style.reset(), style.sub().cls()],
+ [sub], newOptions);
+ }
+
+ // Rule 18a
+ var supShift;
+ var subShift;
+ if (isCharacterBox(group.value.base)) {
+ supShift = 0;
+ subShift = 0;
+ } else {
+ supShift = base.height - style.metrics.supDrop;
+ subShift = base.depth + style.metrics.subDrop;
+ }
+
+ // Rule 18c
+ var minSupShift;
+ if (style === Style.DISPLAY) {
+ minSupShift = style.metrics.sup1;
+ } else if (style.cramped) {
+ minSupShift = style.metrics.sup3;
+ } else {
+ minSupShift = style.metrics.sup2;
+ }
+
+ // scriptspace is a font-size-independent size, so scale it
+ // appropriately
+ var multiplier = Style.TEXT.sizeMultiplier *
+ style.sizeMultiplier;
+ var scriptspace =
+ (0.5 / fontMetrics.metrics.ptPerEm) / multiplier + "em";
+
+ var supsub;
+ if (!group.value.sup) {
+ // Rule 18b
+ subShift = Math.max(
+ subShift, style.metrics.sub1,
+ sub.height - 0.8 * style.metrics.xHeight);
+
+ supsub = buildCommon.makeVList([
+ {type: "elem", elem: submid}
+ ], "shift", subShift, options);
+
+ supsub.children[0].style.marginRight = scriptspace;
+
+ // Subscripts shouldn't be shifted by the base's italic correction.
+ // Account for that by shifting the subscript back the appropriate
+ // amount. Note we only do this when the base is a single symbol.
+ if (base instanceof domTree.symbolNode) {
+ supsub.children[0].style.marginLeft = -base.italic + "em";
+ }
+ } else if (!group.value.sub) {
+ // Rule 18c, d
+ supShift = Math.max(supShift, minSupShift,
+ sup.depth + 0.25 * style.metrics.xHeight);
+
+ supsub = buildCommon.makeVList([
+ {type: "elem", elem: supmid}
+ ], "shift", -supShift, options);
+
+ supsub.children[0].style.marginRight = scriptspace;
+ } else {
+ supShift = Math.max(
+ supShift, minSupShift, sup.depth + 0.25 * style.metrics.xHeight);
+ subShift = Math.max(subShift, style.metrics.sub2);
+
+ var ruleWidth = fontMetrics.metrics.defaultRuleThickness;
+
+ // Rule 18e
+ if ((supShift - sup.depth) - (sub.height - subShift) <
+ 4 * ruleWidth) {
+ subShift = 4 * ruleWidth - (supShift - sup.depth) + sub.height;
+ var psi = 0.8 * style.metrics.xHeight - (supShift - sup.depth);
+ if (psi > 0) {
+ supShift += psi;
+ subShift -= psi;
+ }
+ }
+
+ supsub = buildCommon.makeVList([
+ {type: "elem", elem: submid, shift: subShift},
+ {type: "elem", elem: supmid, shift: -supShift}
+ ], "individualShift", null, options);
+
+ // See comment above about subscripts not being shifted
+ if (base instanceof domTree.symbolNode) {
+ supsub.children[0].style.marginLeft = -base.italic + "em";
+ }
+
+ supsub.children[0].style.marginRight = scriptspace;
+ supsub.children[1].style.marginRight = scriptspace;
+ }
+
+ // We ensure to wrap the supsub vlist in a span.msupsub to reset text-align
+ var mclass = getTypeOfDomTree(base) || "mord";
+ return makeSpan([mclass],
+ [base, makeSpan(["msupsub"], [supsub])],
+ options);
+};
+
+groupTypes.genfrac = function(group, options) {
+ // Fractions are handled in the TeXbook on pages 444-445, rules 15(a-e).
+ // Figure out what style this fraction should be in based on the
+ // function used
+ var style = options.style;
+ if (group.value.size === "display") {
+ style = Style.DISPLAY;
+ } else if (group.value.size === "text") {
+ style = Style.TEXT;
+ }
+
+ var nstyle = style.fracNum();
+ var dstyle = style.fracDen();
+ var newOptions;
+
+ newOptions = options.withStyle(nstyle);
+ var numer = buildGroup(group.value.numer, newOptions);
+ var numerreset = makeSpan([style.reset(), nstyle.cls()],
+ [numer], newOptions);
+
+ newOptions = options.withStyle(dstyle);
+ var denom = buildGroup(group.value.denom, newOptions);
+ var denomreset = makeSpan([style.reset(), dstyle.cls()],
+ [denom], newOptions);
+
+ var ruleWidth;
+ if (group.value.hasBarLine) {
+ ruleWidth = fontMetrics.metrics.defaultRuleThickness /
+ options.style.sizeMultiplier;
+ } else {
+ ruleWidth = 0;
+ }
+
+ // Rule 15b
+ var numShift;
+ var clearance;
+ var denomShift;
+ if (style.size === Style.DISPLAY.size) {
+ numShift = style.metrics.num1;
+ if (ruleWidth > 0) {
+ clearance = 3 * ruleWidth;
+ } else {
+ clearance = 7 * fontMetrics.metrics.defaultRuleThickness;
+ }
+ denomShift = style.metrics.denom1;
+ } else {
+ if (ruleWidth > 0) {
+ numShift = style.metrics.num2;
+ clearance = ruleWidth;
+ } else {
+ numShift = style.metrics.num3;
+ clearance = 3 * fontMetrics.metrics.defaultRuleThickness;
+ }
+ denomShift = style.metrics.denom2;
+ }
+
+ var frac;
+ if (ruleWidth === 0) {
+ // Rule 15c
+ var candidateClearance =
+ (numShift - numer.depth) - (denom.height - denomShift);
+ if (candidateClearance < clearance) {
+ numShift += 0.5 * (clearance - candidateClearance);
+ denomShift += 0.5 * (clearance - candidateClearance);
+ }
+
+ frac = buildCommon.makeVList([
+ {type: "elem", elem: denomreset, shift: denomShift},
+ {type: "elem", elem: numerreset, shift: -numShift}
+ ], "individualShift", null, options);
+ } else {
+ // Rule 15d
+ var axisHeight = style.metrics.axisHeight;
+
+ if ((numShift - numer.depth) - (axisHeight + 0.5 * ruleWidth) <
+ clearance) {
+ numShift +=
+ clearance - ((numShift - numer.depth) -
+ (axisHeight + 0.5 * ruleWidth));
+ }
+
+ if ((axisHeight - 0.5 * ruleWidth) - (denom.height - denomShift) <
+ clearance) {
+ denomShift +=
+ clearance - ((axisHeight - 0.5 * ruleWidth) -
+ (denom.height - denomShift));
+ }
+
+ var mid = makeSpan(
+ [options.style.reset(), Style.TEXT.cls(), "frac-line"]);
+ // Manually set the height of the line because its height is
+ // created in CSS
+ mid.height = ruleWidth;
+
+ var midShift = -(axisHeight - 0.5 * ruleWidth);
+
+ frac = buildCommon.makeVList([
+ {type: "elem", elem: denomreset, shift: denomShift},
+ {type: "elem", elem: mid, shift: midShift},
+ {type: "elem", elem: numerreset, shift: -numShift}
+ ], "individualShift", null, options);
+ }
+
+ // Since we manually change the style sometimes (with \dfrac or \tfrac),
+ // account for the possible size change here.
+ frac.height *= style.sizeMultiplier / options.style.sizeMultiplier;
+ frac.depth *= style.sizeMultiplier / options.style.sizeMultiplier;
+
+ // Rule 15e
+ var delimSize;
+ if (style.size === Style.DISPLAY.size) {
+ delimSize = style.metrics.delim1;
+ } else {
+ delimSize = style.metrics.delim2;
+ }
+
+ var leftDelim;
+ var rightDelim;
+ if (group.value.leftDelim == null) {
+ leftDelim = makeNullDelimiter(options, ["mopen"]);
+ } else {
+ leftDelim = delimiter.customSizedDelim(
+ group.value.leftDelim, delimSize, true,
+ options.withStyle(style), group.mode, ["mopen"]);
+ }
+ if (group.value.rightDelim == null) {
+ rightDelim = makeNullDelimiter(options, ["mclose"]);
+ } else {
+ rightDelim = delimiter.customSizedDelim(
+ group.value.rightDelim, delimSize, true,
+ options.withStyle(style), group.mode, ["mclose"]);
+ }
+
+ return makeSpan(
+ ["mord", options.style.reset(), style.cls()],
+ [leftDelim, makeSpan(["mfrac"], [frac]), rightDelim],
+ options);
+};
+
+var calculateSize = function(sizeValue, style) {
+ var x = sizeValue.number;
+ if (sizeValue.unit === "ex") {
+ x *= style.metrics.emPerEx;
+ } else if (sizeValue.unit === "mu") {
+ x /= 18;
+ }
+ return x;
+};
+
+groupTypes.array = function(group, options) {
+ var r;
+ var c;
+ var nr = group.value.body.length;
+ var nc = 0;
+ var body = new Array(nr);
+
+ var style = options.style;
+
+ // Horizontal spacing
+ var pt = 1 / fontMetrics.metrics.ptPerEm;
+ var arraycolsep = 5 * pt; // \arraycolsep in article.cls
+
+ // Vertical spacing
+ var baselineskip = 12 * pt; // see size10.clo
+ // Default \arraystretch from lttab.dtx
+ // TODO(gagern): may get redefined once we have user-defined macros
+ var arraystretch = utils.deflt(group.value.arraystretch, 1);
+ var arrayskip = arraystretch * baselineskip;
+ var arstrutHeight = 0.7 * arrayskip; // \strutbox in ltfsstrc.dtx and
+ var arstrutDepth = 0.3 * arrayskip; // \@arstrutbox in lttab.dtx
+
+ var totalHeight = 0;
+ for (r = 0; r < group.value.body.length; ++r) {
+ var inrow = group.value.body[r];
+ var height = arstrutHeight; // \@array adds an \@arstrut
+ var depth = arstrutDepth; // to each tow (via the template)
+
+ if (nc < inrow.length) {
+ nc = inrow.length;
+ }
+
+ var outrow = new Array(inrow.length);
+ for (c = 0; c < inrow.length; ++c) {
+ var elt = buildGroup(inrow[c], options);
+ if (depth < elt.depth) {
+ depth = elt.depth;
+ }
+ if (height < elt.height) {
+ height = elt.height;
+ }
+ outrow[c] = elt;
+ }
+
+ var gap = 0;
+ if (group.value.rowGaps[r]) {
+ gap = calculateSize(group.value.rowGaps[r].value, style);
+ if (gap > 0) { // \@argarraycr
+ gap += arstrutDepth;
+ if (depth < gap) {
+ depth = gap; // \@xargarraycr
+ }
+ gap = 0;
+ }
+ }
+
+ outrow.height = height;
+ outrow.depth = depth;
+ totalHeight += height;
+ outrow.pos = totalHeight;
+ totalHeight += depth + gap; // \@yargarraycr
+ body[r] = outrow;
+ }
+
+ var offset = totalHeight / 2 + style.metrics.axisHeight;
+ var colDescriptions = group.value.cols || [];
+ var cols = [];
+ var colSep;
+ var colDescrNum;
+ for (c = 0, colDescrNum = 0;
+ // Continue while either there are more columns or more column
+ // descriptions, so trailing separators don't get lost.
+ c < nc || colDescrNum < colDescriptions.length;
+ ++c, ++colDescrNum) {
+
+ var colDescr = colDescriptions[colDescrNum] || {};
+
+ var firstSeparator = true;
+ while (colDescr.type === "separator") {
+ // If there is more than one separator in a row, add a space
+ // between them.
+ if (!firstSeparator) {
+ colSep = makeSpan(["arraycolsep"], []);
+ colSep.style.width =
+ fontMetrics.metrics.doubleRuleSep + "em";
+ cols.push(colSep);
+ }
+
+ if (colDescr.separator === "|") {
+ var separator = makeSpan(
+ ["vertical-separator"],
+ []);
+ separator.style.height = totalHeight + "em";
+ separator.style.verticalAlign =
+ -(totalHeight - offset) + "em";
+
+ cols.push(separator);
+ } else {
+ throw new ParseError(
+ "Invalid separator type: " + colDescr.separator);
+ }
+
+ colDescrNum++;
+ colDescr = colDescriptions[colDescrNum] || {};
+ firstSeparator = false;
+ }
+
+ if (c >= nc) {
+ continue;
+ }
+
+ var sepwidth;
+ if (c > 0 || group.value.hskipBeforeAndAfter) {
+ sepwidth = utils.deflt(colDescr.pregap, arraycolsep);
+ if (sepwidth !== 0) {
+ colSep = makeSpan(["arraycolsep"], []);
+ colSep.style.width = sepwidth + "em";
+ cols.push(colSep);
+ }
+ }
+
+ var col = [];
+ for (r = 0; r < nr; ++r) {
+ var row = body[r];
+ var elem = row[c];
+ if (!elem) {
+ continue;
+ }
+ var shift = row.pos - offset;
+ elem.depth = row.depth;
+ elem.height = row.height;
+ col.push({type: "elem", elem: elem, shift: shift});
+ }
+
+ col = buildCommon.makeVList(col, "individualShift", null, options);
+ col = makeSpan(
+ ["col-align-" + (colDescr.align || "c")],
+ [col]);
+ cols.push(col);
+
+ if (c < nc - 1 || group.value.hskipBeforeAndAfter) {
+ sepwidth = utils.deflt(colDescr.postgap, arraycolsep);
+ if (sepwidth !== 0) {
+ colSep = makeSpan(["arraycolsep"], []);
+ colSep.style.width = sepwidth + "em";
+ cols.push(colSep);
+ }
+ }
+ }
+ body = makeSpan(["mtable"], cols);
+ return makeSpan(["mord"], [body], options);
+};
+
+groupTypes.spacing = function(group, options) {
+ if (group.value === "\\ " || group.value === "\\space" ||
+ group.value === " " || group.value === "~") {
+ // Spaces are generated by adding an actual space. Each of these
+ // things has an entry in the symbols table, so these will be turned
+ // into appropriate outputs.
+ if (group.mode === "text") {
+ return buildCommon.makeOrd(group, options, "textord");
+ } else {
+ return makeSpan(["mspace"],
+ [buildCommon.mathsym(group.value, group.mode, options)],
+ options);
+ }
+ } else {
+ // Other kinds of spaces are of arbitrary width. We use CSS to
+ // generate these.
+ return makeSpan(
+ ["mspace",
+ buildCommon.spacingFunctions[group.value].className],
+ [], options);
+ }
+};
+
+groupTypes.llap = function(group, options) {
+ var inner = makeSpan(
+ ["inner"], [buildGroup(group.value.body, options.reset())]);
+ var fix = makeSpan(["fix"], []);
+ return makeSpan(
+ ["mord", "llap", options.style.cls()], [inner, fix], options);
+};
+
+groupTypes.rlap = function(group, options) {
+ var inner = makeSpan(
+ ["inner"], [buildGroup(group.value.body, options.reset())]);
+ var fix = makeSpan(["fix"], []);
+ return makeSpan(
+ ["mord", "rlap", options.style.cls()], [inner, fix], options);
+};
+
+groupTypes.op = function(group, options) {
+ // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+ var supGroup;
+ var subGroup;
+ var hasLimits = false;
+ if (group.type === "supsub") {
+ // If we have limits, supsub will pass us its group to handle. Pull
+ // out the superscript and subscript and set the group to the op in
+ // its base.
+ supGroup = group.value.sup;
+ subGroup = group.value.sub;
+ group = group.value.base;
+ hasLimits = true;
+ }
+
+ var style = options.style;
+
+ // Most operators have a large successor symbol, but these don't.
+ var noSuccessor = [
+ "\\smallint"
+ ];
+
+ var large = false;
+ if (style.size === Style.DISPLAY.size &&
+ group.value.symbol &&
+ !utils.contains(noSuccessor, group.value.body)) {
+
+ // Most symbol operators get larger in displaystyle (rule 13)
+ large = true;
+ }
+
+ var base;
+ var baseShift = 0;
+ var slant = 0;
+ if (group.value.symbol) {
+ // If this is a symbol, create the symbol.
+ var fontName = large ? "Size2-Regular" : "Size1-Regular";
+ base = buildCommon.makeSymbol(
+ group.value.body, fontName, "math", options,
+ ["mop", "op-symbol", large ? "large-op" : "small-op"]);
+
+ // Shift the symbol so its center lies on the axis (rule 13). It
+ // appears that our fonts have the centers of the symbols already
+ // almost on the axis, so these numbers are very small. Note we
+ // don't actually apply this here, but instead it is used either in
+ // the vlist creation or separately when there are no limits.
+ baseShift = (base.height - base.depth) / 2 -
+ style.metrics.axisHeight * style.sizeMultiplier;
+
+ // The slant of the symbol is just its italic correction.
+ slant = base.italic;
+ } else if (group.value.value) {
+ // If this is a list, compose that list.
+ var inner = buildExpression(group.value.value, options, true);
+
+ base = makeSpan(["mop"], inner, options);
+ } else {
+ // Otherwise, this is a text operator. Build the text from the
+ // operator's name.
+ // TODO(emily): Add a space in the middle of some of these
+ // operators, like \limsup
+ var output = [];
+ for (var i = 1; i < group.value.body.length; i++) {
+ output.push(buildCommon.mathsym(group.value.body[i], group.mode));
+ }
+ base = makeSpan(["mop"], output, options);
+ }
+
+ if (hasLimits) {
+ // IE 8 clips \int if it is in a display: inline-block. We wrap it
+ // in a new span so it is an inline, and works.
+ base = makeSpan([], [base]);
+
+ var supmid;
+ var supKern;
+ var submid;
+ var subKern;
+ var newOptions;
+ // We manually have to handle the superscripts and subscripts. This,
+ // aside from the kern calculations, is copied from supsub.
+ if (supGroup) {
+ newOptions = options.withStyle(style.sup());
+ var sup = buildGroup(supGroup, newOptions);
+ supmid = makeSpan([style.reset(), style.sup().cls()],
+ [sup], newOptions);
+
+ supKern = Math.max(
+ fontMetrics.metrics.bigOpSpacing1,
+ fontMetrics.metrics.bigOpSpacing3 - sup.depth);
+ }
+
+ if (subGroup) {
+ newOptions = options.withStyle(style.sub());
+ var sub = buildGroup(subGroup, newOptions);
+ submid = makeSpan([style.reset(), style.sub().cls()],
+ [sub], newOptions);
+
+ subKern = Math.max(
+ fontMetrics.metrics.bigOpSpacing2,
+ fontMetrics.metrics.bigOpSpacing4 - sub.height);
+ }
+
+ // Build the final group as a vlist of the possible subscript, base,
+ // and possible superscript.
+ var finalGroup;
+ var top;
+ var bottom;
+ if (!supGroup) {
+ top = base.height - baseShift;
+
+ finalGroup = buildCommon.makeVList([
+ {type: "kern", size: fontMetrics.metrics.bigOpSpacing5},
+ {type: "elem", elem: submid},
+ {type: "kern", size: subKern},
+ {type: "elem", elem: base}
+ ], "top", top, options);
+
+ // Here, we shift the limits by the slant of the symbol. Note
+ // that we are supposed to shift the limits by 1/2 of the slant,
+ // but since we are centering the limits adding a full slant of
+ // margin will shift by 1/2 that.
+ finalGroup.children[0].style.marginLeft = -slant + "em";
+ } else if (!subGroup) {
+ bottom = base.depth + baseShift;
+
+ finalGroup = buildCommon.makeVList([
+ {type: "elem", elem: base},
+ {type: "kern", size: supKern},
+ {type: "elem", elem: supmid},
+ {type: "kern", size: fontMetrics.metrics.bigOpSpacing5}
+ ], "bottom", bottom, options);
+
+ // See comment above about slants
+ finalGroup.children[1].style.marginLeft = slant + "em";
+ } else if (!supGroup && !subGroup) {
+ // This case probably shouldn't occur (this would mean the
+ // supsub was sending us a group with no superscript or
+ // subscript) but be safe.
+ return base;
+ } else {
+ bottom = fontMetrics.metrics.bigOpSpacing5 +
+ submid.height + submid.depth +
+ subKern +
+ base.depth + baseShift;
+
+ finalGroup = buildCommon.makeVList([
+ {type: "kern", size: fontMetrics.metrics.bigOpSpacing5},
+ {type: "elem", elem: submid},
+ {type: "kern", size: subKern},
+ {type: "elem", elem: base},
+ {type: "kern", size: supKern},
+ {type: "elem", elem: supmid},
+ {type: "kern", size: fontMetrics.metrics.bigOpSpacing5}
+ ], "bottom", bottom, options);
+
+ // See comment above about slants
+ finalGroup.children[0].style.marginLeft = -slant + "em";
+ finalGroup.children[2].style.marginLeft = slant + "em";
+ }
+
+ return makeSpan(["mop", "op-limits"], [finalGroup], options);
+ } else {
+ if (group.value.symbol) {
+ base.style.top = baseShift + "em";
+ }
+
+ return base;
+ }
+};
+
+groupTypes.mod = function(group, options) {
+ var inner = [];
+
+ if (group.value.modType === "bmod") {
+ // “\nonscript\mskip-\medmuskip\mkern5mu”
+ if (!options.style.isTight()) {
+ inner.push(makeSpan(
+ ["mspace", "negativemediumspace"], [], options));
+ }
+ inner.push(makeSpan(["mspace", "thickspace"], [], options));
+ } else if (options.style.size === Style.DISPLAY.size) {
+ inner.push(makeSpan(["mspace", "quad"], [], options));
+ } else if (group.value.modType === "mod") {
+ inner.push(makeSpan(["mspace", "twelvemuspace"], [], options));
+ } else {
+ inner.push(makeSpan(["mspace", "eightmuspace"], [], options));
+ }
+
+ if (group.value.modType === "pod" || group.value.modType === "pmod") {
+ inner.push(buildCommon.mathsym("(", group.mode));
+ }
+
+ if (group.value.modType !== "pod") {
+ var modInner = [
+ buildCommon.mathsym("m", group.mode),
+ buildCommon.mathsym("o", group.mode),
+ buildCommon.mathsym("d", group.mode)];
+ if (group.value.modType === "bmod") {
+ inner.push(makeSpan(["mbin"], modInner, options));
+ // “\mkern5mu\nonscript\mskip-\medmuskip”
+ inner.push(makeSpan(["mspace", "thickspace"], [], options));
+ if (!options.style.isTight()) {
+ inner.push(makeSpan(
+ ["mspace", "negativemediumspace"], [], options));
+ }
+ } else {
+ Array.prototype.push.apply(inner, modInner);
+ inner.push(makeSpan(["mspace", "sixmuspace"], [], options));
+ }
+ }
+
+ if (group.value.value) {
+ Array.prototype.push.apply(inner,
+ buildExpression(group.value.value, options, false));
+ }
+
+ if (group.value.modType === "pod" || group.value.modType === "pmod") {
+ inner.push(buildCommon.mathsym(")", group.mode));
+ }
+
+ return buildCommon.makeFragment(inner);
+};
+
+groupTypes.katex = function(group, options) {
+ // The KaTeX logo. The offsets for the K and a were chosen to look
+ // good, but the offsets for the T, E, and X were taken from the
+ // definition of \TeX in TeX (see TeXbook pg. 356)
+ var k = makeSpan(
+ ["k"], [buildCommon.mathsym("K", group.mode)], options);
+ var a = makeSpan(
+ ["a"], [buildCommon.mathsym("A", group.mode)], options);
+
+ a.height = (a.height + 0.2) * 0.75;
+ a.depth = (a.height - 0.2) * 0.75;
+
+ var t = makeSpan(
+ ["t"], [buildCommon.mathsym("T", group.mode)], options);
+ var e = makeSpan(
+ ["e"], [buildCommon.mathsym("E", group.mode)], options);
+
+ e.height = (e.height - 0.2155);
+ e.depth = (e.depth + 0.2155);
+
+ var x = makeSpan(
+ ["x"], [buildCommon.mathsym("X", group.mode)], options);
+
+ return makeSpan(
+ ["mord", "katex-logo"], [k, a, t, e, x], options);
+};
+
+groupTypes.overline = function(group, options) {
+ // Overlines are handled in the TeXbook pg 443, Rule 9.
+ var style = options.style;
+
+ // Build the inner group in the cramped style.
+ var innerGroup = buildGroup(group.value.body,
+ options.withStyle(style.cramp()));
+
+ var ruleWidth = fontMetrics.metrics.defaultRuleThickness /
+ style.sizeMultiplier;
+
+ // Create the line above the body
+ var line = makeSpan(
+ [style.reset(), Style.TEXT.cls(), "overline-line"]);
+ line.height = ruleWidth;
+ line.maxFontSize = 1.0;
+
+ // Generate the vlist, with the appropriate kerns
+ var vlist = buildCommon.makeVList([
+ {type: "elem", elem: innerGroup},
+ {type: "kern", size: 3 * ruleWidth},
+ {type: "elem", elem: line},
+ {type: "kern", size: ruleWidth}
+ ], "firstBaseline", null, options);
+
+ return makeSpan(["mord", "overline"], [vlist], options);
+};
+
+groupTypes.underline = function(group, options) {
+ // Underlines are handled in the TeXbook pg 443, Rule 10.
+ var style = options.style;
+
+ // Build the inner group.
+ var innerGroup = buildGroup(group.value.body, options);
+
+ var ruleWidth = fontMetrics.metrics.defaultRuleThickness /
+ style.sizeMultiplier;
+
+ // Create the line above the body
+ var line = makeSpan([style.reset(), Style.TEXT.cls(), "underline-line"]);
+ line.height = ruleWidth;
+ line.maxFontSize = 1.0;
+
+ // Generate the vlist, with the appropriate kerns
+ var vlist = buildCommon.makeVList([
+ {type: "kern", size: ruleWidth},
+ {type: "elem", elem: line},
+ {type: "kern", size: 3 * ruleWidth},
+ {type: "elem", elem: innerGroup}
+ ], "top", innerGroup.height, options);
+
+ return makeSpan(["mord", "underline"], [vlist], options);
+};
+
+groupTypes.sqrt = function(group, options) {
+ // Square roots are handled in the TeXbook pg. 443, Rule 11.
+ var style = options.style;
+
+ // First, we do the same steps as in overline to build the inner group
+ // and line
+ var inner = buildGroup(group.value.body, options.withStyle(style.cramp()));
+
+ var ruleWidth = fontMetrics.metrics.defaultRuleThickness /
+ style.sizeMultiplier;
+
+ var line = makeSpan(
+ [style.reset(), Style.TEXT.cls(), "sqrt-line"], [],
+ options);
+ line.height = ruleWidth;
+ line.maxFontSize = 1.0;
+
+ var phi = ruleWidth;
+ if (style.id < Style.TEXT.id) {
+ phi = style.metrics.xHeight;
+ }
+
+ // Calculate the clearance between the body and line
+ var lineClearance = ruleWidth + phi / 4;
+
+ var innerHeight = (inner.height + inner.depth) * style.sizeMultiplier;
+ var minDelimiterHeight = innerHeight + lineClearance + ruleWidth;
+
+ // Create a \surd delimiter of the required minimum size
+ var delim = makeSpan(["sqrt-sign"], [
+ delimiter.customSizedDelim("\\surd", minDelimiterHeight,
+ false, options, group.mode)],
+ options);
+
+ var delimDepth = (delim.height + delim.depth) - ruleWidth;
+
+ // Adjust the clearance based on the delimiter size
+ if (delimDepth > inner.height + inner.depth + lineClearance) {
+ lineClearance =
+ (lineClearance + delimDepth - inner.height - inner.depth) / 2;
+ }
+
+ // Shift the delimiter so that its top lines up with the top of the line
+ var delimShift = -(inner.height + lineClearance + ruleWidth) + delim.height;
+ delim.style.top = delimShift + "em";
+ delim.height -= delimShift;
+ delim.depth += delimShift;
+
+ // We add a special case here, because even when `inner` is empty, we
+ // still get a line. So, we use a simple heuristic to decide if we
+ // should omit the body entirely. (note this doesn't work for something
+ // like `\sqrt{\rlap{x}}`, but if someone is doing that they deserve for
+ // it not to work.
+ var body;
+ if (inner.height === 0 && inner.depth === 0) {
+ body = makeSpan();
+ } else {
+ body = buildCommon.makeVList([
+ {type: "elem", elem: inner},
+ {type: "kern", size: lineClearance},
+ {type: "elem", elem: line},
+ {type: "kern", size: ruleWidth}
+ ], "firstBaseline", null, options);
+ }
+
+ if (!group.value.index) {
+ return makeSpan(["mord", "sqrt"], [delim, body], options);
+ } else {
+ // Handle the optional root index
+
+ // The index is always in scriptscript style
+ var newOptions = options.withStyle(Style.SCRIPTSCRIPT);
+ var root = buildGroup(group.value.index, newOptions);
+ var rootWrap = makeSpan(
+ [style.reset(), Style.SCRIPTSCRIPT.cls()],
+ [root],
+ newOptions);
+
+ // Figure out the height and depth of the inner part
+ var innerRootHeight = Math.max(delim.height, body.height);
+ var innerRootDepth = Math.max(delim.depth, body.depth);
+
+ // The amount the index is shifted by. This is taken from the TeX
+ // source, in the definition of `\r@@t`.
+ var toShift = 0.6 * (innerRootHeight - innerRootDepth);
+
+ // Build a VList with the superscript shifted up correctly
+ var rootVList = buildCommon.makeVList(
+ [{type: "elem", elem: rootWrap}],
+ "shift", -toShift, options);
+ // Add a class surrounding it so we can add on the appropriate
+ // kerning
+ var rootVListWrap = makeSpan(["root"], [rootVList]);
+
+ return makeSpan(["mord", "sqrt"],
+ [rootVListWrap, delim, body], options);
+ }
+};
+
+groupTypes.sizing = function(group, options) {
+ // Handle sizing operators like \Huge. Real TeX doesn't actually allow
+ // these functions inside of math expressions, so we do some special
+ // handling.
+ var inner = buildExpression(group.value.value,
+ options.withSize(group.value.size), false);
+
+ // Compute the correct maxFontSize.
+ var style = options.style;
+ var fontSize = buildCommon.sizingMultiplier[group.value.size];
+ fontSize = fontSize * style.sizeMultiplier;
+
+ // Add size-resetting classes to the inner list and set maxFontSize
+ // manually. Handle nested size changes.
+ for (var i = 0; i < inner.length; i++) {
+ var pos = utils.indexOf(inner[i].classes, "sizing");
+ if (pos < 0) {
+ inner[i].classes.push("sizing", "reset-" + options.size,
+ group.value.size, style.cls());
+ inner[i].maxFontSize = fontSize;
+ } else if (inner[i].classes[pos + 1] === "reset-" + group.value.size) {
+ // This is a nested size change: e.g., inner[i] is the "b" in
+ // `\Huge a \small b`. Override the old size (the `reset-` class)
+ // but not the new size.
+ inner[i].classes[pos + 1] = "reset-" + options.size;
+ }
+ }
+
+ return buildCommon.makeFragment(inner);
+};
+
+groupTypes.styling = function(group, options) {
+ // Style changes are handled in the TeXbook on pg. 442, Rule 3.
+
+ // Figure out what style we're changing to.
+ var styleMap = {
+ "display": Style.DISPLAY,
+ "text": Style.TEXT,
+ "script": Style.SCRIPT,
+ "scriptscript": Style.SCRIPTSCRIPT
+ };
+
+ var newStyle = styleMap[group.value.style];
+ var newOptions = options.withStyle(newStyle);
+
+ // Build the inner expression in the new style.
+ var inner = buildExpression(
+ group.value.value, newOptions, false);
+
+ // Add style-resetting classes to the inner list. Handle nested changes.
+ for (var i = 0; i < inner.length; i++) {
+ var pos = utils.indexOf(inner[i].classes, newStyle.reset());
+ if (pos < 0) {
+ inner[i].classes.push(options.style.reset(), newStyle.cls());
+ } else {
+ // This is a nested style change, as `\textstyle a\scriptstyle b`.
+ // Only override the old style (the reset class).
+ inner[i].classes[pos] = options.style.reset();
+ }
+ }
+
+ return new buildCommon.makeFragment(inner);
+};
+
+groupTypes.font = function(group, options) {
+ var font = group.value.font;
+ return buildGroup(group.value.body, options.withFont(font));
+};
+
+groupTypes.delimsizing = function(group, options) {
+ var delim = group.value.value;
+
+ if (delim === ".") {
+ // Empty delimiters still count as elements, even though they don't
+ // show anything.
+ return makeSpan([group.value.mclass]);
+ }
+
+ // Use delimiter.sizedDelim to generate the delimiter.
+ return delimiter.sizedDelim(
+ delim, group.value.size, options, group.mode,
+ [group.value.mclass]);
+};
+
+groupTypes.leftright = function(group, options) {
+ // Build the inner expression
+ var inner = buildExpression(group.value.body, options.reset(), true);
+
+ var innerHeight = 0;
+ var innerDepth = 0;
+ var hadMiddle = false;
+
+ // Calculate its height and depth
+ for (var i = 0; i < inner.length; i++) {
+ if (inner[i].isMiddle) {
+ hadMiddle = true;
+ } else {
+ innerHeight = Math.max(inner[i].height, innerHeight);
+ innerDepth = Math.max(inner[i].depth, innerDepth);
+ }
+ }
+
+ var style = options.style;
+
+ // The size of delimiters is the same, regardless of what style we are
+ // in. Thus, to correctly calculate the size of delimiter we need around
+ // a group, we scale down the inner size based on the size.
+ innerHeight *= style.sizeMultiplier;
+ innerDepth *= style.sizeMultiplier;
+
+ var leftDelim;
+ if (group.value.left === ".") {
+ // Empty delimiters in \left and \right make null delimiter spaces.
+ leftDelim = makeNullDelimiter(options, ["mopen"]);
+ } else {
+ // Otherwise, use leftRightDelim to generate the correct sized
+ // delimiter.
+ leftDelim = delimiter.leftRightDelim(
+ group.value.left, innerHeight, innerDepth, options,
+ group.mode, ["mopen"]);
+ }
+ // Add it to the beginning of the expression
+ inner.unshift(leftDelim);
+
+ // Handle middle delimiters
+ if (hadMiddle) {
+ for (i = 1; i < inner.length; i++) {
+ if (inner[i].isMiddle) {
+ // Apply the options that were active when \middle was called
+ inner[i] = delimiter.leftRightDelim(
+ inner[i].isMiddle.value, innerHeight, innerDepth,
+ inner[i].isMiddle.options, group.mode, []);
+ }
+ }
+ }
+
+ var rightDelim;
+ // Same for the right delimiter
+ if (group.value.right === ".") {
+ rightDelim = makeNullDelimiter(options, ["mclose"]);
+ } else {
+ rightDelim = delimiter.leftRightDelim(
+ group.value.right, innerHeight, innerDepth, options,
+ group.mode, ["mclose"]);
+ }
+ // Add it to the end of the expression.
+ inner.push(rightDelim);
+
+ return makeSpan(
+ ["minner", style.cls()], inner, options);
+};
+
+groupTypes.middle = function(group, options) {
+ var middleDelim;
+ if (group.value.value === ".") {
+ middleDelim = makeNullDelimiter(options, []);
+ } else {
+ middleDelim = delimiter.sizedDelim(
+ group.value.value, 1, options,
+ group.mode, []);
+ middleDelim.isMiddle = {value: group.value.value, options: options};
+ }
+ return middleDelim;
+};
+
+groupTypes.rule = function(group, options) {
+ // Make an empty span for the rule
+ var rule = makeSpan(["mord", "rule"], [], options);
+ var style = options.style;
+
+ // Calculate the shift, width, and height of the rule, and account for units
+ var shift = 0;
+ if (group.value.shift) {
+ shift = calculateSize(group.value.shift, style);
+ }
+
+ var width = calculateSize(group.value.width, style);
+ var height = calculateSize(group.value.height, style);
+
+ // The sizes of rules are absolute, so make it larger if we are in a
+ // smaller style.
+ shift /= style.sizeMultiplier;
+ width /= style.sizeMultiplier;
+ height /= style.sizeMultiplier;
+
+ // Style the rule to the right size
+ rule.style.borderRightWidth = width + "em";
+ rule.style.borderTopWidth = height + "em";
+ rule.style.bottom = shift + "em";
+
+ // Record the height and width
+ rule.width = width;
+ rule.height = height + shift;
+ rule.depth = -shift;
+
+ return rule;
+};
+
+groupTypes.kern = function(group, options) {
+ // Make an empty span for the rule
+ var rule = makeSpan(["mord", "rule"], [], options);
+ var style = options.style;
+
+ var dimension = 0;
+ if (group.value.dimension) {
+ dimension = calculateSize(group.value.dimension, style);
+ }
+
+ dimension /= style.sizeMultiplier;
+
+ rule.style.marginLeft = dimension + "em";
+
+ return rule;
+};
+
+groupTypes.accent = function(group, options) {
+ // Accents are handled in the TeXbook pg. 443, rule 12.
+ var base = group.value.base;
+ var style = options.style;
+
+ var supsubGroup;
+ if (group.type === "supsub") {
+ // If our base is a character box, and we have superscripts and
+ // subscripts, the supsub will defer to us. In particular, we want
+ // to attach the superscripts and subscripts to the inner body (so
+ // that the position of the superscripts and subscripts won't be
+ // affected by the height of the accent). We accomplish this by
+ // sticking the base of the accent into the base of the supsub, and
+ // rendering that, while keeping track of where the accent is.
+
+ // The supsub group is the group that was passed in
+ var supsub = group;
+ // The real accent group is the base of the supsub group
+ group = supsub.value.base;
+ // The character box is the base of the accent group
+ base = group.value.base;
+ // Stick the character box into the base of the supsub group
+ supsub.value.base = base;
+
+ // Rerender the supsub group with its new base, and store that
+ // result.
+ supsubGroup = buildGroup(
+ supsub, options.reset());
+ }
+
+ // Build the base group
+ var body = buildGroup(
+ base, options.withStyle(style.cramp()));
+
+ // Calculate the skew of the accent. This is based on the line "If the
+ // nucleus is not a single character, let s = 0; otherwise set s to the
+ // kern amount for the nucleus followed by the \skewchar of its font."
+ // Note that our skew metrics are just the kern between each character
+ // and the skewchar.
+ var skew;
+ if (isCharacterBox(base)) {
+ // If the base is a character box, then we want the skew of the
+ // innermost character. To do that, we find the innermost character:
+ var baseChar = getBaseElem(base);
+ // Then, we render its group to get the symbol inside it
+ var baseGroup = buildGroup(
+ baseChar, options.withStyle(style.cramp()));
+ // Finally, we pull the skew off of the symbol.
+ skew = baseGroup.skew;
+ // Note that we now throw away baseGroup, because the layers we
+ // removed with getBaseElem might contain things like \color which
+ // we can't get rid of.
+ // TODO(emily): Find a better way to get the skew
+ } else {
+ skew = 0;
+ }
+
+ // calculate the amount of space between the body and the accent
+ var clearance = Math.min(
+ body.height,
+ style.metrics.xHeight);
+
+ // Build the accent
+ var accent = buildCommon.makeSymbol(
+ group.value.accent, "Main-Regular", "math", options);
+ // Remove the italic correction of the accent, because it only serves to
+ // shift the accent over to a place we don't want.
+ accent.italic = 0;
+
+ // The \vec character that the fonts use is a combining character, and
+ // thus shows up much too far to the left. To account for this, we add a
+ // specific class which shifts the accent over to where we want it.
+ // TODO(emily): Fix this in a better way, like by changing the font
+ var vecClass = group.value.accent === "\\vec" ? "accent-vec" : null;
+
+ var accentBody = makeSpan(["accent-body", vecClass], [
+ makeSpan([], [accent])]);
+
+ accentBody = buildCommon.makeVList([
+ {type: "elem", elem: body},
+ {type: "kern", size: -clearance},
+ {type: "elem", elem: accentBody}
+ ], "firstBaseline", null, options);
+
+ // Shift the accent over by the skew. Note we shift by twice the skew
+ // because we are centering the accent, so by adding 2*skew to the left,
+ // we shift it to the right by 1*skew.
+ accentBody.children[1].style.marginLeft = 2 * skew + "em";
+
+ var accentWrap = makeSpan(["mord", "accent"], [accentBody], options);
+
+ if (supsubGroup) {
+ // Here, we replace the "base" child of the supsub with our newly
+ // generated accent.
+ supsubGroup.children[0] = accentWrap;
+
+ // Since we don't rerun the height calculation after replacing the
+ // accent, we manually recalculate height.
+ supsubGroup.height = Math.max(accentWrap.height, supsubGroup.height);
+
+ // Accents should always be ords, even when their innards are not.
+ supsubGroup.classes[0] = "mord";
+
+ return supsubGroup;
+ } else {
+ return accentWrap;
+ }
+};
+
+groupTypes.phantom = function(group, options) {
+ var elements = buildExpression(
+ group.value.value,
+ options.withPhantom(),
+ false
+ );
+
+ // \phantom isn't supposed to affect the elements it contains.
+ // See "color" for more details.
+ return new buildCommon.makeFragment(elements);
+};
+
+groupTypes.mclass = function(group, options) {
+ var elements = buildExpression(group.value.value, options, true);
+
+ return makeSpan([group.value.mclass], elements, options);
+};
+
+/**
+ * buildGroup is the function that takes a group and calls the correct groupType
+ * function for it. It also handles the interaction of size and style changes
+ * between parents and children.
+ */
+var buildGroup = function(group, options) {
+ if (!group) {
+ return makeSpan();
+ }
+
+ if (groupTypes[group.type]) {
+ // Call the groupTypes function
+ var groupNode = groupTypes[group.type](group, options);
+ var multiplier;
+
+ // If the style changed between the parent and the current group,
+ // account for the size difference
+ if (options.style !== options.parentStyle) {
+ multiplier = options.style.sizeMultiplier /
+ options.parentStyle.sizeMultiplier;
+
+ groupNode.height *= multiplier;
+ groupNode.depth *= multiplier;
+ }
+
+ // If the size changed between the parent and the current group, account
+ // for that size difference.
+ if (options.size !== options.parentSize) {
+ multiplier = buildCommon.sizingMultiplier[options.size] /
+ buildCommon.sizingMultiplier[options.parentSize];
+
+ groupNode.height *= multiplier;
+ groupNode.depth *= multiplier;
+ }
+
+ return groupNode;
+ } else {
+ throw new ParseError(
+ "Got group of unknown type: '" + group.type + "'");
+ }
+};
+
+/**
+ * Take an entire parse tree, and build it into an appropriate set of HTML
+ * nodes.
+ */
+var buildHTML = function(tree, options) {
+ // buildExpression is destructive, so we need to make a clone
+ // of the incoming tree so that it isn't accidentally changed
+ tree = JSON.parse(JSON.stringify(tree));
+
+ // Build the expression contained in the tree
+ var expression = buildExpression(tree, options, true);
+ var body = makeSpan(["base", options.style.cls()], expression, options);
+
+ // Add struts, which ensure that the top of the HTML element falls at the
+ // height of the expression, and the bottom of the HTML element falls at the
+ // depth of the expression.
+ var topStrut = makeSpan(["strut"]);
+ var bottomStrut = makeSpan(["strut", "bottom"]);
+
+ topStrut.style.height = body.height + "em";
+ bottomStrut.style.height = (body.height + body.depth) + "em";
+ // We'd like to use `vertical-align: top` but in IE 9 this lowers the
+ // baseline of the box to the bottom of this strut (instead staying in the
+ // normal place) so we use an absolute value for vertical-align instead
+ bottomStrut.style.verticalAlign = -body.depth + "em";
+
+ // Wrap the struts and body together
+ var htmlNode = makeSpan(["katex-html"], [topStrut, bottomStrut, body]);
+
+ htmlNode.setAttribute("aria-hidden", "true");
+
+ return htmlNode;
+};
+
+module.exports = buildHTML;
+
+},{"./ParseError":6,"./Style":9,"./buildCommon":10,"./delimiter":14,"./domTree":15,"./fontMetrics":17,"./utils":25}],12:[function(require,module,exports){
+/**
+ * This file converts a parse tree into a cooresponding MathML tree. The main
+ * entry point is the `buildMathML` function, which takes a parse tree from the
+ * parser.
+ */
+
+var buildCommon = require("./buildCommon");
+var fontMetrics = require("./fontMetrics");
+var mathMLTree = require("./mathMLTree");
+var ParseError = require("./ParseError");
+var symbols = require("./symbols");
+var utils = require("./utils");
+
+var makeSpan = buildCommon.makeSpan;
+var fontMap = buildCommon.fontMap;
+
+/**
+ * Takes a symbol and converts it into a MathML text node after performing
+ * optional replacement from symbols.js.
+ */
+var makeText = function(text, mode) {
+ if (symbols[mode][text] && symbols[mode][text].replace) {
+ text = symbols[mode][text].replace;
+ }
+
+ return new mathMLTree.TextNode(text);
+};
+
+/**
+ * Returns the math variant as a string or null if none is required.
+ */
+var getVariant = function(group, options) {
+ var font = options.font;
+ if (!font) {
+ return null;
+ }
+
+ var mode = group.mode;
+ if (font === "mathit") {
+ return "italic";
+ }
+
+ var value = group.value;
+ if (utils.contains(["\\imath", "\\jmath"], value)) {
+ return null;
+ }
+
+ if (symbols[mode][value] && symbols[mode][value].replace) {
+ value = symbols[mode][value].replace;
+ }
+
+ var fontName = fontMap[font].fontName;
+ if (fontMetrics.getCharacterMetrics(value, fontName)) {
+ return fontMap[options.font].variant;
+ }
+
+ return null;
+};
+
+/**
+ * Functions for handling the different types of groups found in the parse
+ * tree. Each function should take a parse group and return a MathML node.
+ */
+var groupTypes = {};
+
+groupTypes.mathord = function(group, options) {
+ var node = new mathMLTree.MathNode(
+ "mi",
+ [makeText(group.value, group.mode)]);
+
+ var variant = getVariant(group, options);
+ if (variant) {
+ node.setAttribute("mathvariant", variant);
+ }
+ return node;
+};
+
+groupTypes.textord = function(group, options) {
+ var text = makeText(group.value, group.mode);
+
+ var variant = getVariant(group, options) || "normal";
+
+ var node;
+ if (/[0-9]/.test(group.value)) {
+ // TODO(kevinb) merge adjacent <mn> nodes
+ // do it as a post processing step
+ node = new mathMLTree.MathNode("mn", [text]);
+ if (options.font) {
+ node.setAttribute("mathvariant", variant);
+ }
+ } else {
+ node = new mathMLTree.MathNode("mi", [text]);
+ node.setAttribute("mathvariant", variant);
+ }
+
+ return node;
+};
+
+groupTypes.bin = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.rel = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.open = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.close = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.inner = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.punct = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ node.setAttribute("separator", "true");
+
+ return node;
+};
+
+groupTypes.ordgroup = function(group, options) {
+ var inner = buildExpression(group.value, options);
+
+ var node = new mathMLTree.MathNode("mrow", inner);
+
+ return node;
+};
+
+groupTypes.text = function(group, options) {
+ var inner = buildExpression(group.value.body, options);
+
+ var node = new mathMLTree.MathNode("mtext", inner);
+
+ return node;
+};
+
+groupTypes.color = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+
+ var node = new mathMLTree.MathNode("mstyle", inner);
+
+ node.setAttribute("mathcolor", group.value.color);
+
+ return node;
+};
+
+groupTypes.supsub = function(group, options) {
+ var children = [buildGroup(group.value.base, options)];
+
+ if (group.value.sub) {
+ children.push(buildGroup(group.value.sub, options));
+ }
+
+ if (group.value.sup) {
+ children.push(buildGroup(group.value.sup, options));
+ }
+
+ var nodeType;
+ if (!group.value.sub) {
+ nodeType = "msup";
+ } else if (!group.value.sup) {
+ nodeType = "msub";
+ } else {
+ nodeType = "msubsup";
+ }
+
+ var node = new mathMLTree.MathNode(nodeType, children);
+
+ return node;
+};
+
+groupTypes.genfrac = function(group, options) {
+ var node = new mathMLTree.MathNode(
+ "mfrac",
+ [buildGroup(group.value.numer, options),
+ buildGroup(group.value.denom, options)]);
+
+ if (!group.value.hasBarLine) {
+ node.setAttribute("linethickness", "0px");
+ }
+
+ if (group.value.leftDelim != null || group.value.rightDelim != null) {
+ var withDelims = [];
+
+ if (group.value.leftDelim != null) {
+ var leftOp = new mathMLTree.MathNode(
+ "mo", [new mathMLTree.TextNode(group.value.leftDelim)]);
+
+ leftOp.setAttribute("fence", "true");
+
+ withDelims.push(leftOp);
+ }
+
+ withDelims.push(node);
+
+ if (group.value.rightDelim != null) {
+ var rightOp = new mathMLTree.MathNode(
+ "mo", [new mathMLTree.TextNode(group.value.rightDelim)]);
+
+ rightOp.setAttribute("fence", "true");
+
+ withDelims.push(rightOp);
+ }
+
+ var outerNode = new mathMLTree.MathNode("mrow", withDelims);
+
+ return outerNode;
+ }
+
+ return node;
+};
+
+groupTypes.array = function(group, options) {
+ return new mathMLTree.MathNode(
+ "mtable", group.value.body.map(function(row) {
+ return new mathMLTree.MathNode(
+ "mtr", row.map(function(cell) {
+ return new mathMLTree.MathNode(
+ "mtd", [buildGroup(cell, options)]);
+ }));
+ }));
+};
+
+groupTypes.sqrt = function(group, options) {
+ var node;
+ if (group.value.index) {
+ node = new mathMLTree.MathNode(
+ "mroot", [
+ buildGroup(group.value.body, options),
+ buildGroup(group.value.index, options)
+ ]);
+ } else {
+ node = new mathMLTree.MathNode(
+ "msqrt", [buildGroup(group.value.body, options)]);
+ }
+
+ return node;
+};
+
+groupTypes.leftright = function(group, options) {
+ var inner = buildExpression(group.value.body, options);
+
+ if (group.value.left !== ".") {
+ var leftNode = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.left, group.mode)]);
+
+ leftNode.setAttribute("fence", "true");
+
+ inner.unshift(leftNode);
+ }
+
+ if (group.value.right !== ".") {
+ var rightNode = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.right, group.mode)]);
+
+ rightNode.setAttribute("fence", "true");
+
+ inner.push(rightNode);
+ }
+
+ var outerNode = new mathMLTree.MathNode("mrow", inner);
+
+ return outerNode;
+};
+
+groupTypes.middle = function(group, options) {
+ var middleNode = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.middle, group.mode)]);
+ middleNode.setAttribute("fence", "true");
+ return middleNode;
+};
+
+groupTypes.accent = function(group, options) {
+ var accentNode = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.accent, group.mode)]);
+
+ var node = new mathMLTree.MathNode(
+ "mover",
+ [buildGroup(group.value.base, options),
+ accentNode]);
+
+ node.setAttribute("accent", "true");
+
+ return node;
+};
+
+groupTypes.spacing = function(group) {
+ var node;
+
+ if (group.value === "\\ " || group.value === "\\space" ||
+ group.value === " " || group.value === "~") {
+ node = new mathMLTree.MathNode(
+ "mtext", [new mathMLTree.TextNode("\u00a0")]);
+ } else {
+ node = new mathMLTree.MathNode("mspace");
+
+ node.setAttribute(
+ "width", buildCommon.spacingFunctions[group.value].size);
+ }
+
+ return node;
+};
+
+groupTypes.op = function(group, options) {
+ var node;
+
+ // TODO(emily): handle big operators using the `largeop` attribute
+
+ if (group.value.symbol) {
+ // This is a symbol. Just add the symbol.
+ node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.body, group.mode)]);
+ } else if (group.value.value) {
+ // This is an operator with children. Add them.
+ node = new mathMLTree.MathNode(
+ "mo", buildExpression(group.value.value, options));
+ } else {
+ // This is a text operator. Add all of the characters from the
+ // operator's name.
+ // TODO(emily): Add a space in the middle of some of these
+ // operators, like \limsup.
+ node = new mathMLTree.MathNode(
+ "mi", [new mathMLTree.TextNode(group.value.body.slice(1))]);
+ }
+
+ return node;
+};
+
+groupTypes.mod = function(group, options) {
+ var inner = [];
+
+ if (group.value.modType === "pod" || group.value.modType === "pmod") {
+ inner.push(new mathMLTree.MathNode(
+ "mo", [makeText("(", group.mode)]));
+ }
+ if (group.value.modType !== "pod") {
+ inner.push(new mathMLTree.MathNode(
+ "mo", [makeText("mod", group.mode)]));
+ }
+ if (group.value.value) {
+ var space = new mathMLTree.MathNode("mspace");
+ space.setAttribute("width", "0.333333em");
+ inner.push(space);
+ inner = inner.concat(buildExpression(group.value.value, options));
+ }
+ if (group.value.modType === "pod" || group.value.modType === "pmod") {
+ inner.push(new mathMLTree.MathNode(
+ "mo", [makeText(")", group.mode)]));
+ }
+
+ return new mathMLTree.MathNode("mo", inner);
+};
+
+groupTypes.katex = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mtext", [new mathMLTree.TextNode("KaTeX")]);
+
+ return node;
+};
+
+groupTypes.font = function(group, options) {
+ var font = group.value.font;
+ return buildGroup(group.value.body, options.withFont(font));
+};
+
+groupTypes.delimsizing = function(group) {
+ var children = [];
+
+ if (group.value.value !== ".") {
+ children.push(makeText(group.value.value, group.mode));
+ }
+
+ var node = new mathMLTree.MathNode("mo", children);
+
+ if (group.value.mclass === "mopen" ||
+ group.value.mclass === "mclose") {
+ // Only some of the delimsizing functions act as fences, and they
+ // return "mopen" or "mclose" mclass.
+ node.setAttribute("fence", "true");
+ } else {
+ // Explicitly disable fencing if it's not a fence, to override the
+ // defaults.
+ node.setAttribute("fence", "false");
+ }
+
+ return node;
+};
+
+groupTypes.styling = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+
+ var node = new mathMLTree.MathNode("mstyle", inner);
+
+ var styleAttributes = {
+ "display": ["0", "true"],
+ "text": ["0", "false"],
+ "script": ["1", "false"],
+ "scriptscript": ["2", "false"]
+ };
+
+ var attr = styleAttributes[group.value.style];
+
+ node.setAttribute("scriptlevel", attr[0]);
+ node.setAttribute("displaystyle", attr[1]);
+
+ return node;
+};
+
+groupTypes.sizing = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+
+ var node = new mathMLTree.MathNode("mstyle", inner);
+
+ // TODO(emily): This doesn't produce the correct size for nested size
+ // changes, because we don't keep state of what style we're currently
+ // in, so we can't reset the size to normal before changing it. Now
+ // that we're passing an options parameter we should be able to fix
+ // this.
+ node.setAttribute(
+ "mathsize", buildCommon.sizingMultiplier[group.value.size] + "em");
+
+ return node;
+};
+
+groupTypes.overline = function(group, options) {
+ var operator = new mathMLTree.MathNode(
+ "mo", [new mathMLTree.TextNode("\u203e")]);
+ operator.setAttribute("stretchy", "true");
+
+ var node = new mathMLTree.MathNode(
+ "mover",
+ [buildGroup(group.value.body, options),
+ operator]);
+ node.setAttribute("accent", "true");
+
+ return node;
+};
+
+groupTypes.underline = function(group, options) {
+ var operator = new mathMLTree.MathNode(
+ "mo", [new mathMLTree.TextNode("\u203e")]);
+ operator.setAttribute("stretchy", "true");
+
+ var node = new mathMLTree.MathNode(
+ "munder",
+ [buildGroup(group.value.body, options),
+ operator]);
+ node.setAttribute("accentunder", "true");
+
+ return node;
+};
+
+groupTypes.rule = function(group) {
+ // TODO(emily): Figure out if there's an actual way to draw black boxes
+ // in MathML.
+ var node = new mathMLTree.MathNode("mrow");
+
+ return node;
+};
+
+groupTypes.kern = function(group) {
+ // TODO(kevin): Figure out if there's a way to add space in MathML
+ var node = new mathMLTree.MathNode("mrow");
+
+ return node;
+};
+
+groupTypes.llap = function(group, options) {
+ var node = new mathMLTree.MathNode(
+ "mpadded", [buildGroup(group.value.body, options)]);
+
+ node.setAttribute("lspace", "-1width");
+ node.setAttribute("width", "0px");
+
+ return node;
+};
+
+groupTypes.rlap = function(group, options) {
+ var node = new mathMLTree.MathNode(
+ "mpadded", [buildGroup(group.value.body, options)]);
+
+ node.setAttribute("width", "0px");
+
+ return node;
+};
+
+groupTypes.phantom = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+ return new mathMLTree.MathNode("mphantom", inner);
+};
+
+groupTypes.mclass = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+ return new mathMLTree.MathNode("mstyle", inner);
+};
+
+/**
+ * Takes a list of nodes, builds them, and returns a list of the generated
+ * MathML nodes. A little simpler than the HTML version because we don't do any
+ * previous-node handling.
+ */
+var buildExpression = function(expression, options) {
+ var groups = [];
+ for (var i = 0; i < expression.length; i++) {
+ var group = expression[i];
+ groups.push(buildGroup(group, options));
+ }
+ return groups;
+};
+
+/**
+ * Takes a group from the parser and calls the appropriate groupTypes function
+ * on it to produce a MathML node.
+ */
+var buildGroup = function(group, options) {
+ if (!group) {
+ return new mathMLTree.MathNode("mrow");
+ }
+
+ if (groupTypes[group.type]) {
+ // Call the groupTypes function
+ return groupTypes[group.type](group, options);
+ } else {
+ throw new ParseError(
+ "Got group of unknown type: '" + group.type + "'");
+ }
+};
+
+/**
+ * Takes a full parse tree and settings and builds a MathML representation of
+ * it. In particular, we put the elements from building the parse tree into a
+ * <semantics> tag so we can also include that TeX source as an annotation.
+ *
+ * Note that we actually return a domTree element with a `<math>` inside it so
+ * we can do appropriate styling.
+ */
+var buildMathML = function(tree, texExpression, options) {
+ var expression = buildExpression(tree, options);
+
+ // Wrap up the expression in an mrow so it is presented in the semantics
+ // tag correctly.
+ var wrapper = new mathMLTree.MathNode("mrow", expression);
+
+ // Build a TeX annotation of the source
+ var annotation = new mathMLTree.MathNode(
+ "annotation", [new mathMLTree.TextNode(texExpression)]);
+
+ annotation.setAttribute("encoding", "application/x-tex");
+
+ var semantics = new mathMLTree.MathNode(
+ "semantics", [wrapper, annotation]);
+
+ var math = new mathMLTree.MathNode("math", [semantics]);
+
+ // You can't style <math> nodes, so we wrap the node in a span.
+ return makeSpan(["katex-mathml"], [math]);
+};
+
+module.exports = buildMathML;
+
+},{"./ParseError":6,"./buildCommon":10,"./fontMetrics":17,"./mathMLTree":20,"./symbols":23,"./utils":25}],13:[function(require,module,exports){
+var buildHTML = require("./buildHTML");
+var buildMathML = require("./buildMathML");
+var buildCommon = require("./buildCommon");
+var Options = require("./Options");
+var Settings = require("./Settings");
+var Style = require("./Style");
+
+var makeSpan = buildCommon.makeSpan;
+
+var buildTree = function(tree, expression, settings) {
+ settings = settings || new Settings({});
+
+ var startStyle = Style.TEXT;
+ if (settings.displayMode) {
+ startStyle = Style.DISPLAY;
+ }
+
+ // Setup the default options
+ var options = new Options({
+ style: startStyle,
+ size: "size5"
+ });
+
+ // `buildHTML` sometimes messes with the parse tree (like turning bins ->
+ // ords), so we build the MathML version first.
+ var mathMLNode = buildMathML(tree, expression, options);
+ var htmlNode = buildHTML(tree, options);
+
+ var katexNode = makeSpan(["katex"], [
+ mathMLNode, htmlNode
+ ]);
+
+ if (settings.displayMode) {
+ return makeSpan(["katex-display"], [katexNode]);
+ } else {
+ return katexNode;
+ }
+};
+
+module.exports = buildTree;
+
+},{"./Options":5,"./Settings":8,"./Style":9,"./buildCommon":10,"./buildHTML":11,"./buildMathML":12}],14:[function(require,module,exports){
+/**
+ * This file deals with creating delimiters of various sizes. The TeXbook
+ * discusses these routines on page 441-442, in the "Another subroutine sets box
+ * x to a specified variable delimiter" paragraph.
+ *
+ * There are three main routines here. `makeSmallDelim` makes a delimiter in the
+ * normal font, but in either text, script, or scriptscript style.
+ * `makeLargeDelim` makes a delimiter in textstyle, but in one of the Size1,
+ * Size2, Size3, or Size4 fonts. `makeStackedDelim` makes a delimiter out of
+ * smaller pieces that are stacked on top of one another.
+ *
+ * The functions take a parameter `center`, which determines if the delimiter
+ * should be centered around the axis.
+ *
+ * Then, there are three exposed functions. `sizedDelim` makes a delimiter in
+ * one of the given sizes. This is used for things like `\bigl`.
+ * `customSizedDelim` makes a delimiter with a given total height+depth. It is
+ * called in places like `\sqrt`. `leftRightDelim` makes an appropriate
+ * delimiter which surrounds an expression of a given height an depth. It is
+ * used in `\left` and `\right`.
+ */
+
+var ParseError = require("./ParseError");
+var Style = require("./Style");
+
+var buildCommon = require("./buildCommon");
+var fontMetrics = require("./fontMetrics");
+var symbols = require("./symbols");
+var utils = require("./utils");
+
+var makeSpan = buildCommon.makeSpan;
+
+/**
+ * Get the metrics for a given symbol and font, after transformation (i.e.
+ * after following replacement from symbols.js)
+ */
+var getMetrics = function(symbol, font) {
+ if (symbols.math[symbol] && symbols.math[symbol].replace) {
+ return fontMetrics.getCharacterMetrics(
+ symbols.math[symbol].replace, font);
+ } else {
+ return fontMetrics.getCharacterMetrics(
+ symbol, font);
+ }
+};
+
+/**
+ * Builds a symbol in the given font size (note size is an integer)
+ */
+var mathrmSize = function(value, size, mode, options) {
+ return buildCommon.makeSymbol(value, "Size" + size + "-Regular",
+ mode, options);
+};
+
+/**
+ * Puts a delimiter span in a given style, and adds appropriate height, depth,
+ * and maxFontSizes.
+ */
+var styleWrap = function(delim, toStyle, options, classes) {
+ classes = classes || [];
+ var span = makeSpan(
+ classes.concat(["style-wrap", options.style.reset(), toStyle.cls()]),
+ [delim], options);
+
+ var multiplier = toStyle.sizeMultiplier / options.style.sizeMultiplier;
+
+ span.height *= multiplier;
+ span.depth *= multiplier;
+ span.maxFontSize = toStyle.sizeMultiplier;
+
+ return span;
+};
+
+/**
+ * Makes a small delimiter. This is a delimiter that comes in the Main-Regular
+ * font, but is restyled to either be in textstyle, scriptstyle, or
+ * scriptscriptstyle.
+ */
+var makeSmallDelim = function(delim, style, center, options, mode, classes) {
+ var text = buildCommon.makeSymbol(delim, "Main-Regular", mode, options);
+
+ var span = styleWrap(text, style, options, classes);
+
+ if (center) {
+ var shift =
+ (1 - options.style.sizeMultiplier / style.sizeMultiplier) *
+ options.style.metrics.axisHeight;
+
+ span.style.top = shift + "em";
+ span.height -= shift;
+ span.depth += shift;
+ }
+
+ return span;
+};
+
+/**
+ * Makes a large delimiter. This is a delimiter that comes in the Size1, Size2,
+ * Size3, or Size4 fonts. It is always rendered in textstyle.
+ */
+var makeLargeDelim = function(delim, size, center, options, mode, classes) {
+ var inner = mathrmSize(delim, size, mode, options);
+
+ var span = styleWrap(
+ makeSpan(["delimsizing", "size" + size], [inner], options),
+ Style.TEXT, options, classes);
+
+ if (center) {
+ var shift = (1 - options.style.sizeMultiplier) *
+ options.style.metrics.axisHeight;
+
+ span.style.top = shift + "em";
+ span.height -= shift;
+ span.depth += shift;
+ }
+
+ return span;
+};
+
+/**
+ * Make an inner span with the given offset and in the given font. This is used
+ * in `makeStackedDelim` to make the stacking pieces for the delimiter.
+ */
+var makeInner = function(symbol, font, mode) {
+ var sizeClass;
+ // Apply the correct CSS class to choose the right font.
+ if (font === "Size1-Regular") {
+ sizeClass = "delim-size1";
+ } else if (font === "Size4-Regular") {
+ sizeClass = "delim-size4";
+ }
+
+ var inner = makeSpan(
+ ["delimsizinginner", sizeClass],
+ [makeSpan([], [buildCommon.makeSymbol(symbol, font, mode)])]);
+
+ // Since this will be passed into `makeVList` in the end, wrap the element
+ // in the appropriate tag that VList uses.
+ return {type: "elem", elem: inner};
+};
+
+/**
+ * Make a stacked delimiter out of a given delimiter, with the total height at
+ * least `heightTotal`. This routine is mentioned on page 442 of the TeXbook.
+ */
+var makeStackedDelim = function(delim, heightTotal, center, options, mode,
+ classes) {
+ // There are four parts, the top, an optional middle, a repeated part, and a
+ // bottom.
+ var top;
+ var middle;
+ var repeat;
+ var bottom;
+ top = repeat = bottom = delim;
+ middle = null;
+ // Also keep track of what font the delimiters are in
+ var font = "Size1-Regular";
+
+ // We set the parts and font based on the symbol. Note that we use
+ // '\u23d0' instead of '|' and '\u2016' instead of '\\|' for the
+ // repeats of the arrows
+ if (delim === "\\uparrow") {
+ repeat = bottom = "\u23d0";
+ } else if (delim === "\\Uparrow") {
+ repeat = bottom = "\u2016";
+ } else if (delim === "\\downarrow") {
+ top = repeat = "\u23d0";
+ } else if (delim === "\\Downarrow") {
+ top = repeat = "\u2016";
+ } else if (delim === "\\updownarrow") {
+ top = "\\uparrow";
+ repeat = "\u23d0";
+ bottom = "\\downarrow";
+ } else if (delim === "\\Updownarrow") {
+ top = "\\Uparrow";
+ repeat = "\u2016";
+ bottom = "\\Downarrow";
+ } else if (delim === "[" || delim === "\\lbrack") {
+ top = "\u23a1";
+ repeat = "\u23a2";
+ bottom = "\u23a3";
+ font = "Size4-Regular";
+ } else if (delim === "]" || delim === "\\rbrack") {
+ top = "\u23a4";
+ repeat = "\u23a5";
+ bottom = "\u23a6";
+ font = "Size4-Regular";
+ } else if (delim === "\\lfloor") {
+ repeat = top = "\u23a2";
+ bottom = "\u23a3";
+ font = "Size4-Regular";
+ } else if (delim === "\\lceil") {
+ top = "\u23a1";
+ repeat = bottom = "\u23a2";
+ font = "Size4-Regular";
+ } else if (delim === "\\rfloor") {
+ repeat = top = "\u23a5";
+ bottom = "\u23a6";
+ font = "Size4-Regular";
+ } else if (delim === "\\rceil") {
+ top = "\u23a4";
+ repeat = bottom = "\u23a5";
+ font = "Size4-Regular";
+ } else if (delim === "(") {
+ top = "\u239b";
+ repeat = "\u239c";
+ bottom = "\u239d";
+ font = "Size4-Regular";
+ } else if (delim === ")") {
+ top = "\u239e";
+ repeat = "\u239f";
+ bottom = "\u23a0";
+ font = "Size4-Regular";
+ } else if (delim === "\\{" || delim === "\\lbrace") {
+ top = "\u23a7";
+ middle = "\u23a8";
+ bottom = "\u23a9";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\}" || delim === "\\rbrace") {
+ top = "\u23ab";
+ middle = "\u23ac";
+ bottom = "\u23ad";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\lgroup") {
+ top = "\u23a7";
+ bottom = "\u23a9";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\rgroup") {
+ top = "\u23ab";
+ bottom = "\u23ad";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\lmoustache") {
+ top = "\u23a7";
+ bottom = "\u23ad";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\rmoustache") {
+ top = "\u23ab";
+ bottom = "\u23a9";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\surd") {
+ top = "\ue001";
+ bottom = "\u23b7";
+ repeat = "\ue000";
+ font = "Size4-Regular";
+ }
+
+ // Get the metrics of the four sections
+ var topMetrics = getMetrics(top, font);
+ var topHeightTotal = topMetrics.height + topMetrics.depth;
+ var repeatMetrics = getMetrics(repeat, font);
+ var repeatHeightTotal = repeatMetrics.height + repeatMetrics.depth;
+ var bottomMetrics = getMetrics(bottom, font);
+ var bottomHeightTotal = bottomMetrics.height + bottomMetrics.depth;
+ var middleHeightTotal = 0;
+ var middleFactor = 1;
+ if (middle !== null) {
+ var middleMetrics = getMetrics(middle, font);
+ middleHeightTotal = middleMetrics.height + middleMetrics.depth;
+ middleFactor = 2; // repeat symmetrically above and below middle
+ }
+
+ // Calcuate the minimal height that the delimiter can have.
+ // It is at least the size of the top, bottom, and optional middle combined.
+ var minHeight = topHeightTotal + bottomHeightTotal + middleHeightTotal;
+
+ // Compute the number of copies of the repeat symbol we will need
+ var repeatCount = Math.ceil(
+ (heightTotal - minHeight) / (middleFactor * repeatHeightTotal));
+
+ // Compute the total height of the delimiter including all the symbols
+ var realHeightTotal =
+ minHeight + repeatCount * middleFactor * repeatHeightTotal;
+
+ // The center of the delimiter is placed at the center of the axis. Note
+ // that in this context, "center" means that the delimiter should be
+ // centered around the axis in the current style, while normally it is
+ // centered around the axis in textstyle.
+ var axisHeight = options.style.metrics.axisHeight;
+ if (center) {
+ axisHeight *= options.style.sizeMultiplier;
+ }
+ // Calculate the depth
+ var depth = realHeightTotal / 2 - axisHeight;
+
+ // Now, we start building the pieces that will go into the vlist
+
+ // Keep a list of the inner pieces
+ var inners = [];
+
+ // Add the bottom symbol
+ inners.push(makeInner(bottom, font, mode));
+
+ var i;
+ if (middle === null) {
+ // Add that many symbols
+ for (i = 0; i < repeatCount; i++) {
+ inners.push(makeInner(repeat, font, mode));
+ }
+ } else {
+ // When there is a middle bit, we need the middle part and two repeated
+ // sections
+ for (i = 0; i < repeatCount; i++) {
+ inners.push(makeInner(repeat, font, mode));
+ }
+ inners.push(makeInner(middle, font, mode));
+ for (i = 0; i < repeatCount; i++) {
+ inners.push(makeInner(repeat, font, mode));
+ }
+ }
+
+ // Add the top symbol
+ inners.push(makeInner(top, font, mode));
+
+ // Finally, build the vlist
+ var inner = buildCommon.makeVList(inners, "bottom", depth, options);
+
+ return styleWrap(
+ makeSpan(["delimsizing", "mult"], [inner], options),
+ Style.TEXT, options, classes);
+};
+
+// There are three kinds of delimiters, delimiters that stack when they become
+// too large
+var stackLargeDelimiters = [
+ "(", ")", "[", "\\lbrack", "]", "\\rbrack",
+ "\\{", "\\lbrace", "\\}", "\\rbrace",
+ "\\lfloor", "\\rfloor", "\\lceil", "\\rceil",
+ "\\surd"
+];
+
+// delimiters that always stack
+var stackAlwaysDelimiters = [
+ "\\uparrow", "\\downarrow", "\\updownarrow",
+ "\\Uparrow", "\\Downarrow", "\\Updownarrow",
+ "|", "\\|", "\\vert", "\\Vert",
+ "\\lvert", "\\rvert", "\\lVert", "\\rVert",
+ "\\lgroup", "\\rgroup", "\\lmoustache", "\\rmoustache"
+];
+
+// and delimiters that never stack
+var stackNeverDelimiters = [
+ "<", ">", "\\langle", "\\rangle", "/", "\\backslash", "\\lt", "\\gt"
+];
+
+// Metrics of the different sizes. Found by looking at TeX's output of
+// $\bigl| // \Bigl| \biggl| \Biggl| \showlists$
+// Used to create stacked delimiters of appropriate sizes in makeSizedDelim.
+var sizeToMaxHeight = [0, 1.2, 1.8, 2.4, 3.0];
+
+/**
+ * Used to create a delimiter of a specific size, where `size` is 1, 2, 3, or 4.
+ */
+var makeSizedDelim = function(delim, size, options, mode, classes) {
+ // < and > turn into \langle and \rangle in delimiters
+ if (delim === "<" || delim === "\\lt") {
+ delim = "\\langle";
+ } else if (delim === ">" || delim === "\\gt") {
+ delim = "\\rangle";
+ }
+
+ // Sized delimiters are never centered.
+ if (utils.contains(stackLargeDelimiters, delim) ||
+ utils.contains(stackNeverDelimiters, delim)) {
+ return makeLargeDelim(delim, size, false, options, mode, classes);
+ } else if (utils.contains(stackAlwaysDelimiters, delim)) {
+ return makeStackedDelim(
+ delim, sizeToMaxHeight[size], false, options, mode, classes);
+ } else {
+ throw new ParseError("Illegal delimiter: '" + delim + "'");
+ }
+};
+
+/**
+ * There are three different sequences of delimiter sizes that the delimiters
+ * follow depending on the kind of delimiter. This is used when creating custom
+ * sized delimiters to decide whether to create a small, large, or stacked
+ * delimiter.
+ *
+ * In real TeX, these sequences aren't explicitly defined, but are instead
+ * defined inside the font metrics. Since there are only three sequences that
+ * are possible for the delimiters that TeX defines, it is easier to just encode
+ * them explicitly here.
+ */
+
+// Delimiters that never stack try small delimiters and large delimiters only
+var stackNeverDelimiterSequence = [
+ {type: "small", style: Style.SCRIPTSCRIPT},
+ {type: "small", style: Style.SCRIPT},
+ {type: "small", style: Style.TEXT},
+ {type: "large", size: 1},
+ {type: "large", size: 2},
+ {type: "large", size: 3},
+ {type: "large", size: 4}
+];
+
+// Delimiters that always stack try the small delimiters first, then stack
+var stackAlwaysDelimiterSequence = [
+ {type: "small", style: Style.SCRIPTSCRIPT},
+ {type: "small", style: Style.SCRIPT},
+ {type: "small", style: Style.TEXT},
+ {type: "stack"}
+];
+
+// Delimiters that stack when large try the small and then large delimiters, and
+// stack afterwards
+var stackLargeDelimiterSequence = [
+ {type: "small", style: Style.SCRIPTSCRIPT},
+ {type: "small", style: Style.SCRIPT},
+ {type: "small", style: Style.TEXT},
+ {type: "large", size: 1},
+ {type: "large", size: 2},
+ {type: "large", size: 3},
+ {type: "large", size: 4},
+ {type: "stack"}
+];
+
+/**
+ * Get the font used in a delimiter based on what kind of delimiter it is.
+ */
+var delimTypeToFont = function(type) {
+ if (type.type === "small") {
+ return "Main-Regular";
+ } else if (type.type === "large") {
+ return "Size" + type.size + "-Regular";
+ } else if (type.type === "stack") {
+ return "Size4-Regular";
+ }
+};
+
+/**
+ * Traverse a sequence of types of delimiters to decide what kind of delimiter
+ * should be used to create a delimiter of the given height+depth.
+ */
+var traverseSequence = function(delim, height, sequence, options) {
+ // Here, we choose the index we should start at in the sequences. In smaller
+ // sizes (which correspond to larger numbers in style.size) we start earlier
+ // in the sequence. Thus, scriptscript starts at index 3-3=0, script starts
+ // at index 3-2=1, text starts at 3-1=2, and display starts at min(2,3-0)=2
+ var start = Math.min(2, 3 - options.style.size);
+ for (var i = start; i < sequence.length; i++) {
+ if (sequence[i].type === "stack") {
+ // This is always the last delimiter, so we just break the loop now.
+ break;
+ }
+
+ var metrics = getMetrics(delim, delimTypeToFont(sequence[i]));
+ var heightDepth = metrics.height + metrics.depth;
+
+ // Small delimiters are scaled down versions of the same font, so we
+ // account for the style change size.
+
+ if (sequence[i].type === "small") {
+ heightDepth *= sequence[i].style.sizeMultiplier;
+ }
+
+ // Check if the delimiter at this size works for the given height.
+ if (heightDepth > height) {
+ return sequence[i];
+ }
+ }
+
+ // If we reached the end of the sequence, return the last sequence element.
+ return sequence[sequence.length - 1];
+};
+
+/**
+ * Make a delimiter of a given height+depth, with optional centering. Here, we
+ * traverse the sequences, and create a delimiter that the sequence tells us to.
+ */
+var makeCustomSizedDelim = function(delim, height, center, options, mode,
+ classes) {
+ if (delim === "<" || delim === "\\lt") {
+ delim = "\\langle";
+ } else if (delim === ">" || delim === "\\gt") {
+ delim = "\\rangle";
+ }
+
+ // Decide what sequence to use
+ var sequence;
+ if (utils.contains(stackNeverDelimiters, delim)) {
+ sequence = stackNeverDelimiterSequence;
+ } else if (utils.contains(stackLargeDelimiters, delim)) {
+ sequence = stackLargeDelimiterSequence;
+ } else {
+ sequence = stackAlwaysDelimiterSequence;
+ }
+
+ // Look through the sequence
+ var delimType = traverseSequence(delim, height, sequence, options);
+
+ // Depending on the sequence element we decided on, call the appropriate
+ // function.
+ if (delimType.type === "small") {
+ return makeSmallDelim(delim, delimType.style, center, options, mode,
+ classes);
+ } else if (delimType.type === "large") {
+ return makeLargeDelim(delim, delimType.size, center, options, mode,
+ classes);
+ } else if (delimType.type === "stack") {
+ return makeStackedDelim(delim, height, center, options, mode, classes);
+ }
+};
+
+/**
+ * Make a delimiter for use with `\left` and `\right`, given a height and depth
+ * of an expression that the delimiters surround.
+ */
+var makeLeftRightDelim = function(delim, height, depth, options, mode,
+ classes) {
+ // We always center \left/\right delimiters, so the axis is always shifted
+ var axisHeight =
+ options.style.metrics.axisHeight * options.style.sizeMultiplier;
+
+ // Taken from TeX source, tex.web, function make_left_right
+ var delimiterFactor = 901;
+ var delimiterExtend = 5.0 / fontMetrics.metrics.ptPerEm;
+
+ var maxDistFromAxis = Math.max(
+ height - axisHeight, depth + axisHeight);
+
+ var totalHeight = Math.max(
+ // In real TeX, calculations are done using integral values which are
+ // 65536 per pt, or 655360 per em. So, the division here truncates in
+ // TeX but doesn't here, producing different results. If we wanted to
+ // exactly match TeX's calculation, we could do
+ // Math.floor(655360 * maxDistFromAxis / 500) *
+ // delimiterFactor / 655360
+ // (To see the difference, compare
+ // x^{x^{\left(\rule{0.1em}{0.68em}\right)}}
+ // in TeX and KaTeX)
+ maxDistFromAxis / 500 * delimiterFactor,
+ 2 * maxDistFromAxis - delimiterExtend);
+
+ // Finally, we defer to `makeCustomSizedDelim` with our calculated total
+ // height
+ return makeCustomSizedDelim(delim, totalHeight, true, options, mode,
+ classes);
+};
+
+module.exports = {
+ sizedDelim: makeSizedDelim,
+ customSizedDelim: makeCustomSizedDelim,
+ leftRightDelim: makeLeftRightDelim
+};
+
+},{"./ParseError":6,"./Style":9,"./buildCommon":10,"./fontMetrics":17,"./symbols":23,"./utils":25}],15:[function(require,module,exports){
+/**
+ * These objects store the data about the DOM nodes we create, as well as some
+ * extra data. They can then be transformed into real DOM nodes with the
+ * `toNode` function or HTML markup using `toMarkup`. They are useful for both
+ * storing extra properties on the nodes, as well as providing a way to easily
+ * work with the DOM.
+ *
+ * Similar functions for working with MathML nodes exist in mathMLTree.js.
+ */
+var unicodeRegexes = require("./unicodeRegexes");
+var utils = require("./utils");
+
+/**
+ * Create an HTML className based on a list of classes. In addition to joining
+ * with spaces, we also remove null or empty classes.
+ */
+var createClass = function(classes) {
+ classes = classes.slice();
+ for (var i = classes.length - 1; i >= 0; i--) {
+ if (!classes[i]) {
+ classes.splice(i, 1);
+ }
+ }
+
+ return classes.join(" ");
+};
+
+/**
+ * This node represents a span node, with a className, a list of children, and
+ * an inline style. It also contains information about its height, depth, and
+ * maxFontSize.
+ */
+function span(classes, children, options) {
+ this.classes = classes || [];
+ this.children = children || [];
+ this.height = 0;
+ this.depth = 0;
+ this.maxFontSize = 0;
+ this.style = {};
+ this.attributes = {};
+ if (options) {
+ if (options.style.isTight()) {
+ this.classes.push("mtight");
+ }
+ if (options.getColor()) {
+ this.style.color = options.getColor();
+ }
+ }
+}
+
+/**
+ * Sets an arbitrary attribute on the span. Warning: use this wisely. Not all
+ * browsers support attributes the same, and having too many custom attributes
+ * is probably bad.
+ */
+span.prototype.setAttribute = function(attribute, value) {
+ this.attributes[attribute] = value;
+};
+
+span.prototype.tryCombine = function(sibling) {
+ return false;
+};
+
+/**
+ * Convert the span into an HTML node
+ */
+span.prototype.toNode = function() {
+ var span = document.createElement("span");
+
+ // Apply the class
+ span.className = createClass(this.classes);
+
+ // Apply inline styles
+ for (var style in this.style) {
+ if (Object.prototype.hasOwnProperty.call(this.style, style)) {
+ span.style[style] = this.style[style];
+ }
+ }
+
+ // Apply attributes
+ for (var attr in this.attributes) {
+ if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+ span.setAttribute(attr, this.attributes[attr]);
+ }
+ }
+
+ // Append the children, also as HTML nodes
+ for (var i = 0; i < this.children.length; i++) {
+ span.appendChild(this.children[i].toNode());
+ }
+
+ return span;
+};
+
+/**
+ * Convert the span into an HTML markup string
+ */
+span.prototype.toMarkup = function() {
+ var markup = "<span";
+
+ // Add the class
+ if (this.classes.length) {
+ markup += " class=\"";
+ markup += utils.escape(createClass(this.classes));
+ markup += "\"";
+ }
+
+ var styles = "";
+
+ // Add the styles, after hyphenation
+ for (var style in this.style) {
+ if (this.style.hasOwnProperty(style)) {
+ styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+ }
+ }
+
+ if (styles) {
+ markup += " style=\"" + utils.escape(styles) + "\"";
+ }
+
+ // Add the attributes
+ for (var attr in this.attributes) {
+ if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+ markup += " " + attr + "=\"";
+ markup += utils.escape(this.attributes[attr]);
+ markup += "\"";
+ }
+ }
+
+ markup += ">";
+
+ // Add the markup of the children, also as markup
+ for (var i = 0; i < this.children.length; i++) {
+ markup += this.children[i].toMarkup();
+ }
+
+ markup += "</span>";
+
+ return markup;
+};
+
+/**
+ * This node represents a document fragment, which contains elements, but when
+ * placed into the DOM doesn't have any representation itself. Thus, it only
+ * contains children and doesn't have any HTML properties. It also keeps track
+ * of a height, depth, and maxFontSize.
+ */
+function documentFragment(children) {
+ this.children = children || [];
+ this.height = 0;
+ this.depth = 0;
+ this.maxFontSize = 0;
+}
+
+/**
+ * Convert the fragment into a node
+ */
+documentFragment.prototype.toNode = function() {
+ // Create a fragment
+ var frag = document.createDocumentFragment();
+
+ // Append the children
+ for (var i = 0; i < this.children.length; i++) {
+ frag.appendChild(this.children[i].toNode());
+ }
+
+ return frag;
+};
+
+/**
+ * Convert the fragment into HTML markup
+ */
+documentFragment.prototype.toMarkup = function() {
+ var markup = "";
+
+ // Simply concatenate the markup for the children together
+ for (var i = 0; i < this.children.length; i++) {
+ markup += this.children[i].toMarkup();
+ }
+
+ return markup;
+};
+
+var iCombinations = {
+ 'î': '\u0131\u0302',
+ 'ï': '\u0131\u0308',
+ 'í': '\u0131\u0301',
+ // 'ī': '\u0131\u0304', // enable when we add Extended Latin
+ 'ì': '\u0131\u0300'
+};
+
+/**
+ * A symbol node contains information about a single symbol. It either renders
+ * to a single text node, or a span with a single text node in it, depending on
+ * whether it has CSS classes, styles, or needs italic correction.
+ */
+function symbolNode(value, height, depth, italic, skew, classes, style) {
+ this.value = value || "";
+ this.height = height || 0;
+ this.depth = depth || 0;
+ this.italic = italic || 0;
+ this.skew = skew || 0;
+ this.classes = classes || [];
+ this.style = style || {};
+ this.maxFontSize = 0;
+
+ // Mark CJK characters with specific classes so that we can specify which
+ // fonts to use. This allows us to render these characters with a serif
+ // font in situations where the browser would either default to a sans serif
+ // or render a placeholder character.
+ if (unicodeRegexes.cjkRegex.test(value)) {
+ // I couldn't find any fonts that contained Hangul as well as all of
+ // the other characters we wanted to test there for it gets its own
+ // CSS class.
+ if (unicodeRegexes.hangulRegex.test(value)) {
+ this.classes.push('hangul_fallback');
+ } else {
+ this.classes.push('cjk_fallback');
+ }
+ }
+
+ if (/[îïíì]/.test(this.value)) { // add ī when we add Extended Latin
+ this.value = iCombinations[this.value];
+ }
+}
+
+symbolNode.prototype.tryCombine = function(sibling) {
+ if (!sibling
+ || !(sibling instanceof symbolNode)
+ || this.italic > 0
+ || createClass(this.classes) !== createClass(sibling.classes)
+ || this.skew !== sibling.skew
+ || this.maxFontSize !== sibling.maxFontSize) {
+ return false;
+ }
+ for (var style in this.style) {
+ if (this.style.hasOwnProperty(style)
+ && this.style[style] !== sibling.style[style]) {
+ return false;
+ }
+ }
+ for (style in sibling.style) {
+ if (sibling.style.hasOwnProperty(style)
+ && this.style[style] !== sibling.style[style]) {
+ return false;
+ }
+ }
+ this.value += sibling.value;
+ this.height = Math.max(this.height, sibling.height);
+ this.depth = Math.max(this.depth, sibling.depth);
+ this.italic = sibling.italic;
+ return true;
+};
+
+/**
+ * Creates a text node or span from a symbol node. Note that a span is only
+ * created if it is needed.
+ */
+symbolNode.prototype.toNode = function() {
+ var node = document.createTextNode(this.value);
+ var span = null;
+
+ if (this.italic > 0) {
+ span = document.createElement("span");
+ span.style.marginRight = this.italic + "em";
+ }
+
+ if (this.classes.length > 0) {
+ span = span || document.createElement("span");
+ span.className = createClass(this.classes);
+ }
+
+ for (var style in this.style) {
+ if (this.style.hasOwnProperty(style)) {
+ span = span || document.createElement("span");
+ span.style[style] = this.style[style];
+ }
+ }
+
+ if (span) {
+ span.appendChild(node);
+ return span;
+ } else {
+ return node;
+ }
+};
+
+/**
+ * Creates markup for a symbol node.
+ */
+symbolNode.prototype.toMarkup = function() {
+ // TODO(alpert): More duplication than I'd like from
+ // span.prototype.toMarkup and symbolNode.prototype.toNode...
+ var needsSpan = false;
+
+ var markup = "<span";
+
+ if (this.classes.length) {
+ needsSpan = true;
+ markup += " class=\"";
+ markup += utils.escape(createClass(this.classes));
+ markup += "\"";
+ }
+
+ var styles = "";
+
+ if (this.italic > 0) {
+ styles += "margin-right:" + this.italic + "em;";
+ }
+ for (var style in this.style) {
+ if (this.style.hasOwnProperty(style)) {
+ styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+ }
+ }
+
+ if (styles) {
+ needsSpan = true;
+ markup += " style=\"" + utils.escape(styles) + "\"";
+ }
+
+ var escaped = utils.escape(this.value);
+ if (needsSpan) {
+ markup += ">";
+ markup += escaped;
+ markup += "</span>";
+ return markup;
+ } else {
+ return escaped;
+ }
+};
+
+module.exports = {
+ span: span,
+ documentFragment: documentFragment,
+ symbolNode: symbolNode
+};
+
+},{"./unicodeRegexes":24,"./utils":25}],16:[function(require,module,exports){
+/* eslint no-constant-condition:0 */
+var parseData = require("./parseData");
+var ParseError = require("./ParseError");
+var Style = require("./Style");
+
+var ParseNode = parseData.ParseNode;
+
+/**
+ * Parse the body of the environment, with rows delimited by \\ and
+ * columns delimited by &, and create a nested list in row-major order
+ * with one group per cell.
+ */
+function parseArray(parser, result) {
+ var row = [];
+ var body = [row];
+ var rowGaps = [];
+ while (true) {
+ var cell = parser.parseExpression(false, null);
+ row.push(new ParseNode("ordgroup", cell, parser.mode));
+ var next = parser.nextToken.text;
+ if (next === "&") {
+ parser.consume();
+ } else if (next === "\\end") {
+ break;
+ } else if (next === "\\\\" || next === "\\cr") {
+ var cr = parser.parseFunction();
+ rowGaps.push(cr.value.size);
+ row = [];
+ body.push(row);
+ } else {
+ throw new ParseError("Expected & or \\\\ or \\end",
+ parser.nextToken);
+ }
+ }
+ result.body = body;
+ result.rowGaps = rowGaps;
+ return new ParseNode(result.type, result, parser.mode);
+}
+
+/*
+ * An environment definition is very similar to a function definition:
+ * it is declared with a name or a list of names, a set of properties
+ * and a handler containing the actual implementation.
+ *
+ * The properties include:
+ * - numArgs: The number of arguments after the \begin{name} function.
+ * - argTypes: (optional) Just like for a function
+ * - allowedInText: (optional) Whether or not the environment is allowed inside
+ * text mode (default false) (not enforced yet)
+ * - numOptionalArgs: (optional) Just like for a function
+ * A bare number instead of that object indicates the numArgs value.
+ *
+ * The handler function will receive two arguments
+ * - context: information and references provided by the parser
+ * - args: an array of arguments passed to \begin{name}
+ * The context contains the following properties:
+ * - envName: the name of the environment, one of the listed names.
+ * - parser: the parser object
+ * - lexer: the lexer object
+ * - positions: the positions associated with these arguments from args.
+ * The handler must return a ParseResult.
+ */
+
+function defineEnvironment(names, props, handler) {
+ if (typeof names === "string") {
+ names = [names];
+ }
+ if (typeof props === "number") {
+ props = { numArgs: props };
+ }
+ // Set default values of environments
+ var data = {
+ numArgs: props.numArgs || 0,
+ argTypes: props.argTypes,
+ greediness: 1,
+ allowedInText: !!props.allowedInText,
+ numOptionalArgs: props.numOptionalArgs || 0,
+ handler: handler
+ };
+ for (var i = 0; i < names.length; ++i) {
+ module.exports[names[i]] = data;
+ }
+}
+
+// Arrays are part of LaTeX, defined in lttab.dtx so its documentation
+// is part of the source2e.pdf file of LaTeX2e source documentation.
+defineEnvironment("array", {
+ numArgs: 1
+}, function(context, args) {
+ var colalign = args[0];
+ colalign = colalign.value.map ? colalign.value : [colalign];
+ var cols = colalign.map(function(node) {
+ var ca = node.value;
+ if ("lcr".indexOf(ca) !== -1) {
+ return {
+ type: "align",
+ align: ca
+ };
+ } else if (ca === "|") {
+ return {
+ type: "separator",
+ separator: "|"
+ };
+ }
+ throw new ParseError(
+ "Unknown column alignment: " + node.value,
+ node);
+ });
+ var res = {
+ type: "array",
+ cols: cols,
+ hskipBeforeAndAfter: true // \@preamble in lttab.dtx
+ };
+ res = parseArray(context.parser, res);
+ return res;
+});
+
+// The matrix environments of amsmath builds on the array environment
+// of LaTeX, which is discussed above.
+defineEnvironment([
+ "matrix",
+ "pmatrix",
+ "bmatrix",
+ "Bmatrix",
+ "vmatrix",
+ "Vmatrix"
+], {
+}, function(context) {
+ var delimiters = {
+ "matrix": null,
+ "pmatrix": ["(", ")"],
+ "bmatrix": ["[", "]"],
+ "Bmatrix": ["\\{", "\\}"],
+ "vmatrix": ["|", "|"],
+ "Vmatrix": ["\\Vert", "\\Vert"]
+ }[context.envName];
+ var res = {
+ type: "array",
+ hskipBeforeAndAfter: false // \hskip -\arraycolsep in amsmath
+ };
+ res = parseArray(context.parser, res);
+ if (delimiters) {
+ res = new ParseNode("leftright", {
+ body: [res],
+ left: delimiters[0],
+ right: delimiters[1]
+ }, context.mode);
+ }
+ return res;
+});
+
+// A cases environment (in amsmath.sty) is almost equivalent to
+// \def\arraystretch{1.2}%
+// \left\{\begin{array}{@{}l@{\quad}l@{}} … \end{array}\right.
+defineEnvironment("cases", {
+}, function(context) {
+ var res = {
+ type: "array",
+ arraystretch: 1.2,
+ cols: [{
+ type: "align",
+ align: "l",
+ pregap: 0,
+ // TODO(kevinb) get the current style.
+ // For now we use the metrics for TEXT style which is what we were
+ // doing before. Before attempting to get the current style we
+ // should look at TeX's behavior especially for \over and matrices.
+ postgap: Style.TEXT.metrics.quad
+ }, {
+ type: "align",
+ align: "l",
+ pregap: 0,
+ postgap: 0
+ }]
+ };
+ res = parseArray(context.parser, res);
+ res = new ParseNode("leftright", {
+ body: [res],
+ left: "\\{",
+ right: "."
+ }, context.mode);
+ return res;
+});
+
+// An aligned environment is like the align* environment
+// except it operates within math mode.
+// Note that we assume \nomallineskiplimit to be zero,
+// so that \strut@ is the same as \strut.
+defineEnvironment("aligned", {
+}, function(context) {
+ var res = {
+ type: "array",
+ cols: []
+ };
+ res = parseArray(context.parser, res);
+ var emptyGroup = new ParseNode("ordgroup", [], context.mode);
+ var numCols = 0;
+ res.value.body.forEach(function(row) {
+ var i;
+ for (i = 1; i < row.length; i += 2) {
+ row[i].value.unshift(emptyGroup);
+ }
+ if (numCols < row.length) {
+ numCols = row.length;
+ }
+ });
+ for (var i = 0; i < numCols; ++i) {
+ var align = "r";
+ var pregap = 0;
+ if (i % 2 === 1) {
+ align = "l";
+ } else if (i > 0) {
+ pregap = 2; // one \qquad between columns
+ }
+ res.value.cols[i] = {
+ type: "align",
+ align: align,
+ pregap: pregap,
+ postgap: 0
+ };
+ }
+ return res;
+});
+
+},{"./ParseError":6,"./Style":9,"./parseData":21}],17:[function(require,module,exports){
+/* eslint no-unused-vars:0 */
+
+var Style = require("./Style");
+var cjkRegex = require("./unicodeRegexes").cjkRegex;
+
+/**
+ * This file contains metrics regarding fonts and individual symbols. The sigma
+ * and xi variables, as well as the metricMap map contain data extracted from
+ * TeX, TeX font metrics, and the TTF files. These data are then exposed via the
+ * `metrics` variable and the getCharacterMetrics function.
+ */
+
+// In TeX, there are actually three sets of dimensions, one for each of
+// textstyle, scriptstyle, and scriptscriptstyle. These are provided in the
+// the arrays below, in that order.
+//
+// The font metrics are stored in fonts cmsy10, cmsy7, and cmsy5 respsectively.
+// This was determined by running the folllowing script:
+//
+// latex -interaction=nonstopmode \
+// '\documentclass{article}\usepackage{amsmath}\begin{document}' \
+// '$a$ \expandafter\show\the\textfont2' \
+// '\expandafter\show\the\scriptfont2' \
+// '\expandafter\show\the\scriptscriptfont2' \
+// '\stop'
+//
+// The metrics themselves were retreived using the following commands:
+//
+// tftopl cmsy10
+// tftopl cmsy7
+// tftopl cmsy5
+//
+// The output of each of these commands is quite lengthy. The only part we
+// care about is the FONTDIMEN section. Each value is measured in EMs.
+var sigmas = {
+ slant: [0.250, 0.250, 0.250], // sigma1
+ space: [0.000, 0.000, 0.000], // sigma2
+ stretch: [0.000, 0.000, 0.000], // sigma3
+ shrink: [0.000, 0.000, 0.000], // sigma4
+ xHeight: [0.431, 0.431, 0.431], // sigma5
+ quad: [1.000, 1.171, 1.472], // sigma6
+ extraSpace: [0.000, 0.000, 0.000], // sigma7
+ num1: [0.677, 0.732, 0.925], // sigma8
+ num2: [0.394, 0.384, 0.387], // sigma9
+ num3: [0.444, 0.471, 0.504], // sigma10
+ denom1: [0.686, 0.752, 1.025], // sigma11
+ denom2: [0.345, 0.344, 0.532], // sigma12
+ sup1: [0.413, 0.503, 0.504], // sigma13
+ sup2: [0.363, 0.431, 0.404], // sigma14
+ sup3: [0.289, 0.286, 0.294], // sigma15
+ sub1: [0.150, 0.143, 0.200], // sigma16
+ sub2: [0.247, 0.286, 0.400], // sigma17
+ supDrop: [0.386, 0.353, 0.494], // sigma18
+ subDrop: [0.050, 0.071, 0.100], // sigma19
+ delim1: [2.390, 1.700, 1.980], // sigma20
+ delim2: [1.010, 1.157, 1.420], // sigma21
+ axisHeight: [0.250, 0.250, 0.250] // sigma22
+};
+
+// These font metrics are extracted from TeX by using
+// \font\a=cmex10
+// \showthe\fontdimenX\a
+// where X is the corresponding variable number. These correspond to the font
+// parameters of the extension fonts (family 3). See the TeXbook, page 441.
+var xi1 = 0;
+var xi2 = 0;
+var xi3 = 0;
+var xi4 = 0;
+var xi5 = 0.431;
+var xi6 = 1;
+var xi7 = 0;
+var xi8 = 0.04;
+var xi9 = 0.111;
+var xi10 = 0.166;
+var xi11 = 0.2;
+var xi12 = 0.6;
+var xi13 = 0.1;
+
+// This value determines how large a pt is, for metrics which are defined in
+// terms of pts.
+// This value is also used in katex.less; if you change it make sure the values
+// match.
+var ptPerEm = 10.0;
+
+// The space between adjacent `|` columns in an array definition. From
+// `\showthe\doublerulesep` in LaTeX.
+var doubleRuleSep = 2.0 / ptPerEm;
+
+/**
+ * This is just a mapping from common names to real metrics
+ */
+var metrics = {
+ defaultRuleThickness: xi8,
+ bigOpSpacing1: xi9,
+ bigOpSpacing2: xi10,
+ bigOpSpacing3: xi11,
+ bigOpSpacing4: xi12,
+ bigOpSpacing5: xi13,
+ ptPerEm: ptPerEm,
+ doubleRuleSep: doubleRuleSep
+};
+
+// This map contains a mapping from font name and character code to character
+// metrics, including height, depth, italic correction, and skew (kern from the
+// character to the corresponding \skewchar)
+// This map is generated via `make metrics`. It should not be changed manually.
+var metricMap = require("./fontMetricsData");
+
+// These are very rough approximations. We default to Times New Roman which
+// should have Latin-1 and Cyrillic characters, but may not depending on the
+// operating system. The metrics do not account for extra height from the
+// accents. In the case of Cyrillic characters which have both ascenders and
+// descenders we prefer approximations with ascenders, primarily to prevent
+// the fraction bar or root line from intersecting the glyph.
+// TODO(kevinb) allow union of multiple glyph metrics for better accuracy.
+var extraCharacterMap = {
+ // Latin-1
+ 'À': 'A',
+ 'Á': 'A',
+ 'Â': 'A',
+ 'Ã': 'A',
+ 'Ä': 'A',
+ 'Å': 'A',
+ 'Æ': 'A',
+ 'Ç': 'C',
+ 'È': 'E',
+ 'É': 'E',
+ 'Ê': 'E',
+ 'Ë': 'E',
+ 'Ì': 'I',
+ 'Í': 'I',
+ 'Î': 'I',
+ 'Ï': 'I',
+ 'Ð': 'D',
+ 'Ñ': 'N',
+ 'Ò': 'O',
+ 'Ó': 'O',
+ 'Ô': 'O',
+ 'Õ': 'O',
+ 'Ö': 'O',
+ 'Ø': 'O',
+ 'Ù': 'U',
+ 'Ú': 'U',
+ 'Û': 'U',
+ 'Ü': 'U',
+ 'Ý': 'Y',
+ 'Þ': 'o',
+ 'ß': 'B',
+ 'à': 'a',
+ 'á': 'a',
+ 'â': 'a',
+ 'ã': 'a',
+ 'ä': 'a',
+ 'å': 'a',
+ 'æ': 'a',
+ 'ç': 'c',
+ 'è': 'e',
+ 'é': 'e',
+ 'ê': 'e',
+ 'ë': 'e',
+ 'ì': 'i',
+ 'í': 'i',
+ 'î': 'i',
+ 'ï': 'i',
+ 'ð': 'd',
+ 'ñ': 'n',
+ 'ò': 'o',
+ 'ó': 'o',
+ 'ô': 'o',
+ 'õ': 'o',
+ 'ö': 'o',
+ 'ø': 'o',
+ 'ù': 'u',
+ 'ú': 'u',
+ 'û': 'u',
+ 'ü': 'u',
+ 'ý': 'y',
+ 'þ': 'o',
+ 'ÿ': 'y',
+
+ // Cyrillic
+ 'А': 'A',
+ 'Б': 'B',
+ 'В': 'B',
+ 'Г': 'F',
+ 'Д': 'A',
+ 'Е': 'E',
+ 'Ж': 'K',
+ 'З': '3',
+ 'И': 'N',
+ 'Й': 'N',
+ 'К': 'K',
+ 'Л': 'N',
+ 'М': 'M',
+ 'Н': 'H',
+ 'О': 'O',
+ 'П': 'N',
+ 'Р': 'P',
+ 'С': 'C',
+ 'Т': 'T',
+ 'У': 'y',
+ 'Ф': 'O',
+ 'Х': 'X',
+ 'Ц': 'U',
+ 'Ч': 'h',
+ 'Ш': 'W',
+ 'Щ': 'W',
+ 'Ъ': 'B',
+ 'Ы': 'X',
+ 'Ь': 'B',
+ 'Э': '3',
+ 'Ю': 'X',
+ 'Я': 'R',
+ 'а': 'a',
+ 'б': 'b',
+ 'в': 'a',
+ 'г': 'r',
+ 'д': 'y',
+ 'е': 'e',
+ 'ж': 'm',
+ 'з': 'e',
+ 'и': 'n',
+ 'й': 'n',
+ 'к': 'n',
+ 'л': 'n',
+ 'м': 'm',
+ 'н': 'n',
+ 'о': 'o',
+ 'п': 'n',
+ 'р': 'p',
+ 'с': 'c',
+ 'т': 'o',
+ 'у': 'y',
+ 'ф': 'b',
+ 'х': 'x',
+ 'ц': 'n',
+ 'ч': 'n',
+ 'ш': 'w',
+ 'щ': 'w',
+ 'ъ': 'a',
+ 'ы': 'm',
+ 'ь': 'a',
+ 'э': 'e',
+ 'ю': 'm',
+ 'я': 'r'
+};
+
+/**
+ * This function is a convenience function for looking up information in the
+ * metricMap table. It takes a character as a string, and a style.
+ *
+ * Note: the `width` property may be undefined if fontMetricsData.js wasn't
+ * built using `Make extended_metrics`.
+ */
+var getCharacterMetrics = function(character, style) {
+ var ch = character.charCodeAt(0);
+ if (character[0] in extraCharacterMap) {
+ ch = extraCharacterMap[character[0]].charCodeAt(0);
+ } else if (cjkRegex.test(character[0])) {
+ ch = 'M'.charCodeAt(0);
+ }
+ var metrics = metricMap[style][ch];
+ if (metrics) {
+ return {
+ depth: metrics[0],
+ height: metrics[1],
+ italic: metrics[2],
+ skew: metrics[3],
+ width: metrics[4]
+ };
+ }
+};
+
+module.exports = {
+ metrics: metrics,
+ sigmas: sigmas,
+ getCharacterMetrics: getCharacterMetrics
+};
+
+},{"./Style":9,"./fontMetricsData":18,"./unicodeRegexes":24}],18:[function(require,module,exports){
+module.exports = {
+ "AMS-Regular": {
+ "65": [0, 0.68889, 0, 0],
+ "66": [0, 0.68889, 0, 0],
+ "67": [0, 0.68889, 0, 0],
+ "68": [0, 0.68889, 0, 0],
+ "69": [0, 0.68889, 0, 0],
+ "70": [0, 0.68889, 0, 0],
+ "71": [0, 0.68889, 0, 0],
+ "72": [0, 0.68889, 0, 0],
+ "73": [0, 0.68889, 0, 0],
+ "74": [0.16667, 0.68889, 0, 0],
+ "75": [0, 0.68889, 0, 0],
+ "76": [0, 0.68889, 0, 0],
+ "77": [0, 0.68889, 0, 0],
+ "78": [0, 0.68889, 0, 0],
+ "79": [0.16667, 0.68889, 0, 0],
+ "80": [0, 0.68889, 0, 0],
+ "81": [0.16667, 0.68889, 0, 0],
+ "82": [0, 0.68889, 0, 0],
+ "83": [0, 0.68889, 0, 0],
+ "84": [0, 0.68889, 0, 0],
+ "85": [0, 0.68889, 0, 0],
+ "86": [0, 0.68889, 0, 0],
+ "87": [0, 0.68889, 0, 0],
+ "88": [0, 0.68889, 0, 0],
+ "89": [0, 0.68889, 0, 0],
+ "90": [0, 0.68889, 0, 0],
+ "107": [0, 0.68889, 0, 0],
+ "165": [0, 0.675, 0.025, 0],
+ "174": [0.15559, 0.69224, 0, 0],
+ "240": [0, 0.68889, 0, 0],
+ "295": [0, 0.68889, 0, 0],
+ "710": [0, 0.825, 0, 0],
+ "732": [0, 0.9, 0, 0],
+ "770": [0, 0.825, 0, 0],
+ "771": [0, 0.9, 0, 0],
+ "989": [0.08167, 0.58167, 0, 0],
+ "1008": [0, 0.43056, 0.04028, 0],
+ "8245": [0, 0.54986, 0, 0],
+ "8463": [0, 0.68889, 0, 0],
+ "8487": [0, 0.68889, 0, 0],
+ "8498": [0, 0.68889, 0, 0],
+ "8502": [0, 0.68889, 0, 0],
+ "8503": [0, 0.68889, 0, 0],
+ "8504": [0, 0.68889, 0, 0],
+ "8513": [0, 0.68889, 0, 0],
+ "8592": [-0.03598, 0.46402, 0, 0],
+ "8594": [-0.03598, 0.46402, 0, 0],
+ "8602": [-0.13313, 0.36687, 0, 0],
+ "8603": [-0.13313, 0.36687, 0, 0],
+ "8606": [0.01354, 0.52239, 0, 0],
+ "8608": [0.01354, 0.52239, 0, 0],
+ "8610": [0.01354, 0.52239, 0, 0],
+ "8611": [0.01354, 0.52239, 0, 0],
+ "8619": [0, 0.54986, 0, 0],
+ "8620": [0, 0.54986, 0, 0],
+ "8621": [-0.13313, 0.37788, 0, 0],
+ "8622": [-0.13313, 0.36687, 0, 0],
+ "8624": [0, 0.69224, 0, 0],
+ "8625": [0, 0.69224, 0, 0],
+ "8630": [0, 0.43056, 0, 0],
+ "8631": [0, 0.43056, 0, 0],
+ "8634": [0.08198, 0.58198, 0, 0],
+ "8635": [0.08198, 0.58198, 0, 0],
+ "8638": [0.19444, 0.69224, 0, 0],
+ "8639": [0.19444, 0.69224, 0, 0],
+ "8642": [0.19444, 0.69224, 0, 0],
+ "8643": [0.19444, 0.69224, 0, 0],
+ "8644": [0.1808, 0.675, 0, 0],
+ "8646": [0.1808, 0.675, 0, 0],
+ "8647": [0.1808, 0.675, 0, 0],
+ "8648": [0.19444, 0.69224, 0, 0],
+ "8649": [0.1808, 0.675, 0, 0],
+ "8650": [0.19444, 0.69224, 0, 0],
+ "8651": [0.01354, 0.52239, 0, 0],
+ "8652": [0.01354, 0.52239, 0, 0],
+ "8653": [-0.13313, 0.36687, 0, 0],
+ "8654": [-0.13313, 0.36687, 0, 0],
+ "8655": [-0.13313, 0.36687, 0, 0],
+ "8666": [0.13667, 0.63667, 0, 0],
+ "8667": [0.13667, 0.63667, 0, 0],
+ "8669": [-0.13313, 0.37788, 0, 0],
+ "8672": [-0.064, 0.437, 0, 0],
+ "8674": [-0.064, 0.437, 0, 0],
+ "8705": [0, 0.825, 0, 0],
+ "8708": [0, 0.68889, 0, 0],
+ "8709": [0.08167, 0.58167, 0, 0],
+ "8717": [0, 0.43056, 0, 0],
+ "8722": [-0.03598, 0.46402, 0, 0],
+ "8724": [0.08198, 0.69224, 0, 0],
+ "8726": [0.08167, 0.58167, 0, 0],
+ "8733": [0, 0.69224, 0, 0],
+ "8736": [0, 0.69224, 0, 0],
+ "8737": [0, 0.69224, 0, 0],
+ "8738": [0.03517, 0.52239, 0, 0],
+ "8739": [0.08167, 0.58167, 0, 0],
+ "8740": [0.25142, 0.74111, 0, 0],
+ "8741": [0.08167, 0.58167, 0, 0],
+ "8742": [0.25142, 0.74111, 0, 0],
+ "8756": [0, 0.69224, 0, 0],
+ "8757": [0, 0.69224, 0, 0],
+ "8764": [-0.13313, 0.36687, 0, 0],
+ "8765": [-0.13313, 0.37788, 0, 0],
+ "8769": [-0.13313, 0.36687, 0, 0],
+ "8770": [-0.03625, 0.46375, 0, 0],
+ "8774": [0.30274, 0.79383, 0, 0],
+ "8776": [-0.01688, 0.48312, 0, 0],
+ "8778": [0.08167, 0.58167, 0, 0],
+ "8782": [0.06062, 0.54986, 0, 0],
+ "8783": [0.06062, 0.54986, 0, 0],
+ "8785": [0.08198, 0.58198, 0, 0],
+ "8786": [0.08198, 0.58198, 0, 0],
+ "8787": [0.08198, 0.58198, 0, 0],
+ "8790": [0, 0.69224, 0, 0],
+ "8791": [0.22958, 0.72958, 0, 0],
+ "8796": [0.08198, 0.91667, 0, 0],
+ "8806": [0.25583, 0.75583, 0, 0],
+ "8807": [0.25583, 0.75583, 0, 0],
+ "8808": [0.25142, 0.75726, 0, 0],
+ "8809": [0.25142, 0.75726, 0, 0],
+ "8812": [0.25583, 0.75583, 0, 0],
+ "8814": [0.20576, 0.70576, 0, 0],
+ "8815": [0.20576, 0.70576, 0, 0],
+ "8816": [0.30274, 0.79383, 0, 0],
+ "8817": [0.30274, 0.79383, 0, 0],
+ "8818": [0.22958, 0.72958, 0, 0],
+ "8819": [0.22958, 0.72958, 0, 0],
+ "8822": [0.1808, 0.675, 0, 0],
+ "8823": [0.1808, 0.675, 0, 0],
+ "8828": [0.13667, 0.63667, 0, 0],
+ "8829": [0.13667, 0.63667, 0, 0],
+ "8830": [0.22958, 0.72958, 0, 0],
+ "8831": [0.22958, 0.72958, 0, 0],
+ "8832": [0.20576, 0.70576, 0, 0],
+ "8833": [0.20576, 0.70576, 0, 0],
+ "8840": [0.30274, 0.79383, 0, 0],
+ "8841": [0.30274, 0.79383, 0, 0],
+ "8842": [0.13597, 0.63597, 0, 0],
+ "8843": [0.13597, 0.63597, 0, 0],
+ "8847": [0.03517, 0.54986, 0, 0],
+ "8848": [0.03517, 0.54986, 0, 0],
+ "8858": [0.08198, 0.58198, 0, 0],
+ "8859": [0.08198, 0.58198, 0, 0],
+ "8861": [0.08198, 0.58198, 0, 0],
+ "8862": [0, 0.675, 0, 0],
+ "8863": [0, 0.675, 0, 0],
+ "8864": [0, 0.675, 0, 0],
+ "8865": [0, 0.675, 0, 0],
+ "8872": [0, 0.69224, 0, 0],
+ "8873": [0, 0.69224, 0, 0],
+ "8874": [0, 0.69224, 0, 0],
+ "8876": [0, 0.68889, 0, 0],
+ "8877": [0, 0.68889, 0, 0],
+ "8878": [0, 0.68889, 0, 0],
+ "8879": [0, 0.68889, 0, 0],
+ "8882": [0.03517, 0.54986, 0, 0],
+ "8883": [0.03517, 0.54986, 0, 0],
+ "8884": [0.13667, 0.63667, 0, 0],
+ "8885": [0.13667, 0.63667, 0, 0],
+ "8888": [0, 0.54986, 0, 0],
+ "8890": [0.19444, 0.43056, 0, 0],
+ "8891": [0.19444, 0.69224, 0, 0],
+ "8892": [0.19444, 0.69224, 0, 0],
+ "8901": [0, 0.54986, 0, 0],
+ "8903": [0.08167, 0.58167, 0, 0],
+ "8905": [0.08167, 0.58167, 0, 0],
+ "8906": [0.08167, 0.58167, 0, 0],
+ "8907": [0, 0.69224, 0, 0],
+ "8908": [0, 0.69224, 0, 0],
+ "8909": [-0.03598, 0.46402, 0, 0],
+ "8910": [0, 0.54986, 0, 0],
+ "8911": [0, 0.54986, 0, 0],
+ "8912": [0.03517, 0.54986, 0, 0],
+ "8913": [0.03517, 0.54986, 0, 0],
+ "8914": [0, 0.54986, 0, 0],
+ "8915": [0, 0.54986, 0, 0],
+ "8916": [0, 0.69224, 0, 0],
+ "8918": [0.0391, 0.5391, 0, 0],
+ "8919": [0.0391, 0.5391, 0, 0],
+ "8920": [0.03517, 0.54986, 0, 0],
+ "8921": [0.03517, 0.54986, 0, 0],
+ "8922": [0.38569, 0.88569, 0, 0],
+ "8923": [0.38569, 0.88569, 0, 0],
+ "8926": [0.13667, 0.63667, 0, 0],
+ "8927": [0.13667, 0.63667, 0, 0],
+ "8928": [0.30274, 0.79383, 0, 0],
+ "8929": [0.30274, 0.79383, 0, 0],
+ "8934": [0.23222, 0.74111, 0, 0],
+ "8935": [0.23222, 0.74111, 0, 0],
+ "8936": [0.23222, 0.74111, 0, 0],
+ "8937": [0.23222, 0.74111, 0, 0],
+ "8938": [0.20576, 0.70576, 0, 0],
+ "8939": [0.20576, 0.70576, 0, 0],
+ "8940": [0.30274, 0.79383, 0, 0],
+ "8941": [0.30274, 0.79383, 0, 0],
+ "8994": [0.19444, 0.69224, 0, 0],
+ "8995": [0.19444, 0.69224, 0, 0],
+ "9416": [0.15559, 0.69224, 0, 0],
+ "9484": [0, 0.69224, 0, 0],
+ "9488": [0, 0.69224, 0, 0],
+ "9492": [0, 0.37788, 0, 0],
+ "9496": [0, 0.37788, 0, 0],
+ "9585": [0.19444, 0.68889, 0, 0],
+ "9586": [0.19444, 0.74111, 0, 0],
+ "9632": [0, 0.675, 0, 0],
+ "9633": [0, 0.675, 0, 0],
+ "9650": [0, 0.54986, 0, 0],
+ "9651": [0, 0.54986, 0, 0],
+ "9654": [0.03517, 0.54986, 0, 0],
+ "9660": [0, 0.54986, 0, 0],
+ "9661": [0, 0.54986, 0, 0],
+ "9664": [0.03517, 0.54986, 0, 0],
+ "9674": [0.11111, 0.69224, 0, 0],
+ "9733": [0.19444, 0.69224, 0, 0],
+ "10003": [0, 0.69224, 0, 0],
+ "10016": [0, 0.69224, 0, 0],
+ "10731": [0.11111, 0.69224, 0, 0],
+ "10846": [0.19444, 0.75583, 0, 0],
+ "10877": [0.13667, 0.63667, 0, 0],
+ "10878": [0.13667, 0.63667, 0, 0],
+ "10885": [0.25583, 0.75583, 0, 0],
+ "10886": [0.25583, 0.75583, 0, 0],
+ "10887": [0.13597, 0.63597, 0, 0],
+ "10888": [0.13597, 0.63597, 0, 0],
+ "10889": [0.26167, 0.75726, 0, 0],
+ "10890": [0.26167, 0.75726, 0, 0],
+ "10891": [0.48256, 0.98256, 0, 0],
+ "10892": [0.48256, 0.98256, 0, 0],
+ "10901": [0.13667, 0.63667, 0, 0],
+ "10902": [0.13667, 0.63667, 0, 0],
+ "10933": [0.25142, 0.75726, 0, 0],
+ "10934": [0.25142, 0.75726, 0, 0],
+ "10935": [0.26167, 0.75726, 0, 0],
+ "10936": [0.26167, 0.75726, 0, 0],
+ "10937": [0.26167, 0.75726, 0, 0],
+ "10938": [0.26167, 0.75726, 0, 0],
+ "10949": [0.25583, 0.75583, 0, 0],
+ "10950": [0.25583, 0.75583, 0, 0],
+ "10955": [0.28481, 0.79383, 0, 0],
+ "10956": [0.28481, 0.79383, 0, 0],
+ "57350": [0.08167, 0.58167, 0, 0],
+ "57351": [0.08167, 0.58167, 0, 0],
+ "57352": [0.08167, 0.58167, 0, 0],
+ "57353": [0, 0.43056, 0.04028, 0],
+ "57356": [0.25142, 0.75726, 0, 0],
+ "57357": [0.25142, 0.75726, 0, 0],
+ "57358": [0.41951, 0.91951, 0, 0],
+ "57359": [0.30274, 0.79383, 0, 0],
+ "57360": [0.30274, 0.79383, 0, 0],
+ "57361": [0.41951, 0.91951, 0, 0],
+ "57366": [0.25142, 0.75726, 0, 0],
+ "57367": [0.25142, 0.75726, 0, 0],
+ "57368": [0.25142, 0.75726, 0, 0],
+ "57369": [0.25142, 0.75726, 0, 0],
+ "57370": [0.13597, 0.63597, 0, 0],
+ "57371": [0.13597, 0.63597, 0, 0]
+ },
+ "Caligraphic-Regular": {
+ "48": [0, 0.43056, 0, 0],
+ "49": [0, 0.43056, 0, 0],
+ "50": [0, 0.43056, 0, 0],
+ "51": [0.19444, 0.43056, 0, 0],
+ "52": [0.19444, 0.43056, 0, 0],
+ "53": [0.19444, 0.43056, 0, 0],
+ "54": [0, 0.64444, 0, 0],
+ "55": [0.19444, 0.43056, 0, 0],
+ "56": [0, 0.64444, 0, 0],
+ "57": [0.19444, 0.43056, 0, 0],
+ "65": [0, 0.68333, 0, 0.19445],
+ "66": [0, 0.68333, 0.03041, 0.13889],
+ "67": [0, 0.68333, 0.05834, 0.13889],
+ "68": [0, 0.68333, 0.02778, 0.08334],
+ "69": [0, 0.68333, 0.08944, 0.11111],
+ "70": [0, 0.68333, 0.09931, 0.11111],
+ "71": [0.09722, 0.68333, 0.0593, 0.11111],
+ "72": [0, 0.68333, 0.00965, 0.11111],
+ "73": [0, 0.68333, 0.07382, 0],
+ "74": [0.09722, 0.68333, 0.18472, 0.16667],
+ "75": [0, 0.68333, 0.01445, 0.05556],
+ "76": [0, 0.68333, 0, 0.13889],
+ "77": [0, 0.68333, 0, 0.13889],
+ "78": [0, 0.68333, 0.14736, 0.08334],
+ "79": [0, 0.68333, 0.02778, 0.11111],
+ "80": [0, 0.68333, 0.08222, 0.08334],
+ "81": [0.09722, 0.68333, 0, 0.11111],
+ "82": [0, 0.68333, 0, 0.08334],
+ "83": [0, 0.68333, 0.075, 0.13889],
+ "84": [0, 0.68333, 0.25417, 0],
+ "85": [0, 0.68333, 0.09931, 0.08334],
+ "86": [0, 0.68333, 0.08222, 0],
+ "87": [0, 0.68333, 0.08222, 0.08334],
+ "88": [0, 0.68333, 0.14643, 0.13889],
+ "89": [0.09722, 0.68333, 0.08222, 0.08334],
+ "90": [0, 0.68333, 0.07944, 0.13889]
+ },
+ "Fraktur-Regular": {
+ "33": [0, 0.69141, 0, 0],
+ "34": [0, 0.69141, 0, 0],
+ "38": [0, 0.69141, 0, 0],
+ "39": [0, 0.69141, 0, 0],
+ "40": [0.24982, 0.74947, 0, 0],
+ "41": [0.24982, 0.74947, 0, 0],
+ "42": [0, 0.62119, 0, 0],
+ "43": [0.08319, 0.58283, 0, 0],
+ "44": [0, 0.10803, 0, 0],
+ "45": [0.08319, 0.58283, 0, 0],
+ "46": [0, 0.10803, 0, 0],
+ "47": [0.24982, 0.74947, 0, 0],
+ "48": [0, 0.47534, 0, 0],
+ "49": [0, 0.47534, 0, 0],
+ "50": [0, 0.47534, 0, 0],
+ "51": [0.18906, 0.47534, 0, 0],
+ "52": [0.18906, 0.47534, 0, 0],
+ "53": [0.18906, 0.47534, 0, 0],
+ "54": [0, 0.69141, 0, 0],
+ "55": [0.18906, 0.47534, 0, 0],
+ "56": [0, 0.69141, 0, 0],
+ "57": [0.18906, 0.47534, 0, 0],
+ "58": [0, 0.47534, 0, 0],
+ "59": [0.12604, 0.47534, 0, 0],
+ "61": [-0.13099, 0.36866, 0, 0],
+ "63": [0, 0.69141, 0, 0],
+ "65": [0, 0.69141, 0, 0],
+ "66": [0, 0.69141, 0, 0],
+ "67": [0, 0.69141, 0, 0],
+ "68": [0, 0.69141, 0, 0],
+ "69": [0, 0.69141, 0, 0],
+ "70": [0.12604, 0.69141, 0, 0],
+ "71": [0, 0.69141, 0, 0],
+ "72": [0.06302, 0.69141, 0, 0],
+ "73": [0, 0.69141, 0, 0],
+ "74": [0.12604, 0.69141, 0, 0],
+ "75": [0, 0.69141, 0, 0],
+ "76": [0, 0.69141, 0, 0],
+ "77": [0, 0.69141, 0, 0],
+ "78": [0, 0.69141, 0, 0],
+ "79": [0, 0.69141, 0, 0],
+ "80": [0.18906, 0.69141, 0, 0],
+ "81": [0.03781, 0.69141, 0, 0],
+ "82": [0, 0.69141, 0, 0],
+ "83": [0, 0.69141, 0, 0],
+ "84": [0, 0.69141, 0, 0],
+ "85": [0, 0.69141, 0, 0],
+ "86": [0, 0.69141, 0, 0],
+ "87": [0, 0.69141, 0, 0],
+ "88": [0, 0.69141, 0, 0],
+ "89": [0.18906, 0.69141, 0, 0],
+ "90": [0.12604, 0.69141, 0, 0],
+ "91": [0.24982, 0.74947, 0, 0],
+ "93": [0.24982, 0.74947, 0, 0],
+ "94": [0, 0.69141, 0, 0],
+ "97": [0, 0.47534, 0, 0],
+ "98": [0, 0.69141, 0, 0],
+ "99": [0, 0.47534, 0, 0],
+ "100": [0, 0.62119, 0, 0],
+ "101": [0, 0.47534, 0, 0],
+ "102": [0.18906, 0.69141, 0, 0],
+ "103": [0.18906, 0.47534, 0, 0],
+ "104": [0.18906, 0.69141, 0, 0],
+ "105": [0, 0.69141, 0, 0],
+ "106": [0, 0.69141, 0, 0],
+ "107": [0, 0.69141, 0, 0],
+ "108": [0, 0.69141, 0, 0],
+ "109": [0, 0.47534, 0, 0],
+ "110": [0, 0.47534, 0, 0],
+ "111": [0, 0.47534, 0, 0],
+ "112": [0.18906, 0.52396, 0, 0],
+ "113": [0.18906, 0.47534, 0, 0],
+ "114": [0, 0.47534, 0, 0],
+ "115": [0, 0.47534, 0, 0],
+ "116": [0, 0.62119, 0, 0],
+ "117": [0, 0.47534, 0, 0],
+ "118": [0, 0.52396, 0, 0],
+ "119": [0, 0.52396, 0, 0],
+ "120": [0.18906, 0.47534, 0, 0],
+ "121": [0.18906, 0.47534, 0, 0],
+ "122": [0.18906, 0.47534, 0, 0],
+ "8216": [0, 0.69141, 0, 0],
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+ "91": [1.25003, 1.75, 0, 0],
+ "92": [1.25003, 1.75, 0, 0],
+ "93": [1.25003, 1.75, 0, 0],
+ "123": [1.25003, 1.75, 0, 0],
+ "125": [1.25003, 1.75, 0, 0],
+ "710": [0, 0.825, 0, 0],
+ "732": [0, 0.825, 0, 0],
+ "770": [0, 0.825, 0, 0],
+ "771": [0, 0.825, 0, 0],
+ "8730": [1.25003, 1.75, 0, 0],
+ "8968": [1.25003, 1.75, 0, 0],
+ "8969": [1.25003, 1.75, 0, 0],
+ "8970": [1.25003, 1.75, 0, 0],
+ "8971": [1.25003, 1.75, 0, 0],
+ "9115": [0.64502, 1.155, 0, 0],
+ "9116": [1e-05, 0.6, 0, 0],
+ "9117": [0.64502, 1.155, 0, 0],
+ "9118": [0.64502, 1.155, 0, 0],
+ "9119": [1e-05, 0.6, 0, 0],
+ "9120": [0.64502, 1.155, 0, 0],
+ "9121": [0.64502, 1.155, 0, 0],
+ "9122": [-0.00099, 0.601, 0, 0],
+ "9123": [0.64502, 1.155, 0, 0],
+ "9124": [0.64502, 1.155, 0, 0],
+ "9125": [-0.00099, 0.601, 0, 0],
+ "9126": [0.64502, 1.155, 0, 0],
+ "9127": [1e-05, 0.9, 0, 0],
+ "9128": [0.65002, 1.15, 0, 0],
+ "9129": [0.90001, 0, 0, 0],
+ "9130": [0, 0.3, 0, 0],
+ "9131": [1e-05, 0.9, 0, 0],
+ "9132": [0.65002, 1.15, 0, 0],
+ "9133": [0.90001, 0, 0, 0],
+ "9143": [0.88502, 0.915, 0, 0],
+ "10216": [1.25003, 1.75, 0, 0],
+ "10217": [1.25003, 1.75, 0, 0],
+ "57344": [-0.00499, 0.605, 0, 0],
+ "57345": [-0.00499, 0.605, 0, 0],
+ "57680": [0, 0.12, 0, 0],
+ "57681": [0, 0.12, 0, 0],
+ "57682": [0, 0.12, 0, 0],
+ "57683": [0, 0.12, 0, 0]
+ },
+ "Typewriter-Regular": {
+ "33": [0, 0.61111, 0, 0],
+ "34": [0, 0.61111, 0, 0],
+ "35": [0, 0.61111, 0, 0],
+ "36": [0.08333, 0.69444, 0, 0],
+ "37": [0.08333, 0.69444, 0, 0],
+ "38": [0, 0.61111, 0, 0],
+ "39": [0, 0.61111, 0, 0],
+ "40": [0.08333, 0.69444, 0, 0],
+ "41": [0.08333, 0.69444, 0, 0],
+ "42": [0, 0.52083, 0, 0],
+ "43": [-0.08056, 0.53055, 0, 0],
+ "44": [0.13889, 0.125, 0, 0],
+ "45": [-0.08056, 0.53055, 0, 0],
+ "46": [0, 0.125, 0, 0],
+ "47": [0.08333, 0.69444, 0, 0],
+ "48": [0, 0.61111, 0, 0],
+ "49": [0, 0.61111, 0, 0],
+ "50": [0, 0.61111, 0, 0],
+ "51": [0, 0.61111, 0, 0],
+ "52": [0, 0.61111, 0, 0],
+ "53": [0, 0.61111, 0, 0],
+ "54": [0, 0.61111, 0, 0],
+ "55": [0, 0.61111, 0, 0],
+ "56": [0, 0.61111, 0, 0],
+ "57": [0, 0.61111, 0, 0],
+ "58": [0, 0.43056, 0, 0],
+ "59": [0.13889, 0.43056, 0, 0],
+ "60": [-0.05556, 0.55556, 0, 0],
+ "61": [-0.19549, 0.41562, 0, 0],
+ "62": [-0.05556, 0.55556, 0, 0],
+ "63": [0, 0.61111, 0, 0],
+ "64": [0, 0.61111, 0, 0],
+ "65": [0, 0.61111, 0, 0],
+ "66": [0, 0.61111, 0, 0],
+ "67": [0, 0.61111, 0, 0],
+ "68": [0, 0.61111, 0, 0],
+ "69": [0, 0.61111, 0, 0],
+ "70": [0, 0.61111, 0, 0],
+ "71": [0, 0.61111, 0, 0],
+ "72": [0, 0.61111, 0, 0],
+ "73": [0, 0.61111, 0, 0],
+ "74": [0, 0.61111, 0, 0],
+ "75": [0, 0.61111, 0, 0],
+ "76": [0, 0.61111, 0, 0],
+ "77": [0, 0.61111, 0, 0],
+ "78": [0, 0.61111, 0, 0],
+ "79": [0, 0.61111, 0, 0],
+ "80": [0, 0.61111, 0, 0],
+ "81": [0.13889, 0.61111, 0, 0],
+ "82": [0, 0.61111, 0, 0],
+ "83": [0, 0.61111, 0, 0],
+ "84": [0, 0.61111, 0, 0],
+ "85": [0, 0.61111, 0, 0],
+ "86": [0, 0.61111, 0, 0],
+ "87": [0, 0.61111, 0, 0],
+ "88": [0, 0.61111, 0, 0],
+ "89": [0, 0.61111, 0, 0],
+ "90": [0, 0.61111, 0, 0],
+ "91": [0.08333, 0.69444, 0, 0],
+ "92": [0.08333, 0.69444, 0, 0],
+ "93": [0.08333, 0.69444, 0, 0],
+ "94": [0, 0.61111, 0, 0],
+ "95": [0.09514, 0, 0, 0],
+ "96": [0, 0.61111, 0, 0],
+ "97": [0, 0.43056, 0, 0],
+ "98": [0, 0.61111, 0, 0],
+ "99": [0, 0.43056, 0, 0],
+ "100": [0, 0.61111, 0, 0],
+ "101": [0, 0.43056, 0, 0],
+ "102": [0, 0.61111, 0, 0],
+ "103": [0.22222, 0.43056, 0, 0],
+ "104": [0, 0.61111, 0, 0],
+ "105": [0, 0.61111, 0, 0],
+ "106": [0.22222, 0.61111, 0, 0],
+ "107": [0, 0.61111, 0, 0],
+ "108": [0, 0.61111, 0, 0],
+ "109": [0, 0.43056, 0, 0],
+ "110": [0, 0.43056, 0, 0],
+ "111": [0, 0.43056, 0, 0],
+ "112": [0.22222, 0.43056, 0, 0],
+ "113": [0.22222, 0.43056, 0, 0],
+ "114": [0, 0.43056, 0, 0],
+ "115": [0, 0.43056, 0, 0],
+ "116": [0, 0.55358, 0, 0],
+ "117": [0, 0.43056, 0, 0],
+ "118": [0, 0.43056, 0, 0],
+ "119": [0, 0.43056, 0, 0],
+ "120": [0, 0.43056, 0, 0],
+ "121": [0.22222, 0.43056, 0, 0],
+ "122": [0, 0.43056, 0, 0],
+ "123": [0.08333, 0.69444, 0, 0],
+ "124": [0.08333, 0.69444, 0, 0],
+ "125": [0.08333, 0.69444, 0, 0],
+ "126": [0, 0.61111, 0, 0],
+ "127": [0, 0.61111, 0, 0],
+ "305": [0, 0.43056, 0, 0],
+ "567": [0.22222, 0.43056, 0, 0],
+ "768": [0, 0.61111, 0, 0],
+ "769": [0, 0.61111, 0, 0],
+ "770": [0, 0.61111, 0, 0],
+ "771": [0, 0.61111, 0, 0],
+ "772": [0, 0.56555, 0, 0],
+ "774": [0, 0.61111, 0, 0],
+ "776": [0, 0.61111, 0, 0],
+ "778": [0, 0.61111, 0, 0],
+ "780": [0, 0.56597, 0, 0],
+ "915": [0, 0.61111, 0, 0],
+ "916": [0, 0.61111, 0, 0],
+ "920": [0, 0.61111, 0, 0],
+ "923": [0, 0.61111, 0, 0],
+ "926": [0, 0.61111, 0, 0],
+ "928": [0, 0.61111, 0, 0],
+ "931": [0, 0.61111, 0, 0],
+ "933": [0, 0.61111, 0, 0],
+ "934": [0, 0.61111, 0, 0],
+ "936": [0, 0.61111, 0, 0],
+ "937": [0, 0.61111, 0, 0],
+ "2018": [0, 0.61111, 0, 0],
+ "2019": [0, 0.61111, 0, 0],
+ "8242": [0, 0.61111, 0, 0]
+ }
+};
+
+},{}],19:[function(require,module,exports){
+var utils = require("./utils");
+var ParseError = require("./ParseError");
+var parseData = require("./parseData");
+var ParseNode = parseData.ParseNode;
+
+/* This file contains a list of functions that we parse, identified by
+ * the calls to defineFunction.
+ *
+ * The first argument to defineFunction is a single name or a list of names.
+ * All functions named in such a list will share a single implementation.
+ *
+ * Each declared function can have associated properties, which
+ * include the following:
+ *
+ * - numArgs: The number of arguments the function takes.
+ * If this is the only property, it can be passed as a number
+ * instead of an element of a properties object.
+ * - argTypes: (optional) An array corresponding to each argument of the
+ * function, giving the type of argument that should be parsed. Its
+ * length should be equal to `numArgs + numOptionalArgs`. Valid
+ * types:
+ * - "size": A size-like thing, such as "1em" or "5ex"
+ * - "color": An html color, like "#abc" or "blue"
+ * - "original": The same type as the environment that the
+ * function being parsed is in (e.g. used for the
+ * bodies of functions like \color where the first
+ * argument is special and the second argument is
+ * parsed normally)
+ * Other possible types (probably shouldn't be used)
+ * - "text": Text-like (e.g. \text)
+ * - "math": Normal math
+ * If undefined, this will be treated as an appropriate length
+ * array of "original" strings
+ * - greediness: (optional) The greediness of the function to use ungrouped
+ * arguments.
+ *
+ * E.g. if you have an expression
+ * \sqrt \frac 1 2
+ * since \frac has greediness=2 vs \sqrt's greediness=1, \frac
+ * will use the two arguments '1' and '2' as its two arguments,
+ * then that whole function will be used as the argument to
+ * \sqrt. On the other hand, the expressions
+ * \frac \frac 1 2 3
+ * and
+ * \frac \sqrt 1 2
+ * will fail because \frac and \frac have equal greediness
+ * and \sqrt has a lower greediness than \frac respectively. To
+ * make these parse, we would have to change them to:
+ * \frac {\frac 1 2} 3
+ * and
+ * \frac {\sqrt 1} 2
+ *
+ * The default value is `1`
+ * - allowedInText: (optional) Whether or not the function is allowed inside
+ * text mode (default false)
+ * - numOptionalArgs: (optional) The number of optional arguments the function
+ * should parse. If the optional arguments aren't found,
+ * `null` will be passed to the handler in their place.
+ * (default 0)
+ * - infix: (optional) Must be true if the function is an infix operator.
+ *
+ * The last argument is that implementation, the handler for the function(s).
+ * It is called to handle these functions and their arguments.
+ * It receives two arguments:
+ * - context contains information and references provided by the parser
+ * - args is an array of arguments obtained from TeX input
+ * The context contains the following properties:
+ * - funcName: the text (i.e. name) of the function, including \
+ * - parser: the parser object
+ * - lexer: the lexer object
+ * - positions: the positions in the overall string of the function
+ * and the arguments.
+ * The latter three should only be used to produce error messages.
+ *
+ * The function should return an object with the following keys:
+ * - type: The type of element that this is. This is then used in
+ * buildHTML/buildMathML to determine which function
+ * should be called to build this node into a DOM node
+ * Any other data can be added to the object, which will be passed
+ * in to the function in buildHTML/buildMathML as `group.value`.
+ */
+
+function defineFunction(names, props, handler) {
+ if (typeof names === "string") {
+ names = [names];
+ }
+ if (typeof props === "number") {
+ props = { numArgs: props };
+ }
+ // Set default values of functions
+ var data = {
+ numArgs: props.numArgs,
+ argTypes: props.argTypes,
+ greediness: (props.greediness === undefined) ? 1 : props.greediness,
+ allowedInText: !!props.allowedInText,
+ numOptionalArgs: props.numOptionalArgs || 0,
+ infix: !!props.infix,
+ handler: handler
+ };
+ for (var i = 0; i < names.length; ++i) {
+ module.exports[names[i]] = data;
+ }
+}
+
+// Since the corresponding buildHTML/buildMathML function expects a
+// list of elements, we normalize for different kinds of arguments
+var ordargument = function(arg) {
+ if (arg.type === "ordgroup") {
+ return arg.value;
+ } else {
+ return [arg];
+ }
+};
+
+// A normal square root
+defineFunction("\\sqrt", {
+ numArgs: 1,
+ numOptionalArgs: 1
+}, function(context, args) {
+ var index = args[0];
+ var body = args[1];
+ return {
+ type: "sqrt",
+ body: body,
+ index: index
+ };
+});
+
+// Non-mathy text, possibly in a font
+var textFunctionStyles = {
+ "\\text": undefined, "\\textrm": "mathrm", "\\textsf": "mathsf",
+ "\\texttt": "mathtt", "\\textnormal": "mathrm", "\\textbf": "mathbf",
+ "\\textit": "textit"
+};
+
+defineFunction([
+ "\\text", "\\textrm", "\\textsf", "\\texttt", "\\textnormal",
+ "\\textbf", "\\textit"
+], {
+ numArgs: 1,
+ argTypes: ["text"],
+ greediness: 2,
+ allowedInText: true
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "text",
+ body: ordargument(body),
+ style: textFunctionStyles[context.funcName]
+ };
+});
+
+// A two-argument custom color
+defineFunction("\\color", {
+ numArgs: 2,
+ allowedInText: true,
+ greediness: 3,
+ argTypes: ["color", "original"]
+}, function(context, args) {
+ var color = args[0];
+ var body = args[1];
+ return {
+ type: "color",
+ color: color.value,
+ value: ordargument(body)
+ };
+});
+
+// An overline
+defineFunction("\\overline", {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "overline",
+ body: body
+ };
+});
+
+// An underline
+defineFunction("\\underline", {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "underline",
+ body: body
+ };
+});
+
+// A box of the width and height
+defineFunction("\\rule", {
+ numArgs: 2,
+ numOptionalArgs: 1,
+ argTypes: ["size", "size", "size"]
+}, function(context, args) {
+ var shift = args[0];
+ var width = args[1];
+ var height = args[2];
+ return {
+ type: "rule",
+ shift: shift && shift.value,
+ width: width.value,
+ height: height.value
+ };
+});
+
+// TODO: In TeX, \mkern only accepts mu-units, and \kern does not accept
+// mu-units. In current KaTeX we relax this; both commands accept any unit.
+defineFunction(["\\kern", "\\mkern"], {
+ numArgs: 1,
+ argTypes: ["size"]
+}, function(context, args) {
+ return {
+ type: "kern",
+ dimension: args[0].value
+ };
+});
+
+// A KaTeX logo
+defineFunction("\\KaTeX", {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "katex"
+ };
+});
+
+defineFunction("\\phantom", {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "phantom",
+ value: ordargument(body)
+ };
+});
+
+// Math class commands except \mathop
+defineFunction([
+ "\\mathord", "\\mathbin", "\\mathrel", "\\mathopen",
+ "\\mathclose", "\\mathpunct", "\\mathinner"
+], {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "mclass",
+ mclass: "m" + context.funcName.substr(5),
+ value: ordargument(body)
+ };
+});
+
+// Build a relation by placing one symbol on top of another
+defineFunction("\\stackrel", {
+ numArgs: 2
+}, function(context, args) {
+ var top = args[0];
+ var bottom = args[1];
+
+ var bottomop = new ParseNode("op", {
+ type: "op",
+ limits: true,
+ alwaysHandleSupSub: true,
+ symbol: false,
+ value: ordargument(bottom)
+ }, bottom.mode);
+
+ var supsub = new ParseNode("supsub", {
+ base: bottomop,
+ sup: top,
+ sub: null
+ }, top.mode);
+
+ return {
+ type: "mclass",
+ mclass: "mrel",
+ value: [supsub]
+ };
+});
+
+// \mod-type functions
+defineFunction("\\bmod", {
+ numArgs: 0
+}, function(context, args) {
+ return {
+ type: "mod",
+ modType: "bmod",
+ value: null
+ };
+});
+
+defineFunction(["\\pod", "\\pmod", "\\mod"], {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "mod",
+ modType: context.funcName.substr(1),
+ value: ordargument(body)
+ };
+});
+
+// Extra data needed for the delimiter handler down below
+var delimiterSizes = {
+ "\\bigl" : {mclass: "mopen", size: 1},
+ "\\Bigl" : {mclass: "mopen", size: 2},
+ "\\biggl": {mclass: "mopen", size: 3},
+ "\\Biggl": {mclass: "mopen", size: 4},
+ "\\bigr" : {mclass: "mclose", size: 1},
+ "\\Bigr" : {mclass: "mclose", size: 2},
+ "\\biggr": {mclass: "mclose", size: 3},
+ "\\Biggr": {mclass: "mclose", size: 4},
+ "\\bigm" : {mclass: "mrel", size: 1},
+ "\\Bigm" : {mclass: "mrel", size: 2},
+ "\\biggm": {mclass: "mrel", size: 3},
+ "\\Biggm": {mclass: "mrel", size: 4},
+ "\\big" : {mclass: "mord", size: 1},
+ "\\Big" : {mclass: "mord", size: 2},
+ "\\bigg" : {mclass: "mord", size: 3},
+ "\\Bigg" : {mclass: "mord", size: 4}
+};
+
+var delimiters = [
+ "(", ")", "[", "\\lbrack", "]", "\\rbrack",
+ "\\{", "\\lbrace", "\\}", "\\rbrace",
+ "\\lfloor", "\\rfloor", "\\lceil", "\\rceil",
+ "<", ">", "\\langle", "\\rangle", "\\lt", "\\gt",
+ "\\lvert", "\\rvert", "\\lVert", "\\rVert",
+ "\\lgroup", "\\rgroup", "\\lmoustache", "\\rmoustache",
+ "/", "\\backslash",
+ "|", "\\vert", "\\|", "\\Vert",
+ "\\uparrow", "\\Uparrow",
+ "\\downarrow", "\\Downarrow",
+ "\\updownarrow", "\\Updownarrow",
+ "."
+];
+
+var fontAliases = {
+ "\\Bbb": "\\mathbb",
+ "\\bold": "\\mathbf",
+ "\\frak": "\\mathfrak"
+};
+
+// Single-argument color functions
+defineFunction([
+ "\\blue", "\\orange", "\\pink", "\\red",
+ "\\green", "\\gray", "\\purple",
+ "\\blueA", "\\blueB", "\\blueC", "\\blueD", "\\blueE",
+ "\\tealA", "\\tealB", "\\tealC", "\\tealD", "\\tealE",
+ "\\greenA", "\\greenB", "\\greenC", "\\greenD", "\\greenE",
+ "\\goldA", "\\goldB", "\\goldC", "\\goldD", "\\goldE",
+ "\\redA", "\\redB", "\\redC", "\\redD", "\\redE",
+ "\\maroonA", "\\maroonB", "\\maroonC", "\\maroonD", "\\maroonE",
+ "\\purpleA", "\\purpleB", "\\purpleC", "\\purpleD", "\\purpleE",
+ "\\mintA", "\\mintB", "\\mintC",
+ "\\grayA", "\\grayB", "\\grayC", "\\grayD", "\\grayE",
+ "\\grayF", "\\grayG", "\\grayH", "\\grayI",
+ "\\kaBlue", "\\kaGreen"
+], {
+ numArgs: 1,
+ allowedInText: true,
+ greediness: 3
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "color",
+ color: "katex-" + context.funcName.slice(1),
+ value: ordargument(body)
+ };
+});
+
+// There are 2 flags for operators; whether they produce limits in
+// displaystyle, and whether they are symbols and should grow in
+// displaystyle. These four groups cover the four possible choices.
+
+// No limits, not symbols
+defineFunction([
+ "\\arcsin", "\\arccos", "\\arctan", "\\arg", "\\cos", "\\cosh",
+ "\\cot", "\\coth", "\\csc", "\\deg", "\\dim", "\\exp", "\\hom",
+ "\\ker", "\\lg", "\\ln", "\\log", "\\sec", "\\sin", "\\sinh",
+ "\\tan", "\\tanh"
+], {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "op",
+ limits: false,
+ symbol: false,
+ body: context.funcName
+ };
+});
+
+// Limits, not symbols
+defineFunction([
+ "\\det", "\\gcd", "\\inf", "\\lim", "\\liminf", "\\limsup", "\\max",
+ "\\min", "\\Pr", "\\sup"
+], {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "op",
+ limits: true,
+ symbol: false,
+ body: context.funcName
+ };
+});
+
+// No limits, symbols
+defineFunction([
+ "\\int", "\\iint", "\\iiint", "\\oint"
+], {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "op",
+ limits: false,
+ symbol: true,
+ body: context.funcName
+ };
+});
+
+// Limits, symbols
+defineFunction([
+ "\\coprod", "\\bigvee", "\\bigwedge", "\\biguplus", "\\bigcap",
+ "\\bigcup", "\\intop", "\\prod", "\\sum", "\\bigotimes",
+ "\\bigoplus", "\\bigodot", "\\bigsqcup", "\\smallint"
+], {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "op",
+ limits: true,
+ symbol: true,
+ body: context.funcName
+ };
+});
+
+// \mathop class command
+defineFunction("\\mathop", {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "op",
+ limits: false,
+ symbol: false,
+ value: ordargument(body)
+ };
+});
+
+// Fractions
+defineFunction([
+ "\\dfrac", "\\frac", "\\tfrac",
+ "\\dbinom", "\\binom", "\\tbinom",
+ "\\\\atopfrac" // can’t be entered directly
+], {
+ numArgs: 2,
+ greediness: 2
+}, function(context, args) {
+ var numer = args[0];
+ var denom = args[1];
+ var hasBarLine;
+ var leftDelim = null;
+ var rightDelim = null;
+ var size = "auto";
+
+ switch (context.funcName) {
+ case "\\dfrac":
+ case "\\frac":
+ case "\\tfrac":
+ hasBarLine = true;
+ break;
+ case "\\\\atopfrac":
+ hasBarLine = false;
+ break;
+ case "\\dbinom":
+ case "\\binom":
+ case "\\tbinom":
+ hasBarLine = false;
+ leftDelim = "(";
+ rightDelim = ")";
+ break;
+ default:
+ throw new Error("Unrecognized genfrac command");
+ }
+
+ switch (context.funcName) {
+ case "\\dfrac":
+ case "\\dbinom":
+ size = "display";
+ break;
+ case "\\tfrac":
+ case "\\tbinom":
+ size = "text";
+ break;
+ }
+
+ return {
+ type: "genfrac",
+ numer: numer,
+ denom: denom,
+ hasBarLine: hasBarLine,
+ leftDelim: leftDelim,
+ rightDelim: rightDelim,
+ size: size
+ };
+});
+
+// Left and right overlap functions
+defineFunction(["\\llap", "\\rlap"], {
+ numArgs: 1,
+ allowedInText: true
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: context.funcName.slice(1),
+ body: body
+ };
+});
+
+// Delimiter functions
+var checkDelimiter = function(delim, context) {
+ if (utils.contains(delimiters, delim.value)) {
+ return delim;
+ } else {
+ throw new ParseError(
+ "Invalid delimiter: '" + delim.value + "' after '" +
+ context.funcName + "'", delim);
+ }
+};
+
+defineFunction([
+ "\\bigl", "\\Bigl", "\\biggl", "\\Biggl",
+ "\\bigr", "\\Bigr", "\\biggr", "\\Biggr",
+ "\\bigm", "\\Bigm", "\\biggm", "\\Biggm",
+ "\\big", "\\Big", "\\bigg", "\\Bigg"
+], {
+ numArgs: 1
+}, function(context, args) {
+ var delim = checkDelimiter(args[0], context);
+
+ return {
+ type: "delimsizing",
+ size: delimiterSizes[context.funcName].size,
+ mclass: delimiterSizes[context.funcName].mclass,
+ value: delim.value
+ };
+});
+
+defineFunction([
+ "\\left", "\\right"
+], {
+ numArgs: 1
+}, function(context, args) {
+ var delim = checkDelimiter(args[0], context);
+
+ // \left and \right are caught somewhere in Parser.js, which is
+ // why this data doesn't match what is in buildHTML.
+ return {
+ type: "leftright",
+ value: delim.value
+ };
+});
+
+defineFunction("\\middle", {
+ numArgs: 1
+}, function(context, args) {
+ var delim = checkDelimiter(args[0], context);
+ if (!context.parser.leftrightDepth) {
+ throw new ParseError("\\middle without preceding \\left", delim);
+ }
+
+ return {
+ type: "middle",
+ value: delim.value
+ };
+});
+
+// Sizing functions (handled in Parser.js explicitly, hence no handler)
+defineFunction([
+ "\\tiny", "\\scriptsize", "\\footnotesize", "\\small",
+ "\\normalsize", "\\large", "\\Large", "\\LARGE", "\\huge", "\\Huge"
+], 0, null);
+
+// Style changing functions (handled in Parser.js explicitly, hence no
+// handler)
+defineFunction([
+ "\\displaystyle", "\\textstyle", "\\scriptstyle",
+ "\\scriptscriptstyle"
+], 0, null);
+
+defineFunction([
+ // styles
+ "\\mathrm", "\\mathit", "\\mathbf",
+
+ // families
+ "\\mathbb", "\\mathcal", "\\mathfrak", "\\mathscr", "\\mathsf",
+ "\\mathtt",
+
+ // aliases
+ "\\Bbb", "\\bold", "\\frak"
+], {
+ numArgs: 1,
+ greediness: 2
+}, function(context, args) {
+ var body = args[0];
+ var func = context.funcName;
+ if (func in fontAliases) {
+ func = fontAliases[func];
+ }
+ return {
+ type: "font",
+ font: func.slice(1),
+ body: body
+ };
+});
+
+// Accents
+defineFunction([
+ "\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve",
+ "\\check", "\\hat", "\\vec", "\\dot"
+ // We don't support expanding accents yet
+ // "\\widetilde", "\\widehat"
+], {
+ numArgs: 1
+}, function(context, args) {
+ var base = args[0];
+ return {
+ type: "accent",
+ accent: context.funcName,
+ base: base
+ };
+});
+
+// Infix generalized fractions
+defineFunction(["\\over", "\\choose", "\\atop"], {
+ numArgs: 0,
+ infix: true
+}, function(context) {
+ var replaceWith;
+ switch (context.funcName) {
+ case "\\over":
+ replaceWith = "\\frac";
+ break;
+ case "\\choose":
+ replaceWith = "\\binom";
+ break;
+ case "\\atop":
+ replaceWith = "\\\\atopfrac";
+ break;
+ default:
+ throw new Error("Unrecognized infix genfrac command");
+ }
+ return {
+ type: "infix",
+ replaceWith: replaceWith,
+ token: context.token
+ };
+});
+
+// Row breaks for aligned data
+defineFunction(["\\\\", "\\cr"], {
+ numArgs: 0,
+ numOptionalArgs: 1,
+ argTypes: ["size"]
+}, function(context, args) {
+ var size = args[0];
+ return {
+ type: "cr",
+ size: size
+ };
+});
+
+// Environment delimiters
+defineFunction(["\\begin", "\\end"], {
+ numArgs: 1,
+ argTypes: ["text"]
+}, function(context, args) {
+ var nameGroup = args[0];
+ if (nameGroup.type !== "ordgroup") {
+ throw new ParseError("Invalid environment name", nameGroup);
+ }
+ var name = "";
+ for (var i = 0; i < nameGroup.value.length; ++i) {
+ name += nameGroup.value[i].value;
+ }
+ return {
+ type: "environment",
+ name: name,
+ nameGroup: nameGroup
+ };
+});
+
+},{"./ParseError":6,"./parseData":21,"./utils":25}],20:[function(require,module,exports){
+/**
+ * These objects store data about MathML nodes. This is the MathML equivalent
+ * of the types in domTree.js. Since MathML handles its own rendering, and
+ * since we're mainly using MathML to improve accessibility, we don't manage
+ * any of the styling state that the plain DOM nodes do.
+ *
+ * The `toNode` and `toMarkup` functions work simlarly to how they do in
+ * domTree.js, creating namespaced DOM nodes and HTML text markup respectively.
+ */
+
+var utils = require("./utils");
+
+/**
+ * This node represents a general purpose MathML node of any type. The
+ * constructor requires the type of node to create (for example, `"mo"` or
+ * `"mspace"`, corresponding to `<mo>` and `<mspace>` tags).
+ */
+function MathNode(type, children) {
+ this.type = type;
+ this.attributes = {};
+ this.children = children || [];
+}
+
+/**
+ * Sets an attribute on a MathML node. MathML depends on attributes to convey a
+ * semantic content, so this is used heavily.
+ */
+MathNode.prototype.setAttribute = function(name, value) {
+ this.attributes[name] = value;
+};
+
+/**
+ * Converts the math node into a MathML-namespaced DOM element.
+ */
+MathNode.prototype.toNode = function() {
+ var node = document.createElementNS(
+ "http://www.w3.org/1998/Math/MathML", this.type);
+
+ for (var attr in this.attributes) {
+ if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+ node.setAttribute(attr, this.attributes[attr]);
+ }
+ }
+
+ for (var i = 0; i < this.children.length; i++) {
+ node.appendChild(this.children[i].toNode());
+ }
+
+ return node;
+};
+
+/**
+ * Converts the math node into an HTML markup string.
+ */
+MathNode.prototype.toMarkup = function() {
+ var markup = "<" + this.type;
+
+ // Add the attributes
+ for (var attr in this.attributes) {
+ if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+ markup += " " + attr + "=\"";
+ markup += utils.escape(this.attributes[attr]);
+ markup += "\"";
+ }
+ }
+
+ markup += ">";
+
+ for (var i = 0; i < this.children.length; i++) {
+ markup += this.children[i].toMarkup();
+ }
+
+ markup += "</" + this.type + ">";
+
+ return markup;
+};
+
+/**
+ * This node represents a piece of text.
+ */
+function TextNode(text) {
+ this.text = text;
+}
+
+/**
+ * Converts the text node into a DOM text node.
+ */
+TextNode.prototype.toNode = function() {
+ return document.createTextNode(this.text);
+};
+
+/**
+ * Converts the text node into HTML markup (which is just the text itself).
+ */
+TextNode.prototype.toMarkup = function() {
+ return utils.escape(this.text);
+};
+
+module.exports = {
+ MathNode: MathNode,
+ TextNode: TextNode
+};
+
+},{"./utils":25}],21:[function(require,module,exports){
+/**
+ * The resulting parse tree nodes of the parse tree.
+ *
+ * It is possible to provide position information, so that a ParseNode can
+ * fulfil a role similar to a Token in error reporting.
+ * For details on the corresponding properties see Token constructor.
+ * Providing such information can lead to better error reporting.
+ *
+ * @param {string} type type of node, like e.g. "ordgroup"
+ * @param {?object} value type-specific representation of the node
+ * @param {string} mode parse mode in action for this node,
+ * "math" or "text"
+ * @param {Token=} firstToken first token of the input for this node,
+ * will omit position information if unset
+ * @param {Token=} lastToken last token of the input for this node,
+ * will default to firstToken if unset
+ */
+function ParseNode(type, value, mode, firstToken, lastToken) {
+ this.type = type;
+ this.value = value;
+ this.mode = mode;
+ if (firstToken && (!lastToken || lastToken.lexer === firstToken.lexer)) {
+ this.lexer = firstToken.lexer;
+ this.start = firstToken.start;
+ this.end = (lastToken || firstToken).end;
+ }
+}
+
+module.exports = {
+ ParseNode: ParseNode
+};
+
+
+},{}],22:[function(require,module,exports){
+/**
+ * Provides a single function for parsing an expression using a Parser
+ * TODO(emily): Remove this
+ */
+
+var Parser = require("./Parser");
+
+/**
+ * Parses an expression using a Parser, then returns the parsed result.
+ */
+var parseTree = function(toParse, settings) {
+ if (!(typeof toParse === 'string' || toParse instanceof String)) {
+ throw new TypeError('KaTeX can only parse string typed expression');
+ }
+ var parser = new Parser(toParse, settings);
+
+ return parser.parse();
+};
+
+module.exports = parseTree;
+
+},{"./Parser":7}],23:[function(require,module,exports){
+/**
+ * This file holds a list of all no-argument functions and single-character
+ * symbols (like 'a' or ';').
+ *
+ * For each of the symbols, there are three properties they can have:
+ * - font (required): the font to be used for this symbol. Either "main" (the
+ normal font), or "ams" (the ams fonts).
+ * - group (required): the ParseNode group type the symbol should have (i.e.
+ "textord", "mathord", etc).
+ See https://github.com/Khan/KaTeX/wiki/Examining-TeX#group-types
+ * - replace: the character that this symbol or function should be
+ * replaced with (i.e. "\phi" has a replace value of "\u03d5", the phi
+ * character in the main font).
+ *
+ * The outermost map in the table indicates what mode the symbols should be
+ * accepted in (e.g. "math" or "text").
+ */
+
+module.exports = {
+ math: {},
+ text: {}
+};
+
+function defineSymbol(mode, font, group, replace, name) {
+ module.exports[mode][name] = {
+ font: font,
+ group: group,
+ replace: replace
+ };
+}
+
+// Some abbreviations for commonly used strings.
+// This helps minify the code, and also spotting typos using jshint.
+
+// modes:
+var math = "math";
+var text = "text";
+
+// fonts:
+var main = "main";
+var ams = "ams";
+
+// groups:
+var accent = "accent";
+var bin = "bin";
+var close = "close";
+var inner = "inner";
+var mathord = "mathord";
+var op = "op";
+var open = "open";
+var punct = "punct";
+var rel = "rel";
+var spacing = "spacing";
+var textord = "textord";
+
+// Now comes the symbol table
+
+// Relation Symbols
+defineSymbol(math, main, rel, "\u2261", "\\equiv");
+defineSymbol(math, main, rel, "\u227a", "\\prec");
+defineSymbol(math, main, rel, "\u227b", "\\succ");
+defineSymbol(math, main, rel, "\u223c", "\\sim");
+defineSymbol(math, main, rel, "\u22a5", "\\perp");
+defineSymbol(math, main, rel, "\u2aaf", "\\preceq");
+defineSymbol(math, main, rel, "\u2ab0", "\\succeq");
+defineSymbol(math, main, rel, "\u2243", "\\simeq");
+defineSymbol(math, main, rel, "\u2223", "\\mid");
+defineSymbol(math, main, rel, "\u226a", "\\ll");
+defineSymbol(math, main, rel, "\u226b", "\\gg");
+defineSymbol(math, main, rel, "\u224d", "\\asymp");
+defineSymbol(math, main, rel, "\u2225", "\\parallel");
+defineSymbol(math, main, rel, "\u22c8", "\\bowtie");
+defineSymbol(math, main, rel, "\u2323", "\\smile");
+defineSymbol(math, main, rel, "\u2291", "\\sqsubseteq");
+defineSymbol(math, main, rel, "\u2292", "\\sqsupseteq");
+defineSymbol(math, main, rel, "\u2250", "\\doteq");
+defineSymbol(math, main, rel, "\u2322", "\\frown");
+defineSymbol(math, main, rel, "\u220b", "\\ni");
+defineSymbol(math, main, rel, "\u221d", "\\propto");
+defineSymbol(math, main, rel, "\u22a2", "\\vdash");
+defineSymbol(math, main, rel, "\u22a3", "\\dashv");
+defineSymbol(math, main, rel, "\u220b", "\\owns");
+
+// Punctuation
+defineSymbol(math, main, punct, "\u002e", "\\ldotp");
+defineSymbol(math, main, punct, "\u22c5", "\\cdotp");
+
+// Misc Symbols
+defineSymbol(math, main, textord, "\u0023", "\\#");
+defineSymbol(text, main, textord, "\u0023", "\\#");
+defineSymbol(math, main, textord, "\u0026", "\\&");
+defineSymbol(text, main, textord, "\u0026", "\\&");
+defineSymbol(math, main, textord, "\u2135", "\\aleph");
+defineSymbol(math, main, textord, "\u2200", "\\forall");
+defineSymbol(math, main, textord, "\u210f", "\\hbar");
+defineSymbol(math, main, textord, "\u2203", "\\exists");
+defineSymbol(math, main, textord, "\u2207", "\\nabla");
+defineSymbol(math, main, textord, "\u266d", "\\flat");
+defineSymbol(math, main, textord, "\u2113", "\\ell");
+defineSymbol(math, main, textord, "\u266e", "\\natural");
+defineSymbol(math, main, textord, "\u2663", "\\clubsuit");
+defineSymbol(math, main, textord, "\u2118", "\\wp");
+defineSymbol(math, main, textord, "\u266f", "\\sharp");
+defineSymbol(math, main, textord, "\u2662", "\\diamondsuit");
+defineSymbol(math, main, textord, "\u211c", "\\Re");
+defineSymbol(math, main, textord, "\u2661", "\\heartsuit");
+defineSymbol(math, main, textord, "\u2111", "\\Im");
+defineSymbol(math, main, textord, "\u2660", "\\spadesuit");
+
+// Math and Text
+defineSymbol(math, main, textord, "\u2020", "\\dag");
+defineSymbol(math, main, textord, "\u2021", "\\ddag");
+
+// Large Delimiters
+defineSymbol(math, main, close, "\u23b1", "\\rmoustache");
+defineSymbol(math, main, open, "\u23b0", "\\lmoustache");
+defineSymbol(math, main, close, "\u27ef", "\\rgroup");
+defineSymbol(math, main, open, "\u27ee", "\\lgroup");
+
+// Binary Operators
+defineSymbol(math, main, bin, "\u2213", "\\mp");
+defineSymbol(math, main, bin, "\u2296", "\\ominus");
+defineSymbol(math, main, bin, "\u228e", "\\uplus");
+defineSymbol(math, main, bin, "\u2293", "\\sqcap");
+defineSymbol(math, main, bin, "\u2217", "\\ast");
+defineSymbol(math, main, bin, "\u2294", "\\sqcup");
+defineSymbol(math, main, bin, "\u25ef", "\\bigcirc");
+defineSymbol(math, main, bin, "\u2219", "\\bullet");
+defineSymbol(math, main, bin, "\u2021", "\\ddagger");
+defineSymbol(math, main, bin, "\u2240", "\\wr");
+defineSymbol(math, main, bin, "\u2a3f", "\\amalg");
+
+// Arrow Symbols
+defineSymbol(math, main, rel, "\u27f5", "\\longleftarrow");
+defineSymbol(math, main, rel, "\u21d0", "\\Leftarrow");
+defineSymbol(math, main, rel, "\u27f8", "\\Longleftarrow");
+defineSymbol(math, main, rel, "\u27f6", "\\longrightarrow");
+defineSymbol(math, main, rel, "\u21d2", "\\Rightarrow");
+defineSymbol(math, main, rel, "\u27f9", "\\Longrightarrow");
+defineSymbol(math, main, rel, "\u2194", "\\leftrightarrow");
+defineSymbol(math, main, rel, "\u27f7", "\\longleftrightarrow");
+defineSymbol(math, main, rel, "\u21d4", "\\Leftrightarrow");
+defineSymbol(math, main, rel, "\u27fa", "\\Longleftrightarrow");
+defineSymbol(math, main, rel, "\u21a6", "\\mapsto");
+defineSymbol(math, main, rel, "\u27fc", "\\longmapsto");
+defineSymbol(math, main, rel, "\u2197", "\\nearrow");
+defineSymbol(math, main, rel, "\u21a9", "\\hookleftarrow");
+defineSymbol(math, main, rel, "\u21aa", "\\hookrightarrow");
+defineSymbol(math, main, rel, "\u2198", "\\searrow");
+defineSymbol(math, main, rel, "\u21bc", "\\leftharpoonup");
+defineSymbol(math, main, rel, "\u21c0", "\\rightharpoonup");
+defineSymbol(math, main, rel, "\u2199", "\\swarrow");
+defineSymbol(math, main, rel, "\u21bd", "\\leftharpoondown");
+defineSymbol(math, main, rel, "\u21c1", "\\rightharpoondown");
+defineSymbol(math, main, rel, "\u2196", "\\nwarrow");
+defineSymbol(math, main, rel, "\u21cc", "\\rightleftharpoons");
+
+// AMS Negated Binary Relations
+defineSymbol(math, ams, rel, "\u226e", "\\nless");
+defineSymbol(math, ams, rel, "\ue010", "\\nleqslant");
+defineSymbol(math, ams, rel, "\ue011", "\\nleqq");
+defineSymbol(math, ams, rel, "\u2a87", "\\lneq");
+defineSymbol(math, ams, rel, "\u2268", "\\lneqq");
+defineSymbol(math, ams, rel, "\ue00c", "\\lvertneqq");
+defineSymbol(math, ams, rel, "\u22e6", "\\lnsim");
+defineSymbol(math, ams, rel, "\u2a89", "\\lnapprox");
+defineSymbol(math, ams, rel, "\u2280", "\\nprec");
+defineSymbol(math, ams, rel, "\u22e0", "\\npreceq");
+defineSymbol(math, ams, rel, "\u22e8", "\\precnsim");
+defineSymbol(math, ams, rel, "\u2ab9", "\\precnapprox");
+defineSymbol(math, ams, rel, "\u2241", "\\nsim");
+defineSymbol(math, ams, rel, "\ue006", "\\nshortmid");
+defineSymbol(math, ams, rel, "\u2224", "\\nmid");
+defineSymbol(math, ams, rel, "\u22ac", "\\nvdash");
+defineSymbol(math, ams, rel, "\u22ad", "\\nvDash");
+defineSymbol(math, ams, rel, "\u22ea", "\\ntriangleleft");
+defineSymbol(math, ams, rel, "\u22ec", "\\ntrianglelefteq");
+defineSymbol(math, ams, rel, "\u228a", "\\subsetneq");
+defineSymbol(math, ams, rel, "\ue01a", "\\varsubsetneq");
+defineSymbol(math, ams, rel, "\u2acb", "\\subsetneqq");
+defineSymbol(math, ams, rel, "\ue017", "\\varsubsetneqq");
+defineSymbol(math, ams, rel, "\u226f", "\\ngtr");
+defineSymbol(math, ams, rel, "\ue00f", "\\ngeqslant");
+defineSymbol(math, ams, rel, "\ue00e", "\\ngeqq");
+defineSymbol(math, ams, rel, "\u2a88", "\\gneq");
+defineSymbol(math, ams, rel, "\u2269", "\\gneqq");
+defineSymbol(math, ams, rel, "\ue00d", "\\gvertneqq");
+defineSymbol(math, ams, rel, "\u22e7", "\\gnsim");
+defineSymbol(math, ams, rel, "\u2a8a", "\\gnapprox");
+defineSymbol(math, ams, rel, "\u2281", "\\nsucc");
+defineSymbol(math, ams, rel, "\u22e1", "\\nsucceq");
+defineSymbol(math, ams, rel, "\u22e9", "\\succnsim");
+defineSymbol(math, ams, rel, "\u2aba", "\\succnapprox");
+defineSymbol(math, ams, rel, "\u2246", "\\ncong");
+defineSymbol(math, ams, rel, "\ue007", "\\nshortparallel");
+defineSymbol(math, ams, rel, "\u2226", "\\nparallel");
+defineSymbol(math, ams, rel, "\u22af", "\\nVDash");
+defineSymbol(math, ams, rel, "\u22eb", "\\ntriangleright");
+defineSymbol(math, ams, rel, "\u22ed", "\\ntrianglerighteq");
+defineSymbol(math, ams, rel, "\ue018", "\\nsupseteqq");
+defineSymbol(math, ams, rel, "\u228b", "\\supsetneq");
+defineSymbol(math, ams, rel, "\ue01b", "\\varsupsetneq");
+defineSymbol(math, ams, rel, "\u2acc", "\\supsetneqq");
+defineSymbol(math, ams, rel, "\ue019", "\\varsupsetneqq");
+defineSymbol(math, ams, rel, "\u22ae", "\\nVdash");
+defineSymbol(math, ams, rel, "\u2ab5", "\\precneqq");
+defineSymbol(math, ams, rel, "\u2ab6", "\\succneqq");
+defineSymbol(math, ams, rel, "\ue016", "\\nsubseteqq");
+defineSymbol(math, ams, bin, "\u22b4", "\\unlhd");
+defineSymbol(math, ams, bin, "\u22b5", "\\unrhd");
+
+// AMS Negated Arrows
+defineSymbol(math, ams, rel, "\u219a", "\\nleftarrow");
+defineSymbol(math, ams, rel, "\u219b", "\\nrightarrow");
+defineSymbol(math, ams, rel, "\u21cd", "\\nLeftarrow");
+defineSymbol(math, ams, rel, "\u21cf", "\\nRightarrow");
+defineSymbol(math, ams, rel, "\u21ae", "\\nleftrightarrow");
+defineSymbol(math, ams, rel, "\u21ce", "\\nLeftrightarrow");
+
+// AMS Misc
+defineSymbol(math, ams, rel, "\u25b3", "\\vartriangle");
+defineSymbol(math, ams, textord, "\u210f", "\\hslash");
+defineSymbol(math, ams, textord, "\u25bd", "\\triangledown");
+defineSymbol(math, ams, textord, "\u25ca", "\\lozenge");
+defineSymbol(math, ams, textord, "\u24c8", "\\circledS");
+defineSymbol(math, ams, textord, "\u00ae", "\\circledR");
+defineSymbol(math, ams, textord, "\u2221", "\\measuredangle");
+defineSymbol(math, ams, textord, "\u2204", "\\nexists");
+defineSymbol(math, ams, textord, "\u2127", "\\mho");
+defineSymbol(math, ams, textord, "\u2132", "\\Finv");
+defineSymbol(math, ams, textord, "\u2141", "\\Game");
+defineSymbol(math, ams, textord, "\u006b", "\\Bbbk");
+defineSymbol(math, ams, textord, "\u2035", "\\backprime");
+defineSymbol(math, ams, textord, "\u25b2", "\\blacktriangle");
+defineSymbol(math, ams, textord, "\u25bc", "\\blacktriangledown");
+defineSymbol(math, ams, textord, "\u25a0", "\\blacksquare");
+defineSymbol(math, ams, textord, "\u29eb", "\\blacklozenge");
+defineSymbol(math, ams, textord, "\u2605", "\\bigstar");
+defineSymbol(math, ams, textord, "\u2222", "\\sphericalangle");
+defineSymbol(math, ams, textord, "\u2201", "\\complement");
+defineSymbol(math, ams, textord, "\u00f0", "\\eth");
+defineSymbol(math, ams, textord, "\u2571", "\\diagup");
+defineSymbol(math, ams, textord, "\u2572", "\\diagdown");
+defineSymbol(math, ams, textord, "\u25a1", "\\square");
+defineSymbol(math, ams, textord, "\u25a1", "\\Box");
+defineSymbol(math, ams, textord, "\u25ca", "\\Diamond");
+defineSymbol(math, ams, textord, "\u00a5", "\\yen");
+defineSymbol(math, ams, textord, "\u2713", "\\checkmark");
+
+// AMS Hebrew
+defineSymbol(math, ams, textord, "\u2136", "\\beth");
+defineSymbol(math, ams, textord, "\u2138", "\\daleth");
+defineSymbol(math, ams, textord, "\u2137", "\\gimel");
+
+// AMS Greek
+defineSymbol(math, ams, textord, "\u03dd", "\\digamma");
+defineSymbol(math, ams, textord, "\u03f0", "\\varkappa");
+
+// AMS Delimiters
+defineSymbol(math, ams, open, "\u250c", "\\ulcorner");
+defineSymbol(math, ams, close, "\u2510", "\\urcorner");
+defineSymbol(math, ams, open, "\u2514", "\\llcorner");
+defineSymbol(math, ams, close, "\u2518", "\\lrcorner");
+
+// AMS Binary Relations
+defineSymbol(math, ams, rel, "\u2266", "\\leqq");
+defineSymbol(math, ams, rel, "\u2a7d", "\\leqslant");
+defineSymbol(math, ams, rel, "\u2a95", "\\eqslantless");
+defineSymbol(math, ams, rel, "\u2272", "\\lesssim");
+defineSymbol(math, ams, rel, "\u2a85", "\\lessapprox");
+defineSymbol(math, ams, rel, "\u224a", "\\approxeq");
+defineSymbol(math, ams, bin, "\u22d6", "\\lessdot");
+defineSymbol(math, ams, rel, "\u22d8", "\\lll");
+defineSymbol(math, ams, rel, "\u2276", "\\lessgtr");
+defineSymbol(math, ams, rel, "\u22da", "\\lesseqgtr");
+defineSymbol(math, ams, rel, "\u2a8b", "\\lesseqqgtr");
+defineSymbol(math, ams, rel, "\u2251", "\\doteqdot");
+defineSymbol(math, ams, rel, "\u2253", "\\risingdotseq");
+defineSymbol(math, ams, rel, "\u2252", "\\fallingdotseq");
+defineSymbol(math, ams, rel, "\u223d", "\\backsim");
+defineSymbol(math, ams, rel, "\u22cd", "\\backsimeq");
+defineSymbol(math, ams, rel, "\u2ac5", "\\subseteqq");
+defineSymbol(math, ams, rel, "\u22d0", "\\Subset");
+defineSymbol(math, ams, rel, "\u228f", "\\sqsubset");
+defineSymbol(math, ams, rel, "\u227c", "\\preccurlyeq");
+defineSymbol(math, ams, rel, "\u22de", "\\curlyeqprec");
+defineSymbol(math, ams, rel, "\u227e", "\\precsim");
+defineSymbol(math, ams, rel, "\u2ab7", "\\precapprox");
+defineSymbol(math, ams, rel, "\u22b2", "\\vartriangleleft");
+defineSymbol(math, ams, rel, "\u22b4", "\\trianglelefteq");
+defineSymbol(math, ams, rel, "\u22a8", "\\vDash");
+defineSymbol(math, ams, rel, "\u22aa", "\\Vvdash");
+defineSymbol(math, ams, rel, "\u2323", "\\smallsmile");
+defineSymbol(math, ams, rel, "\u2322", "\\smallfrown");
+defineSymbol(math, ams, rel, "\u224f", "\\bumpeq");
+defineSymbol(math, ams, rel, "\u224e", "\\Bumpeq");
+defineSymbol(math, ams, rel, "\u2267", "\\geqq");
+defineSymbol(math, ams, rel, "\u2a7e", "\\geqslant");
+defineSymbol(math, ams, rel, "\u2a96", "\\eqslantgtr");
+defineSymbol(math, ams, rel, "\u2273", "\\gtrsim");
+defineSymbol(math, ams, rel, "\u2a86", "\\gtrapprox");
+defineSymbol(math, ams, bin, "\u22d7", "\\gtrdot");
+defineSymbol(math, ams, rel, "\u22d9", "\\ggg");
+defineSymbol(math, ams, rel, "\u2277", "\\gtrless");
+defineSymbol(math, ams, rel, "\u22db", "\\gtreqless");
+defineSymbol(math, ams, rel, "\u2a8c", "\\gtreqqless");
+defineSymbol(math, ams, rel, "\u2256", "\\eqcirc");
+defineSymbol(math, ams, rel, "\u2257", "\\circeq");
+defineSymbol(math, ams, rel, "\u225c", "\\triangleq");
+defineSymbol(math, ams, rel, "\u223c", "\\thicksim");
+defineSymbol(math, ams, rel, "\u2248", "\\thickapprox");
+defineSymbol(math, ams, rel, "\u2ac6", "\\supseteqq");
+defineSymbol(math, ams, rel, "\u22d1", "\\Supset");
+defineSymbol(math, ams, rel, "\u2290", "\\sqsupset");
+defineSymbol(math, ams, rel, "\u227d", "\\succcurlyeq");
+defineSymbol(math, ams, rel, "\u22df", "\\curlyeqsucc");
+defineSymbol(math, ams, rel, "\u227f", "\\succsim");
+defineSymbol(math, ams, rel, "\u2ab8", "\\succapprox");
+defineSymbol(math, ams, rel, "\u22b3", "\\vartriangleright");
+defineSymbol(math, ams, rel, "\u22b5", "\\trianglerighteq");
+defineSymbol(math, ams, rel, "\u22a9", "\\Vdash");
+defineSymbol(math, ams, rel, "\u2223", "\\shortmid");
+defineSymbol(math, ams, rel, "\u2225", "\\shortparallel");
+defineSymbol(math, ams, rel, "\u226c", "\\between");
+defineSymbol(math, ams, rel, "\u22d4", "\\pitchfork");
+defineSymbol(math, ams, rel, "\u221d", "\\varpropto");
+defineSymbol(math, ams, rel, "\u25c0", "\\blacktriangleleft");
+defineSymbol(math, ams, rel, "\u2234", "\\therefore");
+defineSymbol(math, ams, rel, "\u220d", "\\backepsilon");
+defineSymbol(math, ams, rel, "\u25b6", "\\blacktriangleright");
+defineSymbol(math, ams, rel, "\u2235", "\\because");
+defineSymbol(math, ams, rel, "\u22d8", "\\llless");
+defineSymbol(math, ams, rel, "\u22d9", "\\gggtr");
+defineSymbol(math, ams, bin, "\u22b2", "\\lhd");
+defineSymbol(math, ams, bin, "\u22b3", "\\rhd");
+defineSymbol(math, ams, rel, "\u2242", "\\eqsim");
+defineSymbol(math, main, rel, "\u22c8", "\\Join");
+defineSymbol(math, ams, rel, "\u2251", "\\Doteq");
+
+// AMS Binary Operators
+defineSymbol(math, ams, bin, "\u2214", "\\dotplus");
+defineSymbol(math, ams, bin, "\u2216", "\\smallsetminus");
+defineSymbol(math, ams, bin, "\u22d2", "\\Cap");
+defineSymbol(math, ams, bin, "\u22d3", "\\Cup");
+defineSymbol(math, ams, bin, "\u2a5e", "\\doublebarwedge");
+defineSymbol(math, ams, bin, "\u229f", "\\boxminus");
+defineSymbol(math, ams, bin, "\u229e", "\\boxplus");
+defineSymbol(math, ams, bin, "\u22c7", "\\divideontimes");
+defineSymbol(math, ams, bin, "\u22c9", "\\ltimes");
+defineSymbol(math, ams, bin, "\u22ca", "\\rtimes");
+defineSymbol(math, ams, bin, "\u22cb", "\\leftthreetimes");
+defineSymbol(math, ams, bin, "\u22cc", "\\rightthreetimes");
+defineSymbol(math, ams, bin, "\u22cf", "\\curlywedge");
+defineSymbol(math, ams, bin, "\u22ce", "\\curlyvee");
+defineSymbol(math, ams, bin, "\u229d", "\\circleddash");
+defineSymbol(math, ams, bin, "\u229b", "\\circledast");
+defineSymbol(math, ams, bin, "\u22c5", "\\centerdot");
+defineSymbol(math, ams, bin, "\u22ba", "\\intercal");
+defineSymbol(math, ams, bin, "\u22d2", "\\doublecap");
+defineSymbol(math, ams, bin, "\u22d3", "\\doublecup");
+defineSymbol(math, ams, bin, "\u22a0", "\\boxtimes");
+
+// AMS Arrows
+defineSymbol(math, ams, rel, "\u21e2", "\\dashrightarrow");
+defineSymbol(math, ams, rel, "\u21e0", "\\dashleftarrow");
+defineSymbol(math, ams, rel, "\u21c7", "\\leftleftarrows");
+defineSymbol(math, ams, rel, "\u21c6", "\\leftrightarrows");
+defineSymbol(math, ams, rel, "\u21da", "\\Lleftarrow");
+defineSymbol(math, ams, rel, "\u219e", "\\twoheadleftarrow");
+defineSymbol(math, ams, rel, "\u21a2", "\\leftarrowtail");
+defineSymbol(math, ams, rel, "\u21ab", "\\looparrowleft");
+defineSymbol(math, ams, rel, "\u21cb", "\\leftrightharpoons");
+defineSymbol(math, ams, rel, "\u21b6", "\\curvearrowleft");
+defineSymbol(math, ams, rel, "\u21ba", "\\circlearrowleft");
+defineSymbol(math, ams, rel, "\u21b0", "\\Lsh");
+defineSymbol(math, ams, rel, "\u21c8", "\\upuparrows");
+defineSymbol(math, ams, rel, "\u21bf", "\\upharpoonleft");
+defineSymbol(math, ams, rel, "\u21c3", "\\downharpoonleft");
+defineSymbol(math, ams, rel, "\u22b8", "\\multimap");
+defineSymbol(math, ams, rel, "\u21ad", "\\leftrightsquigarrow");
+defineSymbol(math, ams, rel, "\u21c9", "\\rightrightarrows");
+defineSymbol(math, ams, rel, "\u21c4", "\\rightleftarrows");
+defineSymbol(math, ams, rel, "\u21a0", "\\twoheadrightarrow");
+defineSymbol(math, ams, rel, "\u21a3", "\\rightarrowtail");
+defineSymbol(math, ams, rel, "\u21ac", "\\looparrowright");
+defineSymbol(math, ams, rel, "\u21b7", "\\curvearrowright");
+defineSymbol(math, ams, rel, "\u21bb", "\\circlearrowright");
+defineSymbol(math, ams, rel, "\u21b1", "\\Rsh");
+defineSymbol(math, ams, rel, "\u21ca", "\\downdownarrows");
+defineSymbol(math, ams, rel, "\u21be", "\\upharpoonright");
+defineSymbol(math, ams, rel, "\u21c2", "\\downharpoonright");
+defineSymbol(math, ams, rel, "\u21dd", "\\rightsquigarrow");
+defineSymbol(math, ams, rel, "\u21dd", "\\leadsto");
+defineSymbol(math, ams, rel, "\u21db", "\\Rrightarrow");
+defineSymbol(math, ams, rel, "\u21be", "\\restriction");
+
+defineSymbol(math, main, textord, "\u2018", "`");
+defineSymbol(math, main, textord, "$", "\\$");
+defineSymbol(text, main, textord, "$", "\\$");
+defineSymbol(math, main, textord, "%", "\\%");
+defineSymbol(text, main, textord, "%", "\\%");
+defineSymbol(math, main, textord, "_", "\\_");
+defineSymbol(text, main, textord, "_", "\\_");
+defineSymbol(math, main, textord, "\u2220", "\\angle");
+defineSymbol(math, main, textord, "\u221e", "\\infty");
+defineSymbol(math, main, textord, "\u2032", "\\prime");
+defineSymbol(math, main, textord, "\u25b3", "\\triangle");
+defineSymbol(math, main, textord, "\u0393", "\\Gamma");
+defineSymbol(math, main, textord, "\u0394", "\\Delta");
+defineSymbol(math, main, textord, "\u0398", "\\Theta");
+defineSymbol(math, main, textord, "\u039b", "\\Lambda");
+defineSymbol(math, main, textord, "\u039e", "\\Xi");
+defineSymbol(math, main, textord, "\u03a0", "\\Pi");
+defineSymbol(math, main, textord, "\u03a3", "\\Sigma");
+defineSymbol(math, main, textord, "\u03a5", "\\Upsilon");
+defineSymbol(math, main, textord, "\u03a6", "\\Phi");
+defineSymbol(math, main, textord, "\u03a8", "\\Psi");
+defineSymbol(math, main, textord, "\u03a9", "\\Omega");
+defineSymbol(math, main, textord, "\u00ac", "\\neg");
+defineSymbol(math, main, textord, "\u00ac", "\\lnot");
+defineSymbol(math, main, textord, "\u22a4", "\\top");
+defineSymbol(math, main, textord, "\u22a5", "\\bot");
+defineSymbol(math, main, textord, "\u2205", "\\emptyset");
+defineSymbol(math, ams, textord, "\u2205", "\\varnothing");
+defineSymbol(math, main, mathord, "\u03b1", "\\alpha");
+defineSymbol(math, main, mathord, "\u03b2", "\\beta");
+defineSymbol(math, main, mathord, "\u03b3", "\\gamma");
+defineSymbol(math, main, mathord, "\u03b4", "\\delta");
+defineSymbol(math, main, mathord, "\u03f5", "\\epsilon");
+defineSymbol(math, main, mathord, "\u03b6", "\\zeta");
+defineSymbol(math, main, mathord, "\u03b7", "\\eta");
+defineSymbol(math, main, mathord, "\u03b8", "\\theta");
+defineSymbol(math, main, mathord, "\u03b9", "\\iota");
+defineSymbol(math, main, mathord, "\u03ba", "\\kappa");
+defineSymbol(math, main, mathord, "\u03bb", "\\lambda");
+defineSymbol(math, main, mathord, "\u03bc", "\\mu");
+defineSymbol(math, main, mathord, "\u03bd", "\\nu");
+defineSymbol(math, main, mathord, "\u03be", "\\xi");
+defineSymbol(math, main, mathord, "o", "\\omicron");
+defineSymbol(math, main, mathord, "\u03c0", "\\pi");
+defineSymbol(math, main, mathord, "\u03c1", "\\rho");
+defineSymbol(math, main, mathord, "\u03c3", "\\sigma");
+defineSymbol(math, main, mathord, "\u03c4", "\\tau");
+defineSymbol(math, main, mathord, "\u03c5", "\\upsilon");
+defineSymbol(math, main, mathord, "\u03d5", "\\phi");
+defineSymbol(math, main, mathord, "\u03c7", "\\chi");
+defineSymbol(math, main, mathord, "\u03c8", "\\psi");
+defineSymbol(math, main, mathord, "\u03c9", "\\omega");
+defineSymbol(math, main, mathord, "\u03b5", "\\varepsilon");
+defineSymbol(math, main, mathord, "\u03d1", "\\vartheta");
+defineSymbol(math, main, mathord, "\u03d6", "\\varpi");
+defineSymbol(math, main, mathord, "\u03f1", "\\varrho");
+defineSymbol(math, main, mathord, "\u03c2", "\\varsigma");
+defineSymbol(math, main, mathord, "\u03c6", "\\varphi");
+defineSymbol(math, main, bin, "\u2217", "*");
+defineSymbol(math, main, bin, "+", "+");
+defineSymbol(math, main, bin, "\u2212", "-");
+defineSymbol(math, main, bin, "\u22c5", "\\cdot");
+defineSymbol(math, main, bin, "\u2218", "\\circ");
+defineSymbol(math, main, bin, "\u00f7", "\\div");
+defineSymbol(math, main, bin, "\u00b1", "\\pm");
+defineSymbol(math, main, bin, "\u00d7", "\\times");
+defineSymbol(math, main, bin, "\u2229", "\\cap");
+defineSymbol(math, main, bin, "\u222a", "\\cup");
+defineSymbol(math, main, bin, "\u2216", "\\setminus");
+defineSymbol(math, main, bin, "\u2227", "\\land");
+defineSymbol(math, main, bin, "\u2228", "\\lor");
+defineSymbol(math, main, bin, "\u2227", "\\wedge");
+defineSymbol(math, main, bin, "\u2228", "\\vee");
+defineSymbol(math, main, textord, "\u221a", "\\surd");
+defineSymbol(math, main, open, "(", "(");
+defineSymbol(math, main, open, "[", "[");
+defineSymbol(math, main, open, "\u27e8", "\\langle");
+defineSymbol(math, main, open, "\u2223", "\\lvert");
+defineSymbol(math, main, open, "\u2225", "\\lVert");
+defineSymbol(math, main, close, ")", ")");
+defineSymbol(math, main, close, "]", "]");
+defineSymbol(math, main, close, "?", "?");
+defineSymbol(math, main, close, "!", "!");
+defineSymbol(math, main, close, "\u27e9", "\\rangle");
+defineSymbol(math, main, close, "\u2223", "\\rvert");
+defineSymbol(math, main, close, "\u2225", "\\rVert");
+defineSymbol(math, main, rel, "=", "=");
+defineSymbol(math, main, rel, "<", "<");
+defineSymbol(math, main, rel, ">", ">");
+defineSymbol(math, main, rel, ":", ":");
+defineSymbol(math, main, rel, "\u2248", "\\approx");
+defineSymbol(math, main, rel, "\u2245", "\\cong");
+defineSymbol(math, main, rel, "\u2265", "\\ge");
+defineSymbol(math, main, rel, "\u2265", "\\geq");
+defineSymbol(math, main, rel, "\u2190", "\\gets");
+defineSymbol(math, main, rel, ">", "\\gt");
+defineSymbol(math, main, rel, "\u2208", "\\in");
+defineSymbol(math, main, rel, "\u2209", "\\notin");
+defineSymbol(math, main, rel, "\u2282", "\\subset");
+defineSymbol(math, main, rel, "\u2283", "\\supset");
+defineSymbol(math, main, rel, "\u2286", "\\subseteq");
+defineSymbol(math, main, rel, "\u2287", "\\supseteq");
+defineSymbol(math, ams, rel, "\u2288", "\\nsubseteq");
+defineSymbol(math, ams, rel, "\u2289", "\\nsupseteq");
+defineSymbol(math, main, rel, "\u22a8", "\\models");
+defineSymbol(math, main, rel, "\u2190", "\\leftarrow");
+defineSymbol(math, main, rel, "\u2264", "\\le");
+defineSymbol(math, main, rel, "\u2264", "\\leq");
+defineSymbol(math, main, rel, "<", "\\lt");
+defineSymbol(math, main, rel, "\u2260", "\\ne");
+defineSymbol(math, main, rel, "\u2260", "\\neq");
+defineSymbol(math, main, rel, "\u2192", "\\rightarrow");
+defineSymbol(math, main, rel, "\u2192", "\\to");
+defineSymbol(math, ams, rel, "\u2271", "\\ngeq");
+defineSymbol(math, ams, rel, "\u2270", "\\nleq");
+defineSymbol(math, main, spacing, null, "\\!");
+defineSymbol(math, main, spacing, "\u00a0", "\\ ");
+defineSymbol(math, main, spacing, "\u00a0", "~");
+defineSymbol(math, main, spacing, null, "\\,");
+defineSymbol(math, main, spacing, null, "\\:");
+defineSymbol(math, main, spacing, null, "\\;");
+defineSymbol(math, main, spacing, null, "\\enspace");
+defineSymbol(math, main, spacing, null, "\\qquad");
+defineSymbol(math, main, spacing, null, "\\quad");
+defineSymbol(math, main, spacing, "\u00a0", "\\space");
+defineSymbol(math, main, punct, ",", ",");
+defineSymbol(math, main, punct, ";", ";");
+defineSymbol(math, main, punct, ":", "\\colon");
+defineSymbol(math, ams, bin, "\u22bc", "\\barwedge");
+defineSymbol(math, ams, bin, "\u22bb", "\\veebar");
+defineSymbol(math, main, bin, "\u2299", "\\odot");
+defineSymbol(math, main, bin, "\u2295", "\\oplus");
+defineSymbol(math, main, bin, "\u2297", "\\otimes");
+defineSymbol(math, main, textord, "\u2202", "\\partial");
+defineSymbol(math, main, bin, "\u2298", "\\oslash");
+defineSymbol(math, ams, bin, "\u229a", "\\circledcirc");
+defineSymbol(math, ams, bin, "\u22a1", "\\boxdot");
+defineSymbol(math, main, bin, "\u25b3", "\\bigtriangleup");
+defineSymbol(math, main, bin, "\u25bd", "\\bigtriangledown");
+defineSymbol(math, main, bin, "\u2020", "\\dagger");
+defineSymbol(math, main, bin, "\u22c4", "\\diamond");
+defineSymbol(math, main, bin, "\u22c6", "\\star");
+defineSymbol(math, main, bin, "\u25c3", "\\triangleleft");
+defineSymbol(math, main, bin, "\u25b9", "\\triangleright");
+defineSymbol(math, main, open, "{", "\\{");
+defineSymbol(text, main, textord, "{", "\\{");
+defineSymbol(math, main, close, "}", "\\}");
+defineSymbol(text, main, textord, "}", "\\}");
+defineSymbol(math, main, open, "{", "\\lbrace");
+defineSymbol(math, main, close, "}", "\\rbrace");
+defineSymbol(math, main, open, "[", "\\lbrack");
+defineSymbol(math, main, close, "]", "\\rbrack");
+defineSymbol(math, main, open, "\u230a", "\\lfloor");
+defineSymbol(math, main, close, "\u230b", "\\rfloor");
+defineSymbol(math, main, open, "\u2308", "\\lceil");
+defineSymbol(math, main, close, "\u2309", "\\rceil");
+defineSymbol(math, main, textord, "\\", "\\backslash");
+defineSymbol(math, main, textord, "\u2223", "|");
+defineSymbol(math, main, textord, "\u2223", "\\vert");
+defineSymbol(math, main, textord, "\u2225", "\\|");
+defineSymbol(math, main, textord, "\u2225", "\\Vert");
+defineSymbol(math, main, rel, "\u2191", "\\uparrow");
+defineSymbol(math, main, rel, "\u21d1", "\\Uparrow");
+defineSymbol(math, main, rel, "\u2193", "\\downarrow");
+defineSymbol(math, main, rel, "\u21d3", "\\Downarrow");
+defineSymbol(math, main, rel, "\u2195", "\\updownarrow");
+defineSymbol(math, main, rel, "\u21d5", "\\Updownarrow");
+defineSymbol(math, math, op, "\u2210", "\\coprod");
+defineSymbol(math, math, op, "\u22c1", "\\bigvee");
+defineSymbol(math, math, op, "\u22c0", "\\bigwedge");
+defineSymbol(math, math, op, "\u2a04", "\\biguplus");
+defineSymbol(math, math, op, "\u22c2", "\\bigcap");
+defineSymbol(math, math, op, "\u22c3", "\\bigcup");
+defineSymbol(math, math, op, "\u222b", "\\int");
+defineSymbol(math, math, op, "\u222b", "\\intop");
+defineSymbol(math, math, op, "\u222c", "\\iint");
+defineSymbol(math, math, op, "\u222d", "\\iiint");
+defineSymbol(math, math, op, "\u220f", "\\prod");
+defineSymbol(math, math, op, "\u2211", "\\sum");
+defineSymbol(math, math, op, "\u2a02", "\\bigotimes");
+defineSymbol(math, math, op, "\u2a01", "\\bigoplus");
+defineSymbol(math, math, op, "\u2a00", "\\bigodot");
+defineSymbol(math, math, op, "\u222e", "\\oint");
+defineSymbol(math, math, op, "\u2a06", "\\bigsqcup");
+defineSymbol(math, math, op, "\u222b", "\\smallint");
+defineSymbol(text, main, inner, "\u2026", "\\textellipsis");
+defineSymbol(math, main, inner, "\u2026", "\\mathellipsis");
+defineSymbol(text, main, inner, "\u2026", "\\ldots");
+defineSymbol(math, main, inner, "\u2026", "\\ldots");
+defineSymbol(math, main, inner, "\u22ef", "\\cdots");
+defineSymbol(math, main, inner, "\u22f1", "\\ddots");
+defineSymbol(math, main, textord, "\u22ee", "\\vdots");
+defineSymbol(math, main, accent, "\u00b4", "\\acute");
+defineSymbol(math, main, accent, "\u0060", "\\grave");
+defineSymbol(math, main, accent, "\u00a8", "\\ddot");
+defineSymbol(math, main, accent, "\u007e", "\\tilde");
+defineSymbol(math, main, accent, "\u00af", "\\bar");
+defineSymbol(math, main, accent, "\u02d8", "\\breve");
+defineSymbol(math, main, accent, "\u02c7", "\\check");
+defineSymbol(math, main, accent, "\u005e", "\\hat");
+defineSymbol(math, main, accent, "\u20d7", "\\vec");
+defineSymbol(math, main, accent, "\u02d9", "\\dot");
+defineSymbol(math, main, mathord, "\u0131", "\\imath");
+defineSymbol(math, main, mathord, "\u0237", "\\jmath");
+
+defineSymbol(text, main, textord, "\u2013", "--");
+defineSymbol(text, main, textord, "\u2014", "---");
+defineSymbol(text, main, textord, "\u2018", "`");
+defineSymbol(text, main, textord, "\u2019", "'");
+defineSymbol(text, main, textord, "\u201c", "``");
+defineSymbol(text, main, textord, "\u201d", "''");
+defineSymbol(math, main, textord, "\u00b0", "\\degree");
+defineSymbol(text, main, textord, "\u00b0", "\\degree");
+defineSymbol(math, main, mathord, "\u00a3", "\\pounds");
+defineSymbol(math, ams, textord, "\u2720", "\\maltese");
+defineSymbol(text, ams, textord, "\u2720", "\\maltese");
+
+defineSymbol(text, main, spacing, "\u00a0", "\\ ");
+defineSymbol(text, main, spacing, "\u00a0", " ");
+defineSymbol(text, main, spacing, "\u00a0", "~");
+
+// There are lots of symbols which are the same, so we add them in afterwards.
+var i;
+var ch;
+
+// All of these are textords in math mode
+var mathTextSymbols = "0123456789/@.\"";
+for (i = 0; i < mathTextSymbols.length; i++) {
+ ch = mathTextSymbols.charAt(i);
+ defineSymbol(math, main, textord, ch, ch);
+}
+
+// All of these are textords in text mode
+var textSymbols = "0123456789!@*()-=+[]\";:?/.,";
+for (i = 0; i < textSymbols.length; i++) {
+ ch = textSymbols.charAt(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+// All of these are textords in text mode, and mathords in math mode
+var letters = "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ";
+for (i = 0; i < letters.length; i++) {
+ ch = letters.charAt(i);
+ defineSymbol(math, main, mathord, ch, ch);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+// Latin-1 letters
+for (i = 0x00C0; i <= 0x00D6; i++) {
+ ch = String.fromCharCode(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+for (i = 0x00D8; i <= 0x00F6; i++) {
+ ch = String.fromCharCode(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+for (i = 0x00F8; i <= 0x00FF; i++) {
+ ch = String.fromCharCode(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+// Cyrillic
+for (i = 0x0410; i <= 0x044F; i++) {
+ ch = String.fromCharCode(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+// Unicode versions of existing characters
+defineSymbol(text, main, textord, "\u2013", "–");
+defineSymbol(text, main, textord, "\u2014", "—");
+defineSymbol(text, main, textord, "\u2018", "‘");
+defineSymbol(text, main, textord, "\u2019", "’");
+defineSymbol(text, main, textord, "\u201c", "“");
+defineSymbol(text, main, textord, "\u201d", "”");
+
+},{}],24:[function(require,module,exports){
+var hangulRegex = /[\uAC00-\uD7AF]/;
+
+// This regex combines
+// - Hiragana: [\u3040-\u309F]
+// - Katakana: [\u30A0-\u30FF]
+// - CJK ideograms: [\u4E00-\u9FAF]
+// - Hangul syllables: [\uAC00-\uD7AF]
+// Notably missing are halfwidth Katakana and Romanji glyphs.
+var cjkRegex =
+ /[\u3040-\u309F]|[\u30A0-\u30FF]|[\u4E00-\u9FAF]|[\uAC00-\uD7AF]/;
+
+module.exports = {
+ cjkRegex: cjkRegex,
+ hangulRegex: hangulRegex
+};
+
+},{}],25:[function(require,module,exports){
+/**
+ * This file contains a list of utility functions which are useful in other
+ * files.
+ */
+
+/**
+ * Provide an `indexOf` function which works in IE8, but defers to native if
+ * possible.
+ */
+var nativeIndexOf = Array.prototype.indexOf;
+var indexOf = function(list, elem) {
+ if (list == null) {
+ return -1;
+ }
+ if (nativeIndexOf && list.indexOf === nativeIndexOf) {
+ return list.indexOf(elem);
+ }
+ var i = 0;
+ var l = list.length;
+ for (; i < l; i++) {
+ if (list[i] === elem) {
+ return i;
+ }
+ }
+ return -1;
+};
+
+/**
+ * Return whether an element is contained in a list
+ */
+var contains = function(list, elem) {
+ return indexOf(list, elem) !== -1;
+};
+
+/**
+ * Provide a default value if a setting is undefined
+ */
+var deflt = function(setting, defaultIfUndefined) {
+ return setting === undefined ? defaultIfUndefined : setting;
+};
+
+// hyphenate and escape adapted from Facebook's React under Apache 2 license
+
+var uppercase = /([A-Z])/g;
+var hyphenate = function(str) {
+ return str.replace(uppercase, "-$1").toLowerCase();
+};
+
+var ESCAPE_LOOKUP = {
+ "&": "&",
+ ">": ">",
+ "<": "<",
+ "\"": """,
+ "'": "'"
+};
+
+var ESCAPE_REGEX = /[&><"']/g;
+
+function escaper(match) {
+ return ESCAPE_LOOKUP[match];
+}
+
+/**
+ * Escapes text to prevent scripting attacks.
+ *
+ * @param {*} text Text value to escape.
+ * @return {string} An escaped string.
+ */
+function escape(text) {
+ return ("" + text).replace(ESCAPE_REGEX, escaper);
+}
+
+/**
+ * A function to set the text content of a DOM element in all supported
+ * browsers. Note that we don't define this if there is no document.
+ */
+var setTextContent;
+if (typeof document !== "undefined") {
+ var testNode = document.createElement("span");
+ if ("textContent" in testNode) {
+ setTextContent = function(node, text) {
+ node.textContent = text;
+ };
+ } else {
+ setTextContent = function(node, text) {
+ node.innerText = text;
+ };
+ }
+}
+
+/**
+ * A function to clear a node.
+ */
+function clearNode(node) {
+ setTextContent(node, "");
+}
+
+module.exports = {
+ contains: contains,
+ deflt: deflt,
+ escape: escape,
+ hyphenate: hyphenate,
+ indexOf: indexOf,
+ setTextContent: setTextContent,
+ clearNode: clearNode
+};
+
+},{}]},{},[1])(1)
+});
\ No newline at end of file
diff --git a/specs/es/3.2/katex/katex.min.css b/specs/es/3.2/katex/katex.min.css
new file mode 100644
index 0000000..d6fb837
--- /dev/null
+++ b/specs/es/3.2/katex/katex.min.css
@@ -0,0 +1 @@
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n={funcName:e,parser:this,positions:r,token:i};return a[e].handler(n,t)};h.prototype.parseArguments=function(e,t){var r=t.numArgs+t.numOptionalArgs;if(r===0){return[[this.pos]]}var i=t.greediness;var n=[this.pos];var s=[];for(var l=0;l<r;l++){var o=this.nextToken;var u=t.argTypes&&t.argTypes[l];var h;if(l<t.numOptionalArgs){if(u){h=this.parseGroupOfType(u,true)}else{h=this.parseGroup(true)}if(!h){s.push(null);n.push(this.pos);continue}}else{if(u){h=this.parseGroupOfType(u)}else{h=this.parseGroup()}if(!h){if(!this.settings.throwOnError&&this.nextToken.text[0]==="\\"){h=new m(this.handleUnsupportedCmd(this.nextToken.text),false)}else{throw new p("Expected group after '"+e+"'",o)}}}var c;if(h.isFunction){var f=a[h.result].greediness;if(f>i){c=this.parseFunction(h)}else{throw new p("Got function '"+h.result+"' as "+"argument to '"+e+"'",o)}}else{c=h.result}s.push(c);n.push(this.pos)}s.push(n);return s};h.prototype.parseGroupOfType=function(e,t){var r=this.mode;if(e==="original"){e=r}if(e==="color"){return this.parseColorGroup(t)}if(e==="size"){return this.parseSizeGroup(t)}this.switchMode(e);if(e==="text"){while(this.nextToken.text===" "){this.consume()}}var a=this.parseGroup(t);this.switchMode(r);return a};h.prototype.parseStringGroup=function(e,t){if(t&&this.nextToken.text!=="["){return null}var r=this.mode;this.mode="text";this.expect(t?"[":"{");var a="";var i=this.nextToken;var n=i;while(this.nextToken.text!==(t?"]":"}")){if(this.nextToken.text==="EOF"){throw new p("Unexpected end of input in "+e,i.range(this.nextToken,a))}n=this.nextToken;a+=n.text;this.consume()}this.mode=r;this.expect(t?"]":"}");return i.range(n,a)};h.prototype.parseRegexGroup=function(e,t){var r=this.mode;this.mode="text";var a=this.nextToken;var i=a;var n="";while(this.nextToken.text!=="EOF"&&e.test(n+this.nextToken.text)){i=this.nextToken;n+=i.text;this.consume()}if(n===""){throw new p("Invalid "+t+": '"+a.text+"'",a)}this.mode=r;return a.range(i,n)};h.prototype.parseColorGroup=function(e){var t=this.parseStringGroup("color",e);if(!t){return null}var r=/^(#[a-z0-9]+|[a-z]+)$/i.exec(t.text);if(!r){throw new p("Invalid color: '"+t.text+"'",t)}return new m(new c("color",r[0],this.mode),false)};h.prototype.parseSizeGroup=function(e){var t;if(!e&&this.nextToken.text!=="{"){t=this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2}$/,"size")}else{t=this.parseStringGroup("size",e)}if(!t){return null}var r=/([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(t.text);if(!r){throw new p("Invalid size: '"+t.text+"'",t)}var a={number:+(r[1]+r[2]),unit:r[3]};if(a.unit!=="em"&&a.unit!=="ex"&&a.unit!=="mu"){throw new p("Invalid unit: '"+a.unit+"'",t)}return new m(new c("color",a,this.mode),false)};h.prototype.parseGroup=function(e){var t=this.nextToken;if(this.nextToken.text===(e?"[":"{")){this.consume();var r=this.parseExpression(false,e?"]":null);var a=this.nextToken;this.expect(e?"]":"}");if(this.mode==="text"){this.formLigatures(r)}return new m(new c("ordgroup",r,this.mode,t,a),false)}else{return e?null:this.parseSymbol()}};h.prototype.formLigatures=function(e){var t;var r=e.length-1;for(t=0;t<r;++t){var a=e[t];var i=a.value;if(i==="-"&&e[t+1].value==="-"){if(t+1<r&&e[t+2].value==="-"){e.splice(t,3,new c("textord","---","text",a,e[t+2]));r-=2}else{e.splice(t,2,new c("textord","--","text",a,e[t+1]));r-=1}}if((i==="'"||i==="`")&&e[t+1].value===i){e.splice(t,2,new c("textord",i+i,"text",a,e[t+1]));r-=1}}};h.prototype.parseSymbol=function(){var e=this.nextToken;if(a[e.text]){this.consume();return new m(e.text,true,e)}else if(s[this.mode][e.text]){this.consume();return new m(new c(s[this.mode][e.text].group,e.text,this.mode,e),false,e)}else if(this.mode==="text"&&o.test(e.text)){this.consume();return new m(new c("textord",e.text,this.mode,e),false,e)}else{return null}};h.prototype.ParseNode=c;t.exports=h},{"./MacroExpander":4,"./ParseError":6,"./environments":16,"./functions":19,"./parseData":21,"./symbols":23,"./unicodeRegexes":24,"./utils":25}],8:[function(e,t,r){function a(e,t){return e===undefined?t:e}function i(e){e=e||{};this.displayMode=a(e.displayMode,false);this.throwOnError=a(e.throwOnError,true);this.errorColor=a(e.errorColor,"#cc0000");this.macros=e.macros||{}}t.exports=i},{}],9:[function(e,t,r){var a=e("./fontMetrics.js").sigmas;var i=[{},{},{}];var n;for(var s in a){if(a.hasOwnProperty(s)){for(n=0;n<3;n++){i[n][s]=a[s][n]}}}for(n=0;n<3;n++){i[n].emPerEx=a.xHeight[n]/a.quad[n]}function l(e,t,r,a){this.id=e;this.size=t;this.cramped=a;this.sizeMultiplier=r;this.metrics=i[t>0?t-1:0]}l.prototype.sup=function(){return y[x[this.id]]};l.prototype.sub=function(){return y[b[this.id]]};l.prototype.fracNum=function(){return y[w[this.id]]};l.prototype.fracDen=function(){return y[k[this.id]]};l.prototype.cramp=function(){return y[z[this.id]]};l.prototype.cls=function(){return d[this.size]+(this.cramped?" cramped":" uncramped")};l.prototype.reset=function(){return g[this.size]};l.prototype.isTight=function(){return this.size>=2};var o=0;var u=1;var p=2;var h=3;var c=4;var m=5;var f=6;var v=7;var d=["displaystyle textstyle","textstyle","scriptstyle","scriptscriptstyle"];var g=["reset-textstyle","reset-textstyle","reset-scriptstyle","reset-scriptscriptstyle"];var y=[new l(o,0,1,false),new l(u,0,1,true),new l(p,1,1,false),new l(h,1,1,true),new l(c,2,.7,false),new l(m,2,.7,true),new l(f,3,.5,false),new l(v,3,.5,true)];var x=[c,m,c,m,f,v,f,v];var b=[m,m,m,m,v,v,v,v];var w=[p,h,c,m,f,v,f,v];var k=[h,h,m,m,v,v,v,v];var z=[u,u,h,h,m,m,v,v];t.exports={DISPLAY:y[o],TEXT:y[p],SCRIPT:y[c],SCRIPTSCRIPT:y[f]}},{"./fontMetrics.js":17}],10:[function(e,t,r){var a=e("./domTree");var i=e("./fontMetrics");var n=e("./symbols");var s=e("./utils");var l=["\\Gamma","\\Delta","\\Theta","\\Lambda","\\Xi","\\Pi","\\Sigma","\\Upsilon","\\Phi","\\Psi","\\Omega"];var o=["\u0131","\u0237","\xa3"];var u=function(e,t,r,s,l){if(n[r][e]&&n[r][e].replace){e=n[r][e].replace}var o=i.getCharacterMetrics(e,t);var u;if(o){var p=o.italic;if(r==="text"){p=0}u=new a.symbolNode(e,o.height,o.depth,p,o.skew,l)}else{typeof console!=="undefined"&&console.warn("No character metrics for '"+e+"' in style '"+t+"'");u=new a.symbolNode(e,0,0,0,0,l)}if(s){if(s.style.isTight()){u.classes.push("mtight")}if(s.getColor()){u.style.color=s.getColor()}}return u};var p=function(e,t,r,a){if(e==="\\"||n[t][e].font==="main"){return u(e,"Main-Regular",t,r,a)}else{return u(e,"AMS-Regular",t,r,a.concat(["amsrm"]))}};var h=function(e,t,r,a,i){if(i==="mathord"){return c(e,t,r,a)}else if(i==="textord"){return u(e,"Main-Regular",t,r,a.concat(["mathrm"]))}else{throw new Error("unexpected type: "+i+" in mathDefault")}};var c=function(e,t,r,a){if(/[0-9]/.test(e.charAt(0))||s.contains(o,e)||s.contains(l,e)){return u(e,"Main-Italic",t,r,a.concat(["mainit"]))}else{return u(e,"Math-Italic",t,r,a.concat(["mathit"]))}};var m=function(e,t,r){var a=e.mode;var l=e.value;if(n[a][l]&&n[a][l].replace){l=n[a][l].replace}var p=["mord"];var m=t.font;if(m){if(m==="mathit"||s.contains(o,l)){return c(l,a,t,p)}else{var f=k[m].fontName;if(i.getCharacterMetrics(l,f)){return u(l,f,a,t,p.concat([m]))}else{return h(l,a,t,p,r)}}}else{return h(l,a,t,p,r)}};var f=function(e){var t=0;var r=0;var a=0;if(e.children){for(var i=0;i<e.children.length;i++){if(e.children[i].height>t){t=e.children[i].height}if(e.children[i].depth>r){r=e.children[i].depth}if(e.children[i].maxFontSize>a){a=e.children[i].maxFontSize}}}e.height=t;e.depth=r;e.maxFontSize=a};var v=function(e,t,r){var i=new a.span(e,t,r);f(i);return i};var d=function(e,t){e.children=t.concat(e.children);f(e)};var g=function(e){var t=new a.documentFragment(e);f(t);return t};var y=function(e,t){var r=v([],[new a.symbolNode("\u200b")]);r.style.fontSize=t/e.style.sizeMultiplier+"em";var i=v(["fontsize-ensurer","reset-"+e.size,"size5"],[r]);return i};var x=function(e,t,r,i){var n;var s;var l;if(t==="individualShift"){var o=e;e=[o[0]];n=-o[0].shift-o[0].elem.depth;s=n;for(l=1;l<o.length;l++){var u=-o[l].shift-s-o[l].elem.depth;var p=u-(o[l-1].elem.height+o[l-1].elem.depth);s=s+u;e.push({type:"kern",size:p});e.push(o[l])}}else if(t==="top"){var h=r;for(l=0;l<e.length;l++){if(e[l].type==="kern"){h-=e[l].size}else{h-=e[l].elem.height+e[l].elem.depth}}n=h}else if(t==="bottom"){n=-r}else if(t==="shift"){n=-e[0].elem.depth-r}else if(t==="firstBaseline"){n=-e[0].elem.depth}else{n=0}var c=0;for(l=0;l<e.length;l++){if(e[l].type==="elem"){c=Math.max(c,e[l].elem.maxFontSize)}}var m=y(i,c);var f=[];s=n;for(l=0;l<e.length;l++){if(e[l].type==="kern"){s+=e[l].size}else{var d=e[l].elem;var g=-d.depth-s;s+=d.height+d.depth;var x=v([],[m,d]);x.height-=g;x.depth+=g;x.style.top=g+"em";f.push(x)}}var b=v(["baseline-fix"],[m,new a.symbolNode("\u200b")]);f.push(b);var w=v(["vlist"],f);w.height=Math.max(s,w.height);w.depth=Math.max(-n,w.depth);return w};var b={size1:.5,size2:.7,size3:.8,size4:.9,size5:1,size6:1.2,size7:1.44,size8:1.73,size9:2.07,size10:2.49};var w={"\\qquad":{size:"2em",className:"qquad"},"\\quad":{size:"1em",className:"quad"},"\\enspace":{size:"0.5em",className:"enspace"},"\\;":{size:"0.277778em",className:"thickspace"},"\\:":{size:"0.22222em",className:"mediumspace"},"\\,":{size:"0.16667em",className:"thinspace"},"\\!":{size:"-0.16667em",className:"negativethinspace"}};var k={mathbf:{variant:"bold",fontName:"Main-Bold"},mathrm:{variant:"normal",fontName:"Main-Regular"},textit:{variant:"italic",fontName:"Main-Italic"},mathbb:{variant:"double-struck",fontName:"AMS-Regular"},mathcal:{variant:"script",fontName:"Caligraphic-Regular"},mathfrak:{variant:"fraktur",fontName:"Fraktur-Regular"},mathscr:{variant:"script",fontName:"Script-Regular"},mathsf:{variant:"sans-serif",fontName:"SansSerif-Regular"},mathtt:{variant:"monospace",fontName:"Typewriter-Regular"}};t.exports={fontMap:k,makeSymbol:u,mathsym:p,makeSpan:v,makeFragment:g,makeVList:x,makeOrd:m,prependChildren:d,sizingMultiplier:b,spacingFunctions:w}},{"./domTree":15,"./fontMetrics":17,"./symbols":23,"./utils":25}],11:[function(e,t,r){var a=e("./ParseError");var i=e("./Style");var n=e("./buildCommon");var s=e("./delimiter");var l=e("./domTree");var o=e("./fontMetrics");var u=e("./utils");var p=n.makeSpan;var h=function(e){return e instanceof l.span&&e.classes[0]==="mspace"};var c=function(e){return e&&e.classes[0]==="mbin"};var m=function(e,t){if(e){return u.contains(["mbin","mopen","mrel","mop","mpunct"],e.classes[0])}else{return t}};var f=function(e,t){if(e){return u.contains(["mrel","mclose","mpunct"],e.classes[0])}else{return t}};var v=function(e,t,r){var a=[];for(var i=0;i<e.length;i++){var s=e[i];var o=z(s,t);if(o instanceof l.documentFragment){Array.prototype.push.apply(a,o.children)}else{a.push(o)}}var u=null;for(i=0;i<a.length;i++){if(h(a[i])){u=u||[];u.push(a[i]);a.splice(i,1);i--}else if(u){if(a[i]instanceof l.symbolNode){a[i]=p([].concat(a[i].classes),[a[i]])}n.prependChildren(a[i],u);u=null}}if(u){Array.prototype.push.apply(a,u)}for(i=0;i<a.length;i++){if(c(a[i])&&(m(a[i-1],r)||f(a[i+1],r))){a[i].classes[0]="mord"}}return a};var d=function(e){if(e instanceof l.documentFragment){if(e.children.length){return d(e.children[e.children.length-1])}}else{if(u.contains(["mord","mop","mbin","mrel","mopen","mclose","mpunct","minner"],e.classes[0])){return e.classes[0]}}return null};var g=function(e,t){if(!e){return false}else if(e.type==="op"){return e.value.limits&&(t.style.size===i.DISPLAY.size||e.value.alwaysHandleSupSub)}else if(e.type==="accent"){return x(e.value.base)}else{return null}};var y=function(e){if(!e){return false}else if(e.type==="ordgroup"){if(e.value.length===1){return y(e.value[0])}else{return e}}else if(e.type==="color"){if(e.value.value.length===1){return y(e.value.value[0])}else{return e}}else if(e.type==="font"){return y(e.value.body)}else{return e}};var x=function(e){var t=y(e);return t.type==="mathord"||t.type==="textord"||t.type==="bin"||t.type==="rel"||t.type==="inner"||t.type==="open"||t.type==="close"||t.type==="punct"};var b=function(e,t){return p(t.concat(["sizing","reset-"+e.size,"size5",e.style.reset(),i.TEXT.cls(),"nulldelimiter"]))};var w={};w.mathord=function(e,t){return n.makeOrd(e,t,"mathord")};w.textord=function(e,t){return n.makeOrd(e,t,"textord")};w.bin=function(e,t){return n.mathsym(e.value,e.mode,t,["mbin"])};w.rel=function(e,t){return n.mathsym(e.value,e.mode,t,["mrel"])};w.open=function(e,t){return n.mathsym(e.value,e.mode,t,["mopen"])};w.close=function(e,t){return n.mathsym(e.value,e.mode,t,["mclose"])};w.inner=function(e,t){return n.mathsym(e.value,e.mode,t,["minner"])};w.punct=function(e,t){return n.mathsym(e.value,e.mode,t,["mpunct"])};w.ordgroup=function(e,t){return p(["mord",t.style.cls()],v(e.value,t.reset(),true),t)};w.text=function(e,t){var r=t.withFont(e.value.style);var a=v(e.value.body,r,true);for(var i=0;i<a.length-1;i++){if(a[i].tryCombine(a[i+1])){a.splice(i+1,1);i--}}return p(["mord","text",r.style.cls()],a,r)};w.color=function(e,t){var r=v(e.value.value,t.withColor(e.value.color),false);return new n.makeFragment(r)};w.supsub=function(e,t){if(g(e.value.base,t)){return w[e.value.base.type](e,t)}var r=z(e.value.base,t.reset());var a;var s;var u;var h;var c=t.style;var m;if(e.value.sup){m=t.withStyle(c.sup());u=z(e.value.sup,m);a=p([c.reset(),c.sup().cls()],[u],m)}if(e.value.sub){m=t.withStyle(c.sub());h=z(e.value.sub,m);s=p([c.reset(),c.sub().cls()],[h],m)}var f;var v;if(x(e.value.base)){f=0;v=0}else{f=r.height-c.metrics.supDrop;v=r.depth+c.metrics.subDrop}var y;if(c===i.DISPLAY){y=c.metrics.sup1}else if(c.cramped){y=c.metrics.sup3}else{y=c.metrics.sup2}var b=i.TEXT.sizeMultiplier*c.sizeMultiplier;var k=.5/o.metrics.ptPerEm/b+"em";var S;if(!e.value.sup){v=Math.max(v,c.metrics.sub1,h.height-.8*c.metrics.xHeight);S=n.makeVList([{type:"elem",elem:s}],"shift",v,t);S.children[0].style.marginRight=k;if(r instanceof l.symbolNode){S.children[0].style.marginLeft=-r.italic+"em"}}else if(!e.value.sub){f=Math.max(f,y,u.depth+.25*c.metrics.xHeight);S=n.makeVList([{type:"elem",elem:a}],"shift",-f,t);S.children[0].style.marginRight=k}else{f=Math.max(f,y,u.depth+.25*c.metrics.xHeight);v=Math.max(v,c.metrics.sub2);var 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t=[];if(e.value.value!=="."){t.push(h(e.value.value,e.mode))}var r=new n.MathNode("mo",t);if(e.value.mclass==="mopen"||e.value.mclass==="mclose"){r.setAttribute("fence","true")}else{r.setAttribute("fence","false")}return r};m.styling=function(e,t){var r=f(e.value.value,t);var a=new n.MathNode("mstyle",r);var i={display:["0","true"],text:["0","false"],script:["1","false"],scriptscript:["2","false"]};var s=i[e.value.style];a.setAttribute("scriptlevel",s[0]);a.setAttribute("displaystyle",s[1]);return a};m.sizing=function(e,t){var r=f(e.value.value,t);var i=new n.MathNode("mstyle",r);i.setAttribute("mathsize",a.sizingMultiplier[e.value.size]+"em");return i};m.overline=function(e,t){var r=new n.MathNode("mo",[new n.TextNode("\u203e")]);r.setAttribute("stretchy","true");var a=new n.MathNode("mover",[v(e.value.body,t),r]);a.setAttribute("accent","true");return a};m.underline=function(e,t){var r=new n.MathNode("mo",[new n.TextNode("\u203e")]);r.setAttribute("stretchy","true");var a=new n.MathNode("munder",[v(e.value.body,t),r]);a.setAttribute("accentunder","true");return a};m.rule=function(e){var t=new n.MathNode("mrow");return t};m.kern=function(e){var t=new n.MathNode("mrow");return t};m.llap=function(e,t){var r=new n.MathNode("mpadded",[v(e.value.body,t)]);r.setAttribute("lspace","-1width");r.setAttribute("width","0px");return r};m.rlap=function(e,t){var r=new n.MathNode("mpadded",[v(e.value.body,t)]);r.setAttribute("width","0px");return r};m.phantom=function(e,t){var r=f(e.value.value,t);return new n.MathNode("mphantom",r)};m.mclass=function(e,t){var r=f(e.value.value,t);return new n.MathNode("mstyle",r)};var f=function(e,t){var r=[];for(var a=0;a<e.length;a++){var i=e[a];r.push(v(i,t))}return r};var v=function(e,t){if(!e){return new n.MathNode("mrow")}if(m[e.type]){return m[e.type](e,t)}else{throw new s("Got group of unknown type: '"+e.type+"'")}};var d=function(e,t,r){var a=f(e,r);var i=new n.MathNode("mrow",a);var s=new n.MathNode("annotation",[new n.TextNode(t)]);s.setAttribute("encoding","application/x-tex");var l=new n.MathNode("semantics",[i,s]);var o=new n.MathNode("math",[l]);return u(["katex-mathml"],[o])};t.exports=d},{"./ParseError":6,"./buildCommon":10,"./fontMetrics":17,"./mathMLTree":20,"./symbols":23,"./utils":25}],13:[function(e,t,r){var a=e("./buildHTML");var i=e("./buildMathML");var n=e("./buildCommon");var s=e("./Options");var l=e("./Settings");var o=e("./Style");var u=n.makeSpan;var p=function(e,t,r){r=r||new l({});var n=o.TEXT;if(r.displayMode){n=o.DISPLAY}var p=new s({style:n,size:"size5"});var h=i(e,t,p);var c=a(e,p);var m=u(["katex"],[h,c]);if(r.displayMode){return u(["katex-display"],[m])}else{return m}};t.exports=p},{"./Options":5,"./Settings":8,"./Style":9,"./buildCommon":10,"./buildHTML":11,"./buildMathML":12}],14:[function(e,t,r){var a=e("./ParseError");var i=e("./Style");var n=e("./buildCommon");var s=e("./fontMetrics");var l=e("./symbols");var o=e("./utils");var u=n.makeSpan;var p=function(e,t){if(l.math[e]&&l.math[e].replace){return s.getCharacterMetrics(l.math[e].replace,t)}else{return s.getCharacterMetrics(e,t)}};var h=function(e,t,r,a){return n.makeSymbol(e,"Size"+t+"-Regular",r,a)};var c=function(e,t,r,a){a=a||[];var i=u(a.concat(["style-wrap",r.style.reset(),t.cls()]),[e],r);var n=t.sizeMultiplier/r.style.sizeMultiplier;i.height*=n;i.depth*=n;i.maxFontSize=t.sizeMultiplier;return i};var m=function(e,t,r,a,i,s){var l=n.makeSymbol(e,"Main-Regular",i,a);var o=c(l,t,a,s);if(r){var u=(1-a.style.sizeMultiplier/t.sizeMultiplier)*a.style.metrics.axisHeight;o.style.top=u+"em";o.height-=u;o.depth+=u}return o};var f=function(e,t,r,a,n,s){var l=h(e,t,n,a);var o=c(u(["delimsizing","size"+t],[l],a),i.TEXT,a,s);if(r){var p=(1-a.style.sizeMultiplier)*a.style.metrics.axisHeight;o.style.top=p+"em";o.height-=p;o.depth+=p}return o};var v=function(e,t,r){var a;if(t==="Size1-Regular"){a="delim-size1"}else if(t==="Size4-Regular"){a="delim-size4"}var i=u(["delimsizinginner",a],[u([],[n.makeSymbol(e,t,r)])]);return{type:"elem",elem:i}};var d=function(e,t,r,a,s,l){var o;var h;var m;var f;o=m=f=e;h=null;var d="Size1-Regular";if(e==="\\uparrow"){m=f="\u23d0"}else if(e==="\\Uparrow"){m=f="\u2016"}else if(e==="\\downarrow"){o=m="\u23d0"}else if(e==="\\Downarrow"){o=m="\u2016"}else if(e==="\\updownarrow"){o="\\uparrow";m="\u23d0";f="\\downarrow"}else if(e==="\\Updownarrow"){o="\\Uparrow";m="\u2016";f="\\Downarrow"}else if(e==="["||e==="\\lbrack"){o="\u23a1";m="\u23a2";f="\u23a3";d="Size4-Regular"}else if(e==="]"||e==="\\rbrack"){o="\u23a4";m="\u23a5";f="\u23a6";d="Size4-Regular"}else if(e==="\\lfloor"){m=o="\u23a2";f="\u23a3";d="Size4-Regular"}else if(e==="\\lceil"){o="\u23a1";m=f="\u23a2";d="Size4-Regular"}else if(e==="\\rfloor"){m=o="\u23a5";f="\u23a6";d="Size4-Regular"}else if(e==="\\rceil"){o="\u23a4";m=f="\u23a5";d="Size4-Regular"}else if(e==="("){o="\u239b";m="\u239c";f="\u239d";d="Size4-Regular"}else 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C;if(h===null){for(C=0;C<A;C++){E.push(v(m,d,s))}}else{for(C=0;C<A;C++){E.push(v(m,d,s))}E.push(v(h,d,s));for(C=0;C<A;C++){E.push(v(m,d,s))}}E.push(v(o,d,s));var D=n.makeVList(E,"bottom",R,a);return c(u(["delimsizing","mult"],[D],a),i.TEXT,a,l)};var g=["(",")","[","\\lbrack","]","\\rbrack","\\{","\\lbrace","\\}","\\rbrace","\\lfloor","\\rfloor","\\lceil","\\rceil","\\surd"];var y=["\\uparrow","\\downarrow","\\updownarrow","\\Uparrow","\\Downarrow","\\Updownarrow","|","\\|","\\vert","\\Vert","\\lvert","\\rvert","\\lVert","\\rVert","\\lgroup","\\rgroup","\\lmoustache","\\rmoustache"];var x=["<",">","\\langle","\\rangle","/","\\backslash","\\lt","\\gt"];var b=[0,1.2,1.8,2.4,3];var w=function(e,t,r,i,n){if(e==="<"||e==="\\lt"){e="\\langle"}else if(e===">"||e==="\\gt"){e="\\rangle"}if(o.contains(g,e)||o.contains(x,e)){return f(e,t,false,r,i,n)}else if(o.contains(y,e)){return d(e,b[t],false,r,i,n)}else{throw new a("Illegal delimiter: '"+e+"'")}};var k=[{type:"small",style:i.SCRIPTSCRIPT},{type:"small",style:i.SCRIPT},{type:"small",style:i.TEXT},{type:"large",size:1},{type:"large",size:2},{type:"large",size:3},{type:"large",size:4}];var z=[{type:"small",style:i.SCRIPTSCRIPT},{type:"small",style:i.SCRIPT},{type:"small",style:i.TEXT},{type:"stack"}];var S=[{type:"small",style:i.SCRIPTSCRIPT},{type:"small",style:i.SCRIPT},{type:"small",style:i.TEXT},{type:"large",size:1},{type:"large",size:2},{type:"large",size:3},{type:"large",size:4},{type:"stack"}];var M=function(e){if(e.type==="small"){return"Main-Regular"}else if(e.type==="large"){return"Size"+e.size+"-Regular"}else if(e.type==="stack"){return"Size4-Regular"}};var T=function(e,t,r,a){var i=Math.min(2,3-a.style.size);for(var n=i;n<r.length;n++){if(r[n].type==="stack"){break}var s=p(e,M(r[n]));var l=s.height+s.depth;if(r[n].type==="small"){l*=r[n].style.sizeMultiplier}if(l>t){return r[n]}}return r[r.length-1]};var A=function(e,t,r,a,i,n){if(e==="<"||e==="\\lt"){e="\\langle"}else if(e===">"||e==="\\gt"){e="\\rangle"}var s;if(o.contains(x,e)){s=k}else if(o.contains(g,e)){s=S}else{s=z}var l=T(e,t,s,a);if(l.type==="small"){return m(e,l.style,r,a,i,n)}else if(l.type==="large"){return f(e,l.size,r,a,i,n)}else if(l.type==="stack"){return d(e,t,r,a,i,n)}};var N=function(e,t,r,a,i,n){var l=a.style.metrics.axisHeight*a.style.sizeMultiplier;var o=901;var u=5/s.metrics.ptPerEm;var p=Math.max(t-l,r+l);var h=Math.max(p/500*o,2*p-u);return A(e,h,true,a,i,n)};t.exports={sizedDelim:w,customSizedDelim:A,leftRightDelim:N}},{"./ParseError":6,"./Style":9,"./buildCommon":10,"./fontMetrics":17,"./symbols":23,"./utils":25}],15:[function(e,t,r){var a=e("./unicodeRegexes");var i=e("./utils");var n=function(e){e=e.slice();for(var t=e.length-1;t>=0;t--){if(!e[t]){e.splice(t,1)}}return e.join(" ")};function s(e,t,r){this.classes=e||[];this.children=t||[];this.height=0;this.depth=0;this.maxFontSize=0;this.style={};this.attributes={};if(r){if(r.style.isTight()){this.classes.push("mtight")}if(r.getColor()){this.style.color=r.getColor()}}}s.prototype.setAttribute=function(e,t){this.attributes[e]=t};s.prototype.tryCombine=function(e){return false};s.prototype.toNode=function(){var e=document.createElement("span");e.className=n(this.classes);for(var t in this.style){if(Object.prototype.hasOwnProperty.call(this.style,t)){e.style[t]=this.style[t]}}for(var r in this.attributes){if(Object.prototype.hasOwnProperty.call(this.attributes,r)){e.setAttribute(r,this.attributes[r])}}for(var a=0;a<this.children.length;a++){e.appendChild(this.children[a].toNode())}return e};s.prototype.toMarkup=function(){var e="<span";if(this.classes.length){e+=' class="';e+=i.escape(n(this.classes));e+='"'}var t="";for(var r in this.style){if(this.style.hasOwnProperty(r)){t+=i.hyphenate(r)+":"+this.style[r]+";"}}if(t){e+=' style="'+i.escape(t)+'"'}for(var a in this.attributes){if(Object.prototype.hasOwnProperty.call(this.attributes,a)){e+=" "+a+'="';e+=i.escape(this.attributes[a]);e+='"'}}e+=">";for(var s=0;s<this.children.length;s++){e+=this.children[s].toMarkup()}e+="</span>";return e};function l(e){this.children=e||[];this.height=0;this.depth=0;this.maxFontSize=0}l.prototype.toNode=function(){var e=document.createDocumentFragment();for(var t=0;t<this.children.length;t++){e.appendChild(this.children[t].toNode())}return e};l.prototype.toMarkup=function(){var e="";for(var t=0;t<this.children.length;t++){e+=this.children[t].toMarkup()}return e};var o={"\xee":"\u0131\u0302","\xef":"\u0131\u0308","\xed":"\u0131\u0301","\xec":"\u0131\u0300"};function u(e,t,r,i,n,s,l){this.value=e||"";this.height=t||0;this.depth=r||0;this.italic=i||0;this.skew=n||0;this.classes=s||[];this.style=l||{};this.maxFontSize=0;if(a.cjkRegex.test(e)){if(a.hangulRegex.test(e)){this.classes.push("hangul_fallback")}else{this.classes.push("cjk_fallback")}}if(/[\xee\xef\xed\xec]/.test(this.value)){this.value=o[this.value]}}u.prototype.tryCombine=function(e){if(!e||!(e instanceof u)||this.italic>0||n(this.classes)!==n(e.classes)||this.skew!==e.skew||this.maxFontSize!==e.maxFontSize){return false}for(var t in this.style){if(this.style.hasOwnProperty(t)&&this.style[t]!==e.style[t]){return false}}for(t in e.style){if(e.style.hasOwnProperty(t)&&this.style[t]!==e.style[t]){return false}}this.value+=e.value;this.height=Math.max(this.height,e.height);this.depth=Math.max(this.depth,e.depth);this.italic=e.italic;return true};u.prototype.toNode=function(){var e=document.createTextNode(this.value);var t=null;if(this.italic>0){t=document.createElement("span");t.style.marginRight=this.italic+"em"}if(this.classes.length>0){t=t||document.createElement("span");t.className=n(this.classes)}for(var r in this.style){if(this.style.hasOwnProperty(r)){t=t||document.createElement("span");t.style[r]=this.style[r]}}if(t){t.appendChild(e);return t}else{return e}};u.prototype.toMarkup=function(){var e=false;var t="<span";if(this.classes.length){e=true;t+=' class="';t+=i.escape(n(this.classes));t+='"'}var r="";if(this.italic>0){r+="margin-right:"+this.italic+"em;"}for(var a in this.style){if(this.style.hasOwnProperty(a)){r+=i.hyphenate(a)+":"+this.style[a]+";"}}if(r){e=true;t+=' style="'+i.escape(r)+'"'}var s=i.escape(this.value);if(e){t+=">";t+=s;t+="</span>";return t}else{return s}};t.exports={span:s,documentFragment:l,symbolNode:u}},{"./unicodeRegexes":24,"./utils":25}],16:[function(e,t,r){var a=e("./parseData");var i=e("./ParseError");var n=e("./Style");var s=a.ParseNode;function l(e,t){var r=[];var a=[r];var n=[];while(true){var l=e.parseExpression(false,null);r.push(new s("ordgroup",l,e.mode));var o=e.nextToken.text;if(o==="&"){e.consume()}else if(o==="\\end"){break}else if(o==="\\\\"||o==="\\cr"){var u=e.parseFunction();n.push(u.value.size);r=[];a.push(r)}else{throw new i("Expected & or \\\\ or \\end",e.nextToken)}}t.body=a;t.rowGaps=n;return new s(t.type,t,e.mode)}function o(e,r,a){if(typeof e==="string"){e=[e]}if(typeof r==="number"){r={numArgs:r}}var i={numArgs:r.numArgs||0,argTypes:r.argTypes,greediness:1,allowedInText:!!r.allowedInText,numOptionalArgs:r.numOptionalArgs||0,handler:a};for(var n=0;n<e.length;++n){t.exports[e[n]]=i}}o("array",{numArgs:1},function(e,t){var r=t[0];r=r.value.map?r.value:[r];var a=r.map(function(e){var t=e.value;if("lcr".indexOf(t)!==-1){return{type:"align",align:t}}else if(t==="|"){return{type:"separator",separator:"|"}}throw new i("Unknown column alignment: "+e.value,e)});var 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+<body class="book toc2 toc-left" style="max-width: 100;">
+<div id="header">
+<h1>The OpenGL<sup>®</sup> Shading Language, Version 4.60.6</h1>
+<div class="details">
+<span id="author" class="author">John Kessenich, Google (Editor and Author) ; Dave Baldwin and Randi Rost (Version 1.1 Authors)</span><br>
+<span id="revnumber">version 4.60.6,</span>
+<span id="revdate">Wed, 12 Dec 2018 23:37:49 +0000</span>
+<br><span id="revremark">Git branch information not available</span>
+</div>
+<div id="toc" class="toc2">
+<div id="toctitle">Table of Contents</div>
+<ul class="sectlevel1">
+<li><a href="#introduction">1. Introduction</a>
+<ul class="sectlevel2">
+<li><a href="#changes">1.1. Changes</a></li>
+<li><a href="#overview">1.2. Overview</a></li>
+<li><a href="#error-handling">1.3. Error Handling</a></li>
+<li><a href="#typographical-conventions">1.4. Typographical Conventions</a></li>
+<li><a href="#deprecation">1.5. Deprecation</a></li>
+</ul>
+</li>
+<li><a href="#overview-of-opengl-shading">2. Overview of OpenGL Shading</a>
+<ul class="sectlevel2">
+<li><a href="#vertex-processor">2.1. Vertex Processor</a></li>
+<li><a href="#tessellation-control-processor">2.2. Tessellation Control Processor</a></li>
+<li><a href="#tessellation-evaluation-processor">2.3. Tessellation Evaluation Processor</a></li>
+<li><a href="#geometry-processor">2.4. Geometry Processor</a></li>
+<li><a href="#fragment-processor">2.5. Fragment Processor</a></li>
+<li><a href="#compute-processor">2.6. Compute Processor</a></li>
+</ul>
+</li>
+<li><a href="#basics">3. Basics</a>
+<ul class="sectlevel2">
+<li><a href="#character-set">3.1. Character Set and Phases of Compilation</a></li>
+<li><a href="#source-strings">3.2. Source Strings</a></li>
+<li><a href="#preprocessor">3.3. Preprocessor</a></li>
+<li><a href="#comments">3.4. Comments</a></li>
+<li><a href="#tokens">3.5. Tokens</a></li>
+<li><a href="#keywords">3.6. Keywords</a></li>
+<li><a href="#identifiers">3.7. Identifiers</a></li>
+<li><a href="#definitions">3.8. Definitions</a></li>
+</ul>
+</li>
+<li><a href="#variables-and-types">4. Variables and Types</a>
+<ul class="sectlevel2">
+<li><a href="#basic-types">4.1. Basic Types</a></li>
+<li><a href="#scoping">4.2. Scoping</a></li>
+<li><a href="#storage-qualifiers">4.3. Storage Qualifiers</a></li>
+<li><a href="#layout-qualifiers">4.4. Layout Qualifiers</a></li>
+<li><a href="#interpolation-qualifiers">4.5. Interpolation Qualifiers</a></li>
+<li><a href="#parameter-qualifiers">4.6. Parameter Qualifiers</a></li>
+<li><a href="#precision-and-precision-qualifiers">4.7. Precision and Precision Qualifiers</a></li>
+<li><a href="#variance-and-the-invariant-qualifier">4.8. Variance and the Invariant Qualifier</a></li>
+<li><a href="#the-precise-qualifier">4.9. The Precise Qualifier</a></li>
+<li><a href="#memory-qualifiers">4.10. Memory Qualifiers</a></li>
+<li><a href="#specialization-constant-qualifier">4.11. Specialization-Constant Qualifier</a></li>
+<li><a href="#order-of-qualification">4.12. Order and Repetition of Qualification</a></li>
+<li><a href="#empty-declarations">4.13. Empty Declarations</a></li>
+</ul>
+</li>
+<li><a href="#operators-and-expressions">5. Operators and Expressions</a>
+<ul class="sectlevel2">
+<li><a href="#operators">5.1. Operators</a></li>
+<li><a href="#array-operations">5.2. Array Operations</a></li>
+<li><a href="#function-calls">5.3. Function Calls</a></li>
+<li><a href="#constructors">5.4. Constructors</a></li>
+<li><a href="#vector-components">5.5. Vector and Scalar Components and Length</a></li>
+<li><a href="#matrix-components">5.6. Matrix Components</a></li>
+<li><a href="#structure-and-array-operations">5.7. Structure and Array Operations</a></li>
+<li><a href="#assignments">5.8. Assignments</a></li>
+<li><a href="#expressions">5.9. Expressions</a></li>
+<li><a href="#vector-and-matrix-operations">5.10. Vector and Matrix Operations</a></li>
+<li><a href="#out-of-bounds-accesses">5.11. Out-of-Bounds Accesses</a></li>
+<li><a href="#specialization-constant-operations">5.12. Specialization Constant Operations</a></li>
+</ul>
+</li>
+<li><a href="#statements-and-structure">6. Statements and Structure</a>
+<ul class="sectlevel2">
+<li><a href="#function-definitions">6.1. Function Definitions</a></li>
+<li><a href="#selection">6.2. Selection</a></li>
+<li><a href="#iteration">6.3. Iteration</a></li>
+<li><a href="#jumps">6.4. Jumps</a></li>
+</ul>
+</li>
+<li><a href="#built-in-variables">7. Built-In Variables</a>
+<ul class="sectlevel2">
+<li><a href="#built-in-language-variables">7.1. Built-In Language Variables</a></li>
+<li><a href="#compatibility-profile-vertex-shader-built-in-inputs">7.2. Compatibility Profile Vertex Shader Built-In Inputs</a></li>
+<li><a href="#built-in-constants">7.3. Built-In Constants</a></li>
+<li><a href="#built-in-uniform-state">7.4. Built-In Uniform State</a></li>
+<li><a href="#redeclaring-built-in-blocks">7.5. Redeclaring Built-In Blocks</a></li>
+</ul>
+</li>
+<li><a href="#built-in-functions">8. Built-In Functions</a>
+<ul class="sectlevel2">
+<li><a href="#angle-and-trigonometry-functions">8.1. Angle and Trigonometry Functions</a></li>
+<li><a href="#exponential-functions">8.2. Exponential Functions</a></li>
+<li><a href="#common-functions">8.3. Common Functions</a></li>
+<li><a href="#floating-point-pack-and-unpack-functions">8.4. Floating-Point Pack and Unpack Functions</a></li>
+<li><a href="#geometric-functions">8.5. Geometric Functions</a></li>
+<li><a href="#matrix-functions">8.6. Matrix Functions</a></li>
+<li><a href="#vector-relational-functions">8.7. Vector Relational Functions</a></li>
+<li><a href="#integer-functions">8.8. Integer Functions</a></li>
+<li><a href="#texture-functions">8.9. Texture Functions</a></li>
+<li><a href="#atomic-counter-functions">8.10. Atomic Counter Functions</a></li>
+<li><a href="#atomic-memory-functions">8.11. Atomic Memory Functions</a></li>
+<li><a href="#image-functions">8.12. Image Functions</a></li>
+<li><a href="#geometry-shader-functions">8.13. Geometry Shader Functions</a></li>
+<li><a href="#fragment-processing-functions">8.14. Fragment Processing Functions</a></li>
+<li><a href="#noise-functions">8.15. Noise Functions</a></li>
+<li><a href="#shader-invocation-control-functions">8.16. Shader Invocation Control Functions</a></li>
+<li><a href="#shader-memory-control-functions">8.17. Shader Memory Control Functions</a></li>
+<li><a href="#shader-invocation-group-functions">8.18. Shader Invocation Group Functions</a></li>
+</ul>
+</li>
+<li><a href="#shading-language-grammar">9. Shading Language Grammar</a></li>
+<li><a href="#acknowledgments">10. Acknowledgments</a></li>
+<li><a href="#references">11. Normative References</a></li>
+</ul>
+</div>
+</div>
+<div id="content">
+<div id="preamble">
+<div class="sectionbody">
+<div style="page-break-after: always;"></div>
+<div class="paragraph">
+<p>Copyright © 2008-2018 The Khronos Group Inc. All Rights Reserved.</p>
+</div>
+<div class="paragraph">
+<p>This specification is protected by copyright laws and contains material
+proprietary to the Khronos Group, Inc. It or any components may not be
+reproduced, republished, distributed, transmitted, displayed, broadcast,
+or otherwise exploited in any manner without the express prior written
+permission of Khronos Group. You may use this specification for
+implementing the functionality therein, without altering or removing any
+trademark, copyright or other notice from the specification, but the
+receipt or possession of this specification does not convey any rights
+to reproduce, disclose, or distribute its contents, or to manufacture,
+use, or sell anything that it may describe, in whole or in part.</p>
+</div>
+<div class="paragraph">
+<p>Khronos Group grants express permission to any current Promoter,
+Contributor or Adopter member of Khronos to copy and redistribute
+UNMODIFIED versions of this specification in any fashion, provided that
+NO CHARGE is made for the specification and the latest available update
+of the specification for any version of the API is used whenever
+possible. Such distributed specification may be reformatted AS LONG AS
+the contents of the specification are not changed in any way. The
+specification may be incorporated into a product that is sold as long as
+such product includes significant independent work developed by the
+seller. A link to the current version of this specification on the
+Khronos Group website should be included whenever possible with
+specification distributions.</p>
+</div>
+<div class="paragraph">
+<p>Khronos Group makes no, and expressly disclaims any, representations or
+warranties, express or implied, regarding this specification, including,
+without limitation, any implied warranties of merchantability or fitness
+for a particular purpose or noninfringement of any intellectual
+property. Khronos Group makes no, and expressly disclaims any,
+warranties, express or implied, regarding the correctness, accuracy,
+completeness, timeliness, and reliability of the specification. Under no
+circumstances will the Khronos Group, or any of its Promoters,
+Contributors or Members or their respective partners, officers,
+directors, employees, agents, or representatives be liable for any
+damages, whether direct, indirect, special or consequential damages for
+lost revenues, lost profits, or otherwise, arising from or in connection
+with these materials.</p>
+</div>
+<div class="paragraph">
+<p>Khronos, Vulkan, SYCL, SPIR, WebGL, EGL, COLLADA, StreamInput, OpenVX,
+OpenKCam, glTF, OpenKODE, OpenVG, OpenWF, OpenSL ES, OpenMAX, OpenMAX
+AL, OpenMAX IL and OpenMAX DL are trademarks and WebCL is a
+certification mark of the Khronos Group Inc. OpenCL is a trademark of
+Apple Inc. and OpenGL and OpenML are registered trademarks and the
+OpenGL ES and OpenGL SC logos are trademarks of Silicon Graphics
+International used under license by Khronos. All other product names,
+trademarks, and/or company names are used solely for identification and
+belong to their respective owners.</p>
+</div>
+<div style="page-break-after: always;"></div>
+<!-- toc disabled -->
+</div>
+</div>
+<div class="sect1">
+<h2 id="introduction">1. Introduction</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>This document specifies only version 4.60 of the OpenGL Shading Language.
+It requires __VERSION__ to substitute 460, and requires
+<strong>#version</strong> to accept only
+<code>460</code>.
+If <strong>#version</strong> is declared with a smaller number, the language accepted is a
+previous version of the shading language, which will be supported depending
+on the version and type of context in the OpenGL API.
+See the <a href="#references">OpenGL Specification</a> for details on what language versions are
+supported.</p>
+</div>
+<div class="paragraph">
+<p>Previous versions of the OpenGL Shading Language, as well as the OpenGL ES Shading Language,
+are not strict subsets of the version specified here, particularly with
+respect to precision, name-hiding rules, and treatment of interface
+variables.
+See the specification corresponding to a particular language version for
+details specific to that version of the language.</p>
+</div>
+<div class="paragraph">
+<p>The OpenGL API will specify the commands used to manipulate SPIR-V
+shaders.
+Independent offline tool chains will compile GLSL down to the SPIR-V
+intermediate language.
+SPIR-V generation is not enabled with a <strong>#extension</strong>, <strong>#version</strong>, or a
+profile.
+Instead, use of GLSL for SPIR-V is determined by offline tool-chain use.
+See the documentation of such tools to see how to request generation of
+SPIR-V for OpenGL.</p>
+</div>
+<div class="paragraph">
+<p>GLSL → SPIR-V compilers must be directed as to what SPIR-V <strong>Capabilities</strong>
+are legal at run-time and give errors for GLSL feature use outside those
+capabilities.
+This is also true for implementation-dependent limits that can be error
+checked by the front-end against built-in constants present in the GLSL
+source: the front-end can be informed of such limits, and report errors when
+they are exceeded.</p>
+</div>
+<div class="paragraph">
+<p>All references in this specification to the <a href="#references">OpenGL Specification</a> are to
+the Core profile of version 4.6, unless a different profile is
+specified.</p>
+</div>
+<div class="sect2">
+<h3 id="changes">1.1. Changes</h3>
+<div class="sect3">
+<h4 id="_changes_from_revision_5_of_glsl_4_6">1.1.1. Changes from Revision 5 of GLSL 4.6</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Private GLSL issue #34: Clarify/consolidate implicit conversion rules from int → uint
+to be the same as explicit construction.</p>
+</li>
+<li>
+<p>Private GLSL issue #24: Clarify that <strong>barrier</strong>() by itself is enough to synchronize
+both control flow and memory accesses to <strong>shared</strong> variables and tessellation
+control output variables. For other memory accesses an additional memory
+barrier is still required.</p>
+</li>
+<li>
+<p>Normatively reference IEEE-754 for definitions of floating-point formats.</p>
+</li>
+<li>
+<p>Private GLSL issue 36: <strong>refract</strong> function on <strong>double</strong> types requires eta
+argument to have type <strong>double</strong>.</p>
+</li>
+<li>
+<p>Clarify restrictions on input variables in tessellation and geometry stages.</p>
+</li>
+<li>
+<p>Private GLSL issue 15: Clarify the ordering of bindings for arrays of arrays.</p>
+</li>
+<li>
+<p>Private GLSL issue 14: Uniform variables need only match at link time if they
+are statically used.</p>
+</li>
+<li>
+<p>For <strong>precise</strong> computations, the controlling expressions for
+control flow and ternary operators (<strong>?:</strong>) are not included.</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="_changes_from_revision_4_of_glsl_4_6">1.1.2. Changes from Revision 4 of GLSL 4.6</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Private bug 13012: Clarified that builtin uniform variables might only
+be available in the fragment stage.</p>
+</li>
+<li>
+<p>Private bug 13837: Ternary and sequence operators may operate on <strong>void</strong> types.</p>
+</li>
+<li>
+<p>Clarified that errors arising from preprocessing must be returned at compile time.</p>
+</li>
+<li>
+<p>Clarified that access to any part of a variable constitutes a static use.</p>
+</li>
+<li>
+<p>Private GLSL issue 19: A statement is required following any label at the end of a <strong>switch</strong>.</p>
+</li>
+<li>
+<p>Private GLSL issue 26: <strong>noise</strong> is not valid when compiling for SPIR-V.</p>
+</li>
+<li>
+<p>Private GLSL issue 20: <strong>length</strong>() expressions returning a constant value may not
+contain side effects.</p>
+</li>
+<li>
+<p>Public OpenGL-API issue 7: Variables can be declared as both <strong>readonly</strong>
+and <strong>writeonly</strong>.</p>
+</li>
+<li>
+<p>Private GLSL issue 16: Use of constant expressions within <strong>#line</strong> directives is undefined.</p>
+</li>
+<li>
+<p>Corrected return type of <strong>imageAtomicExchange</strong> on <strong>float</strong> images.</p>
+</li>
+<li>
+<p>Private GLSL issue 32: Remove <strong>length</strong>() method contradiction:
+Non runtime-sized arrays only support <strong>length</strong>() on explicitly
+sized arrays.</p>
+</li>
+<li>
+<p>Private GLSL issue 21: Clarified the l-value restriction on <strong>interpolateAt</strong>.</p>
+</li>
+<li>
+<p>Private OpenGL-API issue 53: Clarified bit-width requirements for location aliasing.</p>
+</li>
+<li>
+<p>Public GLSL issue 15: <strong>gl_in</strong> can be redeclared using unsized-array syntax.</p>
+</li>
+<li>
+<p>Clarification of the formats needed for DEPTH_COMPONENT and
+STENCIL_COMPONENT for depth/stencil textures.</p>
+</li>
+<li>
+<p>Added image formats to the layout-qualifier table in the
+<a href="#layout-qualifiers">Layout Qualifiers</a> section.</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="_changes_from_revision_3_of_glsl_4_6">1.1.3. Changes from Revision 3 of GLSL 4.6</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Private GLSL issue 13: Fix misspelling of <strong>allInvocationsEqual</strong>().
+(The one in the table was incorrectly listed as <strong>anyInvocationsEqual</strong>(),
+other spellings were correct.)</p>
+</li>
+</ul>
+</div>
+</div>
+<div class="sect3">
+<h4 id="_summary_of_changes_from_revision_7_of_glsl_version_4_50">1.1.4. Summary of Changes from Revision 7 of GLSL Version 4.50</h4>
+<div class="ulist">
+<ul>
+<li>
+<p>Incorporated the GL_ARB_shader_atomic_counter_ops extension.</p>
+</li>
+<li>
+<p>Incorporated the GL_ARB_shader_draw_parameters extension.</p>
+</li>
+<li>
+<p>Incorporated the GL_ARB_shader_group_vote extension.</p>
+</li>
+<li>
+<p>Incorporated the GL_ARB_gl_spirv extension.</p>
+</li>
+<li>
+<p>Private Bug 16070: Allow extra semi-colons at global scope.</p>
+</li>
+<li>
+<p>Private GLSL Issue 5: Be explicit that “fail to link” is really
+“compile-time or link-time error”, for some forms of error.</p>
+</li>
+<li>
+<p>Private GLSL Issue 7: Change <em>gl_MaxComputeUniformComponents</em> to 1024.</p>
+</li>
+<li>
+<p>Private OpenGL API Issue 35: Require location on transparent individual
+uniform variables for SPIR-V.</p>
+</li>
+<li>
+<p>Private GLSL Issue 8: Be more clear an <strong>interpolateAt</strong>() interpolant can
+be a structure member.</p>
+</li>
+<li>
+<p>Private GLSL Issue 9: Specify how <strong>xfb_buffer</strong> interacts with a block
+array: the capturing buffer increments for each block array element.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="overview">1.2. Overview</h3>
+<div class="paragraph">
+<p>This document describes <em>The OpenGL Shading Language, version 4.60</em>.</p>
+</div>
+<div class="paragraph">
+<p>Independent compilation units written in this language are called <em>shaders</em>.
+A <em>program</em> is a set of shaders that are compiled and linked
+together,
+completely creating one or more of the programmable stages of the
+OpenGL pipeline.
+All the shaders for a single programmable stage must be within the same
+program.
+A complete set of programmable stages can be put into a single program or
+the stages can be partitioned across multiple programs.
+The aim of this document is to thoroughly specify the programming language.
+The <a href="#references">OpenGL Specification</a> will specify the OpenGL entry points used to
+manipulate and communicate with programs and shaders.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="error-handling">1.3. Error Handling</h3>
+<div class="paragraph">
+<p>Compilers, in general, accept programs that are ill-formed, due to the
+impossibility of detecting all ill-formed programs.
+Portability is only ensured for well-formed programs, which this
+specification describes.
+Compilers are encouraged to detect ill-formed programs and issue diagnostic
+messages, but are not required to do so for all cases.
+Compile-time errors must be returned for lexically or grammatically
+incorrect shaders.
+Other errors are reported at compile time or link time as indicated.
+Code that is “dead” must still be error checked.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">if</span> (<span class="predefined-constant">false</span>) <span class="comment">// changing false to true cannot uncover additional errors</span>
+ statement; <span class="comment">// statement must be error checked regardless</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="typographical-conventions">1.4. Typographical Conventions</h3>
+<div class="paragraph">
+<p>Italic, bold, and font choices have been used in this specification
+primarily to improve readability.
+Code fragments use a fixed width font.
+Identifiers embedded in text are italicized.
+Keywords embedded in text are bold.
+Operators are called by their name, followed by their symbol in bold in
+parentheses.
+The clarifying grammar fragments in the text use bold for literals and
+italics for non-terminals.
+The official grammar in “<a href="#shading-language-grammar">Shading Language
+Grammar</a>” uses all capitals for terminals and lower case for
+non-terminals.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="deprecation">1.5. Deprecation</h3>
+<div class="paragraph">
+<p>The OpenGL Shading Language has deprecated some features.
+These are clearly called out in this specification as “deprecated”.
+They are still present in this version of the language, but are targeted for
+potential removal in a future version of the shading language.
+The OpenGL API has a forward compatibility mode that will disallow use of
+deprecated features.
+If compiling in a mode where use of deprecated features is disallowed, their
+use causes compile-time or link-time errors.
+See the <a href="#references">OpenGL Specification</a> for details on what causes deprecated
+language features to be accepted or to return an error.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="overview-of-opengl-shading">2. Overview of OpenGL Shading</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The OpenGL Shading Language is actually several closely related languages.
+These languages are used to create shaders for each of the programmable
+processors contained in the OpenGL processing pipeline.
+Currently, these processors are the vertex, tessellation control,
+tessellation evaluation, geometry, fragment, and compute processors.</p>
+</div>
+<div class="paragraph">
+<p>Unless otherwise noted in this paper, a language feature applies to all
+languages, and common usage will refer to these languages as a single
+language.
+The specific languages will be referred to by the name of the processor they
+target: vertex, tessellation control, tessellation evaluation, geometry,
+fragment, or compute.</p>
+</div>
+<div class="paragraph">
+<p>Most OpenGL state is not tracked or made available to shaders.
+Typically, user-defined variables will be used for communicating between
+different stages of the OpenGL pipeline.
+However, a small amount of state is still tracked and automatically made
+available to shaders, and there are a few built-in variables for interfaces
+between different stages of the OpenGL pipeline.</p>
+</div>
+<div class="sect2">
+<h3 id="vertex-processor">2.1. Vertex Processor</h3>
+<div class="paragraph">
+<p>The <em>vertex processor</em> is a programmable unit that operates on incoming
+vertices and their associated data.
+Compilation units written in the OpenGL Shading Language to run on this processor are called
+<em>vertex shaders</em>.
+When a set of vertex shaders are successfully compiled and linked, they
+result in a <em>vertex shader executable</em> that runs on the vertex processor.</p>
+</div>
+<div class="paragraph">
+<p>The vertex processor operates on one vertex at a time.
+It does not replace graphics operations that require knowledge of several
+vertices at a time.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="tessellation-control-processor">2.2. Tessellation Control Processor</h3>
+<div class="paragraph">
+<p>The <em>tessellation control processor</em> is a programmable unit that operates on
+a patch of incoming vertices and their associated data, emitting a new
+output patch.
+Compilation units written in the OpenGL Shading Language to run on this processor are called
+tessellation control shaders.
+When a set of tessellation control shaders are successfully compiled and
+linked, they result in a <em>tessellation control shader executable</em> that runs
+on the tessellation control processor.</p>
+</div>
+<div class="paragraph">
+<p>The tessellation control shader is invoked for each vertex of the output
+patch.
+Each invocation can read the attributes of any vertex in the input or output
+patches, but can only write per-vertex attributes for the corresponding
+output patch vertex.
+The shader invocations collectively produce a set of per-patch attributes
+for the output patch.</p>
+</div>
+<div class="paragraph">
+<p>After all tessellation control shader invocations have completed, the output
+vertices and per-patch attributes are assembled to form a patch to be used
+by subsequent pipeline stages.</p>
+</div>
+<div class="paragraph">
+<p>Tessellation control shader invocations run mostly independently, with
+undefined relative execution order.
+However, the built-in function <strong>barrier</strong>() can be used to control execution
+order by synchronizing invocations, effectively dividing tessellation
+control shader execution into a set of phases.
+Tessellation control shaders will get undefined results if one invocation
+reads from a per-vertex or per-patch attribute written by another invocation
+at any point during the same phase, or if two invocations attempt to write
+different values to the same per-patch output
+32-bit component
+in a single phase.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="tessellation-evaluation-processor">2.3. Tessellation Evaluation Processor</h3>
+<div class="paragraph">
+<p>The <em>tessellation evaluation processor</em> is a programmable unit that
+evaluates the position and other attributes of a vertex generated by the
+tessellation primitive generator, using a patch of incoming vertices and
+their associated data.
+Compilation units written in the OpenGL Shading Language to run on this processor are called
+tessellation evaluation shaders.
+When a set of tessellation evaluation shaders are successfully compiled and
+linked, they result in a <em>tessellation evaluation shader executable</em> that
+runs on the tessellation evaluation processor.</p>
+</div>
+<div class="paragraph">
+<p>Each invocation of the tessellation evaluation executable computes the
+position and attributes of a single vertex generated by the tessellation
+primitive generator.
+The executable can read the attributes of any vertex in the input patch,
+plus the tessellation coordinate, which is the relative location of the
+vertex in the primitive being tessellated.
+The executable writes the position and other attributes of the vertex.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="geometry-processor">2.4. Geometry Processor</h3>
+<div class="paragraph">
+<p>The <em>geometry processor</em> is a programmable unit that operates on data for
+incoming vertices for a primitive assembled after vertex processing and
+outputs a sequence of vertices forming output primitives.
+Compilation units written in the OpenGL Shading Language to run on this processor are called
+<em>geometry shaders</em>.
+When a set of geometry shaders are successfully compiled and linked, they
+result in a <em>geometry shader executable</em> that runs on the geometry
+processor.</p>
+</div>
+<div class="paragraph">
+<p>A single invocation of the geometry shader executable on the geometry
+processor will operate on a declared input primitive with a fixed number of
+vertices.
+This single invocation can emit a variable number of vertices that are
+assembled into primitives of a declared output primitive type and passed to
+subsequent pipeline stages.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="fragment-processor">2.5. Fragment Processor</h3>
+<div class="paragraph">
+<p>The <em>fragment processor</em> is a programmable unit that operates on fragment
+values and their associated data.
+Compilation units written in the OpenGL Shading Language to run on this processor are called
+<em>fragment shaders</em>.
+When a set of fragment shaders are successfully compiled and linked, they
+result in a <em>fragment shader executable</em> that runs on the fragment
+processor.</p>
+</div>
+<div class="paragraph">
+<p>A fragment shader cannot change a fragment’s (<em>x</em>, <em>y</em>) position.
+Access to neighboring fragments is not allowed.
+The values computed by the fragment shader are ultimately used to update
+framebuffer memory or texture memory, depending on the current OpenGL
+state and the OpenGL command that caused the fragments to be generated.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="compute-processor">2.6. Compute Processor</h3>
+<div class="paragraph">
+<p>The <em>compute processor</em> is a programmable unit that operates independently
+from the other shader processors.
+Compilation units written in the OpenGL Shading Language to run on this processor are called
+<em>compute shaders</em>.
+When a set of compute shaders are successfully compiled and linked, they
+result in a <em>compute shader executable</em> that runs on the compute processor.</p>
+</div>
+<div class="paragraph">
+<p>A compute shader has access to many of the same resources as fragment and
+other shader processors, such as textures, buffers, image variables, and
+atomic counters.
+It does not have fixed-function outputs.
+It is not part of the graphics pipeline and its visible side effects are
+through changes to images, storage buffers, and atomic counters.</p>
+</div>
+<div class="paragraph">
+<p>A compute shader operates on a group of work items called a <em>workgroup</em>.
+A workgroup is a collection of shader invocations that execute the same
+code, potentially in parallel.
+An invocation within a workgroup may share data with other members of the
+same workgroup through shared variables and issue memory and control flow
+barriers to synchronize with other members of the same workgroup.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="basics">3. Basics</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="character-set">3.1. Character Set and Phases of Compilation</h3>
+<div class="paragraph">
+<p>The source character set used for the OpenGL Shading Language is Unicode in the UTF-8
+encoding scheme.</p>
+</div>
+<div class="paragraph">
+<p>After preprocessing, only the following characters are allowed in the
+resulting stream of GLSL tokens:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The letters <strong>a-z</strong>, <strong>A-Z</strong>, and the underscore (<strong>_</strong>).</p>
+</li>
+<li>
+<p>The numbers <strong>0-9</strong>.</p>
+</li>
+<li>
+<p>The symbols period (<strong>.</strong>), plus (<strong>+</strong>), dash (<strong>-</strong>), slash (<strong>/</strong>), asterisk
+(<strong>*</strong>), percent (<strong>%</strong>), angled brackets (<strong><</strong> and <strong>></strong>), square brackets
+(<strong>[</strong> and <strong>]</strong>), parentheses (<strong>(</strong> and <strong>)</strong>), braces (<strong>{</strong> and <strong>}</strong>), caret
+(<strong>^</strong>), vertical bar (<strong>|</strong>), ampersand (<strong>&</strong>), tilde (<strong>~</strong>), equals (<strong>=</strong>),
+exclamation point (<strong>!</strong>), colon (<strong>:</strong>), semicolon (<strong>;</strong>), comma (<strong>,</strong>), and
+question mark (<strong>?</strong>).</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>A compile-time error will be given if any other character is used in a GLSL
+token.</p>
+</div>
+<div class="paragraph">
+<p>There are no digraphs or trigraphs.
+There are no escape sequences or other uses of the backslash beyond use as
+the line-continuation character.</p>
+</div>
+<div class="paragraph">
+<p>Lines are relevant for compiler diagnostic messages and the preprocessor.
+They are terminated by carriage-return or line-feed.
+If both are used together, it will count as only a single line termination.
+For the remainder of this document, any of these combinations is simply
+referred to as a new-line.
+Lines may be of arbitrary length.</p>
+</div>
+<div class="paragraph">
+<p>In general, the language’s use of this character set is case sensitive.</p>
+</div>
+<div class="paragraph">
+<p>There are no character or string data types, so no quoting characters are
+included.</p>
+</div>
+<div class="paragraph">
+<p>There is no end-of-file character.</p>
+</div>
+<div class="paragraph">
+<p>More formally, compilation happens as if the following logical phases were
+executed in order:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>Source strings are concatenated to form a single input.
+All provided new-lines are retained.</p>
+</li>
+<li>
+<p>Line numbering is noted, based on all present new-lines, and does not
+change when new-lines are later eliminated.</p>
+</li>
+<li>
+<p>Wherever a backslash ('\') occurs immediately before a new-line, both
+are eliminated.
+(Note no white space is substituted, allowing a single token to span a
+new-line.) Any newly formed backslash followed by a new-line is not
+eliminated; only those pairs originally occurring after phase 1 are
+eliminated.</p>
+</li>
+<li>
+<p>All comments are replaced with a single space.
+(Note that '//' style comments end before their terminating new-lines
+and white space is generally relevant to preprocessing.)</p>
+</li>
+<li>
+<p>Preprocessing is done, resulting in a sequence of GLSL tokens, formed
+from the character set stated above.</p>
+</li>
+<li>
+<p>GLSL processing is done on the sequence of GLSL tokens.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Details that fully define source strings, comments, line numbering, new-line
+elimination, and preprocessing are all discussed in upcoming sections.
+Sections beyond those describe GLSL processing.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="source-strings">3.2. Source Strings</h3>
+<div class="paragraph">
+<p>The source for a single shader is an array of strings of characters from the
+character set.
+A single shader is made from the concatenation of these strings.
+Each string can contain multiple lines, separated by new-lines.
+No new-lines need be present in a string; a single line can be formed from
+multiple strings.
+No new-lines or other characters are inserted by the implementation when it
+concatenates the strings to form a single shader.
+Multiple shaders can be linked together to form a single program.</p>
+</div>
+<div class="paragraph">
+<p>Diagnostic messages returned from compiling a shader must identify both the
+line number within a string and which source string the message applies to.
+Source strings are counted sequentially with the first string being string
+0.
+Line numbers are one more than the number of new-lines that have been
+processed, including counting the new-lines that will be removed by the
+line-continuation character (<strong>\</strong>).</p>
+</div>
+<div class="paragraph">
+<p>Lines separated by the line-continuation character preceding a new-line are
+concatenated together before either comment processing or preprocessing.
+This means that no white space is substituted for the line-continuation
+character.
+That is, a single token could be formed by the concatenation by taking the
+characters at the end of one line concatenating them with the characters at
+the beginning of the next line.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> f\
+oo;
+<span class="comment">// forms a single line equivalent to "float foo;"</span>
+<span class="comment">// (assuming '\' is the last character before the new-line and "oo" are</span>
+<span class="comment">// the first two characters of the next line)</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="preprocessor">3.3. Preprocessor</h3>
+<div class="paragraph">
+<p>There is a preprocessor that processes the source strings as part of the
+compilation process.
+Except as noted below, it behaves as the C++ standard preprocessor (see
+“<a href="#references">Normative References</a>”).</p>
+</div>
+<div class="paragraph">
+<p>The complete list of preprocessor directives is as follows.</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#<br>
+#define<br>
+#undef<br></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#if<br>
+#ifdef<br>
+#ifndef<br>
+#else<br>
+#elif<br>
+#endif<br></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#error<br>
+#pragma<br></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#extension<br>
+#version<br></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>#line</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The following
+operators are
+also available:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>defined<br>
+##</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>Each number sign (<strong>#</strong>) can be preceded in its line only by spaces or
+horizontal tabs.
+It may also be followed by spaces and horizontal tabs, preceding the
+directive.
+Each directive is terminated by a new-line.
+Preprocessing does not change the number or relative location of new-lines
+in a source string.
+Preprocessing takes places after new-lines have been removed by the
+line-continuation character.</p>
+</div>
+<div class="paragraph">
+<p>The number sign (<strong>#</strong>) on a line by itself is ignored.
+Any directive not listed above will cause a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p><strong>#define</strong> and <strong>#undef</strong> functionality are defined as is standard for C++
+preprocessors for macro definitions both with and without macro parameters.</p>
+</div>
+<div class="paragraph">
+<p>The following predefined macros are available:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>__LINE__<br>
+__FILE__<br>
+__VERSION__<br></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>__LINE__ will substitute a decimal integer constant that is one more than
+the number of preceding new-lines in the current source string.</p>
+</div>
+<div class="paragraph">
+<p>__FILE__ will substitute a decimal integer constant that says which source
+string number is currently being processed.</p>
+</div>
+<div class="paragraph">
+<p>__VERSION__ will substitute a decimal integer reflecting the version
+number of the OpenGL Shading Language.
+The version of the shading language described in this document will have
+__VERSION__ substitute the decimal integer 460.</p>
+</div>
+<div class="paragraph">
+<p>By convention, all macro names containing two consecutive underscores (__)
+are reserved for use by underlying software layers.
+Defining
+or undefining
+such a name in a shader does not itself result in an error, but may
+result in unintended behaviors that stem from having multiple definitions of
+the same name.
+All macro names prefixed with “GL_” (“GL” followed by a single
+underscore) are also reserved, and defining
+or undefining
+such a name results in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>Implementations must support macro-name lengths of up to 1024 characters.
+Implementations are allowed to generate an error for a macro name of length
+greater than 1024 characters, but are also allowed to support lengths
+greater than 1024.</p>
+</div>
+<div class="paragraph">
+<p><strong>#if</strong>, <strong>#ifdef</strong>, <strong>#ifndef</strong>, <strong>#else</strong>, <strong>#elif</strong>, and <strong>#endif</strong> are defined to
+operate as is standard for C++ preprocessors except for the following:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Expressions following <strong>#if</strong> and <strong>#elif</strong> are
+further restricted to
+expressions operating on literal integer constants, plus identifiers
+consumed by the <strong>defined</strong> operator.</p>
+</li>
+<li>
+<p>Character constants are not supported.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>The operators available are as follows.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Precedence</th>
+<th class="tableblock halign-left valign-top">Operator class</th>
+<th class="tableblock halign-left valign-top">Operators</th>
+<th class="tableblock halign-left valign-top">Associativity</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1 (highest)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">parenthetical grouping</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">( )</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">NA</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">unary</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">defined<br>
+ + - ~ !</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Right to Left</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">multiplicative</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">* / %</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">additive</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">+ -</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise shift</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><< >></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">6</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">relational</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">< > <= >=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">7</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">equality</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">== !=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">8</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise and</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">&</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">9</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise exclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">^</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">10</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise inclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">|</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">11</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical and</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">&&</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">12 (lowest)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical inclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">||</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The <strong>defined</strong> operator can be used in either of the following ways:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">defined identifier
+defined ( identifier )</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Two tokens in a macro can be concatenated into one token using the token
+pasting (<strong>##</strong>) operator, as is standard for C++ preprocessors.
+The result must be a valid single token, which will then be subject to macro
+expansion.
+That is, macro expansion happens only after token pasting.
+There are no other number sign based operators (e.g. no <strong>#</strong> or <strong>#@</strong>), nor is
+there a <strong>sizeof</strong> operator.</p>
+</div>
+<div class="paragraph">
+<p>The semantics of applying operators to integer literals in the preprocessor
+match those standard in the C++ preprocessor, not those in the OpenGL Shading Language.</p>
+</div>
+<div class="paragraph">
+<p>Preprocessor expressions will be evaluated according to the behavior of the
+host processor, not the processor targeted by the shader.</p>
+</div>
+<div class="paragraph">
+<p><strong>#error</strong> will cause the implementation to put a compile-time diagnostic message
+into the shader object’s information log (see section 7.12 “Shader, Program
+and Program Pipeline Queries” of the <a href="#references">OpenGL Specification</a> for how to
+access a shader object’s information log).
+The message will be the tokens following the <strong>#error</strong> directive, up to the
+first new-line.
+The implementation must treat the presence of a <strong>#error</strong> directive as a
+compile-time error.</p>
+</div>
+<div class="paragraph">
+<p><strong>#pragma</strong> allows implementation-dependent compiler control.
+Tokens following <strong>#pragma</strong> are not subject to preprocessor macro expansion.
+If an implementation does not recognize the tokens following <strong>#pragma</strong>, then
+it will ignore that pragma.
+The following pragmas are defined as part of the language.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> STDGL</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>STDGL</strong> pragma is used to reserve pragmas for use by future revisions of
+this language.
+No implementation may use a pragma whose first token is <strong>STDGL</strong>.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> optimize(on)
+<span class="preprocessor">#pragma</span> optimize(off)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>can be used to turn off optimizations as an aid in developing and debugging
+shaders.
+It can only be used outside function definitions.
+By default, optimization is turned on for all shaders.
+The debug pragma</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> debug(on)
+<span class="preprocessor">#pragma</span> debug(off)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>can be used to enable compiling and annotating a shader with debug
+information, so that it can be used with a debugger.
+It can only be used outside function definitions.
+By default, debug is turned off.</p>
+</div>
+<div class="paragraph">
+<p>Shaders should declare the version of the language they are written to.
+The language version a shader is written to is specified by</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#version</span> number profile_opt</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>number</em> must be a version of the language, following the same
+convention as __VERSION__ above.
+The directive “<strong>#version 460</strong>” is required in any shader that
+uses version 4.60 of the language.
+Any <em>number</em> representing a version of the language a compiler does not
+support will cause a compile-time error to be generated.
+Version 1.10 of the language does not require shaders to include this
+directive, and shaders that do not include a <strong>#version</strong> directive will be
+treated as targeting version 1.10.
+Shaders that specify <strong>#version</strong> 100 will be treated as targeting version
+1.00 of the OpenGL ES Shading Language.
+Shaders that specify <strong>#version</strong> 300 will be treated as targeting version
+3.00 of the OpenGL ES Shading Language.
+Shaders that specify <strong>#version</strong> 310 will be treated as targeting version
+3.10 of the OpenGL ES Shading Language.</p>
+</div>
+<div class="paragraph">
+<p>If the optional <em>profile</em> argument is provided, it must be the name of an
+OpenGL profile.
+Currently, there are three choices:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">core
+compatibility
+es</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A <em>profile</em> argument can only be used with version 150 or greater.
+If no profile argument is provided and the version is 150 or greater, the
+default is <strong>core</strong>.
+If version 300 or 310 is specified, the profile argument is not optional and
+must be <strong>es</strong>, or a compile-time error results.
+The Language Specification for the <strong>es</strong> profile is specified in The OpenGL
+ES Shading Language specification.</p>
+</div>
+<div class="paragraph">
+<p>Shaders for the <strong>core</strong> or <strong>compatibility</strong> profiles that declare different
+versions can be linked together.
+However, <strong>es</strong> profile shaders cannot be linked with non-<strong>es</strong> profile shaders
+or with <strong>es</strong> profile shaders of a different version, or a link-time error
+will result.
+When linking shaders of versions allowed by these rules, remaining link-time
+errors will be given as per the linking rules in the GLSL version
+corresponding to the version of the context the shaders are linked under.
+Shader compile-time errors must still be given strictly based on the version
+declared (or defaulted to) within each shader.</p>
+</div>
+<div class="paragraph">
+<p>Unless otherwise specified, this specification is documenting the core
+profile, and everything specified for the core profile is also available in
+the compatibility profile.
+Features specified as belonging specifically to the compatibility profile
+are not available in the core profile.
+Compatibility-profile features are not available when generating SPIR-V.</p>
+</div>
+<div class="paragraph">
+<p>There is a built-in macro definition for each profile the implementation
+supports.
+All implementations provide the following macro:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#define</span> GL_core_profile <span class="integer">1</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Implementations providing the <strong>compatibility</strong> profile provide the following
+macro:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#define</span> GL_compatibility_profile <span class="integer">1</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Implementations providing the <strong>es</strong> profile provide the following macro:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#define</span> GL_es_profile <span class="integer">1</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>#version</strong> directive must occur in a shader before anything else, except
+for comments and white space.</p>
+</div>
+<div class="paragraph">
+<p>By default, compilers of this language must issue compile-time
+lexical
+and
+grammatical errors for shaders that do not conform to this specification.
+Any extended behavior must first be enabled.
+Directives to control the behavior of the compiler with respect to
+extensions are declared with the <strong>#extension</strong> directive</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#extension</span> extension_name : behavior
+<span class="preprocessor">#extension</span> all : behavior</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>extension_name</em> is the name of an extension.
+Extension names are not documented in this specification.
+The token <strong>all</strong> means the behavior applies to all extensions supported by
+the compiler.
+The <em>behavior</em> can be one of the following:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Behavior</th>
+<th class="tableblock halign-left valign-top">Effect</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>require</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Behave as specified by the extension <em>extension_name</em>.<br>
+ Give a compile-time error on the <strong>#extension</strong> if the extension
+ <em>extension_name</em> is not supported, or if <strong>all</strong> is specified.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>enable</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Behave as specified by the extension <em>extension_name</em>.<br>
+ Warn on the <strong>#extension</strong> if the extension <em>extension_name</em> is
+ not supported.<br>
+ Give a compile-time error on the <strong>#extension</strong> if <strong>all</strong> is
+ specified.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>warn</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Behave as specified by the extension <em>extension_name</em>,
+ except issue warnings on any detectable use of that extension,
+ unless such use is supported by other enabled or required
+ extensions.<br>
+ If <strong>all</strong> is specified, then warn on all detectable uses of any
+ extension used.<br>
+ Warn on the <strong>#extension</strong> if the extension <em>extension_name</em> is
+ not supported.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>disable</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Behave (including issuing errors and warnings) as if the
+ extension <em>extension_name</em> is not part of the language
+ definition.<br>
+ If <strong>all</strong> is specified, then behavior must revert back to that
+ of the non-extended core version of the language being
+ compiled to.<br>
+ Warn on the <strong>#extension</strong> if the extension <em>extension_name</em> is
+ not supported.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The <strong>extension</strong> directive is a simple, low-level mechanism to set the
+behavior for each extension.
+It does not define policies such as which combinations are appropriate,
+those must be defined elsewhere.
+Order of directives matters in setting the behavior for each extension:
+Directives that occur later override those seen earlier.
+The <strong>all</strong> variant sets the behavior for all extensions, overriding all
+previously issued <strong>extension</strong> directives, but only for the <em>behaviors</em>
+<strong>warn</strong> and <strong>disable</strong>.</p>
+</div>
+<div class="paragraph">
+<p>The initial state of the compiler is as if the directive</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#extension</span> all : disable</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>was issued, telling the compiler that all error and warning reporting must
+be done according to this specification, ignoring any extensions.</p>
+</div>
+<div class="paragraph">
+<p>Each extension can define its allowed granularity of scope.
+If nothing is said, the granularity is a shader (that is, a single
+compilation unit), and the extension directives must occur before any
+non-preprocessor tokens.
+If necessary, the linker can enforce granularities larger than a single
+compilation unit, in which case each involved shader will have to contain
+the necessary extension directive.</p>
+</div>
+<div class="paragraph">
+<p>Macro expansion is not done on lines containing <strong>#extension</strong> and <strong>#version</strong>
+directives.</p>
+</div>
+<div class="paragraph">
+<p><strong>#line</strong> must have, after macro substitution, one of the following forms:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#line</span> line
+<span class="preprocessor">#line</span> line source-<span class="predefined-type">string</span>-number</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>line</em> and <em>source-string-number</em> are
+constant integer expressions.
+If these constant expressions are not integer literals then behavior is undefined.
+After processing this directive (including its new-line), the implementation
+will behave as if it is compiling at line number <em>line</em> and source string
+number <em>source-string-number</em>.
+Subsequent source strings will be numbered sequentially, until another
+<strong>#line</strong> directive overrides that numbering.</p>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="title">Note</div>
+<div class="paragraph">
+<p>Some implementations have allowed constant expressions in #line directives and
+some have not. Even where expressions are supported the grammar is ambiguous and so
+results are implementation dependent. For example,
++ #line +2 +2 // Line number set to 4, or file to 2 and line to 2</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="paragraph">
+<p>When shaders are compiled for OpenGL SPIR-V, the following predefined
+macro is available:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#define</span> GL_SPIRV <span class="integer">100</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="comments">3.4. Comments</h3>
+<div class="paragraph">
+<p>Comments are delimited by <strong>/*</strong> and <strong>*/</strong>, or by <strong>//</strong> and a new-line.
+The begin comment delimiters (/* or //) are not recognized as comment
+delimiters inside of a comment, hence comments cannot be nested.
+A <strong>/*</strong> comment includes its terminating delimiter (*/).
+However, a <strong>//</strong> comment does not include (or eliminate) its terminating new
+line.</p>
+</div>
+<div class="paragraph">
+<p>Inside comments, any byte values may be used, except a byte whose value is
+0.
+No errors will be given for the content of comments and no validation on the
+content of comments need be done.</p>
+</div>
+<div class="paragraph">
+<p>Removal of new-lines by the line-continuation character (<strong>\</strong>) logically
+occurs before comments are processed.
+That is, a single-line comment ending in the line-continuation character
+(<strong>\</strong>) includes the next line in the comment.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// a single-line comment containing the next line \
+a = b; // this is still in the first comment</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="tokens">3.5. Tokens</h3>
+<div class="paragraph">
+<p>The language, after preprocessing, is a sequence of tokens.
+A token can be</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>token</em> : </dt>
+<dd>
+<p><em>keyword</em><br>
+<em>identifier</em><br>
+<em>integer-constant</em><br>
+<em>floating-constant</em><br>
+<em>operator</em><br>
+<strong>;</strong> <strong>{</strong> <strong>}</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="keywords">3.6. Keywords</h3>
+<div class="paragraph">
+<p>The following are the keywords in the language and (after
+preprocessing) can only be used as described in this specification,
+or a compile-time error results:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>const</strong> <strong>uniform</strong> <strong>buffer</strong> <strong>shared</strong>
+<strong>attribute</strong> <strong>varying</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>coherent</strong> <strong>volatile</strong> <strong>restrict</strong> <strong>readonly</strong> <strong>writeonly</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>atomic_uint</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>layout</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>centroid</strong> <strong>flat</strong> <strong>smooth</strong>
+<strong>noperspective</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>patch</strong> <strong>sample</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>invariant</strong> <strong>precise</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>break</strong> <strong>continue</strong> <strong>do</strong> <strong>for</strong> <strong>while</strong> <strong>switch</strong> <strong>case</strong> <strong>default</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>if</strong> <strong>else</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>subroutine</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>in</strong> <strong>out</strong> <strong>inout</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>int</strong> <strong>void</strong> <strong>bool</strong> <strong>true</strong> <strong>false</strong> <strong>float</strong>
+<strong>double</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>discard</strong> <strong>return</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>vec2</strong> <strong>vec3</strong> <strong>vec4</strong> <strong>ivec2</strong> <strong>ivec3</strong> <strong>ivec4</strong> <strong>bvec2</strong> <strong>bvec3</strong> <strong>bvec4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>uint</strong> <strong>uvec2</strong> <strong>uvec3</strong> <strong>uvec4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dvec2</strong> <strong>dvec3</strong> <strong>dvec4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>mat2</strong> <strong>mat3</strong> <strong>mat4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>mat2x2</strong> <strong>mat2x3</strong> <strong>mat2x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>mat3x2</strong> <strong>mat3x3</strong> <strong>mat3x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>mat4x2</strong> <strong>mat4x3</strong> <strong>mat4x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dmat2</strong> <strong>dmat3</strong> <strong>dmat4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dmat2x2</strong> <strong>dmat2x3</strong> <strong>dmat2x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dmat3x2</strong> <strong>dmat3x3</strong> <strong>dmat3x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>dmat4x2</strong> <strong>dmat4x3</strong> <strong>dmat4x4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>lowp</strong> <strong>mediump</strong> <strong>highp</strong> <strong>precision</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler1D</strong> <strong>sampler1DShadow</strong> <strong>sampler1DArray</strong> <strong>sampler1DArrayShadow</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>isampler1D</strong> <strong>isampler1DArray</strong> <strong>usampler1D</strong> <strong>usampler1DArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler2D</strong> <strong>sampler2DShadow</strong> <strong>sampler2DArray</strong> <strong>sampler2DArrayShadow</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>isampler2D</strong> <strong>isampler2DArray</strong> <strong>usampler2D</strong> <strong>usampler2DArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler2DRect</strong> <strong>sampler2DRectShadow</strong> <strong>isampler2DRect</strong> <strong>usampler2DRect</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler2DMS</strong> <strong>isampler2DMS</strong> <strong>usampler2DMS</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler2DMSArray</strong> <strong>isampler2DMSArray</strong> <strong>usampler2DMSArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler3D</strong> <strong>isampler3D</strong> <strong>usampler3D</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>samplerCube</strong> <strong>samplerCubeShadow</strong> <strong>isamplerCube</strong> <strong>usamplerCube</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>samplerCubeArray</strong> <strong>samplerCubeArrayShadow</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>isamplerCubeArray</strong> <strong>usamplerCubeArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>samplerBuffer</strong> <strong>isamplerBuffer</strong> <strong>usamplerBuffer</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image1D</strong> <strong>iimage1D</strong> <strong>uimage1D</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image1DArray</strong> <strong>iimage1DArray</strong> <strong>uimage1DArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2D</strong> <strong>iimage2D</strong> <strong>uimage2D</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2DArray</strong> <strong>iimage2DArray</strong> <strong>uimage2DArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2DRect</strong> <strong>iimage2DRect</strong> <strong>uimage2DRect</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2DMS</strong> <strong>iimage2DMS</strong> <strong>uimage2DMS</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image2DMSArray</strong> <strong>iimage2DMSArray</strong> <strong>uimage2DMSArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>image3D</strong> <strong>iimage3D</strong> <strong>uimage3D</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>imageCube</strong> <strong>iimageCube</strong> <strong>uimageCube</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>imageCubeArray</strong> <strong>iimageCubeArray</strong> <strong>uimageCubeArray</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>imageBuffer</strong> <strong>iimageBuffer</strong> <strong>uimageBuffer</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>struct</strong></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The following are the keywords reserved for future use.
+Using them will result in a compile-time error:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>common</strong> <strong>partition</strong> <strong>active</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>asm</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>class</strong> <strong>union</strong> <strong>enum</strong> <strong>typedef</strong> <strong>template</strong> <strong>this</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>resource</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>goto</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>inline</strong> <strong>noinline</strong> <strong>public</strong> <strong>static</strong> <strong>extern</strong> <strong>external</strong> <strong>interface</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>long</strong> <strong>short</strong> <strong>half</strong> <strong>fixed</strong> <strong>unsigned</strong> <strong>superp</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>input</strong> <strong>output</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>hvec2</strong> <strong>hvec3</strong> <strong>hvec4</strong> <strong>fvec2</strong> <strong>fvec3</strong> <strong>fvec4</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>filter</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sizeof</strong> <strong>cast</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>namespace</strong> <strong>using</strong></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p><strong>sampler3DRect</strong></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>In addition, all identifiers containing two consecutive underscores (__)
+are reserved for use by underlying software layers.
+Defining such a name in a shader does not itself result in an error, but may
+result in unintended behaviors that stem from having multiple definitions of
+the same name.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="identifiers">3.7. Identifiers</h3>
+<div class="paragraph">
+<p>Identifiers are used for variable names, function names, structure names,
+and field selectors (field selectors select components of
+<code><a href="#vector-components">vectors</a></code> and <code><a href="#matrix-components">matrices</a></code>,
+similarly to structure members).
+Identifiers have the form:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>identifier</em> : </dt>
+<dd>
+<p><em>nondigit</em><br>
+<em>identifier</em> <em>nondigit</em><br>
+<em>identifier</em> <em>digit</em></p>
+</dd>
+<dt class="hdlist1"><em>nondigit</em> : one of </dt>
+<dd>
+<p><strong>_</strong> <strong>a b c d e f g h i j k l m n o p q r s t u v w x y z</strong><br>
+<strong>A B C D E F G H I J K L M N O P Q R S T U V W X Y Z</strong></p>
+</dd>
+<dt class="hdlist1"><em>digit</em> : one of </dt>
+<dd>
+<p><strong>0 1 2 3 4 5 6 7 8 9</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Identifiers starting with “gl_” are reserved for use by OpenGL, and
+in general, may not be declared in a shader;
+this results in a compile-time error.
+However, as noted in the specification, there are some cases where
+previously declared variables can be redeclared, and predeclared “gl_”
+names are allowed to be redeclared in a shader only for these specific
+purposes.</p>
+</div>
+<div class="paragraph">
+<p>Implementations must support identifier lengths of up to 1024 characters.
+Implementations are allowed to generate an error for an identifier of length
+greater than 1024 characters, but are also allowed to support lengths
+greater than 1024.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="definitions">3.8. Definitions</h3>
+<div class="paragraph">
+<p>Some language rules described below depend on the following definitions.</p>
+</div>
+<div class="sect3">
+<h4 id="static-use">3.8.1. Static Use</h4>
+<div class="paragraph">
+<p>A shader contains a <em>static use</em> of a variable <em>x</em> if, after preprocessing,
+the shader contains a statement that would access any part of <em>x</em>,
+whether or not flow of control will cause that statement to be executed.
+Such a variable is referred to as being <em>statically used</em>. If the access is a
+write then <em>x</em> is further said to be <em>statically assigned</em>.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="dynamically-uniform-expressions-and-uniform-control-flow">3.8.2. Dynamically Uniform Expressions and Uniform Control Flow</h4>
+<div class="paragraph">
+<p>Some operations require an expression to be <em>dynamically uniform</em>, or that
+it be located in <em>uniform control flow</em>.
+These requirements are defined by the following set of definitions.</p>
+</div>
+<div class="paragraph">
+<p>An <em>invocation</em> is a single execution of <em>main()</em> for a particular stage,
+operating only on the amount of data explicitly exposed within that stage’s
+shaders.
+(Any implicit operation on additional instances of data would comprise
+additional invocations.) For example, in compute execution models, a single
+invocation operates only on a single work item, or, in a vertex execution
+model, a single invocation operates only on a single vertex.</p>
+</div>
+<div class="paragraph">
+<p>An <em>invocation group</em> is the complete set of invocations collectively
+processing a particular compute workgroup or graphical operation, where the
+scope of a "graphical operation" is implementation-dependent, but at least
+as large as a single triangle or patch, and at most as large as a single
+rendering command, as defined by the client API.</p>
+</div>
+<div class="paragraph">
+<p>Within a single invocation, a single shader statement can be executed
+multiple times, giving multiple <em>dynamic instances</em> of that instruction.
+This can happen when the instruction is executed in a loop, or in a function
+called from multiple call sites, or combinations of multiple of these.
+Different loop iterations and different dynamic function-call-site chains
+yield different dynamic instances of such an instruction.
+Dynamic instances are distinguished by their control-flow path within an
+invocation, not by which invocation executed it.
+That is, different invocations of <em>main()</em> execute the same dynamic
+instances of an instruction when they follow the same control-flow path.</p>
+</div>
+<div class="paragraph">
+<p>An expression is _dynamically uniform for a dynamic instance consuming it
+when its value is the same for all invocations (in the invocation group)
+that execute that dynamic instance.</p>
+</div>
+<div class="paragraph">
+<p><em>Uniform control flow</em> (or converged control flow) occurs when all
+invocations in the invocation group execute the same control-flow path (and
+hence the same sequence of dynamic instances of instructions).
+Uniform control flow is the initial state at the entry into <em>main()</em>, and
+lasts until a conditional branch takes different control paths for different
+invocations (non-uniform or divergent control flow).
+Such divergence can reconverge, with all the invocations once again
+executing the same control-flow path, and this re-establishes the existence
+of uniform control flow.
+If control flow is uniform upon entry into a selection or loop, and all
+invocations in the invocation group subsequently leave that selection or
+loop, then control flow reconverges to be uniform.</p>
+</div>
+<div class="paragraph">
+<p>For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">main()
+{
+ <span class="predefined-type">float</span> a = ...; <span class="comment">// this is uniform control flow</span>
+ <span class="keyword">if</span> (a < b) { <span class="comment">// this expression is true for some fragments, not all</span>
+ ...; <span class="comment">// non-uniform control flow</span>
+ } <span class="keyword">else</span> {
+ ...; <span class="comment">// non-uniform control flow</span>
+ }
+ ...; <span class="comment">// uniform control flow again</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Note that constant expressions are trivially dynamically uniform.
+It follows that typical loop counters based on these are also dynamically
+uniform.</p>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="variables-and-types">4. Variables and Types</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>All variables and functions must be declared before being used.
+Variable and function names are identifiers.</p>
+</div>
+<div class="paragraph">
+<p>There are no default types.
+All variable and function declarations must have a declared type, and
+optionally qualifiers.
+A variable is declared by specifying its type followed by one or more names
+separated by commas.
+In many cases, a variable can be initialized as part of its declaration by
+using the assignment operator (<strong>=</strong>).</p>
+</div>
+<div class="paragraph">
+<p>User-defined types may be defined using <strong>struct</strong> to aggregate a list of
+existing types into a single name.</p>
+</div>
+<div class="paragraph">
+<p>The OpenGL Shading Language is type safe.
+There are some implicit conversions between types.
+Exactly how and when this can occur is described in section
+“<a href="#implicit-conversions">Implicit Conversions</a>” and as referenced by other
+sections in this specification.</p>
+</div>
+<div class="sect2">
+<h3 id="basic-types">4.1. Basic Types</h3>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definition</dt>
+<dd>
+<p>A <em>basic type</em> is a type defined by a keyword in the language.</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The OpenGL Shading Language supports the following basic data types, grouped as follows.</p>
+</div>
+<div class="sect3">
+<h4 id="_transparent_types">4.1.1. Transparent Types</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>void</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for functions that do not return a value</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>bool</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a conditional type, taking on values of true or false</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>int</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a signed integer</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uint</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">an unsigned integer</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>float</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point scalar</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>double</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a double-precision floating-point scalar</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component single-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component single-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component single-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dvec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component double-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dvec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component double-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dvec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component double-precision floating-point vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>bvec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component Boolean vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>bvec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component Boolean vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>bvec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component Boolean vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component signed integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component signed integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component signed integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a two-component unsigned integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a three-component unsigned integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a four-component unsigned integer vector</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 2 × 2 single-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 3 × 3 single-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 4 × 4 single-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>mat2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 2 columns and 3 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 2 columns and 4 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 3 columns and 2 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>mat3</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 3 columns and 4 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 4 columns and 2 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a single-precision floating-point matrix with 4 columns and 3 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>mat4</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 2 × 2 double-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 3 × 3 double-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a 4 × 4 double-precision floating-point matrix</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat2x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>dmat2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat2x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a double-precision floating-point matrix with 2 columns and 3 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat2x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a double-precision floating-point matrix with 2 columns and 4 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat3x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a double-precision floating-point matrix with 3 columns and 2 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat3x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>dmat3</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat3x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a double-precision floating-point matrix with 3 columns and 4 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat4x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a double-precision floating-point matrix with 4 columns and 2 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat4x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a double-precision floating-point matrix with 4 columns and 3 rows</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat4x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as a <strong>dmat4</strong></p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Note that where the following tables say “accessing a texture”, the
+<strong>sampler*</strong> opaque types access textures, and the <strong>image*</strong> opaque types
+access images, of a specified type.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="_floating_point_opaque_types">4.1.2. Floating-Point Opaque Types</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler1D</strong><br>
+ <strong>image1D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 1D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler1DShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 1D depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler1DArray</strong><br>
+ <strong>image1DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 1D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler1DArrayShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 1D array depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2D</strong><br>
+ <strong>image2D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DArray</strong><br>
+ <strong>image2DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DArrayShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D array depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DMS</strong><br>
+ <strong>image2DMS</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D multisample texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DMSArray</strong><br>
+ <strong>image2DMSArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 2D multisample array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DRect</strong><br>
+ <strong>image2DRect</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a rectangle texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler2DRectShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a rectangle texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sampler3D</strong><br>
+ <strong>image3D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a 3D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerCube</strong><br>
+ <strong>imageCube</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a cube mapped texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerCubeShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a cube map depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerCubeArray</strong><br>
+ <strong>imageCubeArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a cube map array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerCubeArrayShadow</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a cube map array depth texture with comparison</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>samplerBuffer</strong><br>
+ <strong>imageBuffer</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing a buffer texture</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="_signed_integer_opaque_types">4.1.3. Signed Integer Opaque Types</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler1D</strong><br>
+ <strong>iimage1D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 1D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler1DArray</strong><br>
+ <strong>iimage1DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 1D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2D</strong><br>
+ <strong>iimage2D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2DArray</strong><br>
+ <strong>iimage2DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2DMS</strong><br>
+ <strong>iimage2DMS</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D multisample texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2DMSArray</strong><br>
+ <strong>iimage2DMSArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D multisample array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler2DRect</strong><br>
+ <strong>iimage2DRect</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 2D rectangle texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isampler3D</strong><br>
+ <strong>iimage3D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer 3D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isamplerCube</strong><br>
+ <strong>iimageCube</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer cube mapped texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isamplerCubeArray</strong><br>
+ <strong>iimageCubeArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer cube map array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>isamplerBuffer</strong><br>
+ <strong>iimageBuffer</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an integer buffer texture</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="_unsigned_integer_opaque_types">4.1.4. Unsigned Integer Opaque Types</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler1D</strong><br>
+ <strong>uimage1D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 1D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler1DArray</strong><br>
+ <strong>uimage1DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 1D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2D</strong><br>
+ <strong>uimage2D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 2D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2DArray</strong><br>
+ <strong>uimage2DArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 2D array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2DMS</strong><br>
+ <strong>uimage2DMS</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 2D multisample texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2DMSArray</strong><br>
+ <strong>uimage2DMSArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 2D multisample array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler2DRect</strong><br>
+ <strong>uimage2DRect</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer rectangle texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usampler3D</strong><br>
+ <strong>uimage3D</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer 3D texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usamplerCube</strong><br>
+ <strong>uimageCube</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer cube mapped texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usamplerCubeArray</strong><br>
+ <strong>uimageCubeArray</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer cube map array texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>usamplerBuffer</strong><br>
+ <strong>uimageBuffer</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer buffer texture</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>atomic_uint</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a handle for accessing an unsigned integer atomic counter</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>In addition, a shader can aggregate these basic types using arrays and
+structures to build more complex types.</p>
+</div>
+<div class="paragraph">
+<p>There are no pointer types.</p>
+</div>
+<div class="paragraph">
+<p>In this specification, an <em>aggregate</em> will mean a structure or array.
+(Matrices and vectors are not by themselves aggregates.) Aggregates,
+matrices, and vectors will collectively be referred to as <em>composites</em>.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="void">4.1.5. Void</h4>
+<div class="paragraph">
+<p>Functions that do not return a value must be declared as <strong>void</strong>.
+There is no default function return type.
+The keyword <strong>void</strong> cannot be used in any other declarations (except for
+empty formal or actual parameter lists), or a compile-time error results.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="booleans">4.1.6. Booleans</h4>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definition</dt>
+<dd>
+<p>A <em>boolean type</em> is any boolean scalar or vector type (<strong>bool</strong>, <strong>bvec2</strong>,
+<strong>bvec3</strong>, <strong>bvec4</strong>)</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>To make conditional execution of code easier to express, the type <strong>bool</strong> is
+supported.
+There is no expectation that hardware directly supports variables of this
+type.
+It is a genuine Boolean type, holding only one of two values meaning either
+true or false.
+Two keywords <strong>true</strong> and <strong>false</strong> can be used as literal Boolean constants.
+Booleans are declared and optionally initialized as in the follow example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">bool</span> success; <span class="comment">// declare "success" to be a Boolean</span>
+<span class="predefined-type">bool</span> done = <span class="predefined-constant">false</span>; <span class="comment">// declare and initialize "done"</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Expressions used for conditional jumps (<strong>if</strong>, <strong>for</strong>, <strong>?:</strong>, <strong>while</strong>,
+<strong>do</strong>-<strong>while</strong>) must evaluate to the type <strong>bool</strong>.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="integers">4.1.7. Integers</h4>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definitions</dt>
+<dd>
+<p>An <em>integral type</em> is any signed or unsigned, scalar or vector integer type.
+It excludes arrays and structures.</p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>A <em>scalar integral type</em> is a scalar signed or unsigned integer type:</p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>A <em>vector integral type</em> is a vector of signed or unsigned integers:</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>Signed and unsigned integer variables are fully supported.
+In this document, the term <em>integer</em> is meant to generally include both
+signed and unsigned integers.
+Unsigned
+integers have exactly 32 bits of precision.
+Signed
+integers use 32 bits, including a sign bit, in two’s complement form.</p>
+</div>
+<div class="paragraph">
+<p>Addition, subtraction, and shift
+operations resulting in overflow or
+underflow will not cause any exception, nor will they saturate, rather they
+will “wrap” to yield the low-order
+32 bits of the result.
+Division and multiplication operations resulting in overflow or underflow
+will not cause any exception but will result in an undefined value.</p>
+</div>
+<div class="paragraph">
+<p>Integers are declared and optionally initialized with integer expressions,
+as in the following example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> i, j = <span class="integer">42</span>; <span class="comment">// default integer literal type is int</span>
+uint k = <span class="integer">3</span>u; <span class="comment">// "u" establishes the type as uint</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Literal integer constants can be expressed in decimal (base 10), octal (base
+8), or hexadecimal (base 16) as follows.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>integer-constant</em> : </dt>
+<dd>
+<p><em>decimal-constant</em> <em>integer-suffix<sub>opt</sub></em><br>
+<em>octal-constant</em> <em>integer-suffix<sub>opt</sub></em><br>
+<em>hexadecimal-constant</em> <em>integer-suffix<sub>opt</sub></em></p>
+</dd>
+<dt class="hdlist1"><em>integer-suffix</em> : one of </dt>
+<dd>
+<p><strong>u</strong> <strong>U</strong></p>
+</dd>
+<dt class="hdlist1"><em>decimal-constant</em> : </dt>
+<dd>
+<p><em>nonzero-digit</em><br>
+<em>decimal-constant</em> <em>digit</em></p>
+</dd>
+<dt class="hdlist1"><em>octal-constant</em> : </dt>
+<dd>
+<p><strong>0</strong><br>
+<em>octal-constant</em> <em>octal-digit</em></p>
+</dd>
+<dt class="hdlist1"><em>hexadecimal-constant</em> : </dt>
+<dd>
+<p><strong>0x</strong> <em>hexadecimal-digit</em><br>
+<strong>0X</strong> <em>hexadecimal-digit</em><br>
+<em>hexadecimal-constant</em> <em>hexadecimal-digit</em></p>
+</dd>
+<dt class="hdlist1"><em>digit</em> : </dt>
+<dd>
+<p><strong>0</strong><br>
+<em>nonzero-digit</em></p>
+</dd>
+<dt class="hdlist1"><em>nonzero-digit</em> : one of </dt>
+<dd>
+<p><strong>1 2 3 4 5 6 7 8 9</strong></p>
+</dd>
+<dt class="hdlist1"><em>octal-digit</em> : one of </dt>
+<dd>
+<p><strong>0 1 2 3 4 5 6 7</strong></p>
+</dd>
+<dt class="hdlist1"><em>hexadecimal-digit</em> : one of </dt>
+<dd>
+<p><strong>0 1 2 3 4 5 6 7 8 9<br>
+a b c d e f<br>
+A B C D E F</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>No white space is allowed between the digits of an integer constant,
+including after the leading <strong>0</strong> or after the leading <strong>0x</strong> or <strong>0X</strong> of a
+constant, or before the suffix <strong>u</strong> or <strong>U</strong>.
+When tokenizing, the maximal token matching the above will be recognized
+before a new token is started.
+When the suffix <strong>u</strong> or <strong>U</strong> is present, the literal has type <strong>uint</strong>,
+otherwise the type is <strong>int</strong>.
+A leading unary minus sign (-) is interpreted as an arithmetic unary
+negation, not as part of the constant.
+Hence, literals themselves are always expressed with non-negative syntax,
+though they could result in a negative value.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to provide a literal integer whose bit pattern
+cannot fit in 32 bits.
+The bit pattern of the literal is always used unmodified.
+So a signed literal whose bit pattern includes a set sign bit creates a
+negative value.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="integer">1</span> <span class="comment">// OK. Signed integer, value 1</span>
+<span class="integer">1</span>u <span class="comment">// OK. Unsigned integer, value 1</span>
+-<span class="integer">1</span> <span class="comment">// OK. Unary minus applied to signed integer.</span>
+ <span class="comment">// result is a signed integer, value -1</span>
+-<span class="integer">1</span>u <span class="comment">// OK. Unary minus applies to unsigned integer.</span>
+ <span class="comment">// Result is an unsigned integer, value 0xffffffff</span>
+<span class="hex">0xA0000000</span> <span class="comment">// OK. 32-bit signed hexadecimal</span>
+<span class="hex">0xABcdEF00</span>u <span class="comment">// OK. 32-bit unsigned hexadecimal</span>
+<span class="hex">0xffffffff</span> <span class="comment">// OK. Signed integer, value -1</span>
+<span class="hex">0x80000000</span> <span class="comment">// OK. Evaluates to -2147483648</span>
+<span class="hex">0xffffffff</span>u <span class="comment">// OK. Unsigned integer, value 0xffffffff</span>
+<span class="hex">0xfffffffff</span> <span class="comment">// Error: needs more than 32 bits</span>
+<span class="integer">3000000000</span> <span class="comment">// OK. A signed decimal literal taking 32 bits.</span>
+ <span class="comment">// It evaluates to -1294967296</span>
+<span class="integer">2147483648</span> <span class="comment">// OK. Evaluates to -2147483648 (the literal set the sign bit)</span>
+<span class="integer">5000000000</span> <span class="comment">// Error: needs more than 32 bits</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="floats">4.1.8. Floats</h4>
+<div class="paragraph">
+<p>Single-precision and double-precision floating-point variables are available
+for use in a variety of scalar calculations.
+Generally, the term <em>floating-point</em> will refer to both single- and
+double-precision floating-point.
+Floating-point variables are defined as in the following examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a, b = <span class="float">1</span><span class="float">.5</span>; <span class="comment">// single-precision floating-point</span>
+<span class="predefined-type">double</span> c, d = <span class="float">2</span><span class="float">.0</span>LF; <span class="comment">// double-precision floating-point</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>As an input value to one of the processing units, a single-precision or
+double-precision floating-point variable is expected to match the
+corresponding IEEE 754 floating-point definition for precision and dynamic
+range.
+Floating-point variables within a shader are also encoded according to the
+IEEE 754 specification for single-precision floating-point values</p>
+</div>
+<div class="paragraph">
+<p>(logically, not necessarily physically).
+While encodings are logically IEEE 754, operations (addition,
+multiplication, etc.) are not necessarily performed as required by IEEE 754.
+See “<a href="#range-and-precision">Range and Precision</a>” for more details on
+precision and usage of NaNs (Not a Number) and Infs (positive or negative
+infinities).</p>
+</div>
+<div class="paragraph">
+<p>Floating-point constants are defined as follows.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>floating-constant</em> : </dt>
+<dd>
+<p><em>fractional-constant</em> <em>exponent-part<sub>opt</sub></em> <em>floating-suffix<sub>opt</sub></em><br>
+<em>digit-sequence</em> <em>exponent-part</em> <em>floating-suffix<sub>opt</sub></em></p>
+</dd>
+<dt class="hdlist1"><em>fractional-constant</em> : </dt>
+<dd>
+<p><em>digit-sequence</em> <strong>.</strong> <em>digit-sequence</em><br>
+<em>digit-sequence</em> <strong>.</strong><br>
+<strong>.</strong> <em>digit-sequence</em></p>
+</dd>
+<dt class="hdlist1"><em>exponent-part</em> : </dt>
+<dd>
+<p><strong>e</strong> <em>sign<sub>opt</sub></em> <em>digit-sequence</em><br>
+<strong>E</strong> <em>sign<sub>opt</sub></em> <em>digit-sequence</em></p>
+</dd>
+<dt class="hdlist1"><em>sign</em> : one of </dt>
+<dd>
+<p><strong>+</strong> <strong>-</strong></p>
+</dd>
+<dt class="hdlist1"><em>digit-sequence</em> : </dt>
+<dd>
+<p><em>digit</em><br>
+<em>digit-sequence</em> <em>digit</em></p>
+</dd>
+<dt class="hdlist1"><em>floating-suffix</em> : one of </dt>
+<dd>
+<p> <strong>f</strong> <strong>F</strong>
+<strong>lf</strong> <strong>LF</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>A decimal point (<strong>.</strong>) is not needed if the exponent part is present.
+No white space may appear anywhere within a floating-point constant,
+including before a suffix.
+When tokenizing, the maximal token matching the above will be recognized
+before a new token is started.
+When the suffix "lf" or "LF" is present, the literal has type <strong>double</strong>.
+Otherwise, the literal has type <strong>float</strong>.
+A leading unary minus sign (<strong>-</strong>) is interpreted as a unary operator and is
+not part of the floating-point constant.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="vectors">4.1.9. Vectors</h4>
+<div class="paragraph">
+<p>The OpenGL Shading Language includes data types for generic 2-, 3-, and 4-component vectors
+of floating-point values, integers, and Booleans.
+Floating-point vector variables can be used to store colors, normals,
+positions, texture coordinates, texture lookup results and the like.
+Boolean vectors can be used for component-wise comparisons of numeric
+vectors.
+Some examples of vector declarations are:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec2 texcoord1, texcoord2;
+vec3 position;
+vec4 myRGBA;
+ivec2 textureLookup;
+bvec3 less;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Initialization of vectors can be done with constructors.
+See “<a href="#vector-and-matrix-constructors">Vector and Matrix Constructors</a>”.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="matrices">4.1.10. Matrices</h4>
+<div class="paragraph">
+<p>The OpenGL Shading Language has built-in types for 2 × 2, 2 × 3, 2 × 4, 3
+× 2, 3 × 3, 3 × 4, 4 × 2, 4 × 3, and 4 ×
+4 matrices of floating-point numbers.
+Matrix types beginning with "<strong>mat</strong>" have single-precision components while
+matrix types beginning with "<strong>dmat</strong>" have double-precision components.
+The first number in the type is the number of columns, the second is the
+number of rows.
+If there is only one number, the matrix is square.
+Example matrix declarations:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat2 mat2D;
+mat3 optMatrix;
+mat4 view, projection;
+mat4x4 view; <span class="comment">// an alternate way of declaring a mat4</span>
+mat3x2 m; <span class="comment">// a matrix with 3 columns and 2 rows</span>
+dmat4 highPrecisionMVP;
+dmat2x4 dm;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Initialization of matrix values is done with constructors (described in
+“<a href="#vector-and-matrix-constructors">Vector and Matrix Constructors</a>”) in
+column-major order.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="opaque-types">4.1.11. Opaque Types</h4>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">Definition</dt>
+<dd>
+<p>An <em>opaque type</em> is a type where the internal structure of the type is
+hidden from the language.</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The opaque types, as listed in the following sections, declare variables
+that are effectively opaque handles to other objects.
+These objects are accessed through built-in functions, not through direct
+reading or writing of the declared variable.
+They can only be declared as function parameters or in <strong>uniform</strong>-qualified
+variables (see “<a href="#uniform-variables">Uniform Variables</a>”).
+The only opaque types that take memory qualifiers are the image types.
+Except for array indexing, structure member selection, and parentheses,
+opaque variables are not allowed to be operands in expressions; such use
+results in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>Opaque variables cannot be treated as l-values; hence cannot be used as
+<strong>out</strong> or <strong>inout</strong> function parameters, nor can they be assigned into.
+Any such use results in a compile-time error.
+However, they can be passed as <strong>in</strong> parameters with matching types and
+memory qualifiers.
+They cannot be declared with an initializer.</p>
+</div>
+<div class="paragraph">
+<p>Because a single opaque type declaration effectively declares two objects,
+the opaque handle itself and the object it is a handle to, there is room for
+both a storage qualifier and a memory qualifier.
+The storage qualifier will qualify the opaque handle, while the memory
+qualifier will qualify the object it is a handle to.</p>
+</div>
+<div class="sect4">
+<h5 id="samplers">Samplers</h5>
+<div class="paragraph">
+<p>Sampler types (e.g. <strong>sampler2D</strong>) are opaque types, declared and behaving as
+described above for opaque types.
+When aggregated into arrays within a shader, samplers can only be indexed
+with a dynamically uniform integral expression, otherwise results are
+undefined.</p>
+</div>
+<div class="paragraph">
+<p>Sampler variables are handles to
+one-,
+two-, and three- dimensional textures, cube maps, depth textures (shadowing),
+etc., as enumerated in the basic types tables.
+There are distinct sampler types for each texture target, and for each of
+float, integer, and unsigned integer data types.
+Texture accesses are done through built-in texture functions (described in
+“<a href="#texture-functions">Texture Functions</a>”) and samplers are used to
+specify which texture to access and how it is to be filtered.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="images">Images</h5>
+<div class="paragraph">
+<p>Image types are opaque types, declared and behaving as described above for
+opaque types.
+They can be further qualified with memory qualifiers.
+When aggregated into arrays within a shader, images can only be indexed with
+a dynamically uniform integral expression, otherwise results are undefined.</p>
+</div>
+<div class="paragraph">
+<p>Image variables are handles to
+one-,
+two-, or three-dimensional images
+corresponding to all or a portion of a single level of a texture image bound
+to an image unit.
+There are distinct image variable types for each texture target, and for
+each of float, integer, and unsigned integer data types.
+Image accesses should use an image type that matches the target of the
+texture whose level is bound to the image unit, or for non-layered bindings
+of 3D or array images should use the image type that matches the
+dimensionality of the layer of the image (i.e., a layer of 3D, 2DArray,
+Cube, or CubeArray should use
+<strong>image2D</strong>, a layer of 1DArray should use <strong>image1D</strong>, and a layer of 2DMSArray
+should use <strong>image2DMS</strong>).
+If the image target type does not match the bound image in this manner, if
+the data type does not match the bound image, or if the format layout
+qualifier does not match the image unit format as described in section
+8.25
+“Texture Image Loads and Stores” of the <a href="#references">OpenGL Specification</a>, the
+results of image accesses are undefined but cannot include program
+termination.</p>
+</div>
+<div class="paragraph">
+<p>Image variables are used in the image load, store, and atomic functions
+described in “<a href="#image-functions">Image Functions</a>” to specify an image to
+access.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="atomic-counters">Atomic Counters</h5>
+<div class="paragraph">
+<p>Atomic counter types (e.g. <strong>atomic_uint</strong>) are opaque handles to counters,
+declared and behaving as described above for opaque types.
+The variables they declare specify which counter to access when using the
+built-in atomic counter functions as described in
+“<a href="#atomic-counter-functions">Atomic Counter Functions</a>”.
+They are bound to buffers as described in
+“<a href="#atomic-counter-layout-qualifiers">Atomic Counter Layout Qualifiers</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Atomic counters aggregated into arrays within a shader can only be indexed
+with dynamically uniform integral expressions, otherwise results are
+undefined.</p>
+</div>
+<div class="paragraph">
+<p>Members of structures cannot be declared as atomic counter types.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="structures">4.1.12. Structures</h4>
+<div class="paragraph">
+<p>User-defined types can be created by aggregating other already defined types
+into a structure using the <strong>struct</strong> keyword.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> light {
+ <span class="predefined-type">float</span> intensity;
+ vec3 position;
+} lightVar;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>In this example, <em>light</em> becomes the name of the new type, and <em>lightVar</em>
+becomes a variable of type <em>light</em>.
+To declare variables of the new type, use its name (without the keyword
+<strong>struct</strong>).</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">light lightVar2;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>More formally, structures are declared as follows.
+However, the definitive grammar is as given in
+“<a href="#shading-language-grammar">Shading Language Grammar</a>”.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>struct-definition</em> : </dt>
+<dd>
+<p><em>qualifier<sub>opt</sub></em> <strong>struct</strong> name<sub>opt</sub>_ <strong>{</strong> <em>member-list</em> <strong>}</strong>
+<em>declarators<sub>opt</sub></em> <strong>;</strong></p>
+</dd>
+<dt class="hdlist1"><em>member-list</em> : </dt>
+<dd>
+<p><em>member-declaration</em> <strong>;</strong><br>
+<em>member-declaration</em> <em>member-list</em> <strong>;</strong></p>
+</dd>
+<dt class="hdlist1"><em>member-declaration</em> : </dt>
+<dd>
+<p><em>basic-type</em> <em>declarators</em> <strong>;</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>name</em> becomes the user-defined type, and can be used to declare
+variables to be of this new type.
+The <em>name</em> shares the same name space as other variables, types, and
+functions.
+All previously visible variables, types, constructors, or functions with
+that name are hidden.
+The optional <em>qualifier</em> only applies to any <em>declarators</em>, and is not part
+of the type being defined for <em>name</em>.</p>
+</div>
+<div class="paragraph">
+<p>Structures must have at least one member declaration.
+Member declarators may contain precision qualifiers, but use of any other
+qualifier results in a compile-time error.
+Bit fields are not supported.
+Member types must be already defined (there are no forward references).
+A compile-time error results if a member declaration contains an
+initializer.
+Member declarators can contain arrays.
+Such arrays must have a size specified, and the size must be a constant
+integral expression that’s greater than zero (see
+“<a href="#constant-expressions">Constant Expressions</a>”).
+Each level of structure has its own name space for names given in member
+declarators; such names need only be unique within that name space.</p>
+</div>
+<div class="paragraph">
+<p>Anonymous structures are not supported.
+Embedded structure definitions are not supported.
+These result in compile-time errors.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> S { <span class="predefined-type">float</span> f; }; <span class="comment">// Allowed: S is defined as a structure.</span>
+
+<span class="keyword">struct</span> T {
+ S; <span class="comment">// Error: anonymous structures disallowed</span>
+ <span class="keyword">struct</span> { ... }; <span class="comment">// Error: embedded structures disallowed</span>
+ S s; <span class="comment">// Allowed: nested structure with a name.</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Structures can be initialized at declaration time using constructors, as
+discussed in “<a href="#structure-constructors">Structure Constructors</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Any restrictions on the usage of a type or qualifier also apply to any
+structure that contains a member of that type or qualifier.
+This also applies to structure members that are structures, recursively.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="arrays">4.1.13. Arrays</h4>
+<div class="paragraph">
+<p>Variables of the same type can be aggregated into arrays by declaring a name
+followed by brackets (<strong>[ ]</strong>) enclosing an optional size.
+When an array size is specified in a declaration, it must be an integral
+constant expression (see “<a href="#constant-expressions">Constant Expressions</a>”)
+greater than zero.
+Except for the last declared member of a shader storage block (see section
+“<a href="#interface-blocks">Interface Blocks</a>”), the size of an array must be
+declared (<em>explicitly sized</em>) before it is indexed with anything other than
+a constant integral expression.
+The size of any array must be declared before passing it as an argument to a
+function.
+Violation of any of these rules result in compile-time errors.
+It is legal to declare an array without a size (<em>unsized</em>) and then later
+redeclare the same name as an array of the same type and specify a size, or
+index it only with constant integral expressions (<em>implicitly sized</em>).
+However, unless noted otherwise, blocks cannot be redeclared; an unsized
+array member in a user-declared block cannot be sized by a block redeclaration.
+It is a compile-time error to declare an array with a size, and then later
+(in the same shader) index the same array with a constant integral
+expression greater than or equal to the declared size.
+It is a compile-time error to redeclare an unsized array with a size equal
+to or smaller than any index used earlier in the shader to index the array.
+It is also a compile-time error to index an array with a negative constant
+expression.
+Arrays declared as formal parameters in a function declaration must specify
+a size.
+Undefined behavior results from indexing an array with a non-constant
+expression that’s greater than or equal to the array’s size or less than 0.
+Arrays only have a single dimension (a single entry within "[ ]"), however,
+arrays of arrays can be declared.
+All types (basic types, structures, arrays) can be formed into an array.</p>
+</div>
+<div class="paragraph">
+<p>All arrays are inherently homogeneous; made of elements all having the same
+type and size, with one exception.
+The exception is a shader storage block having an unsized array as its last
+member (<em>run-time sized</em>); an array can be formed from such a shader storage
+block, even if the storage blocks have differing lengths for their last
+member.</p>
+</div>
+<div class="paragraph">
+<p>Some examples are:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> frequencies[<span class="integer">3</span>];
+uniform vec4 lightPosition[<span class="integer">4</span>];
+light lights[];
+<span class="directive">const</span> <span class="predefined-type">int</span> numLights = <span class="integer">2</span>;
+light lights[numLights];
+
+<span class="comment">// a shader storage block, introduced in section 4.3.7 "Buffer Variables"</span>
+buffer b {
+ <span class="predefined-type">float</span> u[]; <span class="comment">// an error, unless u gets statically sized by link time</span>
+ vec4 v[]; <span class="comment">// okay, v will be sized dynamically, if not statically</span>
+} name[<span class="integer">3</span>]; <span class="comment">// when the block is arrayed, all u will be the same size,</span>
+ <span class="comment">// but not necessarily all v, if sized dynamically</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>An array type can be formed by specifying non-array type followed by an
+array specifier.
+All dimensions of such an array specifier must include a size.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">5</span>] <span class="comment">// an array of size [5] of float</span>
+<span class="predefined-type">float</span>[<span class="integer">2</span>][<span class="integer">3</span>] <span class="comment">// an array of size [2][3] of float, not size [3] of float[2]</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This type can be used anywhere any other type can be used, including as the
+return value from a function</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">5</span>] foo() { }</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>as a constructor of an array:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">5</span>](<span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span>, <span class="float">5</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>as an unnamed parameter:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> foo(<span class="predefined-type">float</span>[<span class="integer">5</span>])</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and as an alternate way of declaring a variable or function parameter:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span>[<span class="integer">5</span>] a;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Arrays can have initializers formed from array constructors:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a[<span class="integer">5</span>] = <span class="predefined-type">float</span>[<span class="integer">5</span>](<span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span>, <span class="float">5</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>);
+<span class="predefined-type">float</span> a[<span class="integer">5</span>] = <span class="predefined-type">float</span>[](<span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span>, <span class="float">5</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>); <span class="comment">// same thing</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>An array of arrays can be declared as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 a[<span class="integer">3</span>][<span class="integer">2</span>]; <span class="comment">// size-3 array of size-2 array of vec4</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>which declares a one-dimensional array of size 3 of one-dimensional arrays
+of size 2 of <strong>vec4</strong>.
+The following declarations do the same thing:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4[<span class="integer">2</span>] a[<span class="integer">3</span>]; <span class="comment">// size-3 array of size-2 array of vec4</span>
+vec4[<span class="integer">3</span>][<span class="integer">2</span>] a; <span class="comment">// size-3 array of size-2 array of vec4</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When in transparent memory (like in a uniform block), the layout is that the
+inner-most (right-most in declaration) dimensions iterate faster than the
+outer dimensions.
+That is, for the above, the order in memory would be:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>Low address : a[0][0] : a[0][1] : a[1][0] : a[1][1] : a[2][0] : a[2][1]
+: High address</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The type of <em>a</em> needed for both constructors and nameless parameters is
+“vec4[3][2]”:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 b[<span class="integer">2</span>] = vec4[<span class="integer">2</span>](vec4(<span class="float">0</span><span class="float">.0</span>), vec4(<span class="float">0</span><span class="float">.1</span>));
+vec4[<span class="integer">3</span>][<span class="integer">2</span>] a = vec4[<span class="integer">3</span>][<span class="integer">2</span>](b, b, b); <span class="comment">// constructor</span>
+<span class="directive">void</span> foo(vec4[<span class="integer">3</span>][<span class="integer">2</span>]); <span class="comment">// prototype with unnamed parameter</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Alternatively, the initializer-list syntax can be used to initialize an
+array of arrays:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 a[<span class="integer">3</span>][<span class="integer">2</span>] = { vec4[<span class="integer">2</span>](vec4(<span class="float">0</span><span class="float">.0</span>), vec4(<span class="float">1</span><span class="float">.0</span>)),
+ vec4[<span class="integer">2</span>](vec4(<span class="float">0</span><span class="float">.0</span>), vec4(<span class="float">1</span><span class="float">.0</span>)),
+ vec4[<span class="integer">2</span>](vec4(<span class="float">0</span><span class="float">.0</span>), vec4(<span class="float">1</span><span class="float">.0</span>)) };</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Unsized arrays can be explicitly sized by an initializer at declaration
+time:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a[<span class="integer">5</span>];
+...
+<span class="predefined-type">float</span> b[] = a; <span class="comment">// b is explicitly size 5</span>
+<span class="predefined-type">float</span> b[<span class="integer">5</span>] = a; <span class="comment">// means the same thing</span>
+<span class="predefined-type">float</span> b[] = <span class="predefined-type">float</span>[](<span class="integer">1</span>,<span class="integer">2</span>,<span class="integer">3</span>,<span class="integer">4</span>,<span class="integer">5</span>); <span class="comment">// also explicitly sizes to 5</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>However, it is a compile-time error to assign to an unsized array.
+Note, this is a rare case that initializers and assignments appear to have
+different semantics.
+For arrays of arrays, any unsized dimension is explicitly sized by the
+initializer:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 a[][] = { vec4[<span class="integer">2</span>](vec4(<span class="float">0</span><span class="float">.0</span>), vec4(<span class="float">1</span><span class="float">.0</span>)), <span class="comment">// okay, size to a[3][2]</span>
+ vec4[<span class="integer">2</span>](vec4(<span class="float">0</span><span class="float">.0</span>), vec4(<span class="float">1</span><span class="float">.0</span>)),
+ vec4[<span class="integer">2</span>](vec4(<span class="float">0</span><span class="float">.0</span>), vec4(<span class="float">1</span><span class="float">.0</span>)) };</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Arrays know the number of elements they contain.
+This can be obtained by using the <strong>length</strong>() method:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a[<span class="integer">5</span>];
+a.length(); <span class="comment">// returns 5</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This returns a type <strong>int</strong>.
+If an array has been explicitly sized, the value returned by the <strong>length</strong>()
+method is a constant expression.
+If an array has not been explicitly sized and is the last declared member of
+a shader storage block, the value returned will not be a constant expression
+and will be determined at runtime based on the size of the buffer object
+providing storage for the block.
+Such arrays are runtime sized.
+For runtime-sized arrays, the value returned by the <strong>length</strong>() method will be
+undefined if the array is contained in an array of shader storage blocks
+that is indexed with a non-constant expression less than zero or greater
+than or equal to the number of blocks in the array.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>length</strong>() method cannot be called on an array that is not runtime sized
+and also has not yet been explicitly sized;
+this results in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>When the <strong>length</strong>() method returns a compile-time constant, the expression the
+<strong>length</strong>() method is applied to cannot contain any side effects, such as writes
+to l-values within the expression, or function calls that themselves have side
+effects: only the compile-time constant length itself need be computed.
+Behavior and results, including any compile-time error reporting,
+are undefined if the expression contains other effects.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a, b;
+<span class="directive">const</span> <span class="predefined-type">int</span> s = <span class="predefined-type">float</span>[<span class="integer">2</span>](a=<span class="float">3</span><span class="float">.0</span>, ++b).length(); <span class="comment">// illegal side effects</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>length</strong>() method works equally well for arrays of arrays:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 a[<span class="integer">3</span>][<span class="integer">2</span>];
+a.length() <span class="comment">// this is 3</span>
+a[x].length() <span class="comment">// this is 2</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When the <strong>length</strong>() method returns a compile-time constant, the expression
+in brackets (<em>x</em> above) will be parsed and subjected to the rules required for
+array indexes, but the array will not be dereferenced.
+Thus, behavior is well defined even if the run-time value of the expression
+is out of bounds, as long as the expression contains no side effects.</p>
+</div>
+<div class="paragraph">
+<p>When the <strong>length</strong>() method returns a run-time value
+(not a compile-time constant), the array will be
+dereferenced.
+E.g., if <em>x</em> is not a compile-time constant and is out of range, an undefined
+value results.
+More generally, all involved expressions are fully evaluated and executed.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// for a compile time-sized array b containing a member array a:</span>
+b[x+<span class="integer">3</span>].a.length(); <span class="comment">// b is never dereferenced, x+3 is evaluated and checked</span>
+b[++x].a.length(); <span class="comment">// not allowed; results are undefined</span>
+
+<span class="comment">// for an array s of a shader storage object (run-time sized) containing a member array a:</span>
+s[++x].a.length(); <span class="comment">// s is dereferenced; ++x needs to be a valid index</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For implicitly-sized or run-time-sized arrays, only the outer-most dimension
+can be lacking a size.
+A type that includes an unknown array size cannot be formed into an array
+until it gets an explicit size, except for shader storage blocks where the
+only unsized array member is the last member of the block.</p>
+</div>
+<div class="paragraph">
+<p>In a shader storage block, the last member may be declared without an
+explicit size.
+In this case, the effective array size is inferred at run-time from the size
+of the data store backing the interface block.
+Such run-time-sized arrays may be indexed with general integer expressions.
+However, it is a compile-time error to pass them as an argument to a
+function or index them with a negative constant expression.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="implicit-conversions">4.1.14. Implicit Conversions</h4>
+<div class="paragraph">
+<p>In some situations, an expression and its type will be implicitly converted
+to a different type.
+The following table shows all allowed implicit conversions:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Type of expression</th>
+<th class="tableblock halign-left valign-top">Can be implicitly converted to</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>int</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uint</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>int</strong><br>
+ <strong>uint</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>float</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>int</strong><br>
+ <strong>uint</strong><br>
+ <strong>float</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>double</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec3</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uvec4</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec2</strong><br>
+ <strong>uvec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec3</strong><br>
+ <strong>uvec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec3</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec4</strong><br>
+ <strong>uvec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vec4</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec2</strong><br>
+ <strong>uvec2</strong><br>
+ <strong>vec2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dvec2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec3</strong><br>
+ <strong>uvec3</strong><br>
+ <strong>vec3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dvec3</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>ivec4</strong><br>
+ <strong>uvec4</strong><br>
+ <strong>vec4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dvec4</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat3</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat4</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat2x3</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat2x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat2x4</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat3x2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat3x4</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat3x4</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4x2</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat4x2</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mat4x3</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>dmat4x3</strong></p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>There are no implicit array or structure conversions.
+For example, an array of <strong>int</strong> cannot be implicitly converted to an array of
+<strong>float</strong>.</p>
+</div>
+<div class="paragraph">
+<p>When an implicit conversion is done, it is the same conversion that would be
+done under explicit conversion, using a constructor.
+The explicit conversions via constructors are described in
+<a href="#conversion-and-scalar-constructors">Conversion and Scalar Constructors</a>.</p>
+</div>
+<div class="paragraph">
+<p>When performing implicit conversion for binary operators, there may be
+multiple data types to which the two operands can be converted.
+For example, when adding an <strong>int</strong> value to a <strong>uint</strong> value, both values can
+be implicitly converted to <strong>uint</strong>, <strong>float</strong>, and <strong>double</strong>.
+In such cases, a floating-point type is chosen if either operand has a
+floating-point type.
+Otherwise, an unsigned integer type is chosen if either operand has an
+unsigned integer type.
+Otherwise, a signed integer type is chosen.
+If operands can be implicitly converted to multiple data types deriving from
+the same base data type, the type with the smallest component size is used.</p>
+</div>
+<div class="paragraph">
+<p>The conversions in the table above are done only as indicated by other
+sections of this specification.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="initializers">4.1.15. Initializers</h4>
+<div class="paragraph">
+<p>At declaration, an initial value for a variable may be provided, specified
+as an equals (=) followed by an initializer.
+The initializer is either an <em>assignment-expression</em> or a list of
+initializers enclosed in curly braces.
+The grammar for the initializer is:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>initializer</em> : </dt>
+<dd>
+<p><em>assignment-expression</em><br>
+<strong>{</strong> <em>initializer-list</em> <strong>}</strong><br>
+<strong>{</strong> <em>initializer-list</em> <strong>,</strong> <strong>}</strong></p>
+</dd>
+<dt class="hdlist1"><em>initializer-list</em> : </dt>
+<dd>
+<p><em>initializer</em><br>
+<em>initializer-list</em> , <em>initializer</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The <em>assignment-expression</em> is a normal expression except that a comma (<strong>,</strong>)
+outside parentheses is interpreted as the end of the initializer, not as the
+sequence operator.
+As explained in more detail below, this allows creation of nested
+initializers: The variable type and its initializer must exactly match in
+terms of nesting, number of components/elements/members present at each
+level, and types of components/elements/members.
+An <em>assignment-expression</em> at global scope can include calls to user-defined
+functions.</p>
+</div>
+<div class="paragraph">
+<p>An <em>assignment-expression</em> in an initializer must be either the same type as
+the object it initializes or be a type that can be converted to the object’s
+type according to “<a href="#implicit-conversions">Implicit Conversions</a>”.
+Since these include constructors, a composite variable can be initialized by
+either a constructor or an initializer list; and an element in an
+initializer list can be a constructor.</p>
+</div>
+<div class="paragraph">
+<p>If an initializer is a list of initializers enclosed in curly braces, the
+variable being declared must be a vector, a matrix, an array, or a
+structure.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> i = { <span class="integer">1</span> }; <span class="comment">// illegal, i is not a composite</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A list of initializers enclosed in a matching set of curly braces is applied
+to one composite.
+This may be the variable being declared or a composite contained in the
+variable being declared.
+Individual initializers from the initializer list are applied to the
+elements/members of the composite, in order.</p>
+</div>
+<div class="paragraph">
+<p>If the composite has a vector type, initializers from the list are applied
+to the components of the vector, in order, starting with component 0.
+The number of initializers must match the number of components.</p>
+</div>
+<div class="paragraph">
+<p>If the composite has a matrix type, initializers from the list must be
+vector initializers and are applied to the columns of the matrix, in order,
+starting with column 0.
+The number of initializers must match the number of columns.</p>
+</div>
+<div class="paragraph">
+<p>If the composite has a structure type, initializers from the list are
+applied to the members of the structure, in the order declared in the
+structure, starting with the first member.
+The number of initializers must match the number of members.</p>
+</div>
+<div class="paragraph">
+<p>Applying these rules, the following matrix declarations are equivalent:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat2x2 a = mat2( vec2( <span class="float">1</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span> ), vec2( <span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span> ) );
+mat2x2 b = { vec2( <span class="float">1</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span> ), vec2( <span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span> ) };
+mat2x2 c = { { <span class="float">1</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span> }, { <span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span> } };</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>All of the following declarations result in a compile-time error.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a[<span class="integer">2</span>] = { <span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span> }; <span class="comment">// illegal</span>
+vec2 b = { <span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span> }; <span class="comment">// illegal</span>
+mat3x3 c = { vec3(<span class="float">0</span><span class="float">.0</span>), vec3(<span class="float">1</span><span class="float">.0</span>), vec3(<span class="float">2</span><span class="float">.0</span>), vec3(<span class="float">3</span><span class="float">.0</span>) }; <span class="comment">// illegal</span>
+mat2x2 d = { <span class="float">1</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span> }; <span class="comment">// illegal, can't flatten nesting</span>
+<span class="keyword">struct</span> {
+ <span class="predefined-type">float</span> a;
+ <span class="predefined-type">int</span> b;
+} e = { <span class="float">1</span><span class="float">.2</span>, <span class="integer">2</span>, <span class="integer">3</span> }; <span class="comment">// illegal</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>In all cases, the inner-most initializer (i.e., not a list of initializers
+enclosed in curly braces) applied to an object must have the same type as
+the object being initialized or be a type that can be converted to the
+object’s type according to “<a href="#implicit-conversions">Implicit
+Conversions</a>”.
+In the latter case, an implicit conversion will be done on the initializer
+before the assignment is done.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> {
+ <span class="predefined-type">float</span> a;
+ <span class="predefined-type">int</span> b;
+} e = { <span class="float">1</span><span class="float">.2</span>, <span class="integer">2</span> }; <span class="comment">// legal, all types match</span>
+<span class="keyword">struct</span> {
+ <span class="predefined-type">float</span> a;
+ <span class="predefined-type">int</span> b;
+} e = { <span class="integer">1</span>, <span class="integer">3</span> }; <span class="comment">// legal, first initializer is converted</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>All of the following declarations result in a compile-time error.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> a = <span class="predefined-constant">true</span>; <span class="comment">// illegal</span>
+vec4 b[<span class="integer">2</span>] = { vec4(<span class="float">0</span><span class="float">.0</span>), <span class="float">1</span><span class="float">.0</span> }; <span class="comment">// illegal</span>
+mat4x2 c = { vec3(<span class="float">0</span><span class="float">.0</span>), vec3(<span class="float">1</span><span class="float">.0</span>) }; <span class="comment">// illegal</span>
+
+<span class="keyword">struct</span> S1 {
+ vec4 a;
+ vec4 b;
+};
+
+<span class="keyword">struct</span> {
+ <span class="predefined-type">float</span> s;
+ <span class="predefined-type">float</span> t;
+} d[] = { S1(vec4(<span class="float">0</span><span class="float">.0</span>), vec4(<span class="float">1</span><span class="float">.1</span>)) }; <span class="comment">// illegal</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>If an initializer (of either form) is provided for an unsized array, the
+size of the array is determined by the number of top-level (non-nested)
+initializers within the initializer.
+All of the following declarations create arrays explicitly sized with five
+elements:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> a[] = <span class="predefined-type">float</span>[](<span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span>, <span class="float">5</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>);
+<span class="predefined-type">float</span> b[] = { <span class="float">3</span><span class="float">.4</span>, <span class="float">4</span><span class="float">.2</span>, <span class="float">5</span><span class="float">.0</span>, <span class="float">5</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span> };
+<span class="predefined-type">float</span> c[] = a; <span class="comment">// c is explicitly size 5</span>
+<span class="predefined-type">float</span> d[<span class="integer">5</span>] = b; <span class="comment">// means the same thing</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to have too few or too many initializers in an
+initializer list for the composite being initialized.
+That is, all elements of an array, all members of a structure, all columns
+of a matrix, and all components of a vector must have exactly one
+initializer expression present, with no unconsumed initializers.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="scoping">4.2. Scoping</h3>
+<div class="paragraph">
+<p>The scope of a variable is determined by where it is declared.
+If it is declared outside all function definitions, it has global scope,
+which starts from where it is declared and persists to the end of the shader
+it is declared in.
+If it is declared in a <strong>while</strong> test or a <strong>for</strong> statement, then it is scoped
+to the end of the following sub-statement.
+If it is declared in an <strong>if</strong> or <strong>else</strong> statement, it is scoped to the end of
+that statement.
+(See “<a href="#selection">Selection</a>” and “<a href="#iteration">Iteration</a>” for the
+location of statements and sub-statements.) Otherwise, if it is declared as
+a statement within a compound statement, it is scoped to the end of that
+compound statement.
+If it is declared as a parameter in a function definition, it is scoped
+until the end of that function definition.
+A function’s parameter declarations and body together form a single scope
+nested in the global scope.
+The <strong>if</strong> statement’s expression does not allow new variables to be declared,
+hence does not form a new scope.</p>
+</div>
+<div class="paragraph">
+<p>Within a declaration, the scope of a name starts immediately after the
+initializer if present or immediately after the name being declared if not.
+Several examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> x = <span class="integer">1</span>;
+{
+ <span class="predefined-type">int</span> x = <span class="integer">2</span>, y = x; <span class="comment">// y is initialized to 2</span>
+}
+
+<span class="keyword">struct</span> S
+{
+ <span class="predefined-type">int</span> x;
+};
+
+{
+ S S = S(<span class="integer">0</span>); <span class="comment">// 'S' is only visible as a struct and constructor</span>
+ S; <span class="comment">// 'S' is now visible as a variable</span>
+}
+
+<span class="predefined-type">int</span> x = x; <span class="comment">// Error if x has not been previously defined.</span>
+ <span class="comment">// If the previous definition of x was in this</span>
+ <span class="comment">// same scope, this causes a redeclaration error.</span>
+
+<span class="predefined-type">int</span> f( <span class="comment">/* nested scope begins here */</span> <span class="predefined-type">int</span> k)
+{
+ <span class="predefined-type">int</span> k = k + <span class="integer">3</span>; <span class="comment">// redeclaration error of the name k</span>
+ ...
+}
+
+<span class="predefined-type">int</span> f(<span class="predefined-type">int</span> k)
+{
+ {
+ <span class="predefined-type">int</span> k = k + <span class="integer">3</span>; <span class="comment">// 2nd k is parameter, initializing nested first k</span>
+ <span class="predefined-type">int</span> m = k; <span class="comment">// use of new k, which is hiding the parameter</span>
+ }
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For both <strong>for</strong> and <strong>while</strong> loops, the sub-statement itself does not
+introduce a new scope for variable names, so the following has a
+redeclaration compile-time error:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">for</span> ( <span class="comment">/* nested scope begins here */</span> <span class="predefined-type">int</span> i = <span class="integer">0</span>; i < <span class="integer">10</span>; i++) {
+ <span class="predefined-type">int</span> i; <span class="comment">// redeclaration error</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The body of a <strong>do</strong>-<strong>while</strong> loop introduces a new scope lasting only between
+the <strong>do</strong> and <strong>while</strong> (not including the while test expression), whether or
+not the body is simple or compound:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span> i = <span class="integer">17</span>;
+<span class="keyword">do</span>
+ <span class="predefined-type">int</span> i = <span class="integer">4</span>; <span class="comment">// okay, in nested scope_</span>
+<span class="keyword">while</span> (i == <span class="integer">0</span>); <span class="comment">// i is 17, scoped outside the do-while body</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The statement following a <strong>switch</strong> (…​) forms a nested scope.</p>
+</div>
+<div class="paragraph">
+<p>All variable names, structure type names, and function names in a given
+scope share the same name space.
+Function names can be redeclared in the same scope, with the same or
+different parameters, without error.
+An implicitly-sized array can be redeclared in the same scope as an array of
+the same base type.
+Otherwise, within one compilation unit, a declared name cannot be redeclared
+in the same scope; doing so results in a redeclaration compile-time error.
+If a nested scope redeclares a name used in an outer scope, it hides all
+existing uses of that name.
+There is no way to access the hidden name or make it unhidden, without
+exiting the scope that hid it.</p>
+</div>
+<div class="paragraph">
+<p>The built-in functions are scoped in a scope outside the global scope that
+users declare global variables in.
+That is, a shader’s global scope, available for user-defined functions and
+global variables, is nested inside the scope containing the built-in
+functions.
+When a function name is redeclared in a nested scope, it hides all functions
+declared with that name in the outer scope.
+Function declarations (prototypes) cannot occur inside of functions; they
+must be at global scope, or for the built-in functions, outside the global
+scope, otherwise a compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>Shared globals are global variables declared with the same name in
+independently compiled units (shaders) within the same language (i.e., same
+stage, e.g. vertex) that are linked together when making a single program.
+(Globals forming the interface between two different shader languages are
+discussed in other sections.) Shared globals share the same name space, and
+must be declared with the same type.
+They will share the same storage.</p>
+</div>
+<div class="paragraph">
+<p>Shared global arrays must have the same base type and the same explicit
+size.
+An array implicitly sized in one shader can be explicitly sized by another
+shader in the same stage.
+If no shader in a stage has an explicit size for the array, the largest
+implicit size (one more than the largest index used) in that stage is used.
+There is no cross-stage array sizing.
+If there is no static access to an implicitly sized array within the stage
+declaring it, then the array is given a size of 1, which is relevant when
+the array is declared within an interface block that is shared with other
+stages or the application (other unused arrays might be eliminated by the
+optimizer).</p>
+</div>
+<div class="paragraph">
+<p>Shared global scalars must have exactly the same type name and type
+definition.
+Structures must have the same name, sequence of type names, and type
+definitions, and member names to be considered the same type.
+This rule applies recursively for nested or embedded types.
+If a shared global has multiple initializers, the initializers must all be
+constant expressions, and they must all have the same value.
+Otherwise, a link-time error will result.
+(A shared global having only one initializer does not require that
+initializer to be a constant expression.)</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="storage-qualifiers">4.3. Storage Qualifiers</h3>
+<div class="paragraph">
+<p>Variable declarations may have at most one storage qualifier specified in
+front of the type.
+These are summarized as</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Storage Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><none: default></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">local read/write memory, or an input parameter to a
+ function</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>const</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">a variable whose value cannot be changed</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>in</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">linkage into a shader from a previous stage, variable
+ is copied in</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>out</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">linkage out of a shader to a subsequent stage,
+ variable is copied out</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>attribute</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">compatibility profile only and vertex language only;
+ same as <strong>in</strong> when in a vertex shader</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uniform</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">value does not change across the primitive being
+ processed, uniforms form the linkage between a shader,
+ OpenGL, and the application</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>varying</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">compatibility profile only and vertex and fragment
+ languages only; same as <strong>out</strong> when in a vertex shader
+ and same as <strong>in</strong> when in a fragment shader</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>buffer</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">value is stored in a buffer object, and can be read or
+ written both by shader invocations and the OpenGL
+ API</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>shared</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">compute shader only; variable storage is shared across
+ all work items in a workgroup</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Some input and output qualified variables can be qualified with at most one
+additional auxiliary storage qualifier:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Auxiliary Storage Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>centroid</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">centroid-based interpolation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sample</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">per-sample interpolation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>patch</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">per-tessellation-patch attributes</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Not all combinations of qualification are allowed.
+Auxiliary storage qualifiers can only be used with the <strong>in</strong> or <strong>out</strong> storage
+qualifiers.
+Additional qualifier rules are defined in upcoming sections.</p>
+</div>
+<div class="paragraph">
+<p>Local variables can only use the <strong>const</strong> storage qualifier (or use no
+storage qualifier).</p>
+</div>
+<div class="paragraph">
+<p>Note that function parameters can use <strong>const</strong>, <strong>in</strong>, and <strong>out</strong> qualifiers,
+but as <em>parameter qualifiers</em>.
+Parameter qualifiers are discussed in
+“<a href="#function-calling-conventions">Function Calling Conventions</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Function return types and structure members do not use storage qualifiers.</p>
+</div>
+<div class="paragraph">
+<p>Initializers in global declarations may only be used in declarations of
+global variables with no storage qualifier, with a <strong>const</strong> qualifier, or
+with a <strong>uniform</strong> qualifier.</p>
+</div>
+<div class="paragraph">
+<p>Global variables without storage qualifiers that are not initialized in
+their declaration or by the application will not be initialized by
+OpenGL, but rather will enter <em>main()</em> with undefined values.</p>
+</div>
+<div class="paragraph">
+<p>When comparing an output from one shader stage to an input of a subsequent
+shader stage, the input and output don’t match if their auxiliary qualifiers
+(or lack thereof) are not the same.</p>
+</div>
+<div class="sect3">
+<h4 id="default-storage-qualifier">4.3.1. Default Storage Qualifier</h4>
+<div class="paragraph">
+<p>If no qualifier is present on a global variable, then the variable has no
+linkage to the application or shaders running on other pipeline stages.
+For either global or local unqualified variables, the declaration will
+appear to allocate memory associated with the processor it targets.
+This variable will provide read/write access to this allocated memory.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="constant-qualifier">4.3.2. Constant Qualifier</h4>
+<div class="paragraph">
+<p>Named compile-time constants
+or read-only variables
+can be declared using
+the <strong>const</strong> qualifier.
+The <strong>const</strong> qualifier can be used with any of the non-void transparent basic
+data types, as well as with structures and arrays of these.
+It is a compile-time error to write to a <strong>const</strong> variable outside of its
+declaration, so they must be initialized when declared.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">const</span> vec3 zAxis = vec3 (<span class="float">0</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span>);
+<span class="directive">const</span> <span class="predefined-type">float</span> ceiling = a + b; <span class="comment">// a and b not necessarily constants</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Structure members may not be qualified with <strong>const</strong>.
+Structure variables can be declared as <strong>const</strong>, and initialized with a
+structure
+constructor or initializer.</p>
+</div>
+<div class="paragraph">
+<p>Initializers for <strong>const</strong> declarations
+at global scope
+must be constant expressions, as defined in
+“<a href="#constant-expressions">Constant Expressions</a>”.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="constant-expressions">4.3.3. Constant Expressions</h4>
+<div class="paragraph">
+<p>SPIR-V specialization constants are expressed in GLSL as <strong>const</strong> with the
+layout qualifier <strong>constant_id</strong>, as described in
+“<a href="#specialization-constant-qualifier">Specialization-Constant
+Qualifier.</a>”</p>
+</div>
+<div class="paragraph">
+<p>A <em>constant expression</em> is one of</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A literal value (e.g. <strong>5</strong> or <strong>true</strong>).</p>
+</li>
+<li>
+<p>A variable declared with the <strong>const</strong> qualifier and an initializer, where
+the initializer is a constant expression.
+This includes both <strong>const</strong> declared with a specialization-constant
+layout qualifier, e.g. <strong>layout</strong>(<strong>constant_id</strong> = …​), and those declared
+without a specialization-constant layout qualifier.</p>
+</li>
+<li>
+<p>An expression formed by an operator on operands that are all constant
+expressions, including getting an element of a constant array, or a
+member of a constant structure, or components of a constant vector.
+However, the lowest precedence operators of the sequence operator (<strong>,</strong>)
+and the assignment operators (<strong>=</strong>, <strong>+=</strong>, <strong>…​</strong>) are not included in the
+operators that can create a constant expression.
+Also, an array access with a specialization constant as an index does
+not result in a constant expression.</p>
+</li>
+<li>
+<p>Valid use of the <strong>length</strong>() method on an explicitly sized object,
+whether or not the object itself is constant (implicitly sized or
+run-time sized arrays do not return a constant expression).</p>
+</li>
+<li>
+<p>A constructor whose arguments are all constant expressions.</p>
+</li>
+<li>
+<p>For non-specialization constants only: the value returned by certain
+built-in function calls whose arguments are all constant expressions,
+including at least the list below.
+Any other built-in function that does not access memory (not the texture
+lookup functions, image access, atomic counter, etc.), that has a
+non-<strong>void</strong> return type, that has no <strong>out</strong> parameter, and is not a noise
+function might also be considered a constant.
+When a function is called with an argument that is a specialization
+constant, the result is not a constant expression.</p>
+<div class="ulist">
+<ul>
+<li>
+<p>Angle and Trigonometric Functions</p>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>radians</strong></p>
+</li>
+<li>
+<p><strong>degrees</strong></p>
+</li>
+<li>
+<p><strong>sin</strong></p>
+</li>
+<li>
+<p><strong>cos</strong></p>
+</li>
+<li>
+<p><strong>asin</strong></p>
+</li>
+<li>
+<p><strong>acos</strong></p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>Exponential Functions</p>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>pow</strong></p>
+</li>
+<li>
+<p><strong>exp</strong></p>
+</li>
+<li>
+<p><strong>log</strong></p>
+</li>
+<li>
+<p><strong>exp2</strong></p>
+</li>
+<li>
+<p><strong>log2</strong></p>
+</li>
+<li>
+<p><strong>sqrt</strong></p>
+</li>
+<li>
+<p><strong>inversesqrt</strong></p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>Common Functions</p>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>abs</strong></p>
+</li>
+<li>
+<p><strong>sign</strong></p>
+</li>
+<li>
+<p><strong>floor</strong></p>
+</li>
+<li>
+<p><strong>trunc</strong></p>
+</li>
+<li>
+<p><strong>round</strong></p>
+</li>
+<li>
+<p><strong>ceil</strong></p>
+</li>
+<li>
+<p><strong>mod</strong></p>
+</li>
+<li>
+<p><strong>min</strong></p>
+</li>
+<li>
+<p><strong>max</strong></p>
+</li>
+<li>
+<p><strong>clamp</strong></p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>Geometric Functions</p>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>length</strong></p>
+</li>
+<li>
+<p><strong>dot</strong></p>
+</li>
+<li>
+<p><strong>normalize</strong></p>
+</li>
+</ul>
+</div>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>Function calls to user-defined functions (non-built-in functions) cannot
+be used to form constant expressions.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>A <em>constant integral expression</em> is a constant expression that evaluates to
+a scalar signed or unsigned integer.</p>
+</div>
+<div class="paragraph">
+<p>Constant expressions will be evaluated in an invariant way so as to create
+the same value in multiple shaders when the same constant expressions appear
+in those shaders.
+See “<a href="#the-invariant-qualifier">The Invariant Qualifier</a>” for more details
+on how to create invariant expressions and
+“<a href="#precision-qualifiers">Precision Qualifiers</a>” for detail on how
+expressions are evaluated.</p>
+</div>
+<div class="paragraph">
+<p>Constant expressions respect the <strong>precise</strong> and <strong>invariant</strong> qualifiers but
+will be always be evaluated in an invariant way, independent of the use of
+such qualification, so as to create the same value in multiple shaders when
+the same constant expressions appear in those shaders.
+See “<a href="#the-invariant-qualifier">The Invariant Qualifier</a>” and
+“<a href="#the-precise-qualifier">The Precise Qualifier</a>” for more details on how
+to create invariant expressions.</p>
+</div>
+<div class="paragraph">
+<p>Non-specialization constant
+expressions may be evaluated by the compiler’s
+host platform, and are therefore not required to compute the same value that
+the same expression would evaluate to on the shader execution target.
+However, the host must use the same or greater precision than the target
+would use.</p>
+</div>
+<div class="paragraph">
+<p>Specialization-constant expressions are never evaluated by the compiler
+front end, but instead retain the expression’s operations needed to evaluate
+them later on the host.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="input-variables">4.3.4. Input Variables</h4>
+<div class="paragraph">
+<p>Shader input variables are declared with the <strong>in</strong> storage qualifier.
+They form the input interface between previous stages of the OpenGL
+pipeline and the declaring shader.
+Input variables must be declared at global scope.
+Values from the previous pipeline stage are copied into input variables at
+the beginning of shader execution.
+It is a compile-time error to write to a variable declared as an input.</p>
+</div>
+<div class="paragraph">
+<p>Only the input variables that are
+statically
+read need to be written by the
+previous stage; it is allowed to have superfluous declarations of input
+variables.
+This is shown in the following table.</p>
+</div>
+<table class="tableblock frame-all grid-all fit-content">
+<colgroup>
+<col>
+<col>
+<col>
+<col>
+<col>
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top" colspan="2" rowspan="2"><p class="tableblock">Treatment of Mismatched Input Variables</p></td>
+<td class="tableblock halign-left valign-top" colspan="3"><p class="tableblock">Consuming Shader (input variables)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No Declaration</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Declared but no Static Use</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Declared and Static Use</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top" rowspan="3"><p class="tableblock">Generating Shader (output variables)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">No Declaration</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Link-Time Error</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Declared but no Static Use</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed (values are undefined)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Declared and Static Use</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Allowed (values are potentially undefined)</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Consumption errors are based on static use only.
+Compilation may generate a warning, but not an error, for any dynamic use
+the compiler can deduce that might cause consumption of undefined values.</p>
+</div>
+<div class="paragraph">
+<p>See “<a href="#built-in-variables">Built-In Variables</a>” for a list of the built-in
+input names.</p>
+</div>
+<div class="paragraph">
+<p>Vertex shader input variables (or attributes) receive per-vertex data.
+It is a compile-time error to use auxiliary storage or interpolation qualifiers
+on a vertex shader input.
+The values copied in are established by the OpenGL API or through the use
+of the layout identifier <strong>location</strong>.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to declare a vertex shader input with, or that
+contains, any of the following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+<li>
+<p>A structure</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Example declarations in a vertex shader:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 position;
+in vec3 normal;
+in vec2 texCoord[<span class="integer">4</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It is expected that graphics hardware will have a small number of fixed
+vector locations for passing vertex inputs.
+Therefore, the OpenGL Shading Language defines each non-matrix input variable as taking up
+one such vector location.
+There is an implementation-dependent limit on the number of locations that
+can be used, and if this is exceeded it will cause a link-time error.
+(Declared input variables that are not statically used do not count against
+this limit.) A scalar input counts the same amount against this limit as a
+<strong>vec4</strong>, so applications may want to consider packing groups of four
+unrelated float inputs together into a vector to better utilize the
+capabilities of the underlying hardware.
+A matrix input will use up multiple locations.
+The number of locations used will equal the number of columns in the matrix.</p>
+</div>
+<div class="paragraph">
+<p>Tessellation control, evaluation, and geometry shader input variables get
+the per-vertex values written out by output variables of the same names in
+the previous active shader stage.
+For these inputs, <strong>centroid</strong> and interpolation qualifiers are allowed, but
+have no effect.
+Since tessellation control, tessellation evaluation, and geometry shaders
+operate on a set of vertices, each input variable (or input block, see
+interface blocks below) needs to be declared as an array.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in <span class="predefined-type">float</span> foo[]; <span class="comment">// geometry shader input for vertex "out float foo"</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Each element of such an array corresponds to one vertex of the primitive
+being processed.
+Each array can optionally have a size declared.
+For geometry shaders, the array size will be set by, (or if provided must be
+consistent with) the input <strong>layout</strong> declaration(s) establishing the type of
+input primitive, as described later in “<a href="#input-layout-qualifiers">Input
+Layout Qualifiers</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Some inputs and outputs are <em>arrayed</em>, meaning that for an interface between
+two shader stages either the input or output declaration requires an extra
+level of array indexing for the declarations to match.
+For example, with the interface between a vertex shader and a geometry
+shader, vertex shader output variables and geometry shader input variables
+of the same name must have matching types, except that the geometry shader
+will have one more array dimension than the vertex shader, to allow for
+vertex indexing.
+If such an arrayed interface variable is not declared with the necessary
+additional input or output array dimension, a link-time error will result.
+Geometry shader inputs, tessellation control shader inputs and outputs, and
+tessellation evaluation inputs all have an additional level of arrayness
+relative to other shader inputs and outputs.
+These inputs and outputs are known as <em>per-vertex-arrayed</em> inputs and
+outputs.
+Component limits for arrayed interfaces (e.g.
+<em>gl_MaxTessControlInputComponents</em>) are limits per vertex, not limits for
+the entire interface.</p>
+</div>
+<div class="paragraph">
+<p>For non-arrayed interfaces (meaning array dimensionally stays the same
+between stages), it is a link-time error if the input variable is not
+declared with the same type, including array dimensionality, as the matching
+output variable.</p>
+</div>
+<div class="paragraph">
+<p>The link-time type-matching rules apply to all declared input and output
+variables, whether or not they are used.</p>
+</div>
+<div class="paragraph">
+<p>Additionally, tessellation evaluation shaders support per-patch input
+variables declared with the <strong>patch</strong> and <strong>in</strong> qualifiers.
+Per-patch input variables are filled with the values of per-patch output
+variables written by the tessellation control shader.
+Per-patch inputs may be declared as one-dimensional arrays, but are not
+indexed by vertex number.
+Applying the <strong>patch</strong> qualifier to inputs can only be done in tessellation
+evaluation shaders.
+As with other input variables, per-patch inputs must be declared using the
+same type and qualification as per-patch outputs from the previous
+(tessellation control) shader stage.
+It is a compile-time error to use <strong>patch</strong> with inputs in any other stage.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to declare a tessellation control, tessellation
+evaluation or geometry shader input with, or that contains, any of the
+following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Fragment shader inputs get per-fragment values, typically interpolated from
+a previous stage’s outputs.
+The auxiliary storage qualifiers <strong>centroid</strong> and <strong>sample</strong> can also be
+applied, as well as the interpolation qualifiers <strong>flat</strong>, <strong>noperspective</strong>,
+and <strong>smooth.</strong></p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to declare a fragment shader input with, or that
+contains, any of the following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Fragment shader inputs that are, or contain, integral
+or double-precision floating-point
+types must be
+qualified with the interpolation qualifier <strong>flat</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Fragment inputs are declared as in the following examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec3 normal;
+centroid in vec2 TexCoord;
+invariant centroid in vec4 Color;
+noperspective in <span class="predefined-type">float</span> temperature;
+flat in vec3 myColor;
+noperspective centroid in vec2 myTexCoord;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The fragment shader inputs form an interface with the last active shader in
+the vertex processing pipeline.
+For this interface, the last active shader stage output variables and
+fragment shader input variables of the same name must match in type and
+qualification, with a few exceptions: The storage qualifiers must, of
+course, differ (one is <strong>in</strong> and one is <strong>out</strong>).
+Also,
+interpolation qualification (e.g. <strong>flat</strong>) and
+auxiliary qualification (e.g. <strong>centroid</strong>) may differ.
+These mismatches are allowed between any pair of stages.
+When
+interpolation or
+auxiliary qualifiers do not match, those provided in
+the fragment shader supersede those provided in previous stages.
+If any such qualifiers are completely missing in the fragment shaders, then
+the default is used, rather than any qualifiers that may have been declared
+in previous stages.
+That is, what matters is what is declared in the fragment shaders, not what
+is declared in shaders in previous stages.</p>
+</div>
+<div class="paragraph">
+<p>When an interface between shader stages is formed using shaders from two
+separate program objects, it is not possible to detect mismatches between
+inputs and outputs when the programs are linked.
+When there are mismatches between inputs and outputs on such interfaces,
+the values passed across the interface will be partially or completely
+undefined.</p>
+</div>
+<div class="paragraph">
+<p>Shaders can ensure matches across such interfaces either by using input and
+output layout qualifiers (sections “<a href="#input-layout-qualifiers">Input Layout
+Qualifiers</a>” and “<a href="#output-layout-qualifiers">Output Layout
+Qualifiers</a>”) or by using identical input and output declarations of
+blocks or variables.
+Complete rules for interface matching are found in section 7.4.1 “Shader
+Interface Matching” of the <a href="#references">OpenGL Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>Compute shaders do not permit user-defined input variables and do not form a
+formal interface with any other shader stage.
+See “<a href="#compute-shader-special-variables">Compute Shader Special
+Variables</a>” for a description of built-in compute shader input variables.
+All other input to a compute shader is retrieved explicitly through image
+loads, texture fetches, loads from uniforms or uniform buffers, or other
+user supplied code.
+Redeclaration of built-in input variables in compute shaders is not
+permitted.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="uniform-variables">4.3.5. Uniform Variables</h4>
+<div class="paragraph">
+<p>The <strong>uniform</strong> qualifier is used to declare global variables whose values are
+the same across the entire primitive being processed.
+All <strong>uniform</strong> variables are read-only and are initialized externally either
+at link time or through the API.
+The link-time initial value is either the value of the variable’s
+initializer, if present, or 0 if no initializer is present.
+Opaque types cannot have initializers, or a compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>Example declarations are:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">uniform vec4 lightPosition;
+uniform vec3 color = vec3(<span class="float">0</span><span class="float">.7</span>, <span class="float">0</span><span class="float">.7</span>, <span class="float">0</span><span class="float">.2</span>); <span class="comment">// value assigned at link time</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>uniform</strong> qualifier can be used with any of the basic data types, or
+when declaring a variable whose type is a structure, or an array of any of
+these.</p>
+</div>
+<div class="paragraph">
+<p>There is an implementation-dependent limit on the amount of storage for
+uniforms that can be used for each type of shader and if this is exceeded it
+will cause a compile-time or link-time error.
+Uniform variables that are declared but not
+used do not count against this limit.
+The number of user-defined uniform variables and the number of built-in
+uniform variables that are used within a shader are added together to
+determine whether available uniform storage has been exceeded.</p>
+</div>
+<div class="paragraph">
+<p>Uniforms in shaders all share a single global name space when linked into a
+program or separable program.
+Hence, the types,
+initializers,
+and any location specifiers of all statically used uniform variables with the
+same name must match across all shaders that are linked into a single program.
+However it is not required to repeat the
+initializer or
+location specifier in all the linked shaders.
+While this single uniform name space is cross stage, a uniform variable
+name’s scope is per stage: If a uniform variable name is declared in one
+stage (e.g. a vertex shader) but not in another (e.g. a fragment shader),
+then that name is still available in the other stage for a different use.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="output-variables">4.3.6. Output Variables</h4>
+<div class="paragraph">
+<p>Shader output variables are declared with the <strong>out</strong> storage qualifier.
+They form the output interface between the declaring shader and the
+subsequent stages of the OpenGL pipeline.
+Output variables must be declared at global scope.
+During shader execution they will behave as normal unqualified global
+variables.
+Their values are copied out to the subsequent pipeline stage on shader exit.
+Only output variables that are read by the subsequent pipeline stage need to
+be written; it is allowed to have superfluous declarations of output
+variables.</p>
+</div>
+<div class="paragraph">
+<p>There is <em>not</em> an <strong>inout</strong> storage qualifier for declaring a single variable
+name as both input and output to a shader.
+Also, a variable cannot be declared with both the <strong>in</strong> and the <strong>out</strong>
+qualifiers, this will result in a compile-time or link-time error.
+Output variables must be declared with different names than input variables.
+However, nesting an input or output inside an interface block with an
+instance name allows the same names with one referenced through a block
+instance name.</p>
+</div>
+<div class="paragraph">
+<p>Vertex, tessellation evaluation, and geometry output variables output
+per-vertex data and are declared using the <strong>out</strong> storage qualifier.
+Applying <strong>patch</strong> to an output can only be done in a tessellation control
+shader.
+It is a compile-time error to use <strong>patch</strong> on outputs in any other stage.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to declare a vertex, tessellation evaluation,
+tessellation control, or geometry shader output with, or that contains, any
+of the following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Individual outputs are declared as in the following examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out vec3 normal;
+centroid out vec2 TexCoord;
+invariant centroid out vec4 Color;
+flat out vec3 myColor;
+sample out vec4 perSampleColor;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>These can also appear in interface blocks, as described in
+“<a href="#interface-blocks">Interface Blocks</a>”.
+Interface blocks allow simpler addition of arrays to the interface from
+vertex to geometry shader.
+They also allow a fragment shader to have the same input interface as a
+geometry shader for a given vertex shader.</p>
+</div>
+<div class="paragraph">
+<p>Tessellation control shader output variables are used to output
+per-vertex and per-patch data.
+Per-vertex output variables are arrayed (see <em>arrayed</em> under
+“<a href="#input-variables">Input Variables</a>”) and declared using the <strong>out</strong>
+qualifier without the <strong>patch</strong> qualifier.
+Per-patch output variables are declared using the <strong>patch</strong> and <strong>out</strong>
+qualifiers.</p>
+</div>
+<div class="paragraph">
+<p>Since tessellation control shaders produce an arrayed primitive comprising
+multiple vertices, each per-vertex output variable (or output block, see
+interface blocks below) needs to be declared as an array.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out <span class="predefined-type">float</span> foo[]; <span class="comment">// feeds next stage input "in float foo[]"</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Each element of such an array corresponds to one vertex of the primitive
+being produced.
+Each array can optionally have a size declared.
+The array size will be set by (or if provided must be consistent with) the
+output layout declaration(s) establishing the number of vertices in the
+output patch, as described later in
+“<a href="#tessellation-control-outputs">Tessellation Control Outputs</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Each tessellation control shader invocation has a corresponding output patch
+vertex, and may assign values to per-vertex outputs only if they belong to
+that corresponding vertex.
+If a per-vertex output variable is used as an l-value, it is a compile-time
+or link-time error if the expression indicating the vertex index is not the
+identifier <em>gl_InvocationID</em>.</p>
+</div>
+<div class="paragraph">
+<p>The order of execution of a tessellation control shader invocation relative
+to the other invocations for the same input patch is undefined unless the
+built-in function <strong>barrier</strong>() is used.
+This provides some control over relative execution order.
+When a shader invocation calls <strong>barrier</strong>(), its execution pauses until all
+other invocations have reached the same point of execution.
+Output variable assignments performed by any invocation executed prior to
+calling <strong>barrier</strong>() will be visible to any other invocation after the call
+to <strong>barrier</strong>() returns.</p>
+</div>
+<div class="paragraph">
+<p>Because tessellation control shader invocations execute in undefined order
+between barriers, the values of per-vertex or per-patch output variables
+will sometimes be undefined.
+Consider the beginning and end of shader execution and each call to
+<strong>barrier</strong>() as synchronization points.
+The value of an output variable will be undefined in any of the three
+following cases:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>At the beginning of execution.</p>
+</li>
+<li>
+<p>At each synchronization point, unless</p>
+<div class="openblock">
+<div class="content">
+<div class="ulist">
+<ul>
+<li>
+<p>the value was well-defined after the previous synchronization point and
+was not written by any invocation since, or</p>
+</li>
+<li>
+<p>the value was written by exactly one shader invocation since the previous
+synchronization point, or</p>
+</li>
+<li>
+<p>the value was written by multiple shader invocations since the previous
+synchronization point, and the last write performed by all such
+invocations wrote the same value.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</li>
+<li>
+<p>When read by a shader invocation, if</p>
+<div class="openblock">
+<div class="content">
+<div class="ulist">
+<ul>
+<li>
+<p>the value was undefined at the previous synchronization point and has not
+been written by the same shader invocation since, or</p>
+</li>
+<li>
+<p>the output variable is written to by any other shader invocation between
+the previous and next synchronization points, even if that assignment
+occurs in code following the read.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Fragment outputs output per-fragment data and are declared using the <strong>out</strong>
+storage qualifier.
+It is a compile-time error to use auxiliary storage qualifiers or
+interpolation qualifiers in a fragment shader output declaration.
+It is a compile-time error to declare a fragment shader output with, or that
+contains, any of the following types:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>A <a href="#booleans">boolean type</a></p>
+</li>
+<li>
+<p>A double-precision scalar or vector (<strong>double</strong>, <strong>dvec2</strong>, <strong>dvec3</strong>,
+<strong>dvec4</strong>)</p>
+</li>
+<li>
+<p>An <a href="#opaque-types">opaque type</a></p>
+</li>
+<li>
+<p>A matrix type</p>
+</li>
+<li>
+<p>A structure</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Fragment outputs are declared as in the following examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out vec4 FragmentColor;
+out uint Luminosity;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Compute shaders have no built-in output variables, do not support
+user-defined output variables and do not form a formal interface with any
+other shader stage.
+All outputs from a compute shader take the form of the side effects such as
+image stores and operations on atomic counters.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="buffer-variables">4.3.7. Buffer Variables</h4>
+<div class="paragraph">
+<p>The <strong>buffer</strong> qualifier is used to declare global variables whose values are
+stored in the data store of a buffer object bound through the OpenGL API.
+Buffer variables can be read and written, with the underlying storage shared
+among all active shader invocations.
+Buffer variable memory reads and writes within a single shader invocation
+are processed in order.
+However, the order of reads and writes performed in one invocation relative
+to those performed by another invocation is largely undefined.
+Buffer variables may be qualified with memory qualifiers affecting how the
+underlying memory is accessed, as described in “<a href="#memory-qualifiers">Memory
+Qualifiers</a>”.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>buffer</strong> qualifier can be used to declare interface blocks (see
+“<a href="#interface-blocks">Interface Blocks</a>”), which are then referred to as
+shader storage blocks.
+It is a compile-time error to declare buffer variables outside a block.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// use buffer to create a buffer block (shader storage block)</span>
+buffer BufferName { <span class="comment">// externally visible name of buffer</span>
+ <span class="predefined-type">int</span> count; <span class="comment">// typed, shared memory...</span>
+ ... <span class="comment">// ...</span>
+ vec4 v[]; <span class="comment">// last member may be an array that is not sized</span>
+ <span class="comment">// until after link time (dynamically sized)</span>
+} Name; <span class="comment">// name of block within the shader</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>There are implementation-dependent limits on the number of shader storage
+blocks used for each type of shader, the combined number of shader storage
+blocks used for a program, and the amount of storage required by each
+individual shader storage block.
+If any of these limits are exceeded, it will cause a compile-time or
+link-time error.</p>
+</div>
+<div class="paragraph">
+<p>If multiple shaders are linked together, then they will share a single
+global buffer variable name space.
+Hence, the types of all declared buffer variables with the same name must
+match across all shaders that are linked into a single program.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="shared-variables">4.3.8. Shared Variables</h4>
+<div class="paragraph">
+<p>The <strong>shared</strong> qualifier is used to declare global variables that have storage
+shared between all work items in a compute shader workgroup.
+Variables declared as <strong>shared</strong> may only be used in compute shaders (see
+“<a href="#compute-processor">Compute Processor</a>”).
+Any other declaration of a <strong>shared</strong> variable is a compile-time error.
+Shared variables are implicitly coherent (see
+“<a href="#memory-qualifiers">Memory Qualifiers</a>”).</p>
+</div>
+<div class="paragraph">
+<p>Variables declared as <strong>shared</strong> may not have initializers and their contents
+are undefined at the beginning of shader execution.
+Any data written to <strong>shared</strong> variables will be visible to other work items
+(executing the same shader) within the same workgroup.</p>
+</div>
+<div class="paragraph">
+<p>In the absence of synchronization, the order of reads and writes to the same
+<strong>shared</strong> variable by different invocations of a shader is not defined.</p>
+</div>
+<div class="paragraph">
+<p>In order to achieve ordering with respect to reads and writes to <strong>shared</strong>
+variables, control flow barriers must be employed using the <strong>barrier</strong>() function
+(see “<a href="#shader-invocation-control-functions">Shader Invocation Control
+Functions</a>”).</p>
+</div>
+<div class="paragraph">
+<p>There is a limit to the total size of all variables declared as <strong>shared</strong> in a
+single program.
+This limit, expressed in units of basic machine units may be determined by
+using the OpenGL API to query the value of
+MAX_COMPUTE_SHARED_MEMORY_SIZE.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="interface-blocks">4.3.9. Interface Blocks</h4>
+<div class="paragraph">
+<p>Input, output, uniform, and buffer variable declarations can be grouped into
+named interface blocks to provide coarser granularity backing than is
+achievable with individual declarations.
+They can have an optional instance name, used in the shader to reference
+their members.
+An output block of one programmable stage is backed by a corresponding input
+block in the subsequent programmable stage.
+A <em>uniform block</em> is backed by the application with a buffer object.
+A block of buffer variables, called a <em>shader storage block</em>, is also backed
+by the application with a buffer object.
+It is a compile-time error to have an input block in a vertex shader or an
+output block in a fragment shader.
+These uses are reserved for future use.</p>
+</div>
+<div class="paragraph">
+<p>An interface block is started by an <strong>in</strong>, <strong>out</strong>, <strong>uniform</strong>, or <strong>buffer</strong>
+keyword, followed by a block name, followed by an open curly brace (<strong>{</strong>) as
+follows:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>interface-block</em> : </dt>
+<dd>
+<p><em>layout-qualifier<sub>opt</sub></em> <em>interface-qualifier</em> <em>block-name</em> <strong>{</strong>
+<em>member-list</em> <strong>}</strong> <em>instance-name<sub>opt</sub></em> <strong>;</strong></p>
+</dd>
+</dl>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>interface-qualifier</em> : </dt>
+<dd>
+<p><strong>in</strong><br>
+<strong>out</strong><br>
+<strong>patch</strong> <strong>in</strong> // Note: Qualifiers can be in any order.<br>
+<strong>patch</strong> <strong>out</strong><br>
+<strong>uniform</strong><br>
+<strong>buffer</strong></p>
+</dd>
+</dl>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>member-list</em> : </dt>
+<dd>
+<p><em>member-declaration</em><br>
+<em>member-declaration</em> <em>member-list</em></p>
+</dd>
+<dt class="hdlist1"><em>member-declaration</em> : </dt>
+<dd>
+<p><em>layout-qualifier<sub>opt</sub></em> <em>qualifiers<sub>opt</sub></em> <em>type</em> <em>declarators</em> <strong>;</strong></p>
+</dd>
+</dl>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>instance-name</em> : </dt>
+<dd>
+<p><em>identifier</em><br>
+<em>identifier</em> <strong>[</strong> <strong>]</strong><br>
+<em>identifier</em> <strong>[</strong> <em>constant-integral-expression</em> <strong>]</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Each of the above elements is discussed below, with the exception of layout
+qualifiers (<em>layout-qualifier</em>), which are defined in the next section.</p>
+</div>
+<div class="paragraph">
+<p>First, an example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">uniform Transform {
+ mat4 ModelViewMatrix;
+ mat4 ModelViewProjectionMatrix;
+ uniform mat3 NormalMatrix; <span class="comment">// allowed restatement of qualifier</span>
+ <span class="predefined-type">float</span> Deformation;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The above establishes a uniform block named “Transform” with four uniforms
+grouped inside it.</p>
+</div>
+<div class="paragraph">
+<p>Types and declarators are the same as for other input, output, uniform, and
+buffer variable declarations outside blocks, with these exceptions:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Initializers are not allowed</p>
+</li>
+<li>
+<p>Opaque types are not allowed</p>
+</li>
+<li>
+<p>Structure definitions cannot be nested inside a block</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Any of these would result in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>If no optional qualifier is used in a member-declaration, the qualification
+of the member includes all <strong>in</strong>, <strong>out</strong>, <strong>patch</strong>, <strong>uniform</strong>, or <strong>buffer</strong> as
+determined by <em>interface-qualifier</em>.
+If optional qualifiers are used, they can include interpolation qualifiers,
+auxiliary storage qualifiers,
+and storage qualifiers and they must declare
+an input, output, or uniform member consistent with the interface qualifier
+of the block: Input variables, output variables, uniform variables, and
+<strong>buffer</strong> members can only be in <strong>in</strong> blocks, <strong>out</strong> blocks, <strong>uniform</strong> blocks,
+and shader storage blocks, respectively.</p>
+</div>
+<div class="paragraph">
+<p>Repeating the <strong>in</strong>, <strong>out</strong>, <strong>patch</strong>, <strong>uniform</strong>, or <strong>buffer</strong> interface
+qualifier for a member’s storage qualifier is optional.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in Material {
+ smooth in vec4 Color1; <span class="comment">// legal, input inside in block</span>
+ smooth vec4 Color2; <span class="comment">// legal, 'in' inherited from 'in Material'</span>
+ vec2 TexCoord; <span class="comment">// legal, TexCoord is an input</span>
+ uniform <span class="predefined-type">float</span> Atten; <span class="comment">// illegal, mismatched storage qualifier</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A <em>shader interface</em> is defined to be one of these:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>All the uniform variables and uniform blocks declared in a program.
+This spans all compilation units linked together within one program.</p>
+</li>
+<li>
+<p>All the <strong>buffer</strong> blocks declared in a program.</p>
+</li>
+<li>
+<p>The boundary between adjacent programmable pipeline stages: This spans
+all the outputs declared in all compilation units of the first stage and
+all the inputs declared in all compilation units of the second stage.
+Note that for the purposes of this definition, the fragment shader and
+the preceding shader are considered to have a shared boundary even
+though in practice, all values passed to the fragment shader first pass
+through the rasterizer and interpolator.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>The block name (<em>block-name</em>) is used to match within shader interfaces: an
+output block of one pipeline stage will be matched to an input block with
+the same name in the subsequent pipeline stage.
+For uniform or shader storage blocks, the application uses the block name to
+identify the block.
+Block names have no other use within a shader beyond interface matching; it
+is a compile-time error
+to use a block name at global scope for anything other than as a
+block name (e.g. use of a block name for a global variable name or function
+name is currently reserved).
+It is a compile-time error to use the same block name for more than one
+block declaration in the same shader interface (as defined above) within one
+shader, even if the block contents are identical.</p>
+</div>
+<div class="paragraph">
+<p>Matched block names within a shader interface (as defined above) must match
+in terms of having the same number of declarations with the same sequence of
+types and the same sequence of member names, as well as having matching
+member-wise layout qualification
+(see next section).
+Matched uniform or shader storage block names (but not input or output block
+names) must also either all be lacking an instance name or all having an
+instance name, putting their members at the same scoping level.
+When instance names are present on matched block names, it is allowed for
+the instance names to differ; they need not match for the blocks to match.
+Furthermore, if a matching block is declared as an array, then the array
+sizes must also match (or follow array matching rules for the shader
+interface between consecutive shader stages).
+Any mismatch will generate a link-time error.
+A block name is allowed to have different definitions in different shader
+interfaces within the same shader, allowing, for example, an input block and
+output block to have the same name.</p>
+</div>
+<div class="paragraph">
+<p>If an instance name (<em>instance-name</em>) is not used, the names declared inside
+the block are scoped at the global level and accessed as if they were
+declared outside the block.
+If an instance name (<em>instance-name</em>) is used, then it puts all the members
+inside a scope within its own name space, accessed with the field selector
+(<strong>.</strong>) operator (analogously to structures).
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in Light {
+ vec4 LightPos;
+ vec3 LightColor;
+};
+in ColoredTexture {
+ vec4 Color;
+ vec2 TexCoord;
+} Material; <span class="comment">// instance name</span>
+vec3 Color; <span class="comment">// different Color than Material.Color</span>
+vec4 LightPos; <span class="comment">// illegal, already defined</span>
+...
+... = LightPos; <span class="comment">// accessing LightPos</span>
+... = Material.Color; <span class="comment">// accessing Color in ColoredTexture block</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Outside the shading language (i.e., in the API), members are similarly
+identified except the block name is always used in place of the instance
+name (API accesses are to shader interfaces, not to shaders).
+If there is no instance name, then the API does not use the block name to
+access a member, just the member name.</p>
+</div>
+<div class="paragraph">
+<p>Within a shader interface, all declarations of the same global name must be
+for the same object and must match in type and in whether they declare a
+variable or member of a block with no instance name.
+The API also needs this name to uniquely identify an object in the shader
+interface.
+It is a link-time error if any particular shader interface contains</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>two different blocks, each having no instance name, and each having a
+member of the same name, or</p>
+</li>
+<li>
+<p>a variable outside a block, and a block with no instance name, where the
+variable has the same name as a member in the block.</p>
+</li>
+</ul>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out Vertex {
+ vec4 Position; <span class="comment">// API transform/feedback will use "Vertex.Position"</span>
+ vec2 Texture;
+} Coords; <span class="comment">// shader will use "Coords.Position"</span>
+out Vertex2 {
+ vec4 Color; <span class="comment">// API will use "Color"</span>
+ <span class="predefined-type">float</span> Color2;
+};
+
+<span class="comment">// in same program as Vertex2 above:</span>
+out Vertex3 {
+ <span class="predefined-type">float</span> Intensity;
+ vec4 Color; <span class="comment">// ERROR, name collision with Color in Vertex2</span>
+};
+<span class="predefined-type">float</span> Color2; <span class="comment">// ERROR, collides with Color2 in Vertex2</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For blocks declared as arrays, the array index must also be included when
+accessing members, as in this example</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">uniform Transform { <span class="comment">// API uses "Transform[2]" to refer to instance 2</span>
+ mat4 ModelViewMatrix;
+ mat4 ModelViewProjectionMatrix;
+ vec4 a[]; <span class="comment">// array will get implicitly sized</span>
+ <span class="predefined-type">float</span> Deformation;
+} transforms[<span class="integer">4</span>];
+...
+... = transforms[<span class="integer">2</span>].ModelViewMatrix; <span class="comment">// shader access of instance 2</span>
+<span class="comment">// API uses "Transform.ModelViewMatrix" to query an offset or other query</span>
+transforms[x].a.length(); <span class="comment">// same length for 'a' for all x</span>
+Transform[x]; <span class="comment">// illegal, must use 'transforms'</span>
+Transform.a.length(); <span class="comment">// illegal, must use 'transforms'</span>
+...transforms[<span class="integer">2</span>].a[<span class="integer">3</span>]... <span class="comment">// if these are the only two dereferences of 'a',</span>
+...transforms[<span class="integer">3</span>].a[<span class="integer">7</span>]... <span class="comment">// then 'a' must be size 8, for all</span>
+transforms[x]</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For uniform or shader storage blocks declared as an array, each individual
+array element corresponds to a separate buffer object bind range, backing
+one instance of the block.
+As the array size indicates the number of buffer objects needed, uniform and
+shader storage block array declarations must specify an array size.
+A uniform or shader storage block array can only be indexed with a
+dynamically uniform integral expression, otherwise results are undefined.</p>
+</div>
+<div class="paragraph">
+<p>When using OpenGL API entry points to identify the name of an individual
+block in an array of blocks, the name string may include an array index
+(e.g. <em>Transform[2]</em>).
+When using OpenGL API entry points to refer to offsets or other
+characteristics of a block member, an array index must not be specified
+(e.g. <em>Transform.ModelViewMatrix</em>).</p>
+</div>
+<div class="paragraph">
+<p>Tessellation control, tessellation evaluation and geometry shader input
+blocks must be declared as arrays and follow the array declaration and
+linking rules for all shader inputs for the respective stages.
+All other input and output block arrays must specify an array size.</p>
+</div>
+<div class="paragraph">
+<p>There are implementation-dependent limits on the number of uniform blocks
+and the number of shader storage blocks that can be used per stage.
+If either limit is exceeded, it will cause a link-time error.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="layout-qualifiers">4.4. Layout Qualifiers</h3>
+<div class="paragraph">
+<p>Layout qualifiers can appear in several forms of declaration.
+They can appear as part of an interface block definition or block member, as
+shown in the grammar in the previous section.
+They can also appear with just an <em>interface-qualifier</em> to establish layouts
+of other declarations made with that qualifier:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="paragraph">
+<p><em>layout-qualifier</em> <em>interface-qualifier</em> <strong>;</strong></p>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Or, they can appear with an individual variable declared with an interface
+qualifier:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="paragraph">
+<p><em>layout-qualifier</em> <em>interface-qualifier</em> <em>declaration</em> <strong>;</strong></p>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Declarations of layouts can only be made at global scope or block members,
+and only where indicated in the following subsections; their details are
+specific to what the interface qualifier is, and are discussed individually.</p>
+</div>
+<div class="paragraph">
+<p>The <em>layout-qualifier</em> expands to:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier</em> : </dt>
+<dd>
+<p><strong>layout</strong> <strong>(</strong> <em>layout-qualifier-id-list</em> <strong>)</strong></p>
+</dd>
+<dt class="hdlist1"><em>layout-qualifier-id-list</em> : </dt>
+<dd>
+<p><em>layout-qualifier-id</em><br>
+<em>layout-qualifier-id</em> <strong>,</strong> <em>layout-qualifier-id-list</em></p>
+</dd>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><em>layout-qualifier-name</em><br>
+<em>layout-qualifier-name</em> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>shared</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-value</em> : </dt>
+<dd>
+<p><em>integer-constant-expression</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The tokens used for <em>layout-qualifier-name</em> are identifiers, not keywords,
+however, the <strong>shared</strong> keyword is allowed as a <em>layout-qualifier-id</em>.
+Generally, they can be listed in any order.
+Order-dependent meanings exist only if explicitly called out below.
+Similarly, these identifiers are not case sensitive, unless explicitly noted
+otherwise.</p>
+</div>
+<div class="paragraph">
+<p>More than one layout qualifier may appear in a single declaration.
+Additionally, the same <em>layout-qualifier-name</em> can occur multiple times
+within a layout qualifier or across multiple layout qualifiers in the same
+declaration.
+When the same <em>layout-qualifier-name</em> occurs multiple times, in a single
+declaration, the last occurrence overrides the former occurrence(s).
+Further, if such a <em>layout-qualifier-name</em> will affect subsequent
+declarations or other observable behavior, it is only the last occurrence
+that will have any effect, behaving as if the earlier occurrence(s) within
+the declaration are not present.
+This is also true for overriding <em>layout-qualifier-name</em>, where one
+overrides the other (e.g. <strong>row_major</strong> vs.
+<strong>column_major</strong>); only the last occurrence has any effect.</p>
+</div>
+<div class="paragraph">
+<p><em>integer-constant-expression</em> is defined in
+“<a href="#constant-expressions">Constant Expressions</a>” as “_constant integral
+expression_”, with it being a compile-time error for
+<em>integer-constant-expression</em> to be a specialization constant.</p>
+</div>
+<div class="paragraph">
+<p>The following table summarizes the use of layout qualifiers.
+It shows for each one what kinds of declarations it may be applied to.
+These are all discussed in detail in the following sections.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.6666%;">
+<col style="width: 16.667%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Layout Qualifier</th>
+<th class="tableblock halign-left valign-top">Qualifier Only</th>
+<th class="tableblock halign-left valign-top">Individual Variable</th>
+<th class="tableblock halign-left valign-top">Block</th>
+<th class="tableblock halign-left valign-top">Block Member</th>
+<th class="tableblock halign-left valign-top">Allowed Interfaces</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>shared</strong><br>
+ <strong>packed</strong><br>
+ <strong>std140</strong><br>
+ <strong>std430</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top" rowspan="5"><p class="tableblock"><strong>uniform</strong> / <strong>buffer</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>row_major</strong><br>
+ <strong>column_major</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>binding</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">opaque types only</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>offset</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">atomic counters only</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>align</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>location</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uniform</strong> / <strong>buffer</strong> and subroutine variables</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>location</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top" rowspan="2"><p class="tableblock">all <strong>in</strong> / <strong>out</strong>, except for compute</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>component</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>index</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">fragment <strong>out</strong> and subroutine functions</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>triangles</strong><br>
+ <strong>quads</strong><br>
+ <strong>isolines</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation evaluation <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>equal_spacing</strong><br>
+ <strong>fractional_even_spacing</strong><br>
+ <strong>fractional_odd_spacing</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation evaluation <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>cw</strong><br>
+ <strong>ccw</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation evaluation <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>point_mode</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation evaluation <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>points</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">geometry <strong>in</strong>/<strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">[ <strong>points</strong> ]<br>
+ <strong>lines</strong><br>
+ <strong>lines_adjacency</strong><br>
+ <strong>triangles</strong><br>
+ <strong>triangles_adjacency</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">geometry <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>invocations</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">geometry <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>origin_upper_left</strong><br>
+ <strong>pixel_center_integer</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>gl_FragCoord</em> only</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top" rowspan="2"><p class="tableblock">fragment <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>early_fragment_tests</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>local_size_x</strong> =<br>
+ <strong>local_size_y</strong> =<br>
+ <strong>local_size_z</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">compute <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>local_size_x_id</strong> =<br>
+ <strong>local_size_y_id</strong> =<br>
+ <strong>local_size_z_id</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">compute <strong>in</strong> (SPIR-V generation only)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>xfb_buffer</strong> =<br>
+ <strong>xfb_stride</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top" rowspan="2"><p class="tableblock">vertex, tessellation, and geometry <strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>xfb_offset</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>vertices</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">tessellation control <strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">[ <strong>points</strong> ]<br>
+ <strong>line_strip</strong><br>
+ <strong>triangle_strip</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top" rowspan="3"><p class="tableblock">geometry <strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>max_vertices</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>stream</strong> =</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>depth_any</strong><br>
+ <strong>depth_greater</strong><br>
+ <strong>depth_less</strong><br>
+ <strong>depth_unchanged</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>gl_FragDepth</em> only</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">fragment <strong>out</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>constant_id</strong> =</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">scalar only</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>const</strong> (SPIR-V generation only)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>rgba32f</strong><br>
+ <strong>rgba16f</strong><br>
+ <strong>rg32f</strong><br>
+ <strong>rg16f</strong><br>
+ <strong>r11f_g11f_b10f</strong><br>
+ <strong>r32f</strong><br>
+ <strong>r16f</strong><br>
+ <strong>rgba16</strong><br>
+ <strong>rgb10_a2</strong><br>
+ <strong>rgba8</strong><br>
+ <strong>rg16</strong><br>
+ <strong>rg8</strong><br>
+ <strong>r16</strong><br>
+ <strong>r8</strong><br>
+ <strong>rgba16_snorm</strong><br>
+ <strong>rgba8_snorm</strong><br>
+ <strong>rg16_snorm</strong><br>
+ <strong>rg8_snorm</strong><br>
+ <strong>r16_snorm</strong><br>
+ <strong>r8_snorm</strong><br>
+ <strong>rgba32i</strong><br>
+ <strong>rgba16i</strong><br>
+ <strong>rgba8i</strong><br>
+ <strong>rg32i</strong><br>
+ <strong>rg16i</strong><br>
+ <strong>rg8i</strong><br>
+ <strong>r32i</strong><br>
+ <strong>r16i</strong><br>
+ <strong>r8i</strong><br>
+ <strong>rgba32ui</strong><br>
+ <strong>rgba16ui</strong><br>
+<strong>rgb10_a2ui</strong><br>
+ <strong>rgba8ui</strong><br>
+ <strong>rg32ui</strong><br>
+ <strong>rg16ui</strong><br>
+ <strong>rg8ui</strong><br>
+ <strong>r32ui</strong><br>
+ <strong>r16ui</strong><br>
+ <strong>r8ui</strong></p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">image types only</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>uniform</strong></p></td>
+</tr>
+</tbody>
+</table>
+<div class="sect3">
+<h4 id="input-layout-qualifiers">4.4.1. Input Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Layout qualifiers specific to a particular shader language are discussed in
+separate sections below.</p>
+</div>
+<div class="paragraph">
+<p>All shaders except compute shaders allow <strong>location</strong> layout qualifiers on
+input variable declarations, input block declarations, and input block
+member declarations.
+Of these, variables and block members (but not blocks) additionally allow
+the <strong>component</strong> layout qualifier.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>location</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>component</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">3</span>) in vec4 normal;
+<span class="directive">const</span> <span class="predefined-type">int</span> start = <span class="integer">6</span>;
+layout(location = start + <span class="integer">2</span>) <span class="predefined-type">int</span> vec4 v;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the shader input <em>normal</em> is assigned to vector location
+number
+3 and <em>v</em> is assigned location number 8.
+For vertex shader inputs, the location specifies the number of the
+vertex attribute from which input values are taken.
+For inputs of all other shader types, the location specifies a vector number
+that can be used to match against outputs from a previous shader stage, even
+if that shader is in a different program object.</p>
+</div>
+<div class="paragraph">
+<p>The following language describes how many locations are consumed by a given
+type.
+However, geometry shader inputs, tessellation control shader inputs and
+outputs, and tessellation evaluation inputs all have an additional level of
+arrayness relative to other shader inputs and outputs.
+This outer array level is removed from the type before considering how many
+locations the type consumes.</p>
+</div>
+<div class="paragraph">
+<p>If a vertex shader input is any scalar or vector type, it will consume a
+single location.
+If a non-vertex shader input is a scalar or vector type other than <strong>dvec3</strong>
+or <strong>dvec4</strong>, it will consume a single location, while types <strong>dvec3</strong> or
+<strong>dvec4</strong> will consume two consecutive locations.
+Inputs of type <strong>double</strong> and <strong>dvec2</strong> will consume only a single location, in
+all stages.</p>
+</div>
+<div class="paragraph">
+<p>If the declared input (after potentially removing an outer array level as
+just described above) is an array of size <em>n</em> and each of the elements takes
+<em>m</em> locations, it will be assigned <em>m</em> * <em>n</em> consecutive locations starting
+with the location specified.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">6</span>) in vec4 colors[<span class="integer">3</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the shader input <em>colors</em> is assigned to vector location
+numbers 6, 7, and 8.</p>
+</div>
+<div class="paragraph">
+<p>If the declared input is an <em>n</em> × <em>m</em>
+matrix, it will be assigned multiple locations starting with the location
+specified.
+The number of locations assigned for each matrix will be the same as for an
+<em>n</em>-element array of <em>m</em>-component vectors.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">9</span>) in mat4 transforms[<span class="integer">2</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that shader input <em>transforms</em> is assigned to vector
+locations 9-16, with <em>transforms[0]</em> being assigned to locations 9-12, and
+<em>transforms[1]</em> being assigned to locations 13-16.</p>
+</div>
+<div class="paragraph">
+<p>If the declared input is a structure or block, its members will be assigned
+consecutive locations in their order of declaration, with the first member
+assigned the location provided in the layout qualifier.
+For a structure, this process applies to the entire structure.
+It is a compile-time error to use a <strong>location</strong> qualifier on a member of a
+structure.
+For a block, this process applies to the entire block, or until the first
+member is reached that has a <strong>location</strong> layout qualifier.</p>
+</div>
+<div class="paragraph">
+<p>When a block member is declared with a <strong>location</strong> qualifier, its location
+comes from that qualifier; the member’s <strong>location</strong> qualifier overrides the
+block-level declaration.
+Subsequent members are again assigned consecutive locations, based on the
+newest location, until the next member declared with a <strong>location</strong> qualifier.
+The values used for locations do not have to be declared in increasing
+order.</p>
+</div>
+<div class="paragraph">
+<p>If a block has no block-level <strong>location</strong> layout qualifier, it is required
+that either all or none of its members have a <strong>location</strong> layout qualifier,
+or a compile-time error results.
+For some blocks declared as arrays, the <strong>location</strong> can only be applied at
+the block level: When a block is declared as an array where additional
+locations are needed for each member for each block array element, it is a
+compile-time error to specify locations on the block members.
+That is, when locations would be under specified by applying them on block
+members, they are not allowed on block members.
+For <em>arrayed</em> interfaces (those generally having an extra level of arrayness
+due to interface expansion), the outer array is stripped before applying
+this rule.</p>
+</div>
+<div class="paragraph">
+<p>When generating SPIR-V, all <strong>in</strong> and <strong>out</strong> qualified user-declared (non
+built-in) variables and blocks (or all their members) must have a
+shader-specified <strong>location</strong>.
+Otherwise, a compile-time error is generated.</p>
+</div>
+<div class="paragraph">
+<p>The locations consumed by block and structure members are determined by
+applying the rules above recursively as though the structure member were
+declared as an input variable of the same type.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">3</span>) in <span class="keyword">struct</span> S
+{
+ vec3 a; <span class="comment">// gets location 3</span>
+ mat2 b; <span class="comment">// gets locations 4 and 5</span>
+ vec4 c[<span class="integer">2</span>]; <span class="comment">// gets locations 6 and 7</span>
+ layout(location = <span class="integer">8</span>) vec2 A; <span class="comment">// ERROR, can't use on struct member</span>
+} s;
+layout(location = <span class="integer">4</span>) in block
+{
+ vec4 d; <span class="comment">// gets location 4</span>
+ vec4 e; <span class="comment">// gets location 5</span>
+ layout(location = <span class="integer">7</span>) vec4 f; <span class="comment">// gets location 7</span>
+ vec4 g; <span class="comment">// gets location 8</span>
+ layout(location = <span class="integer">1</span>) vec4 h; <span class="comment">// gets location 1</span>
+ vec4 i; <span class="comment">// gets location 2</span>
+ vec4 j; <span class="comment">// gets location 3</span>
+ vec4 k; <span class="comment">// ERROR, location 4 already used</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The number of input locations available to a shader is limited.
+For vertex shaders, the limit is the advertised number of vertex attributes.
+For all other shaders, the limit is implementation-dependent and must be no
+less than one fourth of the advertised maximum input component count.</p>
+</div>
+<div class="paragraph">
+<p>A program will fail to link if any attached shader uses a location greater
+than or equal to the number of supported locations, unless device-dependent
+optimizations are able to make the program fit within available hardware
+resources.</p>
+</div>
+<div class="paragraph">
+<p>A program will fail to link if explicit location assignments leave the
+linker unable to find space for other variables without explicit
+assignments.</p>
+</div>
+<div class="paragraph">
+<p>For the purposes of determining if a non-vertex input matches an output from
+a previous shader stage, the <strong>location</strong> layout qualifier (if any) must
+match.</p>
+</div>
+<div class="paragraph">
+<p>If a vertex shader input variable with no location assigned in the shader
+text has a location specified through the OpenGL API, the API-assigned
+location will be used.
+Otherwise, such variables will be assigned a location by the linker.
+See section 11.1.1 “Vertex Attributes” of the <a href="#references">OpenGL Specification</a> for
+more details.
+A link-time error will occur if an input variable is declared in multiple
+shaders of the same language with conflicting locations.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>component</strong> qualifier allows the location to be more finely specified
+for scalars and vectors, down to the individual components within a location
+that are consumed.
+It is a compile-time error to use <strong>component</strong> without also specifying the
+<strong>location</strong> qualifier (order does not matter).
+The components within a location are 0, 1, 2, and 3.
+A variable or block member starting at component <em>N</em> will consume components
+<em>N</em>, <em>N+1</em>, <em>N+2</em>, …​
+up through its size.
+It is a compile-time error if this sequence of components gets larger than
+3.
+A scalar <strong>double</strong> will consume two of these components, and a <strong>dvec2</strong> will
+consume all four components available within a location.
+A <strong>dvec3</strong> or <strong>dvec4</strong> can only be declared without specifying a <strong>component</strong>.
+A <strong>dvec3</strong> will consume all four components of the first location and
+components 0 and 1 of the second location.
+This leaves components 2 and 3 available for other component-qualified
+declarations.</p>
+</div>
+<div class="paragraph">
+<p>For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// a consumes components 2 and 3 of location 4</span>
+layout(location = <span class="integer">4</span>, component = <span class="integer">2</span>) in vec2 a;
+
+<span class="comment">// b consumes component 1 of location 4</span>
+layout(location = <span class="integer">4</span>, component = <span class="integer">1</span>) in <span class="predefined-type">float</span> b;
+
+<span class="comment">// ERROR: c overflows component 3</span>
+layout(location = <span class="integer">3</span>, component = <span class="integer">2</span>) in vec3 c;
+
+<span class="comment">// d consumes components 2 and 3 of location 5</span>
+layout(location = <span class="integer">5</span>, component = <span class="integer">2</span>) in <span class="predefined-type">double</span> d;
+
+<span class="comment">// ERROR: e overflows component 3 of location 6</span>
+layout(location = <span class="integer">6</span>, component = <span class="integer">2</span>) in dvec2 e;
+
+<span class="comment">// ERROR: f overlaps with g</span>
+layout(location = <span class="integer">7</span>, component = <span class="integer">0</span>) <span class="predefined-type">double</span> f;
+layout(location = <span class="integer">7</span>, component = <span class="integer">1</span>) <span class="predefined-type">float</span> g;
+
+layout(location = <span class="integer">8</span>) in dvec3 h; <span class="comment">// components 0,1,2 and 3 of location 8</span>
+ <span class="comment">// and components 0 and 1 of location 9</span>
+layout(location = <span class="integer">9</span>, component = <span class="integer">2</span>) in <span class="predefined-type">float</span> i; <span class="comment">// okay, compts 2 and 3</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>If the variable is an array, each element of the array, in order, is
+assigned to consecutive locations, but all at the same specified component
+within each location.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// component 3 in 6 locations are consumed</span>
+layout(location = <span class="integer">2</span>, component = <span class="integer">3</span>) in <span class="predefined-type">float</span> d[<span class="integer">6</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>That is, location 2 component 3 will hold <em>d[0]</em>, location 3 component 3
+will hold <em>d[1]</em>, …​, up through location 7 component 3 holding <em>d[5]</em>.</p>
+</div>
+<div class="paragraph">
+<p>This allows packing of two arrays into the same set of locations:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// e consumes beginning (components 0, 1 and 2) of each of 6 slots</span>
+layout(location = <span class="integer">0</span>, component = <span class="integer">0</span>) in vec3 e[<span class="integer">6</span>];
+
+<span class="comment">// f consumes last component of the same 6 slots</span>
+layout(location = <span class="integer">0</span>, component = <span class="integer">3</span>) in <span class="predefined-type">float</span> f[<span class="integer">6</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>If applying this to an array of arrays, all levels of arrayness are removed
+to get to the elements that are assigned per location to the specified
+component.
+These non-arrayed elements will fill the locations in the order specified
+for arrays of arrays in “<a href="#arrays">Arrays</a>”.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to apply the <strong>component</strong> qualifier to a matrix, a
+structure, a block, or an array containing any of these.
+It is a compile-time error to use <strong>component</strong> 1 or 3 as the beginning of a
+<strong>double</strong> or <strong>dvec2</strong>.
+It is a link-time error to specify different components for the same
+variable within a program.</p>
+</div>
+<div class="paragraph">
+<p><em>Location aliasing</em> is causing two variables or block members to have the
+same location number.
+<em>Component aliasing</em> is assigning the same (or overlapping) component
+numbers for two location aliases.
+(Recall if <strong>component</strong> is not used, components are assigned starting with
+0.) With one exception, location aliasing is allowed only if it does not
+cause component aliasing; it is a compile-time or link-time error to cause
+component aliasing.
+Further, when location aliasing, the aliases sharing the location must have
+the same underlying numerical type and bit width (floating-point or integer,
+32-bit versus 64-bit, etc.) and the same
+auxiliary storage and interpolation qualification.
+The one exception where component aliasing is permitted is for two input
+variables (not block members) to a vertex shader, which are allowed to have
+component aliasing.
+This vertex-variable component aliasing is intended only to support vertex
+shaders where each execution path accesses at most one input per each
+aliased component.
+Implementations are permitted, but not required, to generate link-time
+errors if they detect that every path through the vertex shader executable
+accesses multiple inputs aliased to any single component.</p>
+</div>
+<div class="sect4">
+<h5 id="tessellation-evaluation-inputs">Tessellation Evaluation Inputs</h5>
+<div class="paragraph">
+<p>Additional input layout qualifier identifiers allowed for tessellation
+evaluation shaders are described below.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><em>primitive_mode</em><br>
+<em>vertex_spacing</em><br>
+<em>ordering</em><br>
+<em>point_mode</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>primitive-mode</strong> is used to specify a tessellation primitive mode to be
+used by the tessellation primitive generator.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>primitive-mode</em>: </dt>
+<dd>
+<p><strong>triangles</strong><br>
+<strong>quads</strong><br>
+<strong>isolines</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>If present, the <em>primitive-mode</em> specifies that the tessellation primitive
+generator should subdivide a triangle into smaller triangles, a quad into
+triangles, or a quad into a collection of lines, respectively.</p>
+</div>
+<div class="paragraph">
+<p>A second group of layout identifiers, <em>vertex spacing</em>, is used to specify
+the spacing used by the tessellation primitive generator when subdividing an
+edge.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>vertex-spacing</em>: </dt>
+<dd>
+<p><strong>equal_spacing</strong><br>
+<strong>fractional_even_spacing</strong><br>
+<strong>fractional_odd_spacing</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p><strong>equal_spacing</strong> specifies that edges should be divided into a collection of
+equal-sized segments;</p>
+</div>
+<div class="paragraph">
+<p><strong>fractional_even_spacing</strong> specifies that edges should be divided into an
+even number of equal-length segments plus two additional shorter
+“fractional” segments; or</p>
+</div>
+<div class="paragraph">
+<p><strong>fractional_odd_spacing</strong> specifies that edges should be divided into an odd
+number of equal-length segments plus two additional shorter “fractional”
+segments.</p>
+</div>
+<div class="paragraph">
+<p>A third group of layout identifiers, <em>ordering</em>, specifies whether the
+tessellation primitive generator produces triangles in clockwise or
+counter-clockwise order, according to the coordinate system depicted in the
+<a href="#references">OpenGL Specification</a>.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>ordering</em>: </dt>
+<dd>
+<p><strong>cw</strong><br>
+<strong>ccw</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The identifiers <strong>cw</strong> and <strong>ccw</strong> indicate clockwise and counter-clockwise
+triangles, respectively.
+If the tessellation primitive generator does not produce triangles, the
+order is ignored.</p>
+</div>
+<div class="paragraph">
+<p>Finally, <em>point mode</em> indicates that the tessellation primitive generator
+should produce one point for each distinct vertex in the subdivided
+primitive, rather than generating lines or triangles.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>point-mode</em>: </dt>
+<dd>
+<p><strong>point_mode</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Any or all of these identifiers may be specified one or more times in a
+single input layout declaration.
+If primitive mode, vertex spacing, or ordering is declared more than once in
+the tessellation evaluation shaders of a program, all such declarations must
+use the same identifier.</p>
+</div>
+<div class="paragraph">
+<p>At least one tessellation evaluation shader (compilation unit) in a program
+must declare a primitive mode in its input layout.
+Declaring vertex spacing, ordering, or point mode identifiers is optional.
+It is not required that all tessellation evaluation shaders in a program
+declare a primitive mode.
+If spacing or vertex ordering declarations are omitted, the tessellation
+primitive generator will use equal spacing or counter-clockwise vertex
+ordering, respectively.
+If a point mode declaration is omitted, the tessellation primitive generator
+will produce lines or triangles according to the primitive mode.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="geometry-shader-inputs">Geometry Shader Inputs</h5>
+<div class="paragraph">
+<p>Additional layout qualifier identifiers for geometry shader inputs include
+<em>primitive</em> identifiers and an <em>invocation count</em> identifier:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>points</strong><br>
+<strong>lines</strong><br>
+<strong>lines_adjacency</strong><br>
+<strong>triangles</strong><br>
+<strong>triangles_adjacency</strong><br>
+<strong>invocations</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The identifiers <strong>points</strong>, <strong>lines</strong>, <strong>lines_adjacency</strong>, <strong>triangles</strong>, and
+<strong>triangles_adjacency</strong> are used to specify the type of input primitive
+accepted by the geometry shader, and only one of these is accepted.
+At least one geometry shader (compilation unit) in a program must declare
+this input primitive layout, and all geometry shader input layout
+declarations in a program must declare the same layout.
+It is not required that all geometry shaders in a program declare an input
+primitive layout.</p>
+</div>
+<div class="paragraph">
+<p>The identifier <strong>invocations</strong> is used to specify the number of times the
+geometry shader executable is invoked for each input primitive received.
+Invocation count declarations are optional.
+If no invocation count is declared in any geometry shader in a program, the
+geometry shader will be run once for each input primitive.
+If an invocation count is declared, all such declarations must specify the
+same count.
+If a shader specifies an invocation count greater than the
+implementation-dependent maximum, or less than or equal to zero,
+a compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(triangles, invocations = <span class="integer">6</span>) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that all inputs to the geometry shader are triangles and that
+the geometry shader executable is run six times for each triangle processed.</p>
+</div>
+<div class="paragraph">
+<p>All geometry shader input unsized array declarations will be sized by an
+earlier input primitive layout qualifier, when present, as per the following
+table.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Layout</th>
+<th class="tableblock halign-left valign-top">Size of Input Arrays</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>points</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lines</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lines_adjacency</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>triangles</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>triangles_adjacency</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">6</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The intrinsically declared input array <em>gl_in[]</em> will also be sized by any
+input primitive-layout declaration.
+Hence, the expression</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_in.length()</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will return the value from the table above.</p>
+</div>
+<div class="paragraph">
+<p>For inputs declared without an array size, including intrinsically declared
+inputs (i.e., <em>gl_in</em>), a layout must be declared before any use of the
+method <strong>length</strong>() or other any array use that requires the array size to be
+known.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error if a layout declaration’s array size (from the
+table above) does not match all the explicit array sizes specified in
+declarations of an input variables in the same shader.
+The following includes examples of compile-time errors:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// code sequence within one shader...</span>
+in vec4 Color1[]; <span class="comment">// legal, size still unknown</span>
+in vec4 Color2[<span class="integer">2</span>]; <span class="comment">// legal, size is 2</span>
+in vec4 Color3[<span class="integer">3</span>]; <span class="comment">// illegal, input sizes are inconsistent</span>
+layout(lines) in; <span class="comment">// legal for Color2, input size is 2, matching Color2</span>
+in vec4 Color4[<span class="integer">3</span>]; <span class="comment">// illegal, contradicts layout of lines</span>
+layout(lines) in; <span class="comment">// legal, matches other layout() declaration</span>
+layout(triangles) in; <span class="comment">// illegal, does not match earlier layout() declaration</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It is a link-time error if not all provided sizes (sized input arrays and
+layout size) match across all geometry shaders in a program.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="fragment-shader-inputs">Fragment Shader Inputs</h5>
+<div class="paragraph">
+<p>Additional fragment layout qualifier identifiers include the following for
+<em>gl_FragCoord</em> :</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>origin_upper_left</strong><br>
+<strong>pixel_center_integer</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>By default, <em>gl_FragCoord</em> assumes a lower-left origin for window
+coordinates and assumes pixel centers are located at half-pixel coordinates.
+For example, the (<em>x, y</em>) location (0.5, 0.5) is returned for the
+lower-left-most pixel in a window.
+The origin can be changed by redeclaring <em>gl_FragCoord</em> with the
+<strong>origin_upper_left</strong> qualifier, moving the origin of <em>gl_FragCoord</em> to the
+upper left of the window, with <em>y</em> increasing in value toward the bottom of
+the window.
+The values returned can also be shifted by half a pixel in both <em>x</em> and <em>y</em>
+by <strong>pixel_center_integer</strong> so it appears the pixels are centered at whole
+number pixel offsets.
+This moves the (<em>x</em>, <em>y</em>) value returned by <em>gl_FragCoord</em> of (0.5, 0.5) by
+default, to (0.0, 0.0) with <strong>pixel_center_integer</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Redeclarations are done as follows</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 gl_FragCoord; <span class="comment">// redeclaration that changes nothing is allowed</span>
+
+<span class="comment">// All the following are allowed redeclaration that change behavior</span>
+layout(origin_upper_left) in vec4 gl_FragCoord;
+layout(pixel_center_integer) in vec4 gl_FragCoord;
+layout(origin_upper_left, pixel_center_integer) in vec4 gl_FragCoord;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>If <em>gl_FragCoord</em> is redeclared in any fragment shader in a program, it must
+be redeclared in all the fragment shaders in that program that have a static
+use <em>gl_FragCoord</em>.
+All redeclarations of <em>gl_FragCoord in all fragment shaders in a single
+program must have the same set of qualifiers.
+Within any shader, the first redeclarations of _glFragCoord</em> must appear
+before any use of <em>gl_FragCoord</em>.
+The built-in <em>gl_FragCoord</em> is only predeclared in fragment shaders, so
+redeclaring it in any other shader language results in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>Redeclaring <em>glFragCoord</em> with <strong>origin_upper_left</strong> and/or
+<strong>pixel_center_integer</strong> qualifiers only affects <em>gl_FragCoord.x</em> and
+<em>gl_FragCoord.y</em>.
+It has no effect on rasterization, transformation, or any other part of the
+OpenGL pipeline or language features.</p>
+</div>
+<div class="paragraph">
+<p>Fragment shaders allow the following layout qualifier on <strong>in</strong> only (not with
+variable declarations):</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>early_fragment_tests</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>to request that fragment tests be performed before fragment shader
+execution, as described in section 15.2.4 “Early Fragment Tests” of the
+<a href="#references">OpenGL Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(early_fragment_tests) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Specifying this will make per-fragment tests be performed before fragment
+shader execution.
+If this is not declared, per-fragment tests will be performed after fragment
+shader execution.
+Only one fragment shader (compilation unit) need declare this, though more
+than one can.
+If at least one declares this, then it is enabled.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="compute-shader-inputs">Compute Shader Inputs</h5>
+<div class="paragraph">
+<p>There are no layout location qualifiers for compute shader inputs.</p>
+</div>
+<div class="paragraph">
+<p>Layout qualifier identifiers for compute shader inputs are the workgroup
+size qualifiers:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>local_size_x</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>local_size_y</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>local_size_z</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>local_size_x</strong>, <strong>local_size_y</strong>, and <strong>local_size_z</strong> qualifiers are used
+to declare a fixed workgroup size by the compute shader in the first,
+second, and third dimension, respectively.
+If a shader does not specify a size for one of the dimensions, that
+dimension will have a size of 1.</p>
+</div>
+<div class="paragraph">
+<p>For example, the following declaration in a compute shader</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(local_size_x = <span class="integer">32</span>, local_size_y = <span class="integer">32</span>) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is used to declare a two-dimensional compute shader with a workgroup size of 32
+X 32 elements, which is equivalent to a three-dimensional compute shader
+where the third dimension has size one.</p>
+</div>
+<div class="paragraph">
+<p>As another example, the declaration</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(local_size_x = <span class="integer">8</span>) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>effectively specifies that a one-dimensional compute shader is being
+compiled, and its size is 8 elements.</p>
+</div>
+<div class="paragraph">
+<p>If the fixed workgroup size of the shader in any dimension is less than
+or equal to zero or greater than the maximum size supported by the
+implementation for that dimension, a compile-time error results.
+Also, if such a layout qualifier is declared more than once in the same
+shader, all those declarations must set the same set of workgroup
+sizes and set them to the same values; otherwise a compile-time error
+results.
+If multiple compute shaders attached to a single program object declare a
+fixed workgroup size, the declarations must be identical; otherwise a
+link-time error results.</p>
+</div>
+<div class="paragraph">
+<p>Furthermore, if a program object contains any compute shaders, at least one
+must contain an input layout qualifier specifying a fixed workgroup size
+for the program, or a link-time error will occur.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="output-layout-qualifiers">4.4.2. Output Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Some output layout qualifiers apply to all shader stages and some apply only
+to specific stages.
+The latter are discussed in separate sections below.</p>
+</div>
+<div class="paragraph">
+<p>As with input layout qualifiers, all shaders except compute shaders allow
+<strong>location</strong> layout qualifiers on output variable declarations, output block
+declarations, and output block member declarations.
+Of these, variables and block members (but not blocks) additionally allow
+the <strong>component</strong> layout qualifier.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>location</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>component</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The usage and rules for applying the <strong>location</strong> qualifier
+and the <strong>component</strong> qualifier
+to blocks and structures are exactly as described in
+“<a href="#input-layout-qualifiers">Input Layout Qualifiers</a>”.
+Additionally, for fragment shader outputs, if two variables are placed
+within the same location, they must have the same underlying type
+(floating-point or integer).
+No component aliasing of output variables or members is allowed.</p>
+</div>
+<div class="paragraph">
+<p>Fragment shaders allow an additional <strong>index</strong> output layout qualifier:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>index</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Each of these qualifiers may appear at most once.
+If <strong>index</strong> is specified, <strong>location</strong> must also be specified.
+If <strong>index</strong> is not specified, the value 0 is used.
+For example, in a fragment shader,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">3</span>) out vec4 color;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the fragment shader output <em>color</em> is assigned to
+fragment color 3 as the first (index zero) input to the blend equation.
+And,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">3</span>, index = <span class="integer">1</span>) out vec4 factor;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the fragment shader output <em>factor</em> is assigned to
+fragment color 3 as the second (index one) input to the blend equation.</p>
+</div>
+<div class="paragraph">
+<p>For fragment shader outputs, the location
+and index specify
+the color output number
+and index
+receiving the values of the output.
+For outputs of all other shader stages, the location specifies a vector
+number that can be used to match against inputs in a subsequent shader
+stage, even if that shader is in a different program object.</p>
+</div>
+<div class="paragraph">
+<p>If a declared output is a scalar or vector type other than <strong>dvec3</strong> or
+<strong>dvec4</strong>, it will consume a single location.
+Outputs of type <strong>dvec3</strong> or <strong>dvec4</strong> will consume two consecutive locations.
+Outputs of type <strong>double</strong> and <strong>dvec2</strong> will consume only a single location, in
+all stages.</p>
+</div>
+<div class="paragraph">
+<p>If the declared output is an array, it will be assigned consecutive
+locations starting with the location specified.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(location = <span class="integer">2</span>) out vec4 colors[<span class="integer">3</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that <em>colors</em> is assigned to vector location numbers 2, 3,
+and 4.</p>
+</div>
+<div class="paragraph">
+<p>If the declared output is an <em>n</em> × <em>m</em>
+matrix, it will be assigned multiple locations starting with the location
+specified.
+The number of locations assigned will be the same as for an <em>n</em>-element
+array of <em>m</em>-component vectors.</p>
+</div>
+<div class="paragraph">
+<p>If the declared output is a structure, its members will be assigned
+consecutive locations in the order of declaration, with the first member
+assigned the location specified for the structure.
+The number of locations consumed by a structure member is determined by
+applying the rules above recursively as though the structure member were
+declared as an output variable of the same type.</p>
+</div>
+<div class="paragraph">
+<p><strong>location</strong> layout qualifiers may be used on output variables declared as
+structures.
+However, it is a compile-time error to use a <strong>location</strong> qualifier on a
+structure member.
+Location layout qualifiers may be used on output blocks and output block
+members.</p>
+</div>
+<div class="paragraph">
+<p>The number of output locations available to a shader is limited.
+For fragment shaders, the limit is the advertised number of draw buffers.</p>
+</div>
+<div class="paragraph">
+<p>For all other shaders, the limit is implementation-dependent and must be no
+less than one fourth of the advertised maximum output component count
+(compute shaders have no outputs).
+A program will fail to link if any attached shader uses a location greater
+than or equal to the number of supported locations, unless device-dependent
+optimizations are able to make the program fit within available hardware
+resources.</p>
+</div>
+<div class="paragraph">
+<p>Compile-time errors may also be given if at compile time it is known the
+link will fail.
+A negative output location will result in a compile-time error.
+It is also a compile-time error if a fragment shader sets a layout index to
+less than 0 or greater than 1.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time or link-time error if any of the following occur:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>any two fragment shader output variables are assigned to the same
+location and index.</p>
+</li>
+<li>
+<p>if any two output variables from the same vertex, tessellation or
+geometry shader stage are assigned to the same location.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>For fragment shader outputs, locations can be assigned using either a
+<strong>layout</strong> qualifier or via the OpenGL API.</p>
+</div>
+<div class="paragraph">
+<p>For all shader types, a program will fail to link if explicit location
+assignments leave the linker unable to find space for other variables
+without explicit assignments.</p>
+</div>
+<div class="paragraph">
+<p>If an output variable with no location or index assigned in the shader text
+has a location specified through the OpenGL API, the API-assigned
+location will be used.
+Otherwise, such variables will be assigned a location by the linker.
+All such assignments will have a color index of zero.
+See section 15.2 “Shader Execution” of the <a href="#references">OpenGL Specification</a> for
+more details.
+A link-time error will occur if an output variable is declared in multiple
+shaders of the same language with conflicting location or index values.</p>
+</div>
+<div class="paragraph">
+<p>For the purposes of determining if a non-fragment output matches an input
+from a subsequent shader stage, the <strong>location</strong> layout qualifier (if any)
+must match.</p>
+</div>
+<div class="sect4">
+<h5 id="transform-feedback-layout-qualifiers">Transform Feedback Layout Qualifiers</h5>
+<div class="paragraph">
+<p>The vertex, tessellation, and geometry stages allow shaders to control
+transform feedback.
+When doing this, shaders will dictate which transform feedback buffers are
+in use, which output variables will be written to which buffers, and how
+each buffer is laid out.
+To accomplish this, shaders allow the following layout qualifier identifiers
+on output declarations:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>xfb_buffer</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>xfb_offset</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>xfb_stride</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Any shader making any static use (after preprocessing) of any of these
+<strong>xfb_</strong> qualifiers will cause the shader to be in a transform feedback
+capturing mode and hence responsible for describing the transform feedback
+setup.
+This mode will capture any output selected by <strong>xfb_offset</strong>, directly or
+indirectly, to a transform feedback buffer.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>xfb_buffer</strong> qualifier specifies which transform feedback buffer will
+capture outputs selected with <strong>xfb_offset</strong>.
+The <strong>xfb_buffer</strong> qualifier can be applied to the qualifier <strong>out</strong>, to output
+variables, to output blocks, and to output block members.
+Shaders in the transform feedback capturing mode have an initial global
+default of</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(xfb_buffer = <span class="integer">0</span>) out;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This default can be changed by declaring a different buffer with
+<strong>xfb_buffer</strong> on the interface qualifier <strong>out</strong>.
+This is the only way the global default can be changed.
+When a variable or output block is declared without an <strong>xfb_buffer</strong>
+qualifier, it inherits the global default buffer.
+When a variable or output block is declared with an <strong>xfb_buffer</strong> qualifier,
+it has that declared buffer.
+All members of a block inherit the block’s buffer.
+A member is allowed to declare an <strong>xfb_buffer</strong>, but it must match the buffer
+inherited from its block, or a compile-time error results.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(xfb_buffer=<span class="integer">2</span>, xfb_offset=<span class="integer">0</span>) out block { <span class="comment">// block's buffer is 2</span>
+ layout(xfb_buffer = <span class="integer">2</span>) vec4 v; <span class="comment">// okay, matches the inherited 2</span>
+ layout(xfb_buffer = <span class="integer">3</span>) vec4 u; <span class="comment">// ERROR, mismatched buffer</span>
+ vec4 w; <span class="comment">// inherited</span>
+};
+layout(xfb_offset=<span class="integer">16</span>) out vec4 t; <span class="comment">// initial default is buffer 0</span>
+layout(xfb_buffer=<span class="integer">1</span>) out; <span class="comment">// new global default of 1</span>
+out block { <span class="comment">// block has buffer 1</span>
+ vec4 x; <span class="comment">// x has buffer 1 (not captured)</span>
+ layout(xfb_buffer = <span class="integer">1</span>) vec4 y; <span class="comment">// okay (not captured)</span>
+ layout(xfb_buffer = <span class="integer">0</span>) vec4 z; <span class="comment">// ERROR, mismatched buffer</span>
+};
+layout(xfb_offset=<span class="integer">0</span>) out vec4 g; <span class="comment">// g has buffer 1</span>
+layout(xfb_buffer=<span class="integer">2</span>) out vec4 h; <span class="comment">// does not change global default</span>
+layout(xfb_offset=<span class="integer">16</span>) out vec4 j; <span class="comment">// j has buffer 1</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Note this means all members of a block that go to a transform feedback
+buffer will go to the same buffer.</p>
+</div>
+<div class="paragraph">
+<p>When a block is declared as an array, all members of block array-element 0
+are captured, as previously described, by the declared or inherited
+<strong>xfb_buffer</strong>.
+Generally, an array of size <em>N</em> of blocks is captured by <em>N</em> consecutive
+buffers, with all members of block array-element <em>E</em> captured by buffer <em>B</em>,
+where <em>B</em> equals the declared or inherited <strong>xfb_buffer</strong> plus <em>E</em>.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time or link-time error to specify an <strong>xfb_buffer</strong>,
+including any additional buffers needed to capture an arrays of blocks, that
+is less than zero or greater than or equal to the implementation-dependent
+constant <em>gl_MaxTransformFeedbackBuffers</em>.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>xfb_offset</strong> qualifier assigns a byte offset within a transform feedback
+buffer.
+Only variables, block members, or blocks can be qualified with <strong>xfb_offset</strong>.
+If a block is qualified with <strong>xfb_offset</strong>, all its members are assigned
+transform feedback buffer offsets.
+If a block is not qualified with <strong>xfb_offset</strong>, any members of that block not
+qualified with an <strong>xfb_offset</strong> will not be assigned transform feedback
+buffer offsets.
+Only variables and block members that are assigned offsets will be captured
+(thus, a proper subset of a block can be captured).
+Each time such a variable or block member is written in a shader, the
+written value is captured at the assigned offset.
+If such a block member or variable is not written during a shader
+invocation, the buffer contents at the assigned offset will be undefined.
+Even if there are no static writes to a variable or member that is assigned
+a transform feedback offset, the space is still allocated in the buffer and
+still affects the stride.</p>
+</div>
+<div class="paragraph">
+<p>Variables and block members qualified with <strong>xfb_offset</strong> can be scalars,
+vectors, matrices, structures, and (sized) arrays of these.
+The offset must be a multiple of the size of the first component of the
+first qualified variable or block member, or a compile-time error results.
+Further, if applied to an aggregate containing a <strong>double</strong>, the offset must
+also be a multiple of 8, and the space taken in the buffer will be a
+multiple of 8.
+The given offset applies to the first component of the first member of the
+qualified entity.
+Then, within the qualified entity, subsequent components are each assigned,
+in order, to the next available offset aligned to a multiple of that
+component’s size.
+Aggregate types are flattened down to the component level to get this
+sequence of components.
+It is a compile-time error to apply <strong>xfb_offset</strong> to the declaration of an
+unsized array.</p>
+</div>
+<div class="paragraph">
+<p>No aliasing in output buffers is allowed: It is a compile-time or link-time
+error to specify variables with overlapping transform feedback offsets.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>xfb_stride</strong> qualifier specifies how many bytes are consumed by each
+captured vertex.
+It applies to the transform feedback buffer for that declaration, whether it
+is inherited or explicitly declared.
+It can be applied to variables, blocks, block members, or just the qualifier
+<strong>out</strong>.
+If the buffer is capturing any outputs with double-precision components, the
+stride must be a multiple of 8, otherwise it must be a multiple of 4, or a
+compile-time or link-time error results.
+It is a compile-time or link-time error to have any <strong>xfb_offset</strong> that
+overflows <strong>xfb_stride</strong>, whether stated on declarations before or after the
+<strong>xfb_stride</strong>, or in different compilation units.
+While <strong>xfb_stride</strong> can be declared multiple times for the same buffer, it is
+a compile-time or link-time error to have different values specified for the
+stride for the same buffer.</p>
+</div>
+<div class="paragraph">
+<p>For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// buffer 1 has 32-byte stride</span>
+layout(xfb_buffer = <span class="integer">1</span>, xfb_stride = <span class="integer">32</span>) out;
+
+<span class="comment">// same as previous example; order within layout does not matter</span>
+layout(xfb_stride = <span class="integer">32</span>, xfb_buffer = <span class="integer">1</span>) out;
+
+<span class="comment">// everything in this block goes to buffer 0</span>
+layout(xfb_buffer = <span class="integer">0</span>, xfb_stride = <span class="integer">32</span>) out block1 {
+ layout(xfb_offset = <span class="integer">0</span>) vec4 a; <span class="comment">// a goes to byte offset 0 of buffer 0</span>
+ layout(xfb_offset = <span class="integer">16</span>) vec4 b; <span class="comment">// b goes to offset 16 of buffer 0</span>
+};
+
+layout(xfb_buffer = <span class="integer">3</span>, xfb_offset = <span class="integer">12</span>) out block2 {
+ vec4 v; <span class="comment">// v will be written to byte offsets 12 through 27 of buffer</span>
+ <span class="predefined-type">float</span> u; <span class="comment">// u will be written to offset 28</span>
+ layout(xfb_offset = <span class="integer">40</span>) vec4 w;
+ vec4 x; <span class="comment">// x will be written to offset 56, the next available offset</span>
+};
+
+layout(xfb_buffer = <span class="integer">2</span>, xfb_stride = <span class="integer">32</span>) out block3 {
+ layout(xfb_offset = <span class="integer">12</span>) vec3 c;
+ layout(xfb_offset = <span class="integer">24</span>) vec3 d; <span class="comment">// ERROR, requires stride of 36</span>
+ layout(xfb_offset = <span class="integer">0</span>) vec3 g; <span class="comment">// okay, increasing order not required</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When no <strong>xfb_stride</strong> is specified for a buffer, the stride of the buffer
+will be the smallest needed to hold the variable placed at the highest
+offset, including any required padding.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// if there no other declarations for buffer 3, it has stride 32</span>
+layout(xfb_buffer = <span class="integer">3</span>) out block4 {
+ layout(xfb_offset = <span class="integer">0</span>) vec4 e;
+ layout(xfb_offset = <span class="integer">16</span>) vec4 f;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The resulting stride (implicit or explicit), when divided by 4, must be less
+than or equal to the implementation-dependent constant
+<em>gl_MaxTransformFeedbackInterleavedComponents</em>.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-control-outputs">Tessellation Control Outputs</h5>
+<div class="paragraph">
+<p>Other than for the transform feedback layout qualifiers, tessellation
+control shaders allow output layout qualifiers only on the interface
+qualifier <strong>out</strong>, not on an output block, block member, or variable
+declaration.
+The output layout qualifier identifiers allowed for tessellation control
+shaders are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>vertices</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The identifier <strong>vertices</strong> specifies the number of vertices in the output
+patch produced by the tessellation control shader, which also specifies the
+number of times the tessellation control shader is invoked.
+It is a compile- or link-time error for the output vertex count to be less
+than or equal to zero, or greater than the implementation-dependent maximum
+patch size.</p>
+</div>
+<div class="paragraph">
+<p>The intrinsically declared tessellation control output array <em>gl_out[]</em> will
+also be sized by any output layout declaration.
+Hence, the expression</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_out.length()</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will return the output patch vertex count specified in a previous output
+layout qualifier.
+For outputs declared without an array size, including intrinsically declared
+outputs (i.e., <em>gl_out</em>), a layout must be declared before any use of the
+method <strong>length</strong>() or other array use that requires its size to be known.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error if the output patch vertex count specified in an
+output layout qualifier does not match the array size specified in any
+output variable declaration in the same shader.</p>
+</div>
+<div class="paragraph">
+<p>All tessellation control shader layout declarations in a program must
+specify the same output patch vertex count.
+There must be at least one layout qualifier specifying an output patch
+vertex count in any program containing tessellation control shaders;
+however, such a declaration is not required in all tessellation control
+shaders.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="geometry-outputs">Geometry Outputs</h5>
+<div class="paragraph">
+<p>Geometry shaders can have three additional types of output layout
+identifiers: an output <em>primitive type</em>, a maximum output <em>vertex count</em>,
+and per-output <em>stream</em> numbers.
+The primitive type and vertex count identifiers are allowed only on the
+interface qualifier <strong>out</strong>, not on an output block, block member, or variable
+declaration.
+The stream identifier is allowed on the interface qualifier <strong>out</strong>, on output
+blocks, and on variable declarations.</p>
+</div>
+<div class="paragraph">
+<p>The layout qualifier identifiers for geometry shader outputs are</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>points</strong><br>
+<strong>line_strip</strong><br>
+<strong>triangle_strip</strong><br>
+<strong>max_vertices</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>stream</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The primitive type identifiers <strong>points</strong>, <strong>line_strip</strong>, and <strong>triangle_strip</strong>
+are used to specify the type of output primitive produced by the geometry
+shader, and only one of these is accepted.
+At least one geometry shader (compilation unit) in a program must declare an
+output primitive type, and all geometry shader output primitive type
+declarations in a program must declare the same primitive type.
+It is not required that all geometry shaders in a program declare an output
+primitive type.</p>
+</div>
+<div class="paragraph">
+<p>The vertex count identifier <strong>max_vertices</strong> is used to specify the maximum
+number of vertices the shader will ever emit in a single invocation.
+At least one geometry shader (compilation unit) in a program must declare a
+maximum output vertex count, and all geometry shader output vertex count
+declarations in a program must declare the same count.
+It is not required that all geometry shaders in a program declare a count.</p>
+</div>
+<div class="paragraph">
+<p>In this example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(triangle_strip, max_vertices = <span class="integer">60</span>) out; <span class="comment">// order does not matter</span>
+layout(max_vertices = <span class="integer">60</span>) out; <span class="comment">// redeclaration okay</span>
+layout(triangle_strip) out; <span class="comment">// redeclaration okay</span>
+layout(points) out; <span class="comment">// error, contradicts triangle_strip</span>
+layout(max_vertices = <span class="integer">30</span>) out; <span class="comment">// error, contradicts 60</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>all outputs from the geometry shader are triangles and at most 60 vertices
+will be emitted by the shader.
+It is an error for the maximum number of vertices to be greater than
+<em>gl_MaxGeometryOutputVertices</em>.</p>
+</div>
+<div class="paragraph">
+<p>The identifier <strong>stream</strong> is used to specify that a geometry shader output
+variable or block is associated with a particular vertex stream (numbered
+beginning with zero).
+A default stream number may be declared at global scope by qualifying
+interface qualifier <strong>out</strong> as in this example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(stream = <span class="integer">1</span>) out;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The stream number specified in such a declaration replaces any previous
+default and applies to all subsequent block and variable declarations until
+a new default is established.
+The initial default stream number is zero.</p>
+</div>
+<div class="paragraph">
+<p>Each output block or non-block output variable is associated with a vertex
+stream.
+If the block or variable is declared with the stream identifier, it is
+associated with the specified stream; otherwise, it is associated with the
+current default stream.
+A block member may be declared with a stream identifier, but the specified
+stream must match the stream associated with the containing block.
+One example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(stream=<span class="integer">1</span>) out; <span class="comment">// default is now stream 1</span>
+out vec4 var1; <span class="comment">// var1 gets default stream (1)</span>
+layout(stream=<span class="integer">2</span>) out Block1 { <span class="comment">// "Block1" belongs to stream 2</span>
+ layout(stream=<span class="integer">2</span>) vec4 var2; <span class="comment">// redundant block member stream decl</span>
+ layout(stream=<span class="integer">3</span>) vec2 var3; <span class="comment">// ILLEGAL (must match block stream)</span>
+ vec3 var4; <span class="comment">// belongs to stream 2</span>
+};
+layout(stream=<span class="integer">0</span>) out; <span class="comment">// default is now stream 0</span>
+out vec4 var5; <span class="comment">// var5 gets default stream (0)</span>
+out Block2 { <span class="comment">// "Block2" gets default stream (0)</span>
+ vec4 var6;
+};
+layout(stream=<span class="integer">3</span>) out vec4 var7; <span class="comment">// var7 belongs to stream 3</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Each vertex emitted by the geometry shader is assigned to a specific stream,
+and the attributes of the emitted vertex are taken from the set of output
+blocks and variables assigned to the targeted stream.
+After each vertex is emitted, the values of all output variables become
+undefined.
+Additionally, the output variables associated with each vertex stream may
+share storage.
+Writing to an output variable associated with one stream may overwrite
+output variables associated with any other stream.
+When emitting each vertex, a geometry shader should write to all outputs
+associated with the stream to which the vertex will be emitted and to no
+outputs associated with any other stream.</p>
+</div>
+<div class="paragraph">
+<p>If a geometry shader output block or variable is declared more than once,
+all such declarations must associate the variable with the same vertex
+stream.
+If any stream declaration specifies a non-existent stream number, the shader
+will fail to compile.</p>
+</div>
+<div class="paragraph">
+<p>Built-in geometry shader outputs are always associated with vertex stream
+zero.</p>
+</div>
+<div class="paragraph">
+<p>All geometry shader output layout declarations in a program must declare the
+same layout and same value for <strong>max_vertices</strong>.
+If geometry shaders are in a program, there must be at least one geometry
+output layout declaration somewhere in that
+program, but not all geometry
+shaders (compilation units) are required to declare it.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="fragment-outputs">Fragment Outputs</h5>
+<div class="paragraph">
+<p>The built-in fragment shader variable <em>gl_FragDepth</em> may be redeclared using
+one of the following layout qualifiers.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>depth_any</strong><br>
+<strong>depth_greater</strong><br>
+<strong>depth_less</strong><br>
+<strong>depth_unchanged</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The layout qualifier for <em>gl_FragDepth</em> constrains intentions of the final
+value of <em>gl_FragDepth</em> written by any shader invocation.
+GL implementations are allowed to perform optimizations assuming that the
+depth test fails (or passes) for a given fragment if all values of
+<em>gl_FragDepth</em> consistent with the layout qualifier would fail (or pass).
+This potentially includes skipping shader execution if the fragment is
+discarded because it is occluded and the shader has no side effects.
+If the final value of <em>gl_FragDepth</em> is inconsistent with its layout
+qualifier, the result of the depth test for the corresponding fragment is
+undefined.
+However, no error will be generated in this case.
+If the depth test passes and depth writes are enabled, the value written to
+the depth buffer is always the value of <em>gl_FragDepth</em>, whether or not it is
+consistent with the layout qualifier.</p>
+</div>
+<div class="paragraph">
+<p>By default, <em>gl_FragDepth</em> is qualified as <em>depth_any</em>.
+When the layout qualifier for <em>gl_FragDepth</em> is <em>depth_any</em>, the shader
+compiler will note any assignment to <em>gl_FragDepth</em> modifying it in an
+unknown way, and depth testing will always be performed after the shader has
+executed.
+When the layout qualifier is <em>depth_greater</em>, the GL can assume that the
+final value of <em>gl_FragDepth</em> is greater than or equal to the fragment’s
+interpolated depth value, as given by the <em>z</em> component of <em>gl_FragCoord</em>.
+When the layout qualifier is <em>depth_less</em>, the GL can assume that any
+modification of <em>gl_FragDepth</em> will only decrease its value.
+When the layout qualifier is <em>depth_unchanged</em>, the shader compiler will
+honor any modification to <em>gl_FragDepth</em>, but the rest of the GL can assume
+that <em>gl_FragDepth</em> is not assigned a new value.</p>
+</div>
+<div class="paragraph">
+<p>Redeclarations of <em>gl_FragDepth</em> are performed as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// redeclaration that changes nothing is allowed +</span>
+out <span class="predefined-type">float</span> gl_FragDepth;
+
+<span class="comment">// assume it may be modified in any way</span>
+layout(depth_any) out <span class="predefined-type">float</span> gl_FragDepth;
+
+<span class="comment">// assume it may be modified such that its value will only increase</span>
+layout(depth_greater) out <span class="predefined-type">float</span> gl_FragDepth;
+
+<span class="comment">// assume it may be modified such that its value will only decrease</span>
+layout(depth_less) out <span class="predefined-type">float</span> gl_FragDepth;
+
+<span class="comment">// assume it will not be modified</span>
+layout(depth_unchanged) out <span class="predefined-type">float</span> gl_FragDepth;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>If <em>gl_FragDepth</em> is redeclared in any fragment shader in a program, it must
+be redeclared in all fragment shaders in that program that have static
+assignments to <em>gl_FragDepth</em>.
+All redeclarations of <em>gl_FragDepth</em> in all fragment shaders in a single
+program must have the same set of qualifiers.
+Within any shader, the first redeclarations of <em>gl_FragDepth</em> must appear
+before any use of <em>gl_FragDepth</em>.
+The built-in <em>gl_FragDepth</em> is only predeclared in fragment shaders, so
+redeclaring it in any other shader language results in a compile-time error.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="uniform-variable-layout-qualifiers">4.4.3. Uniform Variable Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Layout qualifiers can be used for uniform variables and subroutine uniforms.
+The layout qualifier identifiers for uniform variables and subroutine
+uniforms are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>location</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The location identifier can be used with default-block uniform variables and
+subroutine uniforms.
+The location specifies the location by which the OpenGL API can reference
+the uniform and update its value.
+Individual elements of a uniform array are assigned consecutive locations
+with the first element taking location <strong>location</strong>.
+No two default-block uniform variables in the program can have the same
+location, even if they are unused, otherwise a compile-time or link-time
+error will be generated.
+No two subroutine uniform variables can have the same location in the same
+shader stage, otherwise a compile-time or link-time error will be generated.
+Valid locations for default-block uniform variable locations are in the
+range of 0 to the implementation-defined maximum number of uniform locations
+minus one.
+Valid locations for subroutine uniforms are in the range of 0 to the
+implementation-defined per-stage maximum number of subroutine uniform
+locations minus one.</p>
+</div>
+<div class="paragraph">
+<p>Locations can be assigned to default-block uniform arrays and structures.
+The first inner-most scalar, vector or matrix member or element takes the
+specified <strong>location</strong> and the compiler assigns the next inner-most member or
+element the next incremental location value.
+Each subsequent inner-most member or element gets incremental locations for
+the entire structure or array.
+This rule applies to nested structures and arrays and gives each inner-most
+scalar, vector, or matrix member a unique location.
+For arrays without an explicit size, the size is calculated based on its
+static usage.
+When the linker generates locations for uniforms without an explicit
+location, it assumes for all uniforms with an explicit location all their
+array elements and structure members are used and the linker will not
+generate a conflicting location, even if that element or member is deemed
+unused.</p>
+</div>
+<div class="paragraph">
+<p>When generating SPIR-V for OpenGL, it is a compile-time error to not
+include a location for individual (default block) non-opaque uniform
+variables.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="subroutine-function-layout-qualifiers">4.4.4. Subroutine Function Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Layout qualifiers can be used for subroutine functions.
+The layout qualifier identifiers for subroutine functions are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>index</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Each subroutine with an index qualifier in the shader must be given a unique
+index, otherwise a compile- or link-time error will be generated.
+The indices must be in the range of 0 to the implementation defined maximum
+number of subroutines minus one.
+It is recommended, but not required, that the shader assigns a range of
+tightly packed <em>index</em> values starting from zero so that the OpenGL
+subroutine function enumeration API returns a non-empty name for all active
+indices.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="uniform-and-shader-storage-block-layout-qualifiers">4.4.5. Uniform and Shader Storage Block Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Layout qualifiers can be used for uniform and shader storage blocks, but not
+for non-block uniform declarations.
+The layout qualifier identifiers (and <strong>shared</strong> keyword) for uniform and
+shader storage blocks are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>shared</strong><br>
+<strong>packed</strong><br>
+<strong>std140</strong><br>
+<strong>std430</strong><br>
+<strong>row_major</strong><br>
+<strong>column_major</strong><br>
+<strong>binding</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>offset</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>align</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>None of these have any semantic effect at all on the usage of the variables
+being declared; they only describe how data is laid out in memory.
+For example, matrix semantics are always column-based, as described in the
+rest of this specification, no matter what layout qualifiers are being used.</p>
+</div>
+<div class="paragraph">
+<p>Uniform and shader storage block layout qualifiers can be declared for
+global scope, on a single uniform or shader storage block, or on a single
+block member declaration.</p>
+</div>
+<div class="paragraph">
+<p>Default layouts are established at global scope for uniform blocks as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(layout-qualifier-id-list) uniform;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and for shader storage blocks as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(layout-qualifier-id-list) buffer;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When this is done, the previous default qualification is first inherited and
+then overridden as per the override rules listed below for each qualifier
+listed in the declaration.
+The result becomes the new default qualification scoped to subsequent
+uniform or shader storage block definitions.</p>
+</div>
+<div class="paragraph">
+<p>The initial state of compilation
+when generating SPIR-V
+is as if the following were declared:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(std140, column_major) uniform;
+layout(std430, column_major) buffer;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The initial state of compilation when not generating SPIR-V is as if the
+following were declared:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(shared, column_major) uniform;
+layout(shared, column_major) buffer;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Uniform and shader storage blocks can be declared with optional layout
+qualifiers, and so can their individual member declarations.
+Such block layout qualification is scoped only to the content of the block.
+As with global layout declarations, block layout qualification first
+inherits from the current default qualification and then overrides it.
+Similarly, individual member layout qualification is scoped just to the
+member declaration, and inherits from and overrides the block’s
+qualification.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>shared</strong> qualifier overrides only the <strong>std140</strong>, <strong>std430</strong>, and <strong>packed</strong>
+qualifiers; other qualifiers are inherited.
+The compiler/linker will ensure that multiple programs and programmable
+stages containing this definition will share the same memory layout for this
+block, as long as all arrays are declared with explicit sizes and all
+matrices have matching <strong>row_major</strong> and/or <strong>column_major</strong> qualifications
+(which may come from a declaration outside the block definition).
+This allows use of the same buffer to back the same block definition across
+different programs.
+It is a compile-time error to use the <strong>shared</strong> qualifier when generating
+SPIR-V.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>packed</strong> qualifier overrides only <strong>std140</strong>, <strong>std430</strong>, and <strong>shared</strong>;
+other qualifiers are inherited.
+When <strong>packed</strong> is used, no shareable layout is guaranteed.
+The compiler and linker can optimize memory use based on what variables
+actively get used and on other criteria.
+Offsets must be queried, as there is no other way of guaranteeing where (and
+which) variables reside within the block.</p>
+</div>
+<div class="paragraph">
+<p>It is a link-time error to access the same packed uniform or shader storage
+block in multiple stages within a program.
+Attempts to access the same packed uniform or shader storage block across
+programs can result in conflicting member offsets and in undefined values
+being read.
+However, implementations may aid application management of packed blocks by
+using canonical layouts for packed blocks.
+It is a compile-time error to use the <strong>packed</strong> qualifier when generating
+SPIR-V.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>std140</strong> and <strong>std430</strong> qualifiers override only the <strong>packed</strong>, <strong>shared</strong>,
+<strong>std140</strong>, and <strong>std430</strong> qualifiers; other qualifiers are inherited.
+The <strong>std430</strong> qualifier is supported only for shader storage blocks; a shader
+using the <strong>std430</strong> qualifier on a uniform block will fail to compile.</p>
+</div>
+<div class="paragraph">
+<p>The layout is explicitly determined by this, as described in section 7.6.2.2
+“Standard Uniform Block Layout” of the <a href="#references">OpenGL Specification</a>.
+Hence, as in <strong>shared</strong> above, the resulting layout is shareable across
+programs.</p>
+</div>
+<div class="paragraph">
+<p>Layout qualifiers on member declarations cannot use the <strong>shared</strong>, <strong>packed</strong>,
+<strong>std140</strong>, or <strong>std430</strong> qualifiers.
+These can only be used at global scope (without an object) or on a block
+declaration, or a compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>row_major</strong> and <strong>column_major</strong> qualifiers only affect the layout of
+matrices, including all matrices contained in structures and arrays they are
+applied to, to all depths of nesting.
+These qualifiers can be applied to other types, but will have no effect.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>row_major</strong> qualifier overrides only the <strong>column_major</strong> qualifier; other
+qualifiers are inherited.
+Elements within a matrix row will be contiguous in memory.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>column_major</strong> qualifier overrides only the <strong>row_major</strong> qualifier; other
+qualifiers are inherited.
+Elements within a matrix column will be contiguous in memory.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>binding</strong> qualifier specifies the uniform buffer binding point
+corresponding to the uniform or shader storage block, which will be used to
+obtain the values of the member variables of the block.
+It is a compile-time error to specify the <strong>binding</strong> qualifier for the global
+scope or for block member declarations.
+Any uniform or shader storage block declared without a <strong>binding</strong> qualifier
+is initially assigned to block binding point zero.
+After a program is linked, the binding points used for uniform
+and shader storage blocks
+declared with or without a <strong>binding</strong> qualifier can be updated
+by the OpenGL API.</p>
+</div>
+<div class="paragraph">
+<p>If the <strong>binding</strong> qualifier is used with a uniform block or shader storage
+block instanced as an array, the first element of the array takes the
+specified block binding and each subsequent element takes the next
+consecutive binding point.
+For an array of arrays, each element (e.g. 6 elements for a[2][3]) gets a
+binding point, and they are ordered per the array of array ordering
+described in “<a href="#arrays.">Arrays.</a>”</p>
+</div>
+<div class="paragraph">
+<p>If the binding point for any uniform or shader storage block instance is
+less than zero, or greater than or equal to the corresponding
+implementation-dependent maximum number of buffer bindings, a compile-time
+error will occur.
+When the <strong>binding</strong> qualifier is used with a uniform or shader storage block
+instanced as an array of size <em>N</em>, all elements of the array from <strong>binding</strong>
+through <em>binding + N - 1</em> must be within this range.
+It is a compile-time or link-time error to use the same binding number for
+more than one uniform block or for more than one buffer block.</p>
+</div>
+<div class="paragraph">
+<p>When multiple arguments are listed in a <strong>layout</strong> declaration, the effect
+will be the same as if they were declared one at a time, in order from left
+to right, each in turn inheriting from and overriding the result from the
+previous qualification.</p>
+</div>
+<div class="paragraph">
+<p>For example</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(row_major, column_major)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>results in the qualification being <strong>column_major</strong>.
+Other examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(shared, row_major) uniform; <span class="comment">// default is now shared and row_major</span>
+
+layout(std140) uniform Transform { <span class="comment">// layout of this block is std140</span>
+ mat4 M1; <span class="comment">// row major</span>
+ layout(column_major) mat4 M2; <span class="comment">// column major</span>
+ mat3 N1; <span class="comment">// row major</span>
+};
+
+uniform T2 { <span class="comment">// layout of this block is shared</span>
+ ...
+};
+
+layout(column_major) uniform T3 { <span class="comment">// shared and column major</span>
+ mat4 M3; <span class="comment">// column major</span>
+ layout(row_major) mat4 m4; <span class="comment">// row major</span>
+ mat3 N2; <span class="comment">// column major</span>
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>offset</strong> qualifier can only be used on block members of blocks declared
+with <strong>std140</strong> or <strong>std430</strong> layouts.
+The <strong>offset</strong> qualifier forces the qualified member to start at or after the
+specified <em>layout-qualifier-value</em>, which will be its byte offset from
+the beginning of the buffer.
+It is a compile-time error to have any offset, explicit or assigned, that
+lies within another member of the block.
+When not generating SPIR-V, it is a compile-time error to specify an offset
+that is smaller than the offset of the previous member in the block.
+Two blocks linked together in the same program with the same block name must
+have the exact same set of members qualified with <strong>offset</strong> and their
+<em>layout-qualifier-value</em> values must be the same, or a link-time error
+results.
+The specified offset must be a multiple of the base alignment of the type of
+the block member it qualifies, or a compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>align</strong> qualifier can only be used on blocks or block members, and only
+for blocks declared with <strong>std140</strong> or <strong>std430</strong> layouts.
+The <strong>align</strong> qualifier makes the start of each block member have a minimum
+byte alignment.
+It does not affect the internal layout within each member, which will still
+follow the <strong>std140</strong> or <strong>std430</strong> rules.
+The specified alignment must be greater than 0 and a power of 2, or a
+compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>The <em>actual alignment</em> of a member will be the greater of the specified
+<strong>align</strong> alignment and the standard (e.g. <strong>std140</strong>) base alignment for the
+member’s type.
+The <em>actual offset</em> of a member is computed as follows: If <strong>offset</strong> was
+declared, start with that offset, otherwise start with the next available
+offset.
+If the resulting offset is not a multiple of the <em>actual alignment</em>,
+increase it to the first offset that is a multiple of the <em>actual
+alignment</em>.
+This results in the <em>actual offset</em> the member will have.</p>
+</div>
+<div class="paragraph">
+<p>When <strong>align</strong> is applied to an array, it affects only the start of the array,
+not the array’s internal stride.
+Both an <strong>offset</strong> and an <strong>align</strong> qualifier can be specified on a declaration.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>align</strong> qualifier, when used on a block, has the same effect as
+qualifying each member with the same <strong>align</strong> value as declared on the block,
+and gets the same compile-time results and errors as if this had been done.
+As described in general earlier, an individual member can specify its own
+<strong>align</strong>, which overrides the block-level <strong>align</strong>, but just for that member.</p>
+</div>
+<div class="paragraph">
+<p>Examples:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(std140) uniform block {
+ vec4 a; <span class="comment">// a takes offsets 0-15</span>
+ layout(offset = <span class="integer">32</span>) vec3 b; <span class="comment">// b takes offsets 32-43</span>
+ layout(offset = <span class="integer">40</span>) vec2 c; <span class="comment">// ERROR, lies within previous member</span>
+ layout(offset = <span class="integer">48</span>) vec2 d; <span class="comment">// d takes offsets 48-55</span>
+ layout(align = <span class="integer">16</span>) <span class="predefined-type">float</span> e; <span class="comment">// e takes offsets 64-67</span>
+ layout(align = <span class="integer">2</span>) <span class="predefined-type">double</span> f; <span class="comment">// f takes offsets 72-79</span>
+ layout(align = <span class="integer">6</span>) <span class="predefined-type">double</span> g; <span class="comment">// ERROR, 6 is not a power of 2</span>
+ layout(offset = <span class="integer">80</span>) <span class="predefined-type">float</span> h; <span class="comment">// h takes offsets 80-83</span>
+ layout(align = <span class="integer">64</span>) dvec3 i; <span class="comment">// i takes offsets 128-151</span>
+ layout(offset = <span class="integer">164</span>, align = <span class="integer">8</span>)
+ <span class="predefined-type">float</span> j; <span class="comment">// j takes offsets 168-171</span>
+};</code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="opaque-uniform-layout-qualifiers">4.4.6. Opaque Uniform Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Uniform layout qualifiers can be used to bind opaque uniform variables to
+specific buffers or units.
+Samplers can be bound to texture image units, images can be bound to image
+units, and atomic counters can be bound to buffers.</p>
+</div>
+<div class="paragraph">
+<p>Sampler, image and atomic counter types take the uniform layout qualifier
+identifier for binding:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>binding</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The identifier <strong>binding</strong> specifies which unit will be bound.
+Any uniform sampler or image
+variable declared without a binding qualifier
+is initially bound to unit zero.
+After a program is linked, the unit referenced by a sampler
+or image
+uniform variable declared with or without a <strong>binding</strong> qualifier can be
+updated by the OpenGL API.</p>
+</div>
+<div class="paragraph">
+<p>If the <strong>binding</strong> qualifier is used with an array, the first element of the
+array takes the specified unit and each subsequent element takes the next
+consecutive unit.
+For an array of arrays, each element (e.g. 6 elements for a[2][3]) gets a
+binding point, and they are ordered per the array of array ordering
+described in “<a href="#arrays.">Arrays.</a>”</p>
+</div>
+<div class="paragraph">
+<p>If the <strong>binding</strong> is less than zero, or greater than or equal to the
+implementation-dependent maximum supported number of units, a compile-time
+error will occur.
+When the <strong>binding</strong> qualifier is used with an array of size <em>N</em>, all elements
+of the array from <strong>binding</strong> through <em>binding + N - 1</em> must be within this
+range.
+It is a compile-time or link-time error to use the same <strong>binding</strong> number for
+more than one atomic counter, unless the <em>offset</em> for the atomic counters
+sharing the same binding are all different.</p>
+</div>
+<div class="paragraph">
+<p>A link-time error will result if two shaders in a program specify different
+<em>layout-qualifier-value</em> bindings for the same opaque-uniform name.
+However, it is not an error to specify a binding on some but not all
+declarations for the same name, as shown in the examples below.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// in one shader...</span>
+layout(binding=<span class="integer">3</span>) uniform sampler2D s; <span class="comment">// s bound to unit 3</span>
+
+<span class="comment">// in another shader...</span>
+uniform sampler2D s; <span class="comment">// okay, s still bound at 3</span>
+
+<span class="comment">// in another shader...</span>
+layout(binding=<span class="integer">4</span>) uniform sampler2D s; <span class="comment">// ERROR: contradictory bindings</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="atomic-counter-layout-qualifiers">4.4.7. Atomic Counter Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Atomic counter layout qualifiers can be used on atomic counter declarations.
+The atomic counter qualifiers are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><strong>binding</strong> <strong>=</strong> <em>layout-qualifier-value</em><br>
+<strong>offset</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding = <span class="integer">2</span>, offset = <span class="integer">4</span>) uniform atomic_uint a;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the opaque handle to the atomic counter <em>a</em> will be
+bound to atomic counter buffer binding point 2 at an offset of 4 basic
+machine units into that buffer.
+The default <em>offset</em> for binding point 2 will be post incremented by 4 (the
+size of an atomic counter).</p>
+</div>
+<div class="paragraph">
+<p>A subsequent atomic counter declaration will inherit the previous (post
+incremented) offset.
+For example, a subsequent declaration of</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding = <span class="integer">2</span>) uniform atomic_uint bar;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will establish that the atomic counter <em>bar</em> has a binding to buffer binding
+point 2 at an offset of 8 basic machine units into that buffer.
+The offset for binding point 2 will again be post-incremented by 4 (the size
+of an atomic counter).</p>
+</div>
+<div class="paragraph">
+<p>When multiple variables are listed in a layout declaration, the effect will
+be the same as if they were declared one at a time, in order from left to
+right.</p>
+</div>
+<div class="paragraph">
+<p>Binding points are not inherited, only offsets.
+Each binding point tracks its own current default <em>offset</em> for inheritance
+of subsequent variables using the same <strong>binding</strong>.
+The initial state of compilation is that all <strong>binding</strong> points have an
+<em>offset</em> of 0.
+The <em>offset</em> can be set per binding point at global scope (without declaring
+a variable).
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding = <span class="integer">2</span>, offset = <span class="integer">4</span>) uniform atomic_uint;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Establishes that the next <strong>atomic_uint</strong> declaration for binding point 2 will
+inherit <em>offset</em> 4 (but does not establish a default <strong>binding</strong>):</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding = <span class="integer">2</span>) uniform atomic_uint bar; <span class="comment">// offset is 4</span>
+layout(offset = <span class="integer">8</span>) uniform atomic_uint bar; <span class="comment">// error, no default binding</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Atomic counters may share the same binding point, but if a binding is
+shared, their offsets must be either explicitly or implicitly (from
+inheritance) unique and non overlapping.</p>
+</div>
+<div class="paragraph">
+<p>Example valid uniform declarations, assuming top of shader:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(binding=<span class="integer">3</span>, offset=<span class="integer">4</span>) uniform atomic_uint a; <span class="comment">// offset = 4</span>
+layout(binding=<span class="integer">2</span>) uniform atomic_uint b; <span class="comment">// offset = 0</span>
+layout(binding=<span class="integer">3</span>) uniform atomic_uint c; <span class="comment">// offset = 8</span>
+layout(binding=<span class="integer">2</span>) uniform atomic_uint d; <span class="comment">// offset = 4</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Example of an invalid uniform declaration:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(offset=<span class="integer">4</span>) ... <span class="comment">// error, must include binding</span>
+layout(binding=<span class="integer">1</span>, offset=<span class="integer">0</span>) ... a; <span class="comment">// okay</span>
+layout(binding=<span class="integer">2</span>, offset=<span class="integer">0</span>) ... b; <span class="comment">// okay</span>
+layout(binding=<span class="integer">1</span>, offset=<span class="integer">0</span>) ... c; <span class="comment">// error, offsets must not be shared</span>
+ <span class="comment">// between a and c</span>
+layout(binding=<span class="integer">1</span>, offset=<span class="integer">2</span>) ... d; <span class="comment">// error, overlaps offset 0 of a</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to bind an atomic counter with a binding value
+greater than or equal to <em>gl_MaxAtomicCounterBindings</em>.
+It is a compile-time error to declare an unsized array of <strong>atomic_uint</strong>.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="format-layout-qualifiers">4.4.8. Format Layout Qualifiers</h4>
+<div class="paragraph">
+<p>Format layout qualifiers can be used on image variable declarations (those
+declared with a basic type having “<strong>image</strong>” in its keyword).
+The format layout qualifier identifiers for image variable declarations are:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>layout-qualifier-id</em> : </dt>
+<dd>
+<p><em>float-image-format-qualifier</em><br>
+<em>int-image-format-qualifier</em><br>
+<em>uint-image-format-qualifier</em><br>
+<strong>binding</strong> <strong>=</strong> <em>layout-qualifier-value</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>float-image-format-qualifier</em> : </dt>
+<dd>
+<p><strong>rgba32f</strong><br>
+<strong>rgba16f</strong><br>
+<strong>rg32f</strong><br>
+<strong>rg16f</strong><br>
+<strong>r11f_g11f_b10f</strong><br>
+<strong>r32f</strong><br>
+<strong>r16f</strong><br>
+<strong>rgba16</strong><br>
+<strong>rgb10_a2</strong><br>
+<strong>rgba8</strong><br>
+<strong>rg16</strong><br>
+<strong>rg8</strong><br>
+<strong>r16</strong><br>
+<strong>r8</strong><br>
+<strong>rgba16_snorm</strong><br>
+<strong>rgba8_snorm</strong><br>
+<strong>rg16_snorm</strong><br>
+<strong>rg8_snorm</strong><br>
+<strong>r16_snorm</strong><br>
+<strong>r8_snorm</strong></p>
+</dd>
+<dt class="hdlist1"><em>int-image-format-qualifier</em> : </dt>
+<dd>
+<p><strong>rgba32i</strong><br>
+<strong>rgba16i</strong><br>
+<strong>rgba8i</strong><br>
+<strong>rg32i</strong><br>
+<strong>rg16i</strong><br>
+<strong>rg8i</strong><br>
+<strong>r32i</strong><br>
+<strong>r16i</strong><br>
+<strong>r8i</strong></p>
+</dd>
+<dt class="hdlist1"><em>uint-image-format-qualifier</em> : </dt>
+<dd>
+<p><strong>rgba32ui</strong><br>
+<strong>rgba16ui</strong><br>
+<strong>rgb10_a2ui</strong><br>
+<strong>rgba8ui</strong><br>
+<strong>rg32ui</strong><br>
+<strong>rg16ui</strong><br>
+<strong>rg8ui</strong><br>
+<strong>r32ui</strong><br>
+<strong>r16ui</strong><br>
+<strong>r8ui</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>A format layout qualifier specifies the image format associated with a
+declared image variable.
+Only one format qualifier may be specified for any image variable
+declaration.
+For image variables with floating-point component types (keywords starting
+with “<strong>image</strong>”), signed integer component types (keywords starting with
+“<strong>iimage</strong>”), or unsigned integer component types (keywords starting with
+“<strong>uimage</strong>”), the format qualifier used must match the
+<em>float-image-format-qualifier</em>, <em>int-image-format-qualifier</em>, or
+<em>uint-image-format-qualifier</em> grammar rules, respectively.
+It is a compile-time error to declare an image variable where the format
+qualifier does not match the image variable type.</p>
+</div>
+<div class="paragraph">
+<p>Any image variable used for image loads or atomic operations must specify a
+format layout qualifier; it is a compile-time error to pass an image uniform
+variable or function parameter declared without a format layout qualifier to
+an image load or atomic function.</p>
+</div>
+<div class="paragraph">
+<p>Uniforms not qualified with <strong>writeonly</strong> must have a format layout qualifier.
+Note that an image variable passed to a function for read access cannot be
+declared as <strong>writeonly</strong> and hence must have been declared with a format
+layout qualifier.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>binding</strong> qualifier was described in
+“<a href="#opaque-uniform-layout-qualifiers">Opaque Uniform Layout Qualifiers</a>”.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="interpolation-qualifiers">4.5. Interpolation Qualifiers</h3>
+<div class="paragraph">
+<p>Inputs and outputs that could be interpolated can be further qualified by at
+most one of the following interpolation qualifiers:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>smooth</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">perspective correct interpolation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>flat</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">no interpolation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>noperspective</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">linear interpolation</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The presence of and type of interpolation is controlled by the above
+interpolation qualifiers as well as the auxiliary storage qualifiers
+<strong>centroid</strong> and <strong>sample</strong>.
+When no interpolation qualifier is present, smooth interpolation is used.
+It is a compile-time error to use more than one interpolation qualifier.
+The auxiliary storage qualifier <strong>patch</strong> is not used for interpolation; it is
+a compile-time error to use interpolation qualifiers with <strong>patch</strong>.</p>
+</div>
+<div class="paragraph">
+<p>A variable qualified as <strong>flat</strong> will not be interpolated.
+Instead, it will have the same value for every fragment within a primitive.
+This value will come from a single provoking vertex, as described by the
+<a href="#references">OpenGL Specification</a>.
+A variable qualified as <strong>flat</strong> may also be qualified as <strong>centroid</strong> or
+<strong>sample</strong>, which will mean the same thing as qualifying it only as <strong>flat</strong>.</p>
+</div>
+<div class="paragraph">
+<p>A variable qualified as <strong>smooth</strong> will be interpolated in a
+perspective-correct manner over the primitive being rendered.
+Interpolation in a perspective correct manner is specified in equation 14.7
+of the <a href="#references">OpenGL Specification</a>, section 14.5 “Line Segments”.</p>
+</div>
+<div class="paragraph">
+<p>A variable qualified as <strong>noperspective</strong> must be interpolated linearly in
+screen space, as described in equation 3.7 of the <a href="#references">OpenGL Specification</a>,
+section 3.5 “Line Segments”.</p>
+</div>
+<div class="paragraph">
+<p>When multisample rasterization is disabled, or for fragment shader input
+variables qualified with neither <strong>centroid</strong> nor <strong>sample</strong>, the value of the
+assigned variable may be interpolated anywhere within the pixel and a single
+value may be assigned to each sample within the pixel, to the extent
+permitted by the <a href="#references">OpenGL Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>When multisample rasterization is enabled, <strong>centroid</strong> and <strong>sample</strong> may be
+used to control the location and frequency of the sampling of the qualified
+fragment shader input.
+If a fragment shader input is qualified with <strong>centroid</strong>, a single value may
+be assigned to that variable for all samples in the pixel, but that value
+must be interpolated at a location that lies in both the pixel and in the
+primitive being rendered, including any of the pixel’s samples covered by
+the primitive.
+Because the location at which the variable is interpolated may be different
+in neighboring pixels, and derivatives may be computed by computing
+differences between neighboring pixels, derivatives of centroid-sampled
+inputs may be less accurate than those for non-centroid interpolated
+variables.
+If a fragment shader input is qualified with <strong>sample</strong>, a separate value must
+be assigned to that variable for each covered sample in the pixel, and that
+value must be sampled at the location of the individual sample.</p>
+</div>
+<div class="paragraph">
+<p>It is a link-time error if, within the same stage, the interpolation
+qualifiers of variables of the same name do not match.</p>
+</div>
+<div class="sect3">
+<h4 id="redeclaring-built-in-interpolation-variables-in-the-compatibility-profile">4.5.1. Redeclaring Built-In Interpolation Variables in the Compatibility Profile</h4>
+<div class="paragraph">
+<p>The following predeclared variables can be redeclared with an interpolation
+qualifier when using the compatibility profile:</p>
+</div>
+<div class="paragraph">
+<p>Vertex, tessellation control, tessellation evaluation, and geometry
+languages:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_FrontColor
+gl_BackColor
+gl_FrontSecondaryColor
+gl_BackSecondaryColor</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Fragment language:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_Color
+gl_SecondaryColor</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 gl_Color; <span class="comment">// predeclared by the fragment language</span>
+flat in vec4 gl_Color; <span class="comment">// redeclared by user to be flat</span>
+flat in vec4 gl_FrontColor; <span class="comment">// input to geometry shader, no "gl_in[]"</span>
+flat out vec4 gl_FrontColor; <span class="comment">// output from geometry shader</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Ideally, these are redeclared as part of the redeclaration of an interface
+block, as described in
+“<a href="#compatibility-profile-built-in-language-variables">Compatibility Profile
+Built-In Language Variables</a>”.
+However, for the above purpose, they can be redeclared as individual
+variables at global scope, outside an interface block.
+Such redeclarations also allow adding the transform-feedback qualifiers
+<strong>xfb_buffer</strong>, <strong>xfb_stride</strong>, and <strong>xfb_offset</strong> to output variables.
+(Using <strong>xfb_buffer</strong> on a variable does not change the global default
+buffer.) A compile-time error will result if a shader has both an interface
+block redeclaration and a separate redeclaration of a member of that
+interface block outside the interface block redeclaration.</p>
+</div>
+<div class="paragraph">
+<p>If <em>gl_Color</em> is redeclared with an interpolation qualifier, then
+<em>gl_FrontColor</em> and <em>gl_BackColor</em> (if they are written to) must also be
+redeclared with the same interpolation qualifier, and vice versa.
+If <em>gl_SecondaryColor</em> is redeclared with an interpolation qualifier, then
+<em>gl_FrontSecondaryColor</em> and _gl_BackSecondaryColor _(if they are written
+to) must also be redeclared with the same interpolation qualifier, and vice
+versa.
+This qualifier matching on predeclared variables is only required for
+variables that are statically used within the shaders in a program.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="parameter-qualifiers">4.6. Parameter Qualifiers</h3>
+<div class="paragraph">
+<p>In addition to precision qualifiers and memory qualifiers, parameters can
+have these parameter qualifiers.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><none: default></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">same as <strong>in</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>const</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for function parameters that cannot be written to</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>in</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for function parameters passed into a function</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>out</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for function parameters passed back out of a function,
+ but not initialized for use when passed in</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>inout</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">for function parameters passed both into and out of a
+ function</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Parameter qualifiers are discussed in more detail in
+“<a href="#function-calling-conventions">Function Calling Conventions</a>”.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="precision-and-precision-qualifiers">4.7. Precision and Precision Qualifiers</h3>
+<div class="paragraph">
+<p>Precision qualifiers are added for code portability with OpenGL ES, not for
+functionality.
+They have the same syntax as in OpenGL ES, as described below, but they have
+no semantic meaning, which includes no effect on the precision used to store
+or operate on variables.</p>
+</div>
+<div class="paragraph">
+<p>If an extension adds in the same semantics and functionality in the OpenGL
+ES 2.0 specification for precision qualifiers, then the extension is allowed
+to reuse the keywords below for that purpose.</p>
+</div>
+<div class="paragraph">
+<p>For the purposes of determining if an output from one shader stage matches
+an input of the next stage, the precision qualifier need not match.</p>
+</div>
+<div class="sect3">
+<h4 id="range-and-precision">4.7.1. Range and Precision</h4>
+<div class="paragraph">
+<p>The precision of stored single- and double-precision floating-point
+variables is defined by the IEEE 754 standard for 32-bit and 64-bit
+floating-point numbers.
+This includes support for NaNs (Not a Number) and Infs (positive or negative
+infinities) and positive and negative zeros.</p>
+</div>
+<div class="paragraph">
+<p>The following rules apply to
+both single and double-precision
+operations:
+Signed infinities and zeros are generated as dictated by IEEE, but subject
+to the precisions allowed in the following table.
+Any subnormal (denormalized) value input into a shader or potentially
+generated by any operation in a shader can be flushed to 0.
+The rounding mode cannot be set and is undefined but must not affect the
+result by more than 1 ULP.
+NaNs are not required to be generated.
+Support for signaling NaNs is not required and exceptions are never raised.
+Operations including built-in functions that operate on a NaN are not
+required to return a NaN as the result.
+However if NaNs are generated, <strong>isnan</strong>() must return the correct value.</p>
+</div>
+<div class="paragraph">
+<p>Precisions are expressed in terms of maximum relative error in units of ULP
+(units in the last place), unless otherwise noted.</p>
+</div>
+<div class="paragraph">
+<p>For single precision operations, precisions are required as follows:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Operation</th>
+<th class="tableblock halign-left valign-top">Precision</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>a</em> + <em>b</em>, <em>a</em> - <em>b</em>, <em>a</em> * <em>b</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Correctly rounded.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><, <=, ==, >, >=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Correct result.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>a</em> / <em>b</em>, 1.0 / <em>b</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2.5 ULP for <span class="eq">|b|</span> in the range <span class="eq">[2<sup>-126</sup>, 2<sup>126</sup>]</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>a</em> * <em>b</em> + <em>c</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Correctly rounded single operation or
+ sequence of two correctly rounded operations.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>fma</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inherited from <em>a</em> * <em>b</em> + <em>c</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>pow</strong>(<em>x</em>, <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inherited from <strong>exp2</strong>(<em>y</em> * <strong>log2</strong>(<em>x</em>)).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>exp</strong>(<em>x</em>), <strong>exp2</strong>(<em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><span class="eq">(3 + 2 · |x|)</span> ULP.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>log</strong>(), <strong>log2</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3 ULP outside the range <span class="eq">[0.5,2.0]</span>.<br>
+ Absolute error < 2<sup>-21</sup> inside the range
+ <span class="eq">[0.5,2.0]</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>sqrt</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inherited from 1.0 / <strong>inversesqrt</strong>().</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>inversesqrt</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2 ULP.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">implicit and explicit
+ conversions between types</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Correctly rounded.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Built-in functions defined in the specification with an equation built from
+the above operations inherit the above errors.
+These include, for example, the geometric functions, the common functions,
+and many of the matrix functions.
+Built-in functions not listed above and not defined as equations of the
+above have undefined precision.
+These include, for example, the trigonometric functions and determinant.</p>
+</div>
+<div class="paragraph">
+<p>The precision of double-precision operations is at least that of single
+precision.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="precision-qualifiers">4.7.2. Precision Qualifiers</h4>
+<div class="paragraph">
+<p>Any
+single-precision
+floating-point, integer, or opaque-type declaration can have the type
+preceded by one of these precision qualifiers:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>highp</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">None.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>mediump</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">None.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>lowp</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">None.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">lowp <span class="predefined-type">float</span> color;
+out mediump vec2 P;
+lowp ivec2 foo(lowp mat3);
+highp mat4 m;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Literal constants do not have precision qualifiers.
+Neither do Boolean variables.
+Neither do constructors.</p>
+</div>
+<div class="paragraph">
+<p>Precision qualifiers, as with other qualifiers, do not affect the basic type
+of the variable.
+In particular, there are no constructors for precision conversions;
+constructors only convert types.
+Similarly, precision qualifiers, as with other qualifiers, do not contribute
+to function overloading based on parameter types.
+As discussed in “<a href="#function-calling-conventions">Function Calling
+Conventions</a>”, function input and output is done through copies, and
+therefore qualifiers do not have to match.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="default-precision-qualifiers">4.7.3. Default Precision Qualifiers</h4>
+<div class="paragraph">
+<p>The precision statement</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precision precision-qualifier type;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>can be used to establish a default precision qualifier.
+The <em>type</em> field can be either <strong>int</strong>, <strong>float</strong>, or any of the opaque types,
+and the <em>precision-qualifier</em> can be <strong>lowp</strong>, <strong>mediump</strong>, or <strong>highp</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Any other types or qualifiers will result in a compile-time error.
+If <em>type</em> is <strong>float</strong>, the directive applies to non-precision-qualified
+single-precision
+floating-point type (scalar, vector, and matrix) declarations.
+If <em>type</em> is <strong>int</strong>, the directive applies to all non-precision-qualified
+integer type (scalar, vector, signed, and unsigned) declarations.
+This includes global variable declarations, function return declarations,
+function parameter declarations, and local variable declarations.</p>
+</div>
+<div class="paragraph">
+<p>Non-precision qualified declarations will use the precision qualifier
+specified in the most recent <strong>precision</strong> statement that is still in scope.
+The <strong>precision</strong> statement has the same scoping rules as variable
+declarations.
+If it is declared inside a compound statement, its effect stops at the end
+of the inner-most statement it was declared in.
+Precision statements in nested scopes override precision statements in outer
+scopes.
+Multiple precision statements for the same basic type can appear inside the
+same scope, with later statements overriding earlier statements within that
+scope.</p>
+</div>
+<div class="paragraph">
+<p>All languages except for the fragment language have the following
+predeclared globally scoped default precision statements:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precision highp <span class="predefined-type">float</span>;
+precision highp <span class="predefined-type">int</span>;
+precision highp atomic_uint;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The fragment language has the following predeclared globally scoped default
+precision statements:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precision mediump <span class="predefined-type">int</span>;
+precision highp <span class="predefined-type">float</span>;
+precision highp atomic_uint;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>There are no errors for omission of a precision qualifier; so the above is
+just for reference of what may happen in OpenGL ES versions of the shading
+languages.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="available-precision-qualifiers">4.7.4. Available Precision Qualifiers</h4>
+<div class="paragraph">
+<p>The built-in macro GL_FRAGMENT_PRECISION_HIGH is defined to one:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#define</span> GL_FRAGMENT_PRECISION_HIGH <span class="integer">1</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This macro is available in all languages except compute.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="variance-and-the-invariant-qualifier">4.8. Variance and the Invariant Qualifier</h3>
+<div class="paragraph">
+<p>In this section, <em>variance</em> refers to the possibility of getting different
+values from the same expression in different programs.
+For example, consider the situation where two vertex shaders, in different
+programs, each set <em>gl_Position</em> with the same expression, and the input
+values into that expression are the same when both shaders run.
+It is possible, due to independent compilation of the two shaders, that the
+values assigned to <em>gl_Position</em> are not exactly the same when the two
+shaders run.
+In this example, this can cause problems with alignment of geometry in a
+multi-pass algorithm.</p>
+</div>
+<div class="paragraph">
+<p>In general, such variance between shaders is allowed.
+When such variance does not exist for a particular output variable, that
+variable is said to be <em>invariant</em>.</p>
+</div>
+<div class="sect3">
+<h4 id="the-invariant-qualifier">4.8.1. The Invariant Qualifier</h4>
+<div class="paragraph">
+<p>To ensure that a particular output variable is invariant, it is necessary to
+use the <strong>invariant</strong> qualifier.
+It can either be used to qualify a previously declared variable as being
+invariant:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">invariant gl_Position; <span class="comment">// make existing gl_Position be invariant</span>
+out vec3 Color;
+invariant Color; <span class="comment">// make existing Color be invariant</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>or as part of a declaration when a variable is declared:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">invariant centroid out vec3 Color;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Only variables output from a shader can be candidates for invariance.
+This includes user-defined output variables and the built-in output
+variables.
+As only outputs can be declared as invariant, an output from one shader
+stage will still match an input of a subsequent stage without the input
+being declared as invariant.</p>
+</div>
+<div class="paragraph">
+<p>Input or output instance names on blocks are not used when redeclaring
+built-in variables.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>invariant</strong> keyword can be followed by a comma separated list of
+previously declared identifiers.
+All uses of <strong>invariant</strong> must be at global scope or on block
+members, and before any use of the variables being declared as invariant.</p>
+</div>
+<div class="paragraph">
+<p>To guarantee invariance of a particular output variable across two programs,
+the following must also be true:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The output variable is declared as invariant in both programs.</p>
+</li>
+<li>
+<p>The same values must be input to all shader input variables consumed by
+expressions and control flow contributing to the value assigned to the
+output variable.</p>
+</li>
+<li>
+<p>The texture formats, texel values, and texture filtering are set the
+same way for any texture function calls contributing to the value of the
+output variable.</p>
+</li>
+<li>
+<p>All input values are all operated on in the same way.
+ All operations in the consuming expressions and any intermediate
+ expressions must be the same, with the same order of operands and same
+ associativity, to give the same order of evaluation.
+ Intermediate variables and functions must be declared as the same type
+ with the same explicit or implicit precision
+qualifiers.
+ Any control flow affecting the output value must be the same, and any
+ expressions consumed to determine this control flow must also follow
+ these invariance rules.</p>
+</li>
+<li>
+<p>All the data flow and control flow leading to setting the invariant
+output variable reside in a single compilation unit.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Essentially, all the data flow and control flow leading to an invariant
+output must match.</p>
+</div>
+<div class="paragraph">
+<p>Initially, by default, all output variables are allowed to be variant.
+To force all output variables to be invariant, use the pragma</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="preprocessor">#pragma</span> STDGL invariant(all)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>before all declarations in a shader.
+If this pragma is used after the declaration of any variables or functions,
+then the set of outputs that behave as invariant is undefined.</p>
+</div>
+<div class="paragraph">
+<p>Generally, invariance is ensured at the cost of flexibility in optimization,
+so performance can be degraded by use of invariance.
+Hence, use of this pragma is intended as a debug aid, to avoid individually
+declaring all output variables as invariant.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="invariance-of-constant-expressions">4.8.2. Invariance of Constant Expressions</h4>
+<div class="paragraph">
+<p>Invariance must be guaranteed for constant expressions.
+A particular constant expression must evaluate to the same result if it
+appears again in the same shader or a different shader.
+This includes the same expression appearing in two shaders of the same
+language or shaders of two different languages.</p>
+</div>
+<div class="paragraph">
+<p>Constant expressions must evaluate to the same result when operated on as
+already described above for invariant variables.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="the-precise-qualifier">4.9. The Precise Qualifier</h3>
+<div class="paragraph">
+<p>Some algorithms require floating-point computations to exactly follow the
+order of operations specified in the source code and to treat all operations
+consistently, even if the implementation supports optimizations that could
+produce nearly equivalent results with higher performance.
+For example, many GL implementations support a “multiply-add” instruction
+that can compute a floating-point expression such as</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">result = (a * b) + (c * d);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>in two operations instead of three operations; one multiply and one
+multiply-add instead of two multiplies and one add.
+The result of a floating-point multiply-add might not always be identical to
+first doing a multiply yielding a floating-point result and then doing a
+floating-point add.
+Hence, in this example, the two multiply operations would not be treated
+consistently; the two multiplies could effectively appear to have differing
+precisions.</p>
+</div>
+<div class="paragraph">
+<p>The key computation that needs to be made consistent appears when
+tessellating, where intermediate points for subdivision are synthesized in
+different directions, yet need to yield the same result, as shown in the
+diagram below.</p>
+</div>
+<div id="img-precise" class="imageblock">
+<div class="content">
+<img src="data:image/svg+xml;base64,<?xml version="1.0" encoding="UTF-8"?>

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       <path fill="none" stroke="rgb(0,0,0)" d="M 10373,7223 L 10416,7254"/>
       <path fill="none" stroke="rgb(0,0,0)" d="M 10459,7285 L 10502,7316"/>
       <path fill="none" stroke="rgb(0,0,0)" d="M 10546,7347 L 10589,7378"/>
       <path fill="none" stroke="rgb(0,0,0)" d="M 10632,7409 L 10675,7439"/>
       <path fill="none" stroke="rgb(0,0,0)" d="M 10718,7470 L 10762,7501"/>
       <path fill="none" stroke="rgb(0,0,0)" d="M 10805,7532 L 10848,7563"/>
       <path fill="none" stroke="rgb(0,0,0)" d="M 10891,7594 L 10900,7600"/>
      </g>
     </g>
    </g>
   </g>
  </g>
 </g>
</svg>" alt="precise">
+</div>
+</div>
+<div class="paragraph">
+<p>Without any qualifiers, implementations are permitted to perform
+optimizations that effectively modify the order or number of operations used
+to evaluate an expression, even if those optimizations may produce slightly
+different results relative to unoptimized code.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>precise</strong> qualifier ensures that operations contributing to a
+variable’s value are done in their stated order and with operator consistency.
+The order is determined by operator precedence and parenthesis, as described in
+“<a href="#operators">Operators</a>”.
+Operator consistency means for each particular operator, for example the
+multiply operator (<strong>*</strong>), its operation is always computed with the same
+precision.
+Specifically, values computed by compiler-generated code must adhere to the
+following identities:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>a + b = b + a</p>
+</li>
+<li>
+<p>a * b = b * a</p>
+</li>
+<li>
+<p>a * b + c * d = b * a + c* d = d * c + b * a = <any other mathematically
+valid combination></p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>While the following are prevented:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>a + (b + c) is not allowed to become (a + b) + c</p>
+</li>
+<li>
+<p>a * (b * c) is not allowed to become (a * b) * c</p>
+</li>
+<li>
+<p>a * b + c is not allowed to become a single operation <strong>fma</strong>(a, b, c)</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>Where <em>a</em>, <em>b</em>, <em>c</em>, and <em>d</em>, are scalars or vectors, not matrices.
+(Matrix multiplication generally does not commute.) It is the shader
+writer’s responsibility to express the computation in terms of these rules
+and the compiler’s responsibility to follow these rules.
+See the description of <em>gl_TessCoord</em> for the rules the tessellation stages
+are responsible for following, which in conjunction with the above allow
+avoiding cracking when subdividing.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">precise out vec4 position;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>declares that operations used to produce the value of <em>position</em> must be
+performed in exactly the order specified in the source code and with all
+operators being treated consistently.
+As with the <strong>invariant</strong> qualifier (see “<a href="#the-invariant-qualifier">The
+Invariant Qualifier</a>”), the <strong>precise</strong> qualifier may be used to qualify a
+built-in or previously declared user-defined variable as being precise:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out vec3 Color;
+precise Color; <span class="comment">// make existing Color be precise</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When applied to a block, a structure type, or a variable of structure type,
+<strong>precise</strong> applies to each contained member, recursively.</p>
+</div>
+<div class="paragraph">
+<p>This qualifier will affect the evaluation of an r-value in a particular
+function if and only if the result is eventually consumed in the same
+function by an l-value qualified as <strong>precise</strong>.
+Any other expressions within a function are not affected, including return
+values and output parameters not declared as <strong>precise</strong> but that are
+eventually consumed outside the function by a variable qualified as
+<strong>precise</strong>. Unaffected expressions also include the controlling expressions
+in selection and iteration statements and the condition in ternary
+operators (<strong>?:</strong>).</p>
+</div>
+<div class="paragraph">
+<p>Some examples of the use of <strong>precise</strong>:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 a, b, c, d;
+precise out vec4 v;
+
+<span class="predefined-type">float</span> func(<span class="predefined-type">float</span> e, <span class="predefined-type">float</span> f, <span class="predefined-type">float</span> g, <span class="predefined-type">float</span> h)
+{
+ <span class="keyword">return</span> (e*f) + (g*h); <span class="comment">// no constraint on order or operator consistency</span>
+}
+
+<span class="predefined-type">float</span> func2(<span class="predefined-type">float</span> e, <span class="predefined-type">float</span> f, <span class="predefined-type">float</span> g, <span class="predefined-type">float</span> h)
+{
+ precise <span class="predefined-type">float</span> result = (e*f) + (g*h); <span class="comment">// ensures same precision for the two multiplies</span>
+ <span class="keyword">return</span> result;
+}
+
+<span class="predefined-type">float</span> func3(<span class="predefined-type">float</span> i, <span class="predefined-type">float</span> j, precise out <span class="predefined-type">float</span> k)
+{
+ k = i * i + j; <span class="comment">// precise, due to <k> declaration</span>
+}
+
+<span class="directive">void</span> main()
+{
+ vec3 r = vec3(a * b); <span class="comment">// precise, used to compute v.xyz</span>
+ vec3 s = vec3(c * d); <span class="comment">// precise, used to compute v.xyz</span>
+ v.xyz = r + s; <span class="comment">// precise</span>
+ v.w = (a.w * b.w) + (c.w * d.w); <span class="comment">// precise</span>
+ v.x = func(a.x, b.x, c.x, d.x); <span class="comment">// values computed in func() are NOT precise</span>
+ v.x = func2(a.x, b.x, c.x, d.x); <span class="comment">// precise!</span>
+ func3(a.x * b.x, c.x * d.x, v.x); <span class="comment">// precise!</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>For the purposes of determining if an output from one shader stage matches
+an input of the next stage, the <strong>precise</strong> qualifier need not match between
+the input and the output.</p>
+</div>
+<div class="paragraph">
+<p>All constant expressions are evaluated as if <strong>precise</strong> was present, whether
+or not it is present.
+However, as described in “<a href="#constant-expressions">Constant Expressions</a>”,
+there is no requirement that a compile-time constant expression evaluates to
+the same value as a corresponding non-constant expression.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="memory-qualifiers">4.10. Memory Qualifiers</h3>
+<div class="paragraph">
+<p>Shader storage blocks, variables declared within shader storage blocks and
+variables declared as image types (the basic opaque types with “<strong>image</strong>”
+in their keyword), can be further qualified with one or more of the
+following memory qualifiers:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Qualifier</th>
+<th class="tableblock halign-left valign-top">Meaning</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>coherent</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable where reads and writes are coherent with
+ reads and writes from other shader invocations</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>volatile</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable whose underlying value may be changed at any
+ point during shader execution by some source other than the
+ current shader invocation</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>restrict</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable where use of that variable is the only way
+ to read and write the underlying memory in the relevant
+ shader stage</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>readonly</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable that can be used to read the underlying
+ memory, but cannot be used to write the underlying memory</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>writeonly</strong></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">memory variable that can be used to write the underlying
+ memory, but cannot be used to read the underlying memory</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>coherent</strong> qualifier
+are performed coherently with accesses to the same location from other
+shader invocations.
+In particular, when reading a variable declared as <strong>coherent</strong>, the values
+returned will reflect the results of previously completed writes performed
+by other shader invocations.
+When writing a variable declared as <strong>coherent</strong>, the values written will be
+reflected in subsequent coherent reads performed by other shader
+invocations.</p>
+</div>
+<div class="paragraph">
+<p>As described in section
+7.12
+“Shader Memory Access” of the
+<a href="#references">OpenGL Specification</a>, shader memory reads and writes complete in a
+largely undefined order.
+The built-in function <strong>memoryBarrier</strong>() can be used if needed to guarantee
+the completion and relative ordering of memory accesses performed by a
+single shader invocation.</p>
+</div>
+<div class="paragraph">
+<p>When accessing memory using variables not declared as <strong>coherent</strong>, the memory
+accessed by a shader may be cached by the implementation to service future
+accesses to the same address.
+Memory stores may be cached in such a way that the values written may not be
+visible to other shader invocations accessing the same memory.
+The implementation may cache the values fetched by memory reads and return
+the same values to any shader invocation accessing the same memory, even if
+the underlying memory has been modified since the first memory read.
+While variables not declared as <strong>coherent</strong> may not be useful for
+communicating between shader invocations, using non-coherent accesses may
+result in higher performance.</p>
+</div>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>volatile</strong> qualifier
+must treat the underlying memory as though it could be read or written at
+any point during shader execution by some source other than the executing
+shader invocation.
+When a volatile variable is read, its value must be re-fetched from the
+underlying memory, even if the shader invocation performing the read had
+previously fetched its value from the same memory.
+When a volatile variable is written, its value must be written to the
+underlying memory, even if the compiler can conclusively determine that its
+value will be overwritten by a subsequent write.
+Since the external source reading or writing a <strong>volatile</strong> variable may be
+another shader invocation, variables declared as <strong>volatile</strong> are
+automatically treated as coherent.</p>
+</div>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>restrict</strong> qualifier
+may be compiled assuming that the variable used to perform the memory access
+is the only way to access the underlying memory using the shader stage in
+question.
+This allows the compiler to coalesce or reorder loads and stores using
+<strong>restrict</strong>-qualified image variables in ways that wouldn’t be permitted
+for image variables not so qualified, because the compiler can assume that
+the underlying image won’t be read or written by other code.
+Applications are responsible for ensuring that image memory referenced by
+variables qualified with <strong>restrict</strong> will not be referenced using other
+variables in the same scope; otherwise, accesses to <strong>restrict</strong>-qualified
+variables will have undefined results.</p>
+</div>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>readonly</strong> qualifier
+may only read the underlying memory, which is treated as read-only memory
+and cannot be written to.
+It is a compile-time error to pass an image variable qualified with
+<strong>readonly</strong> to <strong>imageStore</strong>() or other built-in functions that modify image
+memory.</p>
+</div>
+<div class="paragraph">
+<p>Memory accesses to image variables declared using the <strong>writeonly</strong> qualifier
+may only write the underlying memory; the underlying memory cannot be read.
+It is a compile-time error to pass an image variable qualified with
+<strong>writeonly</strong> to <strong>imageLoad</strong>() or other built-in functions that read image
+memory.</p>
+</div>
+<div class="paragraph">
+<p>A variable could be qualified as both <strong>readonly</strong> and <strong>writeonly</strong>, disallowing
+both read and write. Such variables can still be used with some queries, for
+example <strong>imageSize</strong>() and <strong>.length</strong>().</p>
+</div>
+<div class="paragraph">
+<p>The memory qualifiers <strong>coherent</strong>, <strong>volatile</strong>, <strong>restrict</strong>, <strong>readonly</strong>, and
+<strong>writeonly</strong> may be used in the declaration of buffer variables (i.e.,
+members of shader storage blocks).
+When a buffer variable is declared with a memory qualifier, the behavior
+specified for memory accesses involving image variables described above
+applies identically to memory accesses involving that buffer variable.
+It is a compile-time error to assign to a buffer variable qualified with
+<strong>readonly</strong> or to read from a buffer variable qualified with <strong>writeonly</strong>.
+The combination <strong>readonly</strong> <strong>writeonly</strong> is allowed.</p>
+</div>
+<div class="paragraph">
+<p>Additionally, memory qualifiers may be used at the block-level declaration
+of a shader storage block, including the combination <strong>readonly</strong> <strong>writeonly</strong>.
+When a block declaration is qualified with a memory qualifier, it is as if
+all of its members were declared with the same memory qualifier.
+For example, the block declaration</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">coherent buffer Block {
+ readonly vec4 member1;
+ vec4 member2;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">buffer Block {
+ coherent readonly vec4 member1;
+ coherent vec4 member2;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Memory qualifiers are only supported in the declarations of image variables,
+buffer variables, and shader storage blocks; it is an error to use such
+qualifiers in any other declarations.</p>
+</div>
+<div class="paragraph">
+<p>When calling user-defined functions, variables qualified with <strong>coherent</strong>,
+<strong>volatile</strong>, <strong>readonly</strong>, or <strong>writeonly</strong> may not be passed to functions whose
+formal parameters lack such qualifiers.
+(See “<a href="#function-definitions">Function Definitions</a>” for more detail on
+function calling.) It is legal to have any additional memory qualifiers on a
+formal parameter, but only <strong>restrict</strong> can be taken away from a calling
+argument, by a formal parameter that lacks the <strong>restrict</strong> qualifier.</p>
+</div>
+<div class="paragraph">
+<p>When a built-in function is called, the code generated is to be based on the
+actual qualification of the calling argument, not on the list of memory
+qualifiers specified on the formal parameter in the prototype.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 funcA(restrict image2D a) { ... }
+vec4 funcB(image2D a) { ... }
+layout(rgba32f) uniform image2D img1;
+layout(rgba32f) coherent uniform image2D img2;
+
+funcA(img1); <span class="comment">// OK, adding "restrict" is allowed</span>
+funcB(img2); <span class="comment">// illegal, stripping "coherent" is not</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Layout qualifiers cannot be used on formal function parameters, and layout
+qualification is not included in parameter matching.</p>
+</div>
+<div class="paragraph">
+<p>Note that the use of <strong>const</strong> in an image variable declaration is qualifying
+the const-ness of the variable being declared, not the image it refers to.
+The qualifier <strong>readonly</strong> qualifies the image memory (as accessed through
+that variable) while <strong>const</strong> qualifies the variable itself.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="specialization-constant-qualifier">4.11. Specialization-Constant Qualifier</h3>
+<div class="paragraph">
+<p>Specialization constants are used only for SPIR-V and declared using the
+<strong>constant_id</strong> layout qualifier.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(constant_id = <span class="integer">17</span>) <span class="directive">const</span> <span class="predefined-type">int</span> arraySize = <span class="integer">12</span>;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The above makes a specialization constant with a default value of 12.
+The number 17 is an example author-chosen id by which the API or other tools
+can later refer to this specific specialization constant.
+If it is never changed before final lowering, it will retain the value of
+12.
+It is a compile-time error to use the <strong>constant_id</strong> qualifier on anything
+but SPIR-V generation of a scalar <strong>bool</strong>, <strong>int</strong>, <strong>uint</strong>, <strong>float</strong>, or
+<strong>double</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Built-in constants can be declared to be specialization constants.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(constant_id = <span class="integer">31</span>) gl_MaxClipDistances; <span class="comment">// add specialization_id</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The declaration uses just the name of the previously declared built-in
+variable, with a <strong>constant_id</strong> layout-qualifier declaration.
+It is a compile-time error to do this after the constant has been used:
+Constants are strictly either non-specialization constants or specialization
+constants, not both.</p>
+</div>
+<div class="paragraph">
+<p>The built-in constant vector <em>gl_WorkGroupSize</em> can be specialized using the
+<strong>local_size_{xyz}_id</strong> qualifiers, to individually give the components an id.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(local_size_x_id = <span class="integer">18</span>, local_size_z_id = <span class="integer">19</span>) in;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This leaves <em>gl_WorkGroupSize.y</em> as a non-specialization constant, with
+<em>gl_WorkGroupSize</em> being a partially specialized vector.
+Its <em>x</em> and <em>z</em> components can be later specialized, after generating
+SPIR-V, using the ids 18 and 19.
+These ids are declared independently from declaring the workgroup size:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(local_size_x = <span class="integer">32</span>, local_size_y = <span class="integer">32</span>) in; <span class="comment">// size is (32,32,1)</span>
+layout(local_size_x_id = <span class="integer">18</span>) in; <span class="comment">// constant_id for x</span>
+layout(local_size_z_id = <span class="integer">19</span>) in; <span class="comment">// constant_id for z</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Existing rules for declaring <strong>local_size_x</strong>, <strong>local_size_y</strong>, and
+<strong>local_size_z</strong> are not changed.
+For the local-size ids, it is a compile-time error to provide different id
+values for the same local-size id, or to provide them after any use.
+Otherwise, order, placement, number of statements, and replication do not
+cause errors.</p>
+</div>
+<div class="paragraph">
+<p>Two arrays sized with specialization constants are the same type only if
+sized with the same symbol, and involving no operations.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">layout(constant_id = <span class="integer">51</span>) <span class="directive">const</span> <span class="predefined-type">int</span> aSize = <span class="integer">20</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> pad = <span class="integer">2</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> total = aSize + pad; <span class="comment">// specialization constant</span>
+<span class="predefined-type">int</span> a[total], b[total]; <span class="comment">// a and b have the same type</span>
+<span class="predefined-type">int</span> c[<span class="integer">22</span>]; <span class="comment">// different type than a or b</span>
+<span class="predefined-type">int</span> d[aSize + pad]; <span class="comment">// different type than a, b, or c</span>
+<span class="predefined-type">int</span> e[aSize + <span class="integer">2</span>]; <span class="comment">// different type than a, b, c, or d</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Types containing arrays sized with a specialization constant cannot be
+compared, assigned as aggregates, declared with an initializer, or used as
+an initializer.
+They can, however, be passed as arguments to functions having formal
+parameters of the same type.
+Only the outer-most dimension of a variable declared as an array of arrays
+can be a specialization constant, otherwise a compile-time error results.</p>
+</div>
+<div class="paragraph">
+<p>Arrays inside a block may be sized with a specialization constant, but the
+block will have a static layout.
+Changing the specialized size will not re-layout the block.
+In the absence of explicit offsets, the layout will be based on the default
+size of the array.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="order-of-qualification">4.12. Order and Repetition of Qualification</h3>
+<div class="paragraph">
+<p>When multiple qualifiers are present in a declaration, they may appear in
+any order, but they must all appear before the type.
+The <strong>layout</strong> qualifier is the only qualifier that can appear more than once.
+Further, a declaration can have at most one storage qualifier, at most one
+auxiliary storage qualifier, and at most one interpolation qualifier.
+If <strong>inout</strong> is used, neither <strong>in</strong> nor <strong>out</strong> may be used.
+Multiple memory qualifiers can be used.
+Any violation of these rules will cause a compile-time error.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="empty-declarations">4.13. Empty Declarations</h3>
+<div class="paragraph">
+<p><em>Empty declarations</em> are declarations without a variable name, meaning no
+object is instantiated by the declaration.
+Generally, empty declarations are allowed.
+Some are useful when declaring structures, while many others have no effect.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span>; <span class="comment">// No effect</span>
+<span class="keyword">struct</span> S {<span class="predefined-type">int</span> x;}; <span class="comment">// Defines a struct S</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The combinations of qualifiers that cause compile-time or link-time errors
+are the same whether or not the declaration is empty, for example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">invariant in <span class="predefined-type">float</span> x; <span class="comment">// Error. An input cannot be invariant.</span>
+invariant in <span class="predefined-type">float</span>; <span class="comment">// Error even though no variable is declared.</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="operators-and-expressions">5. Operators and Expressions</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="operators">5.1. Operators</h3>
+<div class="paragraph">
+<p>The OpenGL Shading Language has the following operators.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Precedence</th>
+<th class="tableblock halign-left valign-top">Operator Class</th>
+<th class="tableblock halign-left valign-top">Operators</th>
+<th class="tableblock halign-left valign-top">Associativity</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1 (highest)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">parenthetical grouping</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">( )</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">NA</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">array subscript<br>
+ function call and constructor structure<br>
+ field or method selector, swizzle<br>
+ post fix increment and decrement</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">[ ]<br>
+ ( )<br>
+ .<br>
+ ++ --</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">prefix increment and decrement<br>
+ unary</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">++ --<br>
+ + - ~ !</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Right to Left</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">multiplicative</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">* / %</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">additive</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">+ -</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">6</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise shift</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><< >></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">7</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">relational</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">< > <= >=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">8</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">equality</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">== !=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">9</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise and</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">&</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">10</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise exclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">^</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">11</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bit-wise inclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">|</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">12</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical and</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">&&</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">13</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical exclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">^^</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">14</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">logical inclusive or</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">||</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">15</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">selection</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">? :</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Right to Left</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">16</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Assignment<br>
+ arithmetic assignments</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">=<br>
+ += -=<br>
+ *= /=<br>
+ %= <<= >>=<br>
+ &= ^= |=</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Right to Left</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">17 (lowest)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">sequence</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">,</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Left to Right</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>There is no address-of operator nor a dereference operator.
+There is no typecast operator; constructors are used instead.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="array-operations">5.2. Array Operations</h3>
+<div class="paragraph">
+<p>These are now described in “<a href="#structure-and-array-operations">Structure and
+Array Operations</a>”.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="function-calls">5.3. Function Calls</h3>
+<div class="paragraph">
+<p>If a function returns a value, then a call to that function may be used as
+an expression, whose type will be the type that was used to declare or
+define the function.</p>
+</div>
+<div class="paragraph">
+<p>Function definitions and calling conventions are discussed in
+“<a href="#function-definitions">Function Definitions</a>”.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="constructors">5.4. Constructors</h3>
+<div class="paragraph">
+<p>Constructors use the function call syntax, where the function name is a
+type, and the call makes an object of that type.
+Constructors are used the same way in both initializers and expressions.
+(See “<a href="#shading-language-grammar">Shading Language Grammar</a>” for details.)
+The parameters are used to initialize the constructed value.
+Constructors can be used to request a data type conversion to change from
+one scalar type to another scalar type, or to build larger types out of
+smaller types, or to reduce a larger type to a smaller type.</p>
+</div>
+<div class="paragraph">
+<p>In general, constructors are not built-in functions with predetermined
+prototypes.
+For arrays and structures, there must be exactly one argument in the
+constructor for each element or member.
+For the other types, the arguments must provide a sufficient number of
+components to perform the initialization, and it is a compile-time error to
+include so many arguments that they cannot all be used.
+Detailed rules follow.
+The prototypes actually listed below are merely a subset of examples.</p>
+</div>
+<div class="sect3">
+<h4 id="conversion-and-scalar-constructors">5.4.1. Conversion and Scalar Constructors</h4>
+<div class="paragraph">
+<p>Converting between scalar types is done as the following prototypes
+indicate:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">int</span>(uint) <span class="comment">// converts an unsigned integer to a signed integer</span>
+<span class="predefined-type">int</span>(<span class="predefined-type">bool</span>) <span class="comment">// converts a Boolean value to an int</span>
+<span class="predefined-type">int</span>(<span class="predefined-type">float</span>) <span class="comment">// converts a float value to an int</span>
+<span class="predefined-type">int</span>(<span class="predefined-type">double</span>) <span class="comment">// converts a double value to a signed integer</span>
+uint(<span class="predefined-type">int</span>) <span class="comment">// converts a signed integer value to an unsigned integer</span>
+uint(<span class="predefined-type">bool</span>) <span class="comment">// converts a Boolean value to an unsigned integer</span>
+uint(<span class="predefined-type">float</span>) <span class="comment">// converts a float value to an unsigned integer</span>
+uint(<span class="predefined-type">double</span>) <span class="comment">// converts a double value to an unsigned integer</span>
+<span class="predefined-type">bool</span>(<span class="predefined-type">int</span>) <span class="comment">// converts a signed integer value to a Boolean</span>
+<span class="predefined-type">bool</span>(uint) <span class="comment">// converts an unsigned integer value to a Boolean value</span>
+<span class="predefined-type">bool</span>(<span class="predefined-type">float</span>) <span class="comment">// converts a float value to a Boolean</span>
+<span class="predefined-type">bool</span>(<span class="predefined-type">double</span>) <span class="comment">// converts a double value to a Boolean</span>
+<span class="predefined-type">float</span>(<span class="predefined-type">int</span>) <span class="comment">// converts a signed integer value to a float</span>
+<span class="predefined-type">float</span>(uint) <span class="comment">// converts an unsigned integer value to a float value</span>
+<span class="predefined-type">float</span>(<span class="predefined-type">bool</span>) <span class="comment">// converts a Boolean value to a float</span>
+<span class="predefined-type">float</span>(<span class="predefined-type">double</span>) <span class="comment">// converts a double value to a float</span>
+<span class="predefined-type">double</span>(<span class="predefined-type">int</span>) <span class="comment">// converts a signed integer value to a double</span>
+<span class="predefined-type">double</span>(uint) <span class="comment">// converts an unsigned integer value to a double</span>
+<span class="predefined-type">double</span>(<span class="predefined-type">bool</span>) <span class="comment">// converts a Boolean value to a double</span>
+<span class="predefined-type">double</span>(<span class="predefined-type">float</span>) <span class="comment">// converts a float value to a double</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When constructors are used to convert a floating-point type to an integer
+type, the fractional part of the floating-point value is dropped.
+It is undefined to convert a negative floating-point value to an <strong>uint</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Integer values having more bits of precision than a single-precision
+floating-point mantissa will lose precision when converted to <strong>float</strong>.</p>
+</div>
+<div class="paragraph">
+<p>When a constructor is used to convert any integer or floating-point type to
+a <strong>bool</strong>, 0 and 0.0 are converted to <strong>false</strong>, and non-zero values are
+converted to <strong>true</strong>.
+When a constructor is used to convert a <strong>bool</strong> to any integer or
+floating-point type, <strong>false</strong> is converted to 0 or 0.0, and <strong>true</strong> is
+converted to 1 or 1.0.</p>
+</div>
+<div class="paragraph">
+<p>The constructor <strong>int</strong>(<strong>uint</strong>) preserves the bit pattern in the argument,
+which will change the argument’s value if its sign bit is set.
+The constructor <strong>uint</strong>(<strong>int</strong>) preserves the bit pattern in the argument,
+which will change its value if it is negative.</p>
+</div>
+<div class="paragraph">
+<p>Identity constructors, like <strong>float</strong>(<strong>float</strong>) are also legal, but of little
+use.</p>
+</div>
+<div class="paragraph">
+<p>Scalar constructors with non-scalar parameters can be used to take the first
+element from a non-scalar.
+For example, the constructor <strong>float</strong>(<strong>vec3</strong>) will select the first component
+of the <strong>vec3</strong> parameter.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="vector-and-matrix-constructors">5.4.2. Vector and Matrix Constructors</h4>
+<div class="paragraph">
+<p>Constructors can be used to create vectors or matrices from a set of
+scalars, vectors, or matrices.
+This includes the ability to shorten vectors.</p>
+</div>
+<div class="paragraph">
+<p>If there is a single scalar parameter to a vector constructor, it is used to
+initialize all components of the constructed vector to that scalar’s value.
+If there is a single scalar parameter to a matrix constructor, it is used to
+initialize all the components on the matrix’s diagonal, with the remaining
+components initialized to 0.0.</p>
+</div>
+<div class="paragraph">
+<p>If a vector is constructed from multiple scalars, one or more vectors, or
+one or more matrices, or a mixture of these, the vector’s components will be
+constructed in order from the components of the arguments.
+The arguments will be consumed left to right, and each argument will have
+all its components consumed, in order, before any components from the next
+argument are consumed.
+Similarly for constructing a matrix from multiple scalars or vectors, or a
+mixture of these.
+Matrix components will be constructed and consumed in column major order.
+In these cases, there must be enough components provided in the arguments to
+provide an initializer for every component in the constructed value.
+It is a compile-time error to provide extra arguments beyond this last used
+argument.</p>
+</div>
+<div class="paragraph">
+<p>If a matrix is constructed from a matrix, then each component (column <em>i</em>,
+row <em>j</em>) in the result that has a corresponding component (column <em>i</em>, row
+<em>j</em>) in the argument will be initialized from there.
+All other components will be initialized to the identity matrix.
+If a matrix argument is given to a matrix constructor, it is
+a compile-time error to have any other arguments.</p>
+</div>
+<div class="paragraph">
+<p>If the basic type (<strong>bool</strong>, <strong>int</strong>,
+<strong>float</strong>, or <strong>double</strong>)
+of a parameter to a
+constructor does not match the basic type of the object being constructed,
+the scalar construction rules (above) are used to convert the parameters.</p>
+</div>
+<div class="paragraph">
+<p>Some useful vector constructors are as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3(<span class="predefined-type">float</span>) <span class="comment">// initializes each component of the vec3 with the float</span>
+vec4(ivec4) <span class="comment">// makes a vec4 with component-wise conversion</span>
+vec4(mat2) <span class="comment">// the vec4 is column 0 followed by column 1</span>
+vec2(<span class="predefined-type">float</span>, <span class="predefined-type">float</span>) <span class="comment">// initializes a vec2 with 2 floats</span>
+ivec3(<span class="predefined-type">int</span>, <span class="predefined-type">int</span>, <span class="predefined-type">int</span>) <span class="comment">// initializes an ivec3 with 3 ints</span>
+bvec4(<span class="predefined-type">int</span>, <span class="predefined-type">int</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>) <span class="comment">// uses 4 Boolean conversions</span>
+vec2(vec3) <span class="comment">// drops the third component of a vec3</span>
+vec3(vec4) <span class="comment">// drops the fourth component of a vec4</span>
+vec3(vec2, <span class="predefined-type">float</span>) <span class="comment">// vec3.x = vec2.x, vec3.y = vec2.y, vec3.z = float</span>
+vec3(<span class="predefined-type">float</span>, vec2) <span class="comment">// vec3.x = float, vec3.y = vec2.x, vec3.z = vec2.y</span>
+vec4(vec3, <span class="predefined-type">float</span>)
+vec4(<span class="predefined-type">float</span>, vec3)
+vec4(vec2, vec2)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Some examples of these are:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 color = vec4(<span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span>, <span class="float">0</span><span class="float">.0</span>, <span class="float">1</span><span class="float">.0</span>);
+vec4 rgba = vec4(<span class="float">1</span><span class="float">.0</span>); <span class="comment">// sets each component to 1.0</span>
+vec3 rgb = vec3(color); <span class="comment">// drop the 4th component</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>To initialize the diagonal of a matrix with all other elements set to zero:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat2(<span class="predefined-type">float</span>)
+mat3(<span class="predefined-type">float</span>)
+mat4(<span class="predefined-type">float</span>)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>That is, <em>result[i][j]</em> is set to the <em>float</em> argument for all \(i
+= j\) and set to 0 for all \(i \neq j\).</p>
+</div>
+<div class="paragraph">
+<p>To initialize a matrix by specifying vectors or scalars, the components are
+assigned to the matrix elements in column-major order.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat2(vec2, vec2); <span class="comment">// one column per argument</span>
+mat3(vec3, vec3, vec3); <span class="comment">// one column per argument</span>
+mat4(vec4, vec4, vec4, vec4); <span class="comment">// one column per argument</span>
+mat3x2(vec2, vec2, vec2); <span class="comment">// one column per argument</span>
+dmat2(dvec2, dvec2);
+dmat3(dvec3, dvec3, dvec3);
+dmat4(dvec4, dvec4, dvec4, dvec4);
+mat2(<span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// first column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>); <span class="comment">// second column</span>
+mat3(<span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// first column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// second column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>); <span class="comment">// third column</span>
+mat4(<span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// first column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// second column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="comment">// third column</span>
+ <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>, <span class="predefined-type">float</span>); <span class="comment">// fourth column</span>
+mat2x3(vec2, <span class="predefined-type">float</span>, <span class="comment">// first column</span>
+ vec2, <span class="predefined-type">float</span>); <span class="comment">// second column</span>
+dmat2x4(dvec3, <span class="predefined-type">double</span>, <span class="comment">// first column</span>
+ <span class="predefined-type">double</span>, dvec3); <span class="comment">// second column</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A wide range of other possibilities exist, to construct a matrix from
+vectors and scalars, as long as enough components are present to initialize
+the matrix.
+To construct a matrix from a matrix:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat3x3(mat4x4); <span class="comment">// takes the upper-left 3x3 of the mat4x4</span>
+mat2x3(mat4x2); <span class="comment">// takes the upper-left 2x2 of the mat4x4, last row is 0,0</span>
+mat4x4(mat3x3); <span class="comment">// puts the mat3x3 in the upper-left, sets the lower right</span>
+ <span class="comment">// component to 1, and the rest to 0</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="structure-constructors">5.4.3. Structure Constructors</h4>
+<div class="paragraph">
+<p>Once a structure is defined, and its type is given a name, a constructor is
+available with the same name to construct instances of that structure.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">struct</span> light {
+ <span class="predefined-type">float</span> intensity;
+ vec3 position;
+};
+
+light lightVar = light(<span class="float">3</span><span class="float">.0</span>, vec3(<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>));</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The arguments to the constructor will be used to set the structure’s
+members, in order, using one argument per member.
+Each argument must be the same type as the member it
+sets, or be a type that can be converted to the member’s type according to
+section “<a href="#implicit-conversions">Implicit Conversions</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Structure constructors can be used as initializers or in expressions.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="array-constructors">5.4.4. Array Constructors</h4>
+<div class="paragraph">
+<p>Array types can also be used as constructor names, which can then be used in
+expressions or initializers.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">const</span> <span class="predefined-type">float</span> c[<span class="integer">3</span>] = <span class="predefined-type">float</span>[<span class="integer">3</span>](<span class="float">5</span><span class="float">.0</span>, <span class="float">7</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>);
+<span class="directive">const</span> <span class="predefined-type">float</span> d[<span class="integer">3</span>] = <span class="predefined-type">float</span>[](<span class="float">5</span><span class="float">.0</span>, <span class="float">7</span><span class="float">.2</span>, <span class="float">1</span><span class="float">.1</span>);
+
+<span class="predefined-type">float</span> g;
+...
+<span class="predefined-type">float</span> a[<span class="integer">5</span>] = <span class="predefined-type">float</span>[<span class="integer">5</span>](g, <span class="integer">1</span>, g, <span class="float">2</span><span class="float">.3</span>, g);
+<span class="predefined-type">float</span> b[<span class="integer">3</span>];
+
+b = <span class="predefined-type">float</span>[<span class="integer">3</span>](g, g + <span class="float">1</span><span class="float">.0</span>, g + <span class="float">2</span><span class="float">.0</span>);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>There must be exactly the same number of arguments as the size of the array
+being constructed.
+If no size is present in the constructor, then the array is explicitly sized
+to the number of arguments provided.
+The arguments are assigned in order, starting at element 0, to the elements
+of the constructed array.
+Each argument must be the same type as the element type of the
+array, or be
+a type that can be converted to the element type of the array according to
+“<a href="#implicit-conversions">Implicit Conversions</a>”.</p>
+</div>
+<div class="paragraph">
+<p>Arrays of arrays are similarly constructed, and the size for any dimension
+is <strong class="purple">optional</strong></p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 b[<span class="integer">2</span>] = ...;
+vec4[<span class="integer">3</span>][<span class="integer">2</span>](b, b, b); <span class="comment">// constructor</span>
+vec4[][<span class="integer">2</span>](b, b, b); <span class="comment">// constructor, valid, size deduced</span>
+vec4[<span class="integer">3</span>][](b, b, b); <span class="comment">// constructor, valid, size deduced</span>
+vec4[][](b, b, b); <span class="comment">// constructor, valid, both sizes deduced</span></code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="vector-components">5.5. Vector and Scalar Components and Length</h3>
+<div class="paragraph">
+<p>The names of the components of a vector
+or scalar
+are denoted by a single letter.
+As a notational convenience, several letters are associated with each
+component based on common usage of position, color or texture coordinate
+vectors.
+The individual components can be selected by following the variable name
+with period (<strong>.</strong>) and then the component name.</p>
+</div>
+<div class="paragraph">
+<p>The component names supported are:</p>
+</div>
+<table class="tableblock frame-all grid-all fit-content">
+<colgroup>
+<col>
+<col>
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>{ x, y, z, w }</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Useful when accessing vectors that represent points or normals</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>{ r, g, b, a }</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Useful when accessing vectors that represent colors</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><em>{ s, t, p, q }</em></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Useful when accessing vectors that represent texture coordinates</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The component names <em>x</em>, <em>r</em>, and <em>s</em> are, for example, synonyms for the
+same (first) component in a vector.
+They are also the names of the only component in a scalar.</p>
+</div>
+<div class="paragraph">
+<p>Note that the third component of the texture coordinate set, <em>r</em> in
+OpenGL, has been renamed <em>p</em> so as to avoid the confusion with <em>r</em> (for
+red) in a color.</p>
+</div>
+<div class="paragraph">
+<p>Accessing components beyond those declared for the type is
+a compile-time error so, for example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec2 pos;
+<span class="predefined-type">float</span> height;
+pos.x <span class="comment">// is legal</span>
+pos.z <span class="comment">// is illegal</span>
+height.x <span class="comment">// is legal</span>
+height.y <span class="comment">// is illegal</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The component selection syntax allows multiple components to be selected by
+appending their names (from the same name set) after the period (<strong>.</strong>).</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 v4;
+v4.rgba; <span class="comment">// is a vec4 and the same as just using v4,</span>
+v4.rgb; <span class="comment">// is a vec3,</span>
+v4.b; <span class="comment">// is a float,</span>
+v4.xy; <span class="comment">// is a vec2,</span>
+v4.xgba; <span class="comment">// is illegal - the component names do not come from the same set</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>No more than 4 components can be selected.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 v4;
+v4.xyzwxy; <span class="comment">// is illegal since it has 6 components</span>
+(v4.xyzwxy).xy; <span class="comment">// is illegal since the intermediate value has 6</span>
+components</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The order of the components can be different to swizzle them, or replicated:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 pos = vec4(<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>, <span class="float">4</span><span class="float">.0</span>);
+vec4 swiz = pos.wzyx; <span class="comment">// swiz = (4.0, 3.0, 2.0, 1.0)</span>
+vec4 dup = pos.xxyy; <span class="comment">// dup = (1.0, 1.0, 2.0, 2.0)</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This notation is more concise than the constructor syntax.
+To form an r-value, it can be applied to any expression that results in a
+vector or scalar r-value.</p>
+</div>
+<div class="paragraph">
+<p>The component group notation can occur on the left hand side of an
+expression.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 pos = vec4(<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>, <span class="float">4</span><span class="float">.0</span>);
+pos.xw = vec2(<span class="float">5</span><span class="float">.0</span>, <span class="float">6</span><span class="float">.0</span>); <span class="comment">// pos = (5.0, 2.0, 3.0, 6.0)</span>
+pos.wx = vec2(<span class="float">7</span><span class="float">.0</span>, <span class="float">8</span><span class="float">.0</span>); <span class="comment">// pos = (8.0, 2.0, 3.0, 7.0)</span>
+pos.xx = vec2(<span class="float">3</span><span class="float">.0</span>, <span class="float">4</span><span class="float">.0</span>); <span class="comment">// illegal - 'x' used twice</span>
+pos.xy = vec3(<span class="float">1</span><span class="float">.0</span>, <span class="float">2</span><span class="float">.0</span>, <span class="float">3</span><span class="float">.0</span>); <span class="comment">// illegal - mismatch between vec2 and vec3</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>To form an l-value, swizzling must further be applied to an l-value and
+contain no duplicate components. It results in an l-value of scalar or
+vector type, depending on number of components specified.</p>
+</div>
+<div class="paragraph">
+<p>Array subscripting syntax can also be applied to vectors (but not to
+scalars) to provide numeric indexing.
+So in</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 pos;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p><em>pos[2]</em> refers to the third element of <em>pos</em> and is equivalent to <em>pos.z</em>.
+This allows variable indexing into a vector, as well as a generic way of
+accessing components.
+Any integer expression can be used as the subscript.
+The first component is at index zero.
+Reading from or writing to a vector using a constant integral expression
+with a value that is negative or greater than or equal to the size of the
+vector results in a compile-time error.
+When indexing with non-constant expressions, behavior is undefined if the
+index is negative, or greater than or equal to the size of the vector.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>length</strong>() method may be applied to vectors (but not scalars).
+The result is the number of components in the vector.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3 v;
+<span class="directive">const</span> <span class="predefined-type">int</span> L = v.length();</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>sets the constant <em>L</em> to 3.
+The type returned by <strong>.length</strong>() on a vector is <strong>int</strong>, and the value
+returned is a constant expression.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="matrix-components">5.6. Matrix Components</h3>
+<div class="paragraph">
+<p>The components of a matrix can be accessed using array subscripting syntax.
+Applying a single subscript to a matrix treats the matrix as an array of
+column vectors, and selects a single column, whose type is a vector of the
+same size as the (column size of the) matrix.
+The leftmost column is column 0.
+A second subscript would then operate on the resulting vector, as defined
+earlier for vectors.
+Hence, two subscripts select a column and then a row.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat4 m;
+m[<span class="integer">1</span>] = vec4(<span class="float">2</span><span class="float">.0</span>); <span class="comment">// sets the second column to all 2.0</span>
+m[<span class="integer">0</span>][<span class="integer">0</span>] = <span class="float">1</span><span class="float">.0</span>; <span class="comment">// sets the upper left element to 1.0</span>
+m[<span class="integer">2</span>][<span class="integer">3</span>] = <span class="float">2</span><span class="float">.0</span>; <span class="comment">// sets the 4th element of the third column to 2.0</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Behavior is undefined when accessing a component outside the bounds of a
+matrix with a non-constant expression.
+It is a compile-time error to access a matrix with a constant expression
+that is outside the bounds of the matrix.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>length</strong>() method may be applied to matrices.
+The result is the number of columns of the matrix.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat3x4 v;
+<span class="directive">const</span> <span class="predefined-type">int</span> L = v.length();</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>sets the constant <em>L</em> to 3.
+The type returned by <strong>.length</strong>() on a matrix is <strong>int</strong>, and the value
+returned is a constant expression.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="structure-and-array-operations">5.7. Structure and Array Operations</h3>
+<div class="paragraph">
+<p>The members of a structure and the <strong>length</strong>() method of an array are
+selected using the period (<strong>.</strong>).</p>
+</div>
+<div class="paragraph">
+<p>In total, only the following operators are allowed to operate on arrays and
+structures as whole entities:</p>
+</div>
+<table class="tableblock frame-all grid-all fit-content">
+<colgroup>
+<col>
+<col>
+</colgroup>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">field selector</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>.</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">equality</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>==</strong> <strong>!=</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">assignment</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>=</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Ternary operator</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>?:</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Sequence operator</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>,</strong></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">indexing (arrays only)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock"><strong>[</strong> <strong>]</strong></p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The equality operators and assignment operator are only allowed if the two
+operands are same size and type.
+The operands cannot contain any opaque types.
+Structure types must be of the same declared structure.
+Both array operands must be
+explicitly
+sized.
+When using the equality operators, two structures are equal if and only if
+all the members are component-wise equal, and two arrays are equal if and
+only if all the elements are element-wise equal.</p>
+</div>
+<div class="paragraph">
+<p>Array elements are accessed using the array subscript operator (<strong>[ ]</strong>).
+An example of accessing an array element is</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">diffuseColor += lightIntensity[<span class="integer">3</span>] * NdotL;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Array indices start at zero.
+Array elements are accessed using an expression whose type is <strong>int</strong> or
+<strong>uint</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Behavior is undefined if a shader subscripts an array with an index less
+than 0 or greater than or equal to the size the array was declared with.</p>
+</div>
+<div class="paragraph">
+<p>Arrays can also be accessed with the method operator (<strong>.</strong>) and the <strong>length</strong>
+method to query the size of the array:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">lightIntensity.length() <span class="comment">// return the size of the array</span></code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="assignments">5.8. Assignments</h3>
+<div class="paragraph">
+<p>Assignments of values to variable names are done with the assignment
+operator (<strong>=</strong>):</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><em>lvalue-expression</em> = <em>rvalue-expression</em></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>The <em>lvalue-expression</em> evaluates to an l-value.
+The assignment operator stores the value of <em>rvalue-expression</em> into the
+l-value and returns an r-value with the type and precision of
+<em>lvalue-expression</em>.
+The <em>lvalue-expression</em> and <em>rvalue-expression</em> must have the same
+type, or the expression must have a type in the table in section
+“<a href="#implicit-conversions">Implicit Conversions</a>” that converts to the type
+of <em>lvalue-expression</em>, in which case an implicit conversion will be done on
+the <em>rvalue-expression</em> before the assignment is done.
+Any other desired type-conversions must be specified explicitly via a
+constructor.
+It is a compile-time error if the l-value is not writable.
+Variables that are built-in types, entire structures or arrays, structure
+members, l-values with the field selector (<strong>.</strong>) applied to select components
+or swizzles without repeated fields, l-values within parentheses, and
+l-values dereferenced with the array subscript operator (<strong>[ ]</strong>) are all
+l-values.
+Other binary or unary expressions, function names, swizzles with repeated
+fields, and constants cannot be l-values.
+The ternary operator (<strong>?:</strong>) is also not allowed as an l-value.
+Using an incorrect expression as an l-value results in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>Expressions on the left of an assignment are evaluated before expressions on
+the right of the assignment.</p>
+</div>
+<div class="paragraph">
+<p>The other assignment operators are</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>add into (<strong>+=</strong>)</p>
+</li>
+<li>
+<p>subtract from (<strong>-=</strong>)</p>
+</li>
+<li>
+<p>multiply into (<strong>*=</strong>)</p>
+</li>
+<li>
+<p>divide into (<strong>/=</strong>)</p>
+</li>
+<li>
+<p>modulus into (<strong>%=</strong>)</p>
+</li>
+<li>
+<p>left shift by (<strong><<=</strong>)</p>
+</li>
+<li>
+<p>right shift by (<strong>>>=</strong>)</p>
+</li>
+<li>
+<p>and into (<strong>&=</strong>)</p>
+</li>
+<li>
+<p>inclusive-or into (<strong>|=</strong>)</p>
+</li>
+<li>
+<p>exclusive-or into (<strong>^=</strong>)</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>where the general expression</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><em>lvalue</em> <em>op</em>= <em>expression</em></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><em>lvalue</em> = <em>lvalue</em> <em>op</em> <em>expression</em></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>where <em>lvalue</em> is the value returned by <em>lvalue-expression</em>, <em>op</em> is as
+described below, and the <em>lvalue-expression</em> and <em>expression</em> must satisfy
+the semantic requirements of both <em>op</em> and equals (<strong>=</strong>).</p>
+</div>
+<div class="paragraph">
+<p>Reading a variable before writing (or initializing) it is legal, however the
+value is undefined.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="expressions">5.9. Expressions</h3>
+<div class="paragraph">
+<p>Expressions in the shading language are built from the following:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Constants of type <strong>bool</strong>, all integral types, all floating-point types,
+all vector types, and all matrix types.</p>
+</li>
+<li>
+<p>Constructors of all types.</p>
+</li>
+<li>
+<p>Variable names of all types.</p>
+</li>
+<li>
+<p>An array, vector, or matrix expression with the <strong>length</strong>() method
+applied.</p>
+</li>
+<li>
+<p>Subscripted arrays.</p>
+</li>
+<li>
+<p>Function calls that return values.
+In some cases, function calls returning <strong>void</strong> are also allowed in
+expressions as specified below.</p>
+</li>
+<li>
+<p>Component field selectors and array subscript results.</p>
+</li>
+<li>
+<p>Parenthesized expressions.
+Any expression, including expressions with void type can be
+parenthesized.
+Parentheses can be used to group operations.
+Operations within parentheses are done before operations across
+parentheses.</p>
+</li>
+<li>
+<p>The arithmetic binary operators add (<strong>+</strong>), subtract (<strong>-</strong>), multiply
+(<strong>*</strong>), and divide (<strong>/</strong>) operate on integer and floating-point scalars,
+vectors, and matrices.
+If the fundamental types in the operands do not match, then the
+conversions from “<a href="#implicit-conversions">Implicit Conversions</a>” are
+applied to create matching types.
+All arithmetic binary operators result in the same fundamental type
+(signed integer, unsigned integer, single-precision floating-point, or
+double-precision floating-point) as the operands they operate on, after
+operand type conversion.
+After conversion, the following cases are valid</p>
+<div class="ulist">
+<ul>
+<li>
+<p>The two operands are scalars.
+In this case the operation is applied, resulting in a scalar.</p>
+</li>
+<li>
+<p>One operand is a scalar, and the other is a vector or matrix.
+In this case, the scalar operation is applied independently to each
+component of the vector or matrix, resulting in the same size vector or
+matrix.</p>
+</li>
+<li>
+<p>The two operands are vectors of the same size.
+In this case, the operation is done component-wise resulting in the
+same size vector.</p>
+</li>
+<li>
+<p>The operator is add (<strong>+</strong>), subtract (<strong>-</strong>), or divide (<strong>/</strong>), and the
+operands are matrices with the same number of rows and the same number
+of columns.
+In this case, the operation is done component-wise resulting in the
+same size matrix.</p>
+</li>
+<li>
+<p>The operator is multiply (<strong>*</strong>), where both operands are matrices or one
+operand is a vector and the other a matrix.
+A right vector operand is treated as a column vector and a left vector
+operand as a row vector.
+In all these cases, it is required that the number of columns of the
+left operand is equal to the number of rows of the right operand.
+Then, the multiply (<strong>*</strong>) operation does a linear algebraic multiply,
+yielding an object that has the same number of rows as the left operand
+and the same number of columns as the right operand.
+“<a href="#vector-and-matrix-operations">Vector and Matrix Operations</a>”
+explains in more detail how vectors and matrices are operated on.</p>
+<div class="openblock">
+<div class="content">
+<div class="paragraph">
+<p>All other cases result in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>Use the built-in functions <strong>dot</strong>, <strong>cross</strong>, <strong>matrixCompMult</strong>, and
+<strong>outerProduct</strong>, to get, respectively, vector dot product, vector cross
+product, matrix component-wise multiplication, and the matrix product of a
+column vector times a row vector.</p>
+</div>
+</div>
+</div>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>The operator modulus (<strong>%</strong>) operates on signed or unsigned integers or
+integer vectors.
+If the fundamental types in the operands do not match, then the
+conversions from “<a href="#implicit-conversions">Implicit Conversions</a>” are
+applied to create matching types.
+The operands cannot be vectors of differing size; this is
+a compile-time error.
+If one operand is a scalar and the other vector, then the scalar is
+applied component-wise to the vector, resulting in the same type as the
+vector.
+If both are vectors of the same size, the result is computed
+component-wise.
+The resulting value is undefined for any component computed with a
+second operand that is zero, while results for other components with
+non-zero second operands remain defined.
+If both operands are non-negative, then the remainder is non-negative.
+Results are undefined if one or both operands are negative.
+The operator modulus (<strong>%</strong>) is not defined for any other data types
+(non-integer types).</p>
+</li>
+<li>
+<p>The arithmetic unary operators negate (<strong>-</strong>), post- and pre-increment and
+decrement (<strong>--</strong> and <strong>++</strong>) operate on integer or floating-point values
+(including vectors and matrices).
+All unary operators work component-wise on their operands.
+These result with the same type they operated on.
+For post- and pre-increment and decrement, the expression must be a writable
+l-value.
+Pre-increment and pre-decrement add or subtract 1 or 1.0 to the contents
+of the expression they operate on, and the value of the pre-increment or
+pre-decrement expression is the resulting value of that modification.
+Post-increment and post-decrement expressions add or subtract 1 or 1.0
+to the contents of the expression they operate on, but the resulting
+expression has the expression’s value before the post-increment or
+post-decrement was executed.</p>
+</li>
+<li>
+<p>The relational operators greater than (<strong>></strong>), less than (<strong><</strong>), greater
+than or equal (<strong>>=</strong>), and less than or equal (<strong><=</strong>) operate only on
+scalar integer and scalar floating-point expressions.
+The result is scalar Boolean.
+Either the operands' types must match, or the conversions from section
+“<a href="#implicit-conversions">Implicit Conversions</a>” will be applied to
+obtain matching types.
+To do component-wise relational comparisons on vectors, use the built-in
+functions <strong>lessThan</strong>, <strong>lessThanEqual</strong>, <strong>greaterThan</strong>, and
+<strong>greaterThanEqual.</strong></p>
+</li>
+<li>
+<p>The equality operators <strong>equal</strong> (<strong>==</strong>), and not equal (<strong>!=</strong>) operate on
+all types except opaque types, aggregates that contain opaque types,
+subroutine uniforms, and aggregates that contain subroutine uniforms.
+They result in a scalar Boolean.
+If the operand types do not match, then there must be a conversion from
+“<a href="#implicit-conversions">Implicit Conversions</a>” applied to one operand
+that can make them match, in which case this conversion is done.
+For vectors, matrices, structures, and arrays, all components, members,
+or elements of one operand must equal the corresponding components,
+members, or elements in the other operand for the operands to be
+considered equal.
+To get a vector of component-wise equality results for vectors, use the
+built-in functions <strong>equal</strong> and <strong>notEqual</strong>.</p>
+</li>
+<li>
+<p>The logical binary operators and (<strong>&&</strong>), or (<strong>||</strong>), and exclusive or
+(<strong>^^</strong>) operate only on two Boolean expressions and result in a Boolean
+expression.
+And (<strong>&&</strong>) will only evaluate the right hand operand if the left hand
+operand evaluated to <strong>true</strong>.
+Or (<strong>||</strong>) will only evaluate the right hand operand if the left hand
+operand evaluated to <strong>false</strong>.
+Exclusive or (<strong>^^</strong>) will always evaluate both operands.</p>
+</li>
+<li>
+<p>The logical unary operator not (<strong>!</strong>).
+It operates only on a Boolean expression and results in a Boolean
+expression.
+To operate on a vector, use the built-in function <strong>not</strong>.</p>
+</li>
+<li>
+<p>The sequence (<strong>,</strong>) operator that operates on expressions by returning
+the type and value of the right-most expression in a comma separated
+list of expressions.
+All expressions are evaluated, in order, from left to right.
+The operands to the sequence operator may have <strong>void</strong> type.</p>
+</li>
+<li>
+<p>The ternary selection operator (<strong>?:</strong>).
+It operates on three expressions (<em>exp1</em> <strong>?</strong> <em>exp2</em> <strong>:</strong> <em>exp3</em>).
+This operator evaluates the first expression, which must result in a
+scalar Boolean.
+If the result is true, it selects to evaluate the second expression,
+otherwise it selects to evaluate the third expression.
+Only one of the second and third expressions is evaluated.
+The second and third expressions can be any type, including <strong>void</strong>, as
+long their types match, or there is a conversion in section
+“<a href="#implicit-conversions">Implicit Conversions</a>” that can be applied to
+one of the expressions to make their types match.
+This resulting matching type is the type of the entire expression.</p>
+</li>
+<li>
+<p>The one’s complement operator (<strong>~</strong>).
+The operand must be of type signed or unsigned integer or integer
+vector, and the result is the one’s complement of its operand; each bit
+of each component is complemented, including any sign bits.</p>
+</li>
+<li>
+<p>The shift operators (<strong><<</strong>) and (<strong>>></strong>).
+For both operators, the operands must be signed or unsigned integers or
+integer vectors.
+One operand can be signed while the other is unsigned.
+In all cases, the resulting type will be the same type as the left
+operand.
+If the first operand is a scalar, the second operand has to be a scalar
+as well.
+If the first operand is a vector, the second operand must be a scalar or
+a vector with the same size as the first operand, and the result is
+computed component-wise.
+The result is undefined if the right operand is negative, or greater
+than or equal to the number of bits in the left expression’s base type.
+The value of E1 << E2 is E1 (interpreted as a bit pattern) left-shifted
+by E2 bits.
+The value of E1 >> E2 is E1 right-shifted by E2 bit positions.
+If E1 is a signed integer, the right-shift will extend the sign bit.
+If E1 is an unsigned integer, the right-shift will zero-extend.</p>
+</li>
+<li>
+<p>The bitwise operators and (<strong>&</strong>), exclusive-or (<strong>^</strong>), and inclusive-or
+(<strong>|</strong>).
+The operands must be of type signed or unsigned integers or integer
+vectors.
+The operands cannot be vectors of differing size; this is a compile-time error.
+If one operand is a scalar and the other a vector, the scalar is applied
+component-wise to the vector, resulting in the same type as the vector.
+If the fundamental types in the operands do not match, then the
+conversions from “<a href="#implicit-conversions">Implicit Conversions</a>” are
+applied to create matching types, and this will be the resulting
+fundamental type.
+For and (<strong>&</strong>), the result is the bitwise-and function of the operands.
+For exclusive-or (<strong>^</strong>), the result is the bitwise exclusive-or function
+of the operands.
+For inclusive-or (<strong>|</strong>), the result is the bitwise inclusive-or function
+of the operands.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>For a complete specification of the syntax of expressions, see
+“<a href="#shading-language-grammar">Shading Language Grammar</a>”.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="vector-and-matrix-operations">5.10. Vector and Matrix Operations</h3>
+<div class="paragraph">
+<p>With a few exceptions, operations are component-wise.
+Usually, when an operator operates on a vector or matrix, it is operating
+independently on each component of the vector or matrix, in a component-wise
+fashion.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3 v, u;
+<span class="predefined-type">float</span> f;
+v = u + f;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will be equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">v.x = u.x + f;
+v.y = u.y + f;
+v.z = u.z + f;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>And</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3 v, u, w;
+w = v + u;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>will be equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">w.x = v.x + u.x;
+w.y = v.y + u.y;
+w.z = v.z + u.z;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and likewise for most operators and all integer and floating-point vector
+and matrix types.
+The exceptions are matrix multiplied by vector, vector multiplied by matrix,
+and matrix multiplied by matrix.
+These do not operate component-wise, but rather perform the correct linear
+algebraic multiply.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec3 v, u;
+mat3 m;
+u = v * m;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">u.x = dot(v, m[<span class="integer">0</span>]); <span class="comment">// m[0] is the left column of m</span>
+u.y = dot(v, m[<span class="integer">1</span>]); <span class="comment">// dot(a,b) is the inner (dot) product of a and b</span>
+u.z = dot(v, m[<span class="integer">2</span>]);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>And</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">u = m * v;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">u.x = m[<span class="integer">0</span>].x * v.x + m[<span class="integer">1</span>].x * v.y + m[<span class="integer">2</span>].x * v.z;
+u.y = m[<span class="integer">0</span>].y * v.x + m[<span class="integer">1</span>].y * v.y + m[<span class="integer">2</span>].y * v.z;
+u.z = m[<span class="integer">0</span>].z * v.x + m[<span class="integer">1</span>].z * v.y + m[<span class="integer">2</span>].z * v.z;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>And</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">mat3 m, n, r;
+r = m * n;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>is equivalent to</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">r[<span class="integer">0</span>].x = m[<span class="integer">0</span>].x * n[<span class="integer">0</span>].x + m[<span class="integer">1</span>].x * n[<span class="integer">0</span>].y + m[<span class="integer">2</span>].x * n[<span class="integer">0</span>].z;
+r[<span class="integer">1</span>].x = m[<span class="integer">0</span>].x * n[<span class="integer">1</span>].x + m[<span class="integer">1</span>].x * n[<span class="integer">1</span>].y + m[<span class="integer">2</span>].x * n[<span class="integer">1</span>].z;
+r[<span class="integer">2</span>].x = m[<span class="integer">0</span>].x * n[<span class="integer">2</span>].x + m[<span class="integer">1</span>].x * n[<span class="integer">2</span>].y + m[<span class="integer">2</span>].x * n[<span class="integer">2</span>].z;
+r[<span class="integer">0</span>].y = m[<span class="integer">0</span>].y * n[<span class="integer">0</span>].x + m[<span class="integer">1</span>].y * n[<span class="integer">0</span>].y + m[<span class="integer">2</span>].y * n[<span class="integer">0</span>].z;
+r[<span class="integer">1</span>].y = m[<span class="integer">0</span>].y * n[<span class="integer">1</span>].x + m[<span class="integer">1</span>].y * n[<span class="integer">1</span>].y + m[<span class="integer">2</span>].y * n[<span class="integer">1</span>].z;
+r[<span class="integer">2</span>].y = m[<span class="integer">0</span>].y * n[<span class="integer">2</span>].x + m[<span class="integer">1</span>].y * n[<span class="integer">2</span>].y + m[<span class="integer">2</span>].y * n[<span class="integer">2</span>].z;
+r[<span class="integer">0</span>].z = m[<span class="integer">0</span>].z * n[<span class="integer">0</span>].x + m[<span class="integer">1</span>].z * n[<span class="integer">0</span>].y + m[<span class="integer">2</span>].z * n[<span class="integer">0</span>].z;
+r[<span class="integer">1</span>].z = m[<span class="integer">0</span>].z * n[<span class="integer">1</span>].x + m[<span class="integer">1</span>].z * n[<span class="integer">1</span>].y + m[<span class="integer">2</span>].z * n[<span class="integer">1</span>].z;
+r[<span class="integer">2</span>].z = m[<span class="integer">0</span>].z * n[<span class="integer">2</span>].x + m[<span class="integer">1</span>].z * n[<span class="integer">2</span>].y + m[<span class="integer">2</span>].z * n[<span class="integer">2</span>].z;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and similarly for other sizes of vectors and matrices.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="out-of-bounds-accesses">5.11. Out-of-Bounds Accesses</h3>
+<div class="paragraph">
+<p>In the subsections described above for array, vector, matrix and structure
+accesses, any out-of-bounds access produced undefined behavior.
+However, if robust buffer access is enabled via the OpenGL API, such
+accesses will be bound within the memory extent of the active program.
+It will not be possible to access memory from other programs, and accesses
+will not result in abnormal program termination.
+Out-of-bounds reads return undefined values, which include values from other
+variables of the active program or zero.
+Out-of-bounds writes may be discarded or overwrite other variables of the
+active program, depending on the value of the computed index and how this
+relates to the extent of the active program’s memory.
+Applications that require defined behavior for out-of-bounds accesses should
+range check all computed indices before dereferencing an array.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="specialization-constant-operations">5.12. Specialization Constant Operations</h3>
+<div class="paragraph">
+<p>Only some operations discussed in this section may be applied to a
+specialization constant and still yield a result that is a specialization
+constant.
+The operations that do so are listed below.
+When a specialization constant is operated on with one of these operators
+and with another constant or specialization constant, the result is
+implicitly a specialization constant.</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>int</strong>(), <strong>uint</strong>(), and <strong>bool</strong>() constructors for type conversions from
+any of the following types to any of the following types:</p>
+<div class="ulist">
+<ul>
+<li>
+<p><strong>int</strong></p>
+</li>
+<li>
+<p><strong>uint</strong></p>
+</li>
+<li>
+<p><strong>bool</strong></p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>vector versions of the above conversion constructors</p>
+</li>
+<li>
+<p>allowed implicit conversions of the above</p>
+</li>
+<li>
+<p>swizzles (e.g. <code>foo.yx</code>)</p>
+</li>
+<li>
+<p>the following when applied to integer or unsigned integer types:</p>
+<div class="ulist">
+<ul>
+<li>
+<p>unary negative (<strong>-</strong>)</p>
+</li>
+<li>
+<p>binary operations (<strong>+</strong>, <strong>-</strong>, <strong>*</strong>, <strong>/</strong>, <strong>%</strong>)</p>
+</li>
+<li>
+<p>shift (<strong><<</strong>, <strong>>></strong>)</p>
+</li>
+<li>
+<p>bitwise operations (<strong>&</strong>, <strong>|</strong>, <strong>^</strong>)</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>the following when applied to integer or unsigned integer scalar types:</p>
+<div class="ulist">
+<ul>
+<li>
+<p>comparison (<strong>==</strong>, <strong>!=</strong>, <strong>></strong>, <strong>>=</strong>, <strong><</strong>, <strong>⇐</strong>)</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>The following when applied to the Boolean scalar type:</p>
+<div class="ulist">
+<ul>
+<li>
+<p>not (<strong>!</strong>)</p>
+</li>
+<li>
+<p>logical operations (<strong>&&</strong>, <strong>||</strong>, <strong>^^</strong>)</p>
+</li>
+<li>
+<p>comparison (<strong>==</strong>, <strong>!=</strong>)</p>
+</li>
+</ul>
+</div>
+</li>
+<li>
+<p>the ternary operator (<strong>?:</strong>)</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="statements-and-structure">6. Statements and Structure</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The fundamental building blocks of the OpenGL Shading Language are:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>statements and declarations</p>
+</li>
+<li>
+<p>function definitions</p>
+</li>
+<li>
+<p>selection (<strong>if</strong>-<strong>else</strong> and <strong>switch</strong>-<strong>case</strong>-<strong>default</strong>)</p>
+</li>
+<li>
+<p>iteration (<strong>for</strong>, <strong>while</strong>, and <strong>do</strong>-<strong>while</strong>)</p>
+</li>
+<li>
+<p>jumps (<strong>discard</strong>, <strong>return</strong>, <strong>break</strong>, and <strong>continue</strong>)</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>The overall structure of a shader is as follows</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>translation-unit</em> : </dt>
+<dd>
+<p><em>global-declaration</em><br>
+<em>translation-unit</em> <em>global-declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>global-declaration</em> : </dt>
+<dd>
+<p><em>function-definition</em><br>
+<em>declaration</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>That is, a shader is a sequence of declarations and function bodies.
+Function bodies are defined as</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>function-definition</em> : </dt>
+<dd>
+<p><em>function-prototype</em> <strong>{</strong> <em>statement-list</em> <strong>}</strong></p>
+</dd>
+<dt class="hdlist1"><em>statement-list</em> : </dt>
+<dd>
+<p><em>statement</em><br>
+<em>statement-list</em> <em>statement</em></p>
+</dd>
+<dt class="hdlist1"><em>statement</em> : </dt>
+<dd>
+<p><em>compound-statement</em><br>
+<em>simple-statement</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Curly braces are used to group sequences of statements into compound
+statements.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>compound-statement</em> : </dt>
+<dd>
+<p><strong>{</strong> <em>statement-list</em> <strong>}</strong></p>
+</dd>
+<dt class="hdlist1"><em>simple-statement</em> : </dt>
+<dd>
+<p><em>declaration-statement</em><br>
+<em>expression-statement</em><br>
+<em>selection-statement</em><br>
+<em>iteration-statement</em><br>
+<em>jump-statement</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Simple declaration, expression, and jump statements end in a semi-colon.</p>
+</div>
+<div class="paragraph">
+<p>This above is slightly simplified, and the complete grammar specified in
+“<a href="#shading-language-grammar">Shading Language Grammar</a>” should be used as
+the definitive specification.</p>
+</div>
+<div class="paragraph">
+<p>Declarations and expressions have already been discussed.</p>
+</div>
+<div class="sect2">
+<h3 id="function-definitions">6.1. Function Definitions</h3>
+<div class="paragraph">
+<p>As indicated by the grammar above, a valid shader is a sequence of global
+declarations and function definitions.
+A function is declared as the following example shows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// prototype</span>
+returnType functionName (type0 arg0, type1 arg1, ..., typen argn);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>and a function is defined like</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// definition</span>
+returnType functionName (type0 arg0, type1 arg1, ..., typen argn)
+{
+ <span class="comment">// do some computation</span>
+ <span class="keyword">return</span> returnValue;
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>where <em>returnType</em> must be present and cannot be void, or:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> functionName (type0 arg0, type1 arg1, ..., typen argn)
+{
+ <span class="comment">// do some computation</span>
+ <span class="keyword">return</span>; <span class="comment">// optional</span>
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>If the type of <em>returnValue</em> does not match <em>returnType</em>, there must be an
+implicit conversion in “<a href="#implicit-conversions">Implicit Conversions</a>”
+that converts the type of <em>returnValue</em> to <em>returnType</em>, or a compile-time
+error will result.</p>
+</div>
+<div class="paragraph">
+<p>Each of the <em>typeN</em> must include a type and can optionally include parameter
+qualifiers.
+The formal argument names (<em>args</em> above) in the declarations are optional
+for both the declaration and definition forms.</p>
+</div>
+<div class="paragraph">
+<p>A function is called by using its name followed by a list of arguments in
+parentheses.</p>
+</div>
+<div class="paragraph">
+<p>Arrays are allowed as arguments and as the return type.
+In both cases, the array must be
+explicitly
+sized.
+An array is passed or returned by using just its name, without brackets, and
+the size of the array must match the size specified in the function’s
+declaration.</p>
+</div>
+<div class="paragraph">
+<p>Structures are also allowed as argument types.
+The return type can also be a structure.</p>
+</div>
+<div class="paragraph">
+<p>See “<a href="#shading-language-grammar">Shading Language Grammar</a>” for the
+definitive reference on the syntax to declare and define functions.</p>
+</div>
+<div class="paragraph">
+<p>All functions must be either declared with a prototype or defined with a
+body before they are called.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> myfunc (<span class="predefined-type">float</span> f, <span class="comment">// f is an input parameter</span>
+ out <span class="predefined-type">float</span> g); <span class="comment">// g is an output parameter</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Functions that return no value must be declared as <strong>void</strong>.
+A <strong>void</strong> function can only use <strong>return</strong> without a return argument, even if
+the return argument has <strong>void</strong> type.
+Return statements only accept values:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> func1() { }
+<span class="directive">void</span> func2() { <span class="keyword">return</span> func1(); } <span class="comment">// illegal return statement</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Only a precision qualifier is allowed on the return type of a function.
+Formal parameters can have parameter, precision, and memory qualifiers, but
+no other qualifiers.</p>
+</div>
+<div class="paragraph">
+<p>Functions that accept no input arguments need not use <strong>void</strong> in the argument
+list because prototypes (or definitions) are required and therefore there is
+no ambiguity when an empty argument list “( )” is declared.
+The idiom “(<strong>void</strong>)” as a parameter list is provided for convenience.</p>
+</div>
+<div class="paragraph">
+<p>Function names can be overloaded.
+The same function name can be used for multiple functions, as long as the
+parameter types differ.
+If a function name is declared twice with the same parameter types, then the
+return types and all qualifiers must also match, and it is the same function
+being declared.</p>
+</div>
+<div class="paragraph">
+<p>For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4 f(in vec4 x, out vec4 y); <span class="comment">// (A)</span>
+vec4 f(in vec4 x, out uvec4 y); <span class="comment">// (B) okay, different argument type</span>
+vec4 f(in ivec4 x, out dvec4 y); <span class="comment">// (C) okay, different argument type</span>
+<span class="predefined-type">int</span> f(in vec4 x, out vec4 y); <span class="comment">// error, only return type differs</span>
+vec4 f(in vec4 x, in vec4 y); <span class="comment">// error, only qualifier differs</span>
+vec4 f(<span class="directive">const</span> in vec4 x, out vec4 y); <span class="comment">// error, only qualifier differs</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>When function calls are resolved, an exact type match for all the arguments
+is sought.
+If an exact match is found, all other functions are ignored, and the exact
+match is used.
+If no exact match is found, then the implicit conversions in section
+“<a href="#implicit-conversions">Implicit Conversions</a>” will be applied to find a
+match.
+Mismatched types on input parameters (<strong>in</strong> or <strong>inout</strong> or default) <strong>must</strong>
+have a conversion from the calling argument type to the formal parameter
+type.
+Mismatched types on output parameters (<strong>out</strong> or <strong>inout</strong>) must have a
+conversion from the formal parameter type to the calling argument type.</p>
+</div>
+<div class="paragraph">
+<p>If implicit conversions can be used to find more than one matching function,
+a single best-matching function is sought.
+To determine a best match, the conversions between calling argument and
+formal parameter types are compared for each function argument and pair of
+matching functions.
+After these comparisons are performed, each pair of matching functions are
+compared.
+A function declaration <em>A</em> is considered a better match than function
+declaration <em>B</em> if</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>for at least one function argument, the conversion for that argument in
+<em>A</em> is better than the corresponding conversion in <em>B</em>; and</p>
+</li>
+<li>
+<p>there is no function argument for which the conversion in <em>B</em> is better
+than the corresponding conversion in <em>A</em>.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>If a single function declaration is considered a better match than every
+other matching function declaration, it will be used.
+Otherwise, a compile-time semantic error for an ambiguous overloaded
+function call occurs.</p>
+</div>
+<div class="paragraph">
+<p>To determine whether the conversion for a single argument in one match is
+better than that for another match, the following rules are applied, in
+order:</p>
+</div>
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>An exact match is better than a match involving any implicit conversion.</p>
+</li>
+<li>
+<p>A match involving an implicit conversion from <strong>float</strong> to <strong>double</strong> is
+better than a match involving any other implicit conversion.</p>
+</li>
+<li>
+<p>A match involving an implicit conversion from either <strong>int</strong> or <strong>uint</strong> to
+<strong>float</strong> is better than a match involving an implicit conversion from
+either <strong>int</strong> or <strong>uint</strong> to <strong>double</strong>.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>If none of the rules above apply to a particular pair of conversions,
+neither conversion is considered better than the other.</p>
+</div>
+<div class="paragraph">
+<p>For the example function prototypes (A), (B), and © above, the following
+examples show how the rules apply to different sets of calling argument
+types:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">f(vec4, vec4) <span class="comment">// exact match of vec4 f(in vec4 x, out vec4 y)</span>
+f(vec4, uvec4) <span class="comment">// exact match of vec4 f(in vec4 x, out uvec4 y)</span>
+f(vec4, ivec4) <span class="comment">// matched to vec4 f(in vec4 x, out vec4 y)</span>
+ <span class="comment">// (C) not relevant, can't convert vec4 to</span>
+ <span class="comment">// ivec4. (A) better than (B) for 2nd</span>
+ <span class="comment">// argument (rule 3), same on first argument.</span>
+f(ivec4, vec4); <span class="comment">// NOT matched. All three match by implicit</span>
+ <span class="comment">// conversion. (C) is better than (A) and (B)</span>
+ <span class="comment">// on the first argument. (A) is better than</span>
+ <span class="comment">// (B) and (C).</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>User-defined functions can have multiple declarations, but only one
+definition.</p>
+</div>
+<div class="paragraph">
+<p>A shader can redefine built-in functions.
+If a built-in function is redeclared in a shader (i.e., a prototype is
+visible) before a call to it, then the linker will only attempt to resolve
+that call within the set of shaders that are linked with it.</p>
+</div>
+<div class="paragraph">
+<p>The function <em>main</em> is used as the entry point to a shader executable.
+A shader need not contain a function named <em>main</em>, but one shader in a set
+of shaders linked together to form a single shader executable must, or a
+link-time error results.
+This function takes no arguments, returns no value, and must be declared as
+type <strong>void</strong>:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">void</span> main()
+{
+ ...
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The function <em>main</em> can contain uses of <strong>return</strong>.
+See “<a href="#jumps">Jumps</a>” for more details.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time or link-time error to declare or define a function
+<strong>main</strong> with any other parameters or return type.</p>
+</div>
+<div class="sect3">
+<h4 id="function-calling-conventions">6.1.1. Function Calling Conventions</h4>
+<div class="paragraph">
+<p>Functions are called by value-return.
+This means input arguments are copied into the function at call time, and
+output arguments are copied back to the caller before function exit.
+Because the function works with local copies of parameters, there are no
+issues regarding aliasing of variables within a function.
+To control what parameters are copied in and/or out through a function
+definition or declaration:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>The keyword <strong>in</strong> is used as a qualifier to denote a parameter is to be
+copied in, but not copied out.</p>
+</li>
+<li>
+<p>The keyword <strong>out</strong> is used as a qualifier to denote a parameter is to be
+copied out, but not copied in.
+This should be used whenever possible to avoid unnecessarily copying
+parameters in.</p>
+</li>
+<li>
+<p>The keyword <strong>inout</strong> is used as a qualifier to denote the parameter is to
+be both copied in and copied out.
+It means the same thing as specifying both <strong>in</strong> and <strong>out</strong>.</p>
+</li>
+<li>
+<p>A function parameter declared with no such qualifier means the same
+thing as specifying <strong>in</strong>.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>All arguments are evaluated at call time, exactly once, in order, from left
+to right.
+Evaluation of an <strong>in</strong> parameter results in a value that is copied to the
+formal parameter.
+Evaluation of an <strong>out</strong> parameter results in an l-value that is used to copy
+out a value when the function returns.
+Evaluation of an <strong>inout</strong> parameter results in both a value and an l-value;
+the value is copied to the formal parameter at call time and the l-value is
+used to copy out a value when the function returns.</p>
+</div>
+<div class="paragraph">
+<p>The order in which output parameters are copied back to the caller is
+undefined.</p>
+</div>
+<div class="paragraph">
+<p>If the function matching described in the previous section required argument
+type conversions, these conversions are applied at copy-in and copy-out
+times.</p>
+</div>
+<div class="paragraph">
+<p>In a function, writing to an input-only parameter is allowed.
+Only the function’s copy is modified.
+This can be prevented by declaring a parameter with the <strong>const</strong> qualifier.</p>
+</div>
+<div class="paragraph">
+<p>When calling a function, expressions that do not evaluate to l-values cannot
+be passed to parameters declared as <strong>out</strong> or <strong>inout</strong>, or a compile-time error
+results.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>function-prototype</em> : </dt>
+<dd>
+<p><em>precision-qualifier</em> <em>type</em> <em>function-name</em> <strong>(</strong> <em>parameter-qualifiers</em>
+<em>precision-qualifier</em> <em>type</em> <em>name</em> <em>array-specifier</em> <strong>,</strong> …​
+<strong>)</strong></p>
+</dd>
+<dt class="hdlist1"><em>type</em> : </dt>
+<dd>
+<p>any basic type, array type, structure name, or structure definition</p>
+</dd>
+<dt class="hdlist1"><em>parameter-qualifiers</em> : </dt>
+<dd>
+<p><em>empty</em><br>
+list of <em>parameter-qualifier</em></p>
+</dd>
+<dt class="hdlist1"><em>parameter-qualifier</em> : </dt>
+<dd>
+<p><strong>const</strong><br>
+<strong>in</strong><br>
+<strong>out</strong><br>
+<strong>inout</strong><br>
+<strong>precise</strong><br>
+<em>memory-qualifier</em><br>
+<em>precision-qualifier</em></p>
+</dd>
+<dt class="hdlist1"><em>name</em> : </dt>
+<dd>
+<p>empty<br>
+identifier</p>
+</dd>
+<dt class="hdlist1"><em>array-specifier</em> : </dt>
+<dd>
+<p>empty<br>
+<strong>[</strong> <em>integral-constant-expression</em> <strong>]</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>The <strong>const</strong> qualifier cannot be used with <strong>out</strong> or <strong>inout</strong>, or
+a compile-time error results.
+The above is used both for function declarations (i.e., prototypes) and for
+function definitions.
+Hence, function definitions can have unnamed arguments.</p>
+</div>
+<div class="paragraph">
+<p>Recursion is not allowed, not even statically.
+Static recursion is present if the static function-call graph of a program
+contains cycles.
+This includes all potential function calls through variables declared as
+<strong>subroutine</strong> <strong>uniform</strong> (described below).
+It is a compile-time or link-time error if a single compilation unit
+(shader) contains either static recursion or the potential for recursion
+through subroutine variables.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="subroutines">6.1.2. Subroutines</h4>
+<div class="paragraph">
+<p>Subroutines provide a mechanism allowing shaders to be compiled in a manner
+where the target of one or more function calls can be changed at run-time
+without requiring any shader recompilation.
+For example, a single shader may be compiled with support for multiple
+illumination algorithms to handle different kinds of lights or surface
+materials.
+An application using such a shader may switch illumination algorithms by
+changing the value of its subroutine uniforms.
+To use subroutines, a subroutine type is declared, one or more functions are
+associated with that subroutine type, and a subroutine variable of that type
+is declared.
+The function currently assigned to the variable function is then called by
+using function calling syntax replacing a function name with the name of the
+subroutine variable.
+Subroutine variables are uniforms, and are assigned to specific functions
+only through commands (<strong>UniformSubroutinesuiv</strong>) in the OpenGL API.</p>
+</div>
+<div class="paragraph">
+<p>Subroutine functionality is not available when generating SPIR-V.</p>
+</div>
+<div class="paragraph">
+<p>Subroutine types are declared using a statement similar to a function
+declaration, with the <strong>subroutine</strong> keyword, as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">subroutine returnType subroutineTypeName(type0 arg0, type1 arg1,
+ ..., typen argn);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>As with function declarations, the formal argument names (<em>args</em> above) are
+optional.
+Functions are associated with subroutine types of matching declarations by
+defining the function with the <strong>subroutine</strong> keyword and a list of subroutine
+types the function matches:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">subroutine(subroutineTypeName0, ..., subroutineTypeNameN)
+returnType functionName(type0 arg0, type1 arg1, ..., typen argn)
+{ ... } <span class="comment">// function body</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error if arguments and return type don’t match between
+the function and each associated subroutine type.</p>
+</div>
+<div class="paragraph">
+<p>Functions declared with <strong>subroutine</strong> must include a body.
+An overloaded function cannot be declared with <strong>subroutine</strong>; a program will
+fail to compile or link if any shader or stage contains two or more
+functions with the same name if the name is associated with a subroutine
+type.</p>
+</div>
+<div class="paragraph">
+<p>A function declared with <strong>subroutine</strong> can also be called directly with a
+static use of <em>functionName</em>, as is done with non-subroutine function
+declarations and calls.</p>
+</div>
+<div class="paragraph">
+<p>Subroutine type variables are required to be <em>subroutine uniforms</em>, and are
+declared with a specific subroutine type in a subroutine uniform variable
+declaration:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">subroutine uniform subroutineTypeName subroutineVarName;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Subroutine uniform variables are called the same way functions are called.
+When a subroutine variable (or an element of a subroutine variable array) is
+associated with a particular function, all function calls through that
+variable will call that particular function.</p>
+</div>
+<div class="paragraph">
+<p>Unlike other uniform variables, subroutine uniform variables are scoped to
+the shader execution stage the variable is declared in.</p>
+</div>
+<div class="paragraph">
+<p>Subroutine variables may be declared as explicitly-sized arrays, which can
+be indexed only with dynamically uniform expressions.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time error to use the <strong>subroutine</strong> keyword in any places
+other than (as shown above) to</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>declare a subroutine type at global scope,</p>
+</li>
+<li>
+<p>declare a function as a subroutine, or</p>
+</li>
+<li>
+<p>declare a subroutine variable at global scope.</p>
+</li>
+</ul>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="selection">6.2. Selection</h3>
+<div class="paragraph">
+<p>Conditional control flow in the shading language is done by either <strong>if</strong>,
+<strong>if</strong>-<strong>else</strong>, or <strong>switch</strong> statements:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>selection-statement</em> : </dt>
+<dd>
+<p><strong>if</strong> <strong>(</strong> <em>bool-expression</em> <strong>)</strong> <em>statement</em><br>
+<strong>if</strong> <strong>(</strong> <em>bool-expression</em> <strong>)</strong> <em>statement</em> <strong>else</strong> <em>statement</em><br>
+<strong>switch</strong> <strong>(</strong> <em>init-expression</em> <strong>)</strong> <strong>{</strong> <em>switch-statement-list<sub>opt</sub></em> <strong>}</strong></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Where <em>switch-statement-list</em> is a nested scope containing a list of zero or
+more <em>switch-statement</em> and other statements defined by the language, where
+<em>switch-statement</em> adds some forms of labels.
+That is</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>switch-statement-list</em> : </dt>
+<dd>
+<p><em>switch-statement</em><br>
+<em>switch-statement-list</em> <em>switch-statement</em></p>
+</dd>
+<dt class="hdlist1"><em>switch-statement</em> : </dt>
+<dd>
+<p><strong>case</strong> <em>constant-expression</em> <strong>:</strong><br>
+<strong>default</strong> <strong>:</strong> <em>statement</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>Note the above grammar’s purpose is to aid discussion in this section; the
+normative grammar is in “<a href="#shading-language-grammar">Shading Language
+Grammar</a>”.</p>
+</div>
+<div class="paragraph">
+<p>If an <strong>if</strong>-expression evaluates to <strong>true</strong>, then the first <em>statement</em> is
+executed.
+If it evaluates to <strong>false</strong> and there is an <strong>else</strong> part then the second
+<em>statement</em> is executed.</p>
+</div>
+<div class="paragraph">
+<p>Any expression whose type evaluates to a Boolean can be used as the
+conditional expression <em>bool-expression</em>.
+Vector types are not accepted as the expression to <strong>if</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Conditionals can be nested.</p>
+</div>
+<div class="paragraph">
+<p>The type of <em>init-expression</em> in a <strong>switch</strong> statement must be a scalar
+integer.
+The type of the <em>constant-expression</em> value in a case label also must be a
+scalar integer.
+When any pair of these values is tested for “equal value” and the types do
+not match, an implicit conversion will be done to convert the <strong>int</strong> to a
+<strong>uint</strong> (see “<a href="#implicit-conversions">Implicit Conversions</a>”) before the
+compare is done.
+If a <strong>case</strong> label has a <em>constant-expression</em> of equal value to
+<em>init-expression</em>, execution will continue after that label.
+Otherwise, if there is a <strong>default</strong> label, execution will continue after that
+label.
+Otherwise, execution skips the rest of the switch statement.
+It is a compile-time error to have more than one <strong>default</strong> or a replicated
+<em>constant-expression</em>.
+A <strong>break</strong> statement not nested in a loop or other switch statement (either
+not nested or nested only in <strong>if</strong> or <strong>if</strong>-<strong>else</strong> statements) will also skip
+the rest of the switch statement.
+Fall through labels are allowed, but it is a compile-time error to have no
+statement between a label and the end of the switch statement.
+No statements are allowed in a switch statement before the first <strong>case</strong>
+statement.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>case</strong> and <strong>default</strong> labels can only appear within a <strong>switch</strong> statement.
+No <strong>case</strong> or <strong>default</strong> labels can be nested inside other statements or
+compound statements within their corresponding <strong>switch</strong>.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="iteration">6.3. Iteration</h3>
+<div class="paragraph">
+<p>For, while, and do loops are allowed as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">for</span> (init-expression; condition-expression; loop-expression)
+ sub-statement
+<span class="keyword">while</span> (condition-expression)
+ sub-statement
+<span class="keyword">do</span>
+ statement
+<span class="keyword">while</span> (condition-expression)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>See “<a href="#shading-language-grammar">Shading Language Grammar</a>” for the
+definitive specification of loops.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>for</strong> loop first evaluates the <em>init-expression</em>, then the
+<em>condition-expression</em>.
+If the <em>condition-expression</em> evaluates to <strong>true</strong>, then the body of the loop
+is executed.
+After the body is executed, a <strong>for</strong> loop will then evaluate the
+<em>loop-expression</em>, and then loop back to evaluate the
+<em>condition-expression</em>, repeating until the <em>condition-expression</em> evaluates
+to <strong>false</strong>.
+The loop is then exited, skipping its body and skipping its
+<em>loop-expression</em>.
+Variables modified by the <em>loop-expression</em> maintain their value after the
+loop is exited, provided they are still in scope.
+Variables declared in <em>init-expression</em> or <em>condition-expression</em> are only
+in scope until the end of the sub-statement of the <strong>for</strong> loop.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>while</strong> loop first evaluates the <em>condition-expression</em>.
+If <strong>true</strong>, then the body is executed.
+This is then repeated, until the <em>condition-expression</em> evaluates to
+<strong>false</strong>, exiting the loop and skipping its body.
+Variables declared in the <em>condition-expression</em> are only in scope until the
+end of the sub-statement of the <strong>while</strong> loop.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>do</strong>-<strong>while</strong> loop first executes the body, then executes the
+<em>condition-expression</em>.
+This is repeated until <em>condition-expression</em> evaluates to <strong>false</strong>, and then
+the loop is exited.</p>
+</div>
+<div class="paragraph">
+<p>Expressions for <em>condition-expression</em> must evaluate to a Boolean.</p>
+</div>
+<div class="paragraph">
+<p>Both the <em>condition-expression</em> and the <em>init-expression</em> can declare and
+initialize a variable, except in the <strong>do</strong>-<strong>while</strong> loop, which cannot declare
+a variable in its <em>condition-expression</em>.
+The variable’s scope lasts only until the end of the sub-statement that
+forms the body of the loop.</p>
+</div>
+<div class="paragraph">
+<p>Loops can be nested.</p>
+</div>
+<div class="paragraph">
+<p>Non-terminating loops are allowed.
+The consequences of very long or non-terminating loops are platform
+dependent.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="jumps">6.4. Jumps</h3>
+<div class="paragraph">
+<p>These are the jumps:</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>jump_statement</em> : </dt>
+<dd>
+<p><strong>continue</strong> <strong>;</strong><br>
+<strong>break</strong> <strong>;</strong><br>
+<strong>return</strong> <strong>;</strong><br>
+<strong>return</strong> <em>expression</em> <strong>;</strong><br>
+<strong>discard</strong> <strong>;</strong> // in the fragment shader language only</p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>There is no “goto” or other non-structured flow of control.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>continue</strong> jump is used only in loops.
+It skips the remainder of the body of the inner-most loop of which it is
+inside.
+For <strong>while</strong> and <strong>do</strong>-<strong>while</strong> loops, this jump is to the next evaluation of
+the loop <em>condition-expression</em> from which the loop continues as previously
+defined.
+For <strong>for</strong> loops, the jump is to the <em>loop-expression</em>, followed by the
+<em>condition-expression</em>.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>break</strong> jump can also be used only in loops and <strong>switch</strong> statements.
+It is simply an immediate exit of the inner-most loop or <strong>switch</strong> statements
+containing the <strong>break</strong>.
+No further execution of <em>condition-expression</em>, <em>loop-expression</em>, or
+<em>switch-statement</em> is done.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>discard</strong> keyword is only allowed within fragment shaders.
+It can be used within a fragment shader to abandon the operation on the
+current fragment.
+This keyword causes the fragment to be discarded and no updates to any
+buffers will occur.
+Any prior writes to other buffers such as shader storage buffers are
+unaffected.
+Control flow exits the shader, and subsequent implicit or explicit
+derivatives are undefined when this control flow is non-uniform (meaning
+different fragments within the primitive take different control paths).
+It would typically be used within a conditional statement, for example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="keyword">if</span> (intensity < <span class="float">0</span><span class="float">.0</span>)
+ discard;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>A fragment shader may test a fragment’s alpha value and discard the fragment
+based on that test.
+However, it should be noted that coverage testing occurs after the fragment
+shader runs, and the coverage test can change the alpha value.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>return</strong> jump causes immediate exit of the current function.
+If it has <em>expression</em> then that is the return value for the function.</p>
+</div>
+<div class="paragraph">
+<p>The function <em>main</em> can use <strong>return</strong>.
+This simply causes <em>main</em> to exit in the same way as when the end of the
+function had been reached.
+It does not imply a use of <strong>discard</strong> in a fragment shader.
+Using <strong>return</strong> in <em>main</em> before defining outputs will have the same behavior
+as reaching the end of <em>main</em> before defining outputs.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="built-in-variables">7. Built-In Variables</h2>
+<div class="sectionbody">
+<div class="sect2">
+<h3 id="built-in-language-variables">7.1. Built-In Language Variables</h3>
+<div class="paragraph">
+<p>Some OpenGL operations occur in fixed functionality and need to provide
+values to or receive values from shader executables.
+Shaders communicate with fixed-function OpenGL pipeline stages, and
+optionally with other shader executables, through the use of built-in input
+and output variables.</p>
+</div>
+<div class="sect3">
+<h4 id="vertex-shader-special-variables">7.1.1. Vertex Shader Special Variables</h4>
+<div class="paragraph">
+<p>The built-in vertex shader variables are intrinsically declared as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in <span class="predefined-type">int</span> gl_VertexID;
+in <span class="predefined-type">int</span> gl_InstanceID;
+in <span class="predefined-type">int</span> gl_DrawID;
+in <span class="predefined-type">int</span> gl_BaseVertex;
+in <span class="predefined-type">int</span> gl_BaseInstance;
+
+out gl_PerVertex {
+ vec4 gl_Position;
+ <span class="predefined-type">float</span> gl_PointSize;
+ <span class="predefined-type">float</span> gl_ClipDistance[];
+ <span class="predefined-type">float</span> gl_CullDistance[];
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_Position</em> is intended for writing the homogeneous vertex
+position.
+It can be written at any time during shader execution.
+This value will be used by primitive assembly, clipping, culling, and other
+fixed functionality operations, if present, that operate on primitives after
+vertex processing has occurred.
+Its value is undefined after the vertex processing stage if the vertex
+shader executable does not write <em>gl_Position</em>.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_PointSize</em> is intended for a shader to write the size of
+the point to be rasterized.
+It is measured in pixels.
+If <em>gl_PointSize</em> is not written to, its value is undefined in subsequent
+pipe stages.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_ClipDistance</em> is intended for writing clip distances, and
+provides the forward compatible mechanism for controlling user clipping.
+The element <em>gl_ClipDistance[i]</em> specifies a clip distance for each plane
+<em>i</em>.
+A distance of 0 means the vertex is on the plane, a positive distance means
+the vertex is inside the clip plane, and a negative distance means the point
+is outside the clip plane.
+The clip distances will be linearly interpolated across the primitive and
+the portion of the primitive with interpolated distances less than 0 will be
+clipped.</p>
+</div>
+<div class="paragraph">
+<p>The <em>gl_ClipDistance</em> array is predeclared as unsized and must be explicitly
+sized by the shader either redeclaring it with a size or implicitly sized by
+indexing it only with constant integral expressions.
+This needs to size the array to include all the clip planes that are enabled
+via the OpenGL API; if the size does not include all enabled planes,
+results are undefined.
+The size can be at most <em>gl_MaxClipDistances</em>.
+The number of varying components (see <em>gl_MaxVaryingComponents)</em> consumed by
+<em>gl_ClipDistance</em> will match the size of the array, no matter how many
+planes are enabled.
+The shader must also set all values in <em>gl_ClipDistance</em> that have been
+enabled via the OpenGL API, or results are undefined.
+Values written into <em>gl_ClipDistance</em> for planes that are not enabled have
+no effect.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_CullDistance</em> provides a mechanism for controlling user
+culling.
+The element <em>gl_CullDistance[i]</em> specifies a cull distance for plane <em>i</em>.
+A distance of 0 means the vertex is on the plane, a positive distance means
+the vertex is inside the cull volume, and a negative distance means the
+point is outside the cull volume.
+Primitives whose vertices all have a negative cull distance for plane <em>i</em>
+will be discarded.</p>
+</div>
+<div class="paragraph">
+<p>The <em>gl_CullDistance</em> array is predeclared as unsized and must be sized by
+the shader either redeclaring it with a size or indexing it only with
+constant integral expressions.
+The size determines the number and set of enabled cull distances and can be
+at most <em>gl_MaxCullDistances</em>.
+The number of varying components (see <em>gl_MaxVaryingComponents</em>) consumed by
+<em>gl_CullDistance</em> will match the size of the array.
+Shaders writing <em>gl_CullDistance</em> must write all enabled distances, or
+culling results are undefined.</p>
+</div>
+<div class="paragraph">
+<p>As an output variable, <em>gl_CullDistance</em> provides the place for the shader
+to write these distances.
+As an input in all but the fragment language, it reads the values written in
+the previous shader stage.
+In the fragment language, <em>gl_CullDistance</em> array contains linearly
+interpolated values for the vertex values written by a shader to the
+<em>gl_CullDistance</em> vertex output variable.</p>
+</div>
+<div class="paragraph">
+<p>It is a compile-time or link-time error for the set of shaders forming a
+program to have the sum of the sizes of the <em>gl_ClipDistance</em> and
+<em>gl_CullDistance</em> arrays to be larger than
+<em>gl_MaxCombinedClipAndCullDistances</em>.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_VertexID</em> is a vertex shader input variable that holds an
+integer index for the vertex, as defined under “Shader Inputs” in section
+11.1.3.9 “Shader Inputs” of the <a href="#references">OpenGL Specification</a>.
+While the variable <em>gl_VertexID</em> is always present, its value is not always
+defined.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_InstanceID</em> is a vertex shader input variable that holds
+the instance number of the current primitive in an instanced draw call (see
+“Shader Inputs” in section 11.1.3.9 “Shader Inputs” of the
+<a href="#references">OpenGL Specification</a>).
+If the current primitive does not come from an instanced draw call, the
+value of <em>gl_InstanceID</em> is zero.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_DrawID</em> is a vertex shader input variable that holds the
+integer index of the drawing command to which the current vertex belongs
+(see “Shader Inputs” in section 11.1.3.9 of the <a href="#references">OpenGL Specification</a>).
+If the vertex is not invoked by a <strong>Multi</strong>* form of a draw command, then the
+value of <em>gl_DrawID</em> is zero.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_BaseVertex</em> is a vertex shader input variable that holds
+the integer value passed to the baseVertex parameter of the command that
+resulted in the current shader invocation (see “Shader Inputs” in section
+11.1.3.9 of the <a href="#references">OpenGL Specification</a>).</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_BaseInstance</em> is a vertex shader input variable that holds
+the integer value passed to the baseInstance parameter of the command that
+resulted in the current shader invocation (see “Shader Inputs” in section
+11.1.3.9 of the <a href="#references">OpenGL Specification</a>).</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="tessellation-control-shader-special-variables">7.1.2. Tessellation Control Shader Special Variables</h4>
+<div class="paragraph">
+<p>In the tessellation control shader, built-in variables are intrinsically
+declared as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerVertex {
+ vec4 gl_Position;
+ <span class="predefined-type">float</span> gl_PointSize;
+ <span class="predefined-type">float</span> gl_ClipDistance[];
+ <span class="predefined-type">float</span> gl_CullDistance[];
+} gl_in[gl_MaxPatchVertices];
+
+in <span class="predefined-type">int</span> gl_PatchVerticesIn;
+in <span class="predefined-type">int</span> gl_PrimitiveID;
+in <span class="predefined-type">int</span> gl_InvocationID;
+
+out gl_PerVertex {
+ vec4 gl_Position;
+ <span class="predefined-type">float</span> gl_PointSize;
+ <span class="predefined-type">float</span> gl_ClipDistance[];
+ <span class="predefined-type">float</span> gl_CullDistance[];
+} gl_out[];
+
+patch out <span class="predefined-type">float</span> gl_TessLevelOuter[<span class="integer">4</span>];
+patch out <span class="predefined-type">float</span> gl_TessLevelInner[<span class="integer">2</span>];</code></pre>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-control-input-variables">Tessellation Control Input Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em>, <em>gl_PointSize</em>, <em>gl_ClipDistance</em>, and <em>gl_CullDistance</em>
+contain the values written in the previous shader stage to the corresponding
+outputs.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PatchVerticesIn</em> contains the number of vertices in the input patch
+being processed by the shader.
+A single shader can read patches of differing sizes, so the value of
+<em>gl_PatchVerticesIn</em> may differ between patches.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PrimitiveID</em> contains the number of primitives processed by the shader
+since the current set of rendering primitives was started.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_InvocationID</em> contains the number of the output patch vertex assigned to
+the tessellation control shader invocation.
+It is assigned integer values in the range [0, N-1], where N is the number
+of output patch vertices per primitive.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-control-output-variables">Tessellation Control Output Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em>, <em>gl_PointSize</em>, <em>gl_ClipDistance</em>, and <em>gl_CullDistance</em> are
+used in the same fashion as the corresponding output variables in the vertex
+shader.</p>
+</div>
+<div class="paragraph">
+<p>The values written to <em>gl_TessLevelOuter</em> and <em>gl_TessLevelInner</em> are
+assigned to the corresponding outer and inner tessellation levels of the
+output patch.
+They are used by the tessellation primitive generator to control primitive
+tessellation and may be read by tessellation evaluation shaders.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="tessellation-evaluation-shader-special-variables">7.1.3. Tessellation Evaluation Shader Special Variables</h4>
+<div class="paragraph">
+<p>In the tessellation evaluation shader, built-in variables are intrinsically
+declared as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerVertex {
+ vec4 gl_Position;
+ <span class="predefined-type">float</span> gl_PointSize;
+ <span class="predefined-type">float</span> gl_ClipDistance[];
+ <span class="predefined-type">float</span> gl_CullDistance[];
+} gl_in[gl_MaxPatchVertices];
+
+in <span class="predefined-type">int</span> gl_PatchVerticesIn;
+in <span class="predefined-type">int</span> gl_PrimitiveID;
+in vec3 gl_TessCoord;
+patch in <span class="predefined-type">float</span> gl_TessLevelOuter[<span class="integer">4</span>];
+patch in <span class="predefined-type">float</span> gl_TessLevelInner[<span class="integer">2</span>];
+
+out gl_PerVertex {
+ vec4 gl_Position;
+ <span class="predefined-type">float</span> gl_PointSize;
+ <span class="predefined-type">float</span> gl_ClipDistance[];
+ <span class="predefined-type">float</span> gl_CullDistance[];
+};</code></pre>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-evaluation-input-variables">Tessellation Evaluation Input Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em>, <em>gl_PointSize</em>, <em>gl_ClipDistance</em>, and <em>gl_CullDistance</em>
+contain the values written in the previous shader stage to the corresponding
+outputs.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PatchVerticesIn</em> and <em>gl_PrimitiveID</em> are defined in the same fashion as
+the corresponding input variables in the tessellation control shader.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_TessCoord</em> specifies a three-component <em>(u,v,w)</em> vector identifying the
+position of the vertex being processed by the shader relative to the
+primitive being tessellated.
+Its values will obey the properties</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_TessCoord.x == <span class="float">1</span><span class="float">.0</span> - (<span class="float">1</span><span class="float">.0</span> - gl_TessCoord.x) <span class="comment">// two operations performed</span>
+gl_TessCoord.y == <span class="float">1</span><span class="float">.0</span> - (<span class="float">1</span><span class="float">.0</span> - gl_TessCoord.y) <span class="comment">// two operations performed</span>
+gl_TessCoord.z == <span class="float">1</span><span class="float">.0</span> - (<span class="float">1</span><span class="float">.0</span> - gl_TessCoord.z) <span class="comment">// two operations performed</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>to aid in replicating subdivision computations.</p>
+</div>
+<div class="paragraph">
+<p>If a tessellation control shader is active, the input variables
+<em>gl_TessLevelOuter</em> and <em>gl_TessLevelInner</em> are filled with the
+corresponding outputs written by the tessellation control shader.
+Otherwise, they are assigned with default tessellation levels specified in
+section 11.2.3.3 “Tessellation Evaluation Shader Inputs” of the
+<a href="#references">OpenGL Specification</a>.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="tessellation-evaluation-output-variables">Tessellation Evaluation Output Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em>, <em>gl_PointSize</em>, <em>gl_ClipDistance</em>, and <em>gl_CullDistance</em> are
+used in the same fashion as the corresponding output variables in the vertex
+shader.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="geometry-shader-special-variables">7.1.4. Geometry Shader Special Variables</h4>
+<div class="paragraph">
+<p>In the geometry shader, built-in variables are intrinsically declared as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerVertex {
+ vec4 gl_Position;
+ <span class="predefined-type">float</span> gl_PointSize;
+ <span class="predefined-type">float</span> gl_ClipDistance[];
+ <span class="predefined-type">float</span> gl_CullDistance[];
+} gl_in[];
+
+in <span class="predefined-type">int</span> gl_PrimitiveIDIn;
+in <span class="predefined-type">int</span> gl_InvocationID;
+
+out gl_PerVertex {
+ vec4 gl_Position;
+ <span class="predefined-type">float</span> gl_PointSize;
+ <span class="predefined-type">float</span> gl_ClipDistance[];
+ <span class="predefined-type">float</span> gl_CullDistance[];
+};
+
+out <span class="predefined-type">int</span> gl_PrimitiveID;
+out <span class="predefined-type">int</span> gl_Layer;
+out <span class="predefined-type">int</span> gl_ViewportIndex;</code></pre>
+</div>
+</div>
+<div class="sect4">
+<h5 id="geometry-shader-input-variables">Geometry Shader Input Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em>, <em>gl_PointSize</em>, <em>gl_ClipDistance</em>, and <em>gl_CullDistance</em>
+contain the values written in the previous shader stage to the corresponding
+outputs.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PrimitiveIDIn</em> contains the number of primitives processed by the shader
+since the current set of rendering primitives was started.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_InvocationID</em> contains the invocation number assigned to the geometry
+shader invocation.
+It is assigned integer values in the range [0, N-1], where N is the number
+of geometry shader invocations per primitive.</p>
+</div>
+</div>
+<div class="sect4">
+<h5 id="geometry-shader-output-variables">Geometry Shader Output Variables</h5>
+<div class="paragraph">
+<p><em>gl_Position</em>, <em>gl_PointSize</em>, <em>gl_ClipDistance</em>, and <em>gl_CullDistance</em> are
+used in the same fashion as the corresponding output variables in the vertex
+shader.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_PrimitiveID</em> is filled with a single integer that serves as a primitive
+identifier to the fragment shader.
+This is then available to fragment shaders, which will select the written
+primitive ID from the provoking vertex of the primitive being shaded.
+If a fragment shader using <em>gl_PrimitiveID</em> is active and a geometry shader
+is also active, the geometry shader must write to <em>gl_PrimitiveID</em> or the
+fragment shader input <em>gl_PrimitiveID</em> is undefined.
+See section 11.3.4.5 “Geometry Shader Outputs” of the
+<a href="#references">OpenGL Specification</a> for more information.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_Layer</em> is used to select a specific layer (or face and layer of a cube
+map) of a multi-layer framebuffer attachment.
+The actual layer used will come from one of the vertices in the primitive
+being shaded.
+Which vertex the layer comes from is determined as discussed in section
+11.3.4.6 “Layer and Viewport Selection” of the <a href="#references">OpenGL Specification</a>
+but may be undefined, so it is best to write the same layer value for all
+vertices of a primitive.
+If a shader statically assigns a value to <em>gl_Layer</em>, layered rendering mode
+is enabled.
+See section 11.3.4.5 “Geometry Shader Outputs” and section 9.4.9 “Layered
+Framebuffers” of the <a href="#references">OpenGL Specification</a> for more information.
+If a shader statically assigns a value to <em>gl_Layer</em>, and there is an
+execution path through the shader that does not set <em>gl_Layer</em>, then the
+value of <em>gl_Layer</em> is undefined for executions of the shader that take that
+path.</p>
+</div>
+<div class="paragraph">
+<p>The output variable <em>gl_Layer</em> takes on a special value when used with an
+array of cube map textures.
+Instead of only referring to the layer, it is used to select a cube map face
+and a layer.
+Setting <em>gl_Layer</em> to the value <em>layer*6+face</em> will render to face <em>face</em> of
+the cube defined in layer <em>layer</em>.
+The face values are defined in table 9.3 of section 9.4.9 “Layered
+Framebuffers” of the <a href="#references">OpenGL Specification</a>,
+but repeated below for clarity.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Face Value</th>
+<th class="tableblock halign-left valign-top">Resulting Target</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">0</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_POSITIVE_X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_NEGATIVE_X</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_POSITIVE_Y</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_NEGATIVE_Y</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_POSITIVE_Z</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">TEXTURE_CUBE_MAP_NEGATIVE_Z</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>For example, to render to the positive <em>y</em> cube map face located in the 5th
+layer of the cube map array, <em>gl_Layer</em> should be set to <em>5 * 6 + 2</em>.</p>
+</div>
+<div class="paragraph">
+<p>The output variable <em>gl_ViewportIndex</em> provides the index of the viewport to
+which the next primitive emitted from the geometry shader should be drawn.
+Primitives generated by the geometry shader will undergo viewport
+transformation and scissor testing using the viewport transformation and
+scissor rectangle selected by the value of <em>gl_ViewportIndex</em>.
+The viewport index used will come from one of the vertices in the primitive
+being shaded.
+However, which vertex the viewport index comes from is
+implementation-dependent, so it is best to use the same viewport index for
+all vertices of the primitive.
+If a geometry shader does not assign a value to <em>gl_ViewportIndex</em>, viewport
+transform and scissor rectangle zero will be used.
+If a geometry shader statically assigns a value to <em>gl_ViewportIndex</em> and
+there is a path through the shader that does not assign a value to
+<em>gl_ViewportIndex</em>, the value of <em>gl_ViewportIndex</em> is undefined for
+executions of the shader that take that path.
+See section 11.3.4.6 “Layer and Viewport Selection” of the
+<a href="#references">OpenGL Specification</a> for more information.</p>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="fragment-shader-special-variables">7.1.5. Fragment Shader Special Variables</h4>
+<div class="paragraph">
+<p>The built-in special variables that are accessible from a fragment shader
+are intrinsically declared as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 gl_FragCoord;
+in <span class="predefined-type">bool</span> gl_FrontFacing;
+in <span class="predefined-type">float</span> gl_ClipDistance[];
+in <span class="predefined-type">float</span> gl_CullDistance[];
+in vec2 gl_PointCoord;
+in <span class="predefined-type">int</span> gl_PrimitiveID;
+in <span class="predefined-type">int</span> gl_SampleID;
+in vec2 gl_SamplePosition;
+in <span class="predefined-type">int</span> gl_SampleMaskIn[];
+in <span class="predefined-type">int</span> gl_Layer;
+in <span class="predefined-type">int</span> gl_ViewportIndex;
+in <span class="predefined-type">bool</span> gl_HelperInvocation;
+
+out <span class="predefined-type">float</span> gl_FragDepth;
+out <span class="predefined-type">int</span> gl_SampleMask[];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The output of the fragment shader executable is processed by the fixed
+function operations at the back end of the OpenGL pipeline.</p>
+</div>
+<div class="paragraph">
+<p>The fixed functionality computed depth for a fragment may be obtained by
+reading <em>gl_FragCoord.z</em>, described below.</p>
+</div>
+<div class="paragraph">
+<p>Writing to <em>gl_FragDepth</em> will establish the depth value for the fragment
+being processed.
+If depth buffering is enabled, and no shader writes <em>gl_FragDepth</em>, then the
+fixed function value for depth will be used as the fragment’s depth value.
+If a shader statically assigns a value to <em>gl_FragDepth</em>, and there is an
+execution path through the shader that does not set <em>gl_FragDepth</em>, then the
+value of the fragment’s depth may be undefined for executions of the shader
+that take that path.
+That is, if the set of linked fragment shaders statically contain a write to
+<em>gl_FragDepth</em>, then it is responsible for always writing it.</p>
+</div>
+<div class="paragraph">
+<p>If a shader executes the <strong>discard</strong> keyword, the fragment is discarded, and
+the values of any user-defined fragment outputs, <em>gl_FragDepth</em>, and
+<em>gl_SampleMask</em> become irrelevant.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_FragCoord</em> is available as an input variable from within
+fragment shaders and it holds the window relative coordinates (<em>x</em>, <em>y</em>,
+<em>z</em>, <em>1/w</em>) values for the fragment.
+If multi-sampling, this value can be for any location within the pixel, or
+one of the fragment samples.
+The use of <strong>centroid</strong> does not further restrict this value to be inside the
+current primitive.
+This value is the result of the fixed functionality that interpolates
+primitives after vertex processing to generate fragments.
+The <em>z</em> component is the depth value that would be used for the fragment’s
+depth if no shader contained any writes to <em>gl_FragDepth</em>.
+This is useful for invariance if a shader conditionally computes
+<em>gl_FragDepth</em> but otherwise wants the fixed functionality fragment depth.</p>
+</div>
+<div class="paragraph">
+<p>Fragment shaders have access to the input built-in variable
+<em>gl_FrontFacing</em>, whose value is <strong>true</strong> if the fragment belongs to a
+front-facing primitive.
+One use of this is to emulate two-sided lighting by selecting one of two
+colors calculated by a vertex or geometry shader.</p>
+</div>
+<div class="paragraph">
+<p>The values in <em>gl_PointCoord</em> are two-dimensional coordinates indicating
+where within a point primitive the current fragment is located, when point
+sprites are enabled.
+They range from 0.0 to 1.0 across the point.
+If the current primitive is not a point, or if point sprites are not
+enabled, then the values read from <em>gl_PointCoord</em> are undefined.</p>
+</div>
+<div class="paragraph">
+<p>For both the input array <em>gl_SampleMaskIn[]</em> and the output array
+<em>gl_SampleMask[]</em>, bit <em>B</em> of mask <em>M</em> (<em>gl_SampleMaskIn[M]</em> or
+<em>gl_SampleMask[M]</em>) corresponds to sample <em>32*M+B</em>.
+These arrays have <strong>ceil</strong>(<em>s</em>/32) elements, where <em>s</em> is the maximum number
+of color samples supported by the implementation.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_SampleMaskIn</em> indicates the set of samples covered by
+the primitive generating the fragment during multisample rasterization.
+It has a sample bit set if and only if the sample is considered covered for
+this fragment shader invocation.</p>
+</div>
+<div class="paragraph">
+<p>The output array <em>gl_SampleMask[]</em> sets the sample mask for the fragment
+being processed.
+Coverage for the current fragment will become the logical AND of the
+coverage mask and the output <em>gl_SampleMask</em>.
+This array must be sized in the fragment shader either implicitly or
+explicitly, to be no larger than the implementation-dependent maximum
+sample-mask (as an array of 32bit elements), determined by the maximum
+number of samples..
+If the fragment shader statically assigns a value to <em>gl_SampleMask</em>, the
+sample mask will be undefined for any array elements of any fragment shader
+invocations that fail to assign a value.
+If a shader does not statically assign a value to <em>gl_SampleMask</em>, the
+sample mask has no effect on the processing of a fragment.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_SampleID</em> is filled with the sample number of the
+sample currently being processed.
+This variable is in the range <em>0</em> to <em>gl_NumSamples-1</em>, where
+<em>gl_NumSamples</em> is the total number of samples in the framebuffer, or 1 if
+rendering to a non-multisample framebuffer.
+Any static use of this variable in a fragment shader causes the entire
+shader to be evaluated per-sample.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_SamplePosition</em> contains the position of the current
+sample within the multisample draw buffer.
+The <em>x</em> and <em>y</em> components of <em>gl_SamplePosition</em> contain the sub-pixel
+coordinate of the current sample and will have values in the range 0.0 to
+1.0.
+Any static use of this variable in a fragment shader causes the entire
+shader to be evaluated per sample.</p>
+</div>
+<div class="paragraph">
+<p>The value <em>gl_HelperInvocation</em> is <strong>true</strong> if the fragment shader invocation
+is considered a <em>helper invocation</em> and is <strong>false</strong> otherwise.
+A helper invocation is a fragment shader invocation that is created solely
+for the purposes of evaluating derivatives for use in non-helper fragment
+shader invocations.
+Such derivatives are computed implicitly in the built-in function
+<strong>texture</strong>() (see “<a href="#texture-functions">Texture Functions</a>”), and
+explicitly in the derivative functions in
+“<a href="#derivative-functions">Derivative Functions</a>”, for example <strong>dFdx</strong>() and
+<strong>dFdy</strong>().</p>
+</div>
+<div class="paragraph">
+<p>Fragment shader helper invocations execute the same shader code as
+non-helper invocations, but will not have side effects that modify the
+framebuffer or other shader-accessible memory.
+In particular:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>Fragments corresponding to helper invocations are discarded when shader
+execution is complete, without updating the framebuffer.</p>
+</li>
+<li>
+<p>Stores to image and buffer variables performed by helper invocations
+have no effect on the underlying image or buffer memory.</p>
+</li>
+<li>
+<p>Atomic operations to image, buffer, or atomic counter variables
+performed by helper invocations have no effect on the underlying image
+or buffer memory.
+The values returned by such atomic operations are undefined.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Helper invocations may be generated for pixels not covered by a primitive
+being rendered.
+While fragment shader inputs qualified with <strong>centroid</strong> are normally required
+to be sampled in the intersection of the pixel and the primitive, the
+requirement is ignored for such pixels since there is no intersection
+between the pixel and primitive.</p>
+</div>
+<div class="paragraph">
+<p>Helper invocations may also be generated for fragments that are covered by a
+primitive being rendered when the fragment is killed by early fragment tests
+(using the <strong>early_fragment_tests</strong> qualifier) or where the implementation is
+able to determine that executing the fragment shader would have no effect
+other than assisting in computing derivatives for other fragment shader
+invocations.</p>
+</div>
+<div class="paragraph">
+<p>The set of helper invocations generated when processing any set of
+primitives is implementation-dependent.</p>
+</div>
+<div class="paragraph">
+<p><em>gl_ClipDistance</em> contains linearly interpolated values for the vertex-
+pipeline values written by a shader to the <em>gl_ClipDistance</em> output
+variable. Only elements in this array that have clipping enabled will
+have defined values.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_PrimitiveID</em> is filled with the value written to the
+<em>gl_PrimitiveID</em> geometry shader output, if a geometry shader is present.
+Otherwise, it is filled with the number of primitives processed by the
+shader since the current set of rendering primitives was started.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_Layer</em> is filled with the value written to the
+<em>gl_Layer</em> geometry shader output, if a geometry shader is present.
+If the geometry stage does not dynamically assign a value to <em>gl_Layer</em>, the
+value of <em>gl_Layer</em> in the fragment stage will be undefined.
+If the geometry stage makes no static assignment to <em>gl_Layer</em>, the input
+value in the fragment stage will be zero.
+Otherwise, the fragment stage will read the same value written by the
+geometry stage, even if that value is out of range.
+If a fragment shader contains a static access to <em>gl_Layer</em>, it will count
+against the implementation defined limit for the maximum number of inputs to
+the fragment stage.</p>
+</div>
+<div class="paragraph">
+<p>The input variable <em>gl_ViewportIndex</em> is filled with the value written to
+the output variable <em>gl_ViewportIndex</em> in the geometry stage, if a geometry
+shader is present.
+If the geometry stage does not dynamically assign a value to
+<em>gl_ViewportIndex</em>, the value of <em>gl_ViewportIndex</em> in the fragment shader
+will be undefined.
+If the geometry stage makes no static assignment to <em>gl_ViewportIndex</em>, the
+fragment stage will read zero.
+Otherwise, the fragment stage will read the same value written by the
+geometry stage, even if that value is out of range.
+If a fragment shader contains a static access to <em>gl_ViewportIndex</em>, it will
+count against the implementation defined limit for the maximum number of
+inputs to the fragment stage.</p>
+</div>
+</div>
+<div class="sect3">
+<h4 id="compute-shader-special-variables">7.1.6. Compute Shader Special Variables</h4>
+<div class="paragraph">
+<p>In the compute shader, built-in variables are declared as follows:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">// workgroup dimensions</span>
+in uvec3 gl_NumWorkGroups;
+<span class="directive">const</span> uvec3 gl_WorkGroupSize;
+
+<span class="comment">// workgroup and invocation IDs</span>
+in uvec3 gl_WorkGroupID;
+in uvec3 gl_LocalInvocationID;
+
+<span class="comment">// derived variables</span>
+in uvec3 gl_GlobalInvocationID;
+in uint gl_LocalInvocationIndex;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_NumWorkGroups</em> is a compute-shader input variable
+containing the number of workgroups in each dimension of the dispatch that
+will execute the compute shader.
+Its content is equal to the values specified in the <em>num_groups_x</em>,
+<em>num_groups_y</em>, and <em>num_groups_z</em> parameters passed to the
+<em>DispatchCompute</em> API entry point.</p>
+</div>
+<div class="paragraph">
+<p>The built-in constant <em>gl_WorkGroupSize</em> is a compute-shader constant
+containing the workgroup size of the shader.
+The size of the workgroup in the <em>X</em>, <em>Y</em>, and <em>Z</em> dimensions is stored in
+the <em>x</em>, <em>y</em>, and <em>z</em> components.
+The constants values in <em>gl_WorkGroupSize</em> will match those specified in the
+required <strong>local_size_x</strong>, <strong>local_size_y</strong>, and <strong>local_size_z</strong> layout
+qualifiers for the current shader.
+This is a constant so that it can be used to size arrays of memory that can
+be shared within the workgroup.
+It is a compile-time error to use <em>gl_WorkGroupSize</em> in a shader that does
+not declare a fixed workgroup size, or before that shader has declared a
+fixed workgroup size, using <strong>local_size_x</strong>, <strong>local_size_y</strong>, and
+<strong>local_size_z</strong>.</p>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_WorkGroupID</em> is a compute-shader input variable
+containing the three-dimensional index of the workgroup that the
+current invocation is executing in.
+The possible values range across the parameters passed into
+<em>DispatchCompute</em>, i.e., from (0, 0, 0) to (<em>gl_NumWorkGroups.x</em> - 1,
+<em>gl_NumWorkGroups.y</em> - 1, <em>gl_NumWorkGroups.z</em> -1).</p>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_LocalInvocationID</em> is a compute-shader input
+variable containing the three-dimensional index of the current work item
+within the workgroup.
+The possible values for this variable range across the workgroup
+size, i.e., (0,0,0) to (<em>gl_WorkGroupSize.x</em> - 1, <em>gl_WorkGroupSize.y</em> - 1,
+<em>gl_WorkGroupSize.z</em> - 1).</p>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_GlobalInvocationID</em> is a compute shader input
+variable containing the global index of the current work item.
+This value uniquely identifies this invocation from all other invocations
+across all workgroups initiated by the current <em>DispatchCompute</em> call.
+This is computed as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_GlobalInvocationID =
+ gl_WorkGroupID * gl_WorkGroupSize + gl_LocalInvocationID;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The built-in variable <em>gl_LocalInvocationIndex</em> is a compute shader input
+variable that contains the one-dimensional representation of the
+<em>gl_LocalInvocationID</em>.
+This is computed as:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">gl_LocalInvocationIndex =
+ gl_LocalInvocationID.z * gl_WorkGroupSize.x * gl_WorkGroupSize.y +
+ gl_LocalInvocationID.y * gl_WorkGroupSize.x +
+ gl_LocalInvocationID.x;</code></pre>
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="compatibility-profile-built-in-language-variables">7.1.7. Compatibility Profile Built-In Language Variables</h4>
+<div class="paragraph">
+<p>When using the compatibility profile, the GL can provide fixed functionality
+behavior for the vertex and fragment programmable pipeline stages.
+For example, mixing a fixed functionality vertex stage with a programmable
+fragment stage.</p>
+</div>
+<div class="paragraph">
+<p>The following built-in vertex, tessellation control, tessellation
+evaluation, and geometry output variables are available to specify inputs
+for the subsequent programmable shader stage or the fixed functionality
+fragment stage.
+A particular one should be written to if any functionality in a
+corresponding fragment shader or fixed pipeline uses it or state derived
+from it.
+Otherwise, behavior is undefined.
+The following members are added to the output <em>gl_PerVertex</em> block in these
+languages:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out gl_PerVertex { <span class="comment">// part of the gl_PerVertex block described in 7.1</span>
+ <span class="comment">// in addition to other gl_PerVertex members...</span>
+ vec4 gl_ClipVertex;
+ vec4 gl_FrontColor;
+ vec4 gl_BackColor;
+ vec4 gl_FrontSecondaryColor;
+ vec4 gl_BackSecondaryColor;
+ vec4 gl_TexCoord[];
+ <span class="predefined-type">float</span> gl_FogFragCoord;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The output variable <em>gl_ClipVertex</em> provides a place for vertex and geometry
+shaders to write the coordinate to be used with the user clipping planes.
+Writing to <em>gl_ClipDistance</em> is the preferred method for user clipping.
+It is a compile-time or link-time error for the set of shaders forming a
+program to statically read or write both <em>gl_ClipVertex</em> and either
+<em>gl_ClipDistance</em> or <em>gl_CullDistance</em>.
+If neither <em>gl_ClipVertex</em> nor <em>gl_ClipDistance</em> is written, their values
+are undefined and any clipping against user clip planes is also undefined.</p>
+</div>
+<div class="paragraph">
+<p>Similarly to what was previously described for the core profile, the
+<em>gl_PerVertex</em> block can be redeclared in a shader to explicitly include
+these additional members.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out gl_PerVertex {
+ vec4 gl_Position; <span class="comment">// will use gl_Position</span>
+ vec4 gl_FrontColor; <span class="comment">// will consume gl_color in the fragment shader</span>
+ vec4 gl_BackColor;
+ vec4 gl_TexCoord[<span class="integer">3</span>]; <span class="comment">// 3 elements of gl_TexCoord will be used</span>
+}; <span class="comment">// no other aspects of the fixed interface will be used</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The user must ensure the clip vertex and user clipping planes are defined in
+the same coordinate space.
+User clip planes work properly only under linear transform.
+It is undefined what happens under non-linear transform.</p>
+</div>
+<div class="paragraph">
+<p>The output variables <em>gl_FrontColor</em>, <em>glFrontSecondaryColor</em>,
+<em>gl_BackColor</em>, and <em>glBackSecondaryColor</em> assign primary and secondary
+colors for front and back faces of primitives containing the vertex being
+processed.
+The output variable <em>gl_TexCoord</em> assigns texture coordinates for the vertex
+being processed.</p>
+</div>
+<div class="paragraph">
+<p>For <em>gl_FogFragCoord</em>, the value written will be used as the “c” value in
+section 16.4 “Fog” of the Compatibility profile of the
+<a href="#references">OpenGL Specification</a>, by the fixed functionality pipeline.
+For example, if the z-coordinate of the fragment in eye space is desired as
+“c”, then that’s what the vertex shader executable should write into
+<em>gl_FogFragCoord</em>.</p>
+</div>
+<div class="paragraph">
+<p>As with all arrays, indices used to subscript <em>gl_TexCoord</em> must either be a
+constant integral expressions, or this array must be redeclared by the
+shader with a size.
+The size can be at most <em>gl_MaxTextureCoords</em>.
+Using indexes close to 0 may aid the implementation in preserving varying
+resources.
+The redeclaration of <em>gl_TexCoord</em> can also be done at global scope as, for
+example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 gl_TexCoord[<span class="integer">3</span>];
+out vec4 gl_TexCoord[<span class="integer">4</span>];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>(This treatment is a special case for <em>gl_TexCoord[]</em>, not a general method
+for redeclaring members of blocks.) It is a compile-time error to redeclare
+<em>gl_TexCoord[]</em> at global scope if there is a redeclaration of the
+corresponding built-in block; only one form of redeclaration is allowed
+within a shader (and hence within a stage, as block redeclarations must
+match across all shaders using it).</p>
+</div>
+<div class="paragraph">
+<p>In the tessellation control, evaluation, and geometry shaders, the outputs
+of the previous stage described above are also available in the input
+<em>gl_PerVertex</em> block in these languages.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerVertex { <span class="comment">// part of the gl_PerVertex block described in 7.1</span>
+ <span class="comment">// in addition to other gl_PerVertex members...</span>
+ vec4 gl_ClipVertex;
+ vec4 gl_FrontColor;
+ vec4 gl_BackColor;
+ vec4 gl_FrontSecondaryColor;
+ vec4 gl_BackSecondaryColor;
+ vec4 gl_TexCoord[];
+ <span class="predefined-type">float</span> gl_FogFragCoord;
+} gl_in[];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>These can be redeclared to establish an explicit pipeline interface, the
+same way as described above for the output block <em>gl_PerVertex</em>, and the
+input redeclaration must match the output redeclaration of the previous
+stage.
+However, when a built-in interface block with an instance name is redeclared
+(e.g. <em>gl_in</em>), the instance name must be included in the redeclaration.
+It is a compile-time error to not include the built-in instance name or to
+change its name.
+For example,</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerVertex {
+ vec4 gl_ClipVertex;
+ vec4 gl_FrontColor;
+} gl_in[]; <span class="comment">// must be present and must be "gl_in[]"</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Built-in block arrays predeclared with a size can be redeclared with unsized syntax.
+This keeps their size equal to the original predeclared size.</p>
+</div>
+<div class="paragraph">
+<p>Treatment of <em>gl_TexCoord[]</em> redeclaration is also identical to that
+described for the output block <em>gl_TexCoord[]</em> redeclaration.</p>
+</div>
+<div class="paragraph">
+<p>The following fragment input block is also available in a fragment shader
+when using the compatibility profile:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in gl_PerFragment {
+ in <span class="predefined-type">float</span> gl_FogFragCoord;
+ in vec4 gl_TexCoord[];
+ in vec4 gl_Color;
+ in vec4 gl_SecondaryColor;
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The values in <em>gl_Color</em> and <em>gl_SecondaryColor</em> will be derived
+automatically by the system from <em>gl_FrontColor</em>, <em>gl_BackColor</em>,
+<em>gl_FrontSecondaryColor</em>, and <em>gl_BackSecondaryColor</em> based on which face is
+visible in the primitive producing the fragment.
+If fixed functionality is used for vertex processing, then <em>gl_FogFragCoord</em>
+will either be the z-coordinate of the fragment in eye space, or the
+interpolation of the fog coordinate, as described in section 16.4 “Fog” of
+the Compatibility profile of the <a href="#references">OpenGL Specification</a>.
+The <em>gl_TexCoord[]</em> values are the interpolated <em>gl_TexCoord[]</em> values from
+a vertex shader or the texture coordinates of any fixed pipeline based
+vertex functionality.</p>
+</div>
+<div class="paragraph">
+<p>Indices to the fragment shader <em>gl_TexCoord</em> array are as described above in
+the vertex shader text.</p>
+</div>
+<div class="paragraph">
+<p>As described above for the input and output <em>gl_PerVertex</em> blocks, the
+<em>gl_PerFragment</em> block can be redeclared to create an explicit interface to
+another program.
+When matching these interfaces between separate programs, members in the
+<em>gl_PerVertex</em> output block must be declared if and only if the
+corresponding fragment shader members generated from them are present in the
+<em>gl_PerFragment</em> input block.
+These matches are described in detail in section 7.4.1 “Shader Interface
+Matching” of the <a href="#references">OpenGL Specification</a>.
+If they don’t match within a program, a link-time error will result.
+If the mismatch is between two programs, values passed between programs are
+undefined.
+Unlike with all other block matching, the order of declaration within
+<em>gl_PerFragment</em> does not have to match across shaders and does not have to
+correspond with order of declaration in a matching <em>gl_PerVertex</em>
+redeclaration.</p>
+</div>
+<div class="paragraph">
+<p>The following fragment output variables are available in a fragment shader
+when using the compatibility profile:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out vec4 gl_FragColor;
+out vec4 gl_FragData[gl_MaxDrawBuffers];</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Writing to <em>gl_FragColor</em> specifies the fragment color that will be used by
+the subsequent fixed functionality pipeline.
+If subsequent fixed functionality consumes fragment color and an execution
+of the fragment shader executable does not write a value to <em>gl_FragColor</em>
+then the fragment color consumed is undefined.</p>
+</div>
+<div class="paragraph">
+<p>The variable <em>gl_FragData</em> is an array.
+Writing to <em>gl_FragData[n]</em> specifies the fragment data that will be used by
+the subsequent fixed functionality pipeline for data <em>n</em>.
+If subsequent fixed functionality consumes fragment data and an execution of
+a fragment shader executable does not write a value to it, then the fragment
+data consumed is undefined.</p>
+</div>
+<div class="paragraph">
+<p>If a shader statically assigns a value to <em>gl_FragColor</em>, it may not assign
+a value to any element of <em>gl_FragData</em>.
+If a shader statically writes a value to any element of <em>gl_FragData</em>, it
+may not assign a value to <em>gl_FragColor</em>.
+That is, a shader may assign values to either <em>gl_FragColor</em> or
+<em>gl_FragData</em>, but not both.
+Multiple shaders linked together must also consistently write just one of
+these variables.
+Similarly, if user-declared output variables are in use (statically assigned
+to), then the built-in variables <em>gl_FragColor</em> and <em>gl_FragData</em> may not be
+assigned to.
+These incorrect usages all generate compile-time or link-time errors.</p>
+</div>
+<div class="paragraph">
+<p>If a shader executes the <strong>discard</strong> keyword, the fragment is discarded, and
+the values of <em>gl_FragDepth</em> and <em>gl_FragColor</em> become irrelevant.</p>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="compatibility-profile-vertex-shader-built-in-inputs">7.2. Compatibility Profile Vertex Shader Built-In Inputs</h3>
+<div class="paragraph">
+<p>The following predeclared input names can be used from within a vertex
+shader to access the current values of OpenGL state when using the
+compatibility profile.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">in vec4 gl_Color;
+in vec4 gl_SecondaryColor;
+in vec3 gl_Normal;
+in vec4 gl_Vertex;
+in vec4 gl_MultiTexCoord0;
+in vec4 gl_MultiTexCoord1;
+in vec4 gl_MultiTexCoord2;
+in vec4 gl_MultiTexCoord3;
+in vec4 gl_MultiTexCoord4;
+in vec4 gl_MultiTexCoord5;
+in vec4 gl_MultiTexCoord6;
+in vec4 gl_MultiTexCoord7;
+in <span class="predefined-type">float</span> gl_FogCoord;</code></pre>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="built-in-constants">7.3. Built-In Constants</h3>
+<div class="paragraph">
+<p>The following built-in constants are provided to all shaders.
+The actual values used are implementation-dependent, but must be at least
+the value shown.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">//</span>
+<span class="comment">// Implementation-dependent constants. The example values below</span>
+<span class="comment">// are the minimum values allowed for these maximums.</span>
+<span class="comment">//</span></code></pre>
+</div>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVertexAttribs = <span class="integer">16</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVertexUniformVectors = <span class="integer">256</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVertexUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVertexOutputComponents = <span class="integer">64</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVaryingComponents = <span class="integer">60</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVaryingVectors = <span class="integer">15</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVertexTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVertexImageUniforms = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVertexAtomicCounters = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVertexAtomicCounterBuffers = <span class="integer">0</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessPatchComponents = <span class="integer">120</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxPatchVertices = <span class="integer">32</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessGenLevel = <span class="integer">64</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessControlInputComponents = <span class="integer">128</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessControlOutputComponents = <span class="integer">128</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessControlTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessControlUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessControlTotalOutputComponents = <span class="integer">4096</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessControlImageUniforms = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessControlAtomicCounters = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessControlAtomicCounterBuffers = <span class="integer">0</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessEvaluationInputComponents = <span class="integer">128</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessEvaluationOutputComponents = <span class="integer">128</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessEvaluationTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessEvaluationUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessEvaluationImageUniforms = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessEvaluationAtomicCounters = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTessEvaluationAtomicCounterBuffers = <span class="integer">0</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryInputComponents = <span class="integer">64</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryOutputComponents = <span class="integer">128</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryImageUniforms = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryOutputVertices = <span class="integer">256</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryTotalOutputComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryVaryingComponents = <span class="integer">64</span>; <span class="comment">// deprecated</span>
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryAtomicCounters = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxGeometryAtomicCounterBuffers = <span class="integer">0</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxFragmentImageUniforms = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxFragmentInputComponents = <span class="integer">128</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxFragmentUniformVectors = <span class="integer">256</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxFragmentUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxFragmentAtomicCounters = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxFragmentAtomicCounterBuffers = <span class="integer">1</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxDrawBuffers = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MinProgramTexelOffset = -<span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxProgramTexelOffset = <span class="integer">7</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxImageUnits = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxSamples = <span class="integer">4</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxImageSamples = <span class="integer">0</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxClipDistances = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxCullDistances = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxViewports = <span class="integer">16</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxComputeImageUniforms = <span class="integer">8</span>;
+<span class="directive">const</span> ivec3 gl_MaxComputeWorkGroupCount = { <span class="integer">65535</span>, <span class="integer">65535</span>, <span class="integer">65535</span> };
+<span class="directive">const</span> ivec3 gl_MaxComputeWorkGroupSize = { <span class="integer">1024</span>, <span class="integer">1024</span>, <span class="integer">64</span> };
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxComputeUniformComponents = <span class="integer">1024</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxComputeTextureImageUnits = <span class="integer">16</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxComputeAtomicCounters = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxComputeAtomicCounterBuffers = <span class="integer">8</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxCombinedTextureImageUnits = <span class="integer">96</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxCombinedImageUniforms = <span class="integer">48</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxCombinedImageUnitsAndFragmentOutputs = <span class="integer">8</span>; <span class="comment">// deprecated</span>
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxCombinedShaderOutputResources = <span class="integer">16</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxCombinedAtomicCounters = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxCombinedAtomicCounterBuffers = <span class="integer">1</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxCombinedClipAndCullDistances = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxAtomicCounterBindings = <span class="integer">1</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxAtomicCounterBufferSize = <span class="integer">32</span>;
+
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTransformFeedbackBuffers = <span class="integer">4</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTransformFeedbackInterleavedComponents = <span class="integer">64</span>;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The constant <em>gl_MaxVaryingFloats</em> is removed in the core profile, use
+<em>gl_MaxVaryingComponents</em> instead.</p>
+</div>
+<div class="sect3">
+<h4 id="compatibility-profile-built-in-constants">7.3.1. Compatibility Profile Built-In Constants</h4>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTextureUnits = <span class="integer">2</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxTextureCoords = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxClipPlanes = <span class="integer">8</span>;
+<span class="directive">const</span> <span class="predefined-type">int</span> gl_MaxVaryingFloats = <span class="integer">60</span>;</code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="built-in-uniform-state">7.4. Built-In Uniform State</h3>
+<div class="paragraph">
+<p>Built-in uniform state is not available when generating SPIR-V.</p>
+</div>
+<div class="paragraph">
+<p>Otherwise, as an aid to accessing OpenGL processing state, the following
+uniform variables are built into the OpenGL Shading Language.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">//</span>
+<span class="comment">// Depth range in window coordinates,</span>
+<span class="comment">// section 13.6.1 "Controlling the Viewport" in the</span>
+<span class="comment">// OpenGL Specification.</span>
+<span class="comment">//</span>
+<span class="comment">// Note: Depth-range state is only for viewport 0.</span>
+<span class="comment">//</span>
+<span class="keyword">struct</span> gl_DepthRangeParameters {
+ <span class="predefined-type">float</span> near; <span class="comment">// n</span>
+ <span class="predefined-type">float</span> far; <span class="comment">// f</span>
+ <span class="predefined-type">float</span> diff; <span class="comment">// f - n</span>
+};
+uniform gl_DepthRangeParameters gl_DepthRange;
+uniform <span class="predefined-type">int</span> gl_NumSamples;</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>These variables are only guaranteed to be available in the fragment stage.
+In other stages, their presence and function is implementation-defined.</p>
+</div>
+<div class="sect3">
+<h4 id="compatibility-profile-state">7.4.1. Compatibility Profile State</h4>
+<div class="paragraph">
+<p>These variables are present only in the compatibility profile.
+They are not available to compute shaders, but are available to all other
+shaders.</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+uniform mat4 gl_ModelViewMatrix;
+uniform mat4 gl_ProjectionMatrix;
+uniform mat4 gl_ModelViewProjectionMatrix;
+uniform mat4 gl_TextureMatrix[gl_MaxTextureCoords];
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+uniform mat3 gl_NormalMatrix; <span class="comment">// transpose of the inverse of the</span>
+ <span class="comment">// upper leftmost 3x3 of gl_ModelViewMatrix</span>
+
+uniform mat4 gl_ModelViewMatrixInverse;
+uniform mat4 gl_ProjectionMatrixInverse;
+uniform mat4 gl_ModelViewProjectionMatrixInverse;
+uniform mat4 gl_TextureMatrixInverse[gl_MaxTextureCoords];
+
+uniform mat4 gl_ModelViewMatrixTranspose;
+uniform mat4 gl_ProjectionMatrixTranspose;
+uniform mat4 gl_ModelViewProjectionMatrixTranspose;
+uniform mat4 gl_TextureMatrixTranspose[gl_MaxTextureCoords];
+
+uniform mat4 gl_ModelViewMatrixInverseTranspose;
+uniform mat4 gl_ProjectionMatrixInverseTranspose;
+uniform mat4 gl_ModelViewProjectionMatrixInverseTranspose;
+uniform mat4 gl_TextureMatrixInverseTranspose[gl_MaxTextureCoords];
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+uniform <span class="predefined-type">float</span> gl_NormalScale;
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+uniform vec4 gl_ClipPlane[gl_MaxClipPlanes];
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+<span class="keyword">struct</span> gl_PointParameters {
+ <span class="predefined-type">float</span> size;
+ <span class="predefined-type">float</span> sizeMin;
+ <span class="predefined-type">float</span> sizeMax;
+ <span class="predefined-type">float</span> fadeThresholdSize;
+ <span class="predefined-type">float</span> distanceConstantAttenuation;
+ <span class="predefined-type">float</span> distanceLinearAttenuation;
+ <span class="predefined-type">float</span> distanceQuadraticAttenuation;
+};
+
+uniform gl_PointParameters gl_Point;
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+<span class="keyword">struct</span> gl_MaterialParameters {
+ vec4 emission; <span class="comment">// Ecm</span>
+ vec4 ambient; <span class="comment">// Acm</span>
+ vec4 diffuse; <span class="comment">// Dcm</span>
+ vec4 specular; <span class="comment">// Scm</span>
+ <span class="predefined-type">float</span> shininess; <span class="comment">// Srm</span>
+};
+uniform gl_MaterialParameters gl_FrontMaterial;
+uniform gl_MaterialParameters gl_BackMaterial;
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+<span class="keyword">struct</span> gl_LightSourceParameters {
+ vec4 ambient; <span class="comment">// Acli</span>
+ vec4 diffuse; <span class="comment">// Dcli</span>
+ vec4 specular; <span class="comment">// Scli</span>
+ vec4 position; <span class="comment">// Ppli</span>
+ vec4 halfVector; <span class="comment">// Derived: Hi</span>
+ vec3 spotDirection; <span class="comment">// Sdli</span>
+ <span class="predefined-type">float</span> spotExponent; <span class="comment">// Srli</span>
+ <span class="predefined-type">float</span> spotCutoff; <span class="comment">// Crli</span>
+ <span class="comment">// (range: [0.0,90.0], 180.0)</span>
+ <span class="predefined-type">float</span> spotCosCutoff; <span class="comment">// Derived: cos(Crli)</span>
+ <span class="comment">// (range: [1.0,0.0],-1.0)</span>
+ <span class="predefined-type">float</span> constantAttenuation; <span class="comment">// K0</span>
+ <span class="predefined-type">float</span> linearAttenuation; <span class="comment">// K1</span>
+ <span class="predefined-type">float</span> quadraticAttenuation; <span class="comment">// K2</span>
+};
+
+uniform gl_LightSourceParameters gl_LightSource[gl_MaxLights];
+
+<span class="keyword">struct</span> gl_LightModelParameters {
+ vec4 ambient; <span class="comment">// Acs</span>
+};
+
+uniform gl_LightModelParameters gl_LightModel;
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+<span class="comment">// Derived state from products of light and material.</span>
+<span class="comment">//</span>
+
+<span class="keyword">struct</span> gl_LightModelProducts {
+ vec4 sceneColor; <span class="comment">// Derived. Ecm + Acm * Acs</span>
+};
+
+uniform gl_LightModelProducts gl_FrontLightModelProduct;
+uniform gl_LightModelProducts gl_BackLightModelProduct;
+
+<span class="keyword">struct</span> gl_LightProducts {
+ vec4 ambient; <span class="comment">// Acm * Acli</span>
+ vec4 diffuse; <span class="comment">// Dcm * Dcli</span>
+ vec4 specular; <span class="comment">// Scm * Scli</span>
+};
+
+uniform gl_LightProducts gl_FrontLightProduct[gl_MaxLights];
+uniform gl_LightProducts gl_BackLightProduct[gl_MaxLights];
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+uniform vec4 gl_TextureEnvColor[gl_MaxTextureUnits];
+uniform vec4 gl_EyePlaneS[gl_MaxTextureCoords];
+uniform vec4 gl_EyePlaneT[gl_MaxTextureCoords];
+uniform vec4 gl_EyePlaneR[gl_MaxTextureCoords];
+uniform vec4 gl_EyePlaneQ[gl_MaxTextureCoords];
+uniform vec4 gl_ObjectPlaneS[gl_MaxTextureCoords];
+uniform vec4 gl_ObjectPlaneT[gl_MaxTextureCoords];
+uniform vec4 gl_ObjectPlaneR[gl_MaxTextureCoords];
+uniform vec4 gl_ObjectPlaneQ[gl_MaxTextureCoords];
+
+<span class="comment">//</span>
+<span class="comment">// compatibility profile only</span>
+<span class="comment">//</span>
+<span class="keyword">struct</span> gl_FogParameters {
+ vec4 color;
+ <span class="predefined-type">float</span> density;
+ <span class="predefined-type">float</span> start;
+ <span class="predefined-type">float</span> end;
+ <span class="predefined-type">float</span> scale; <span class="comment">// Derived: 1.0 / (end - start)</span>
+};
+
+uniform gl_FogParameters gl_Fog;</code></pre>
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="redeclaring-built-in-blocks">7.5. Redeclaring Built-In Blocks</h3>
+<div class="paragraph">
+<p>The <em>gl_PerVertex</em> block can be redeclared in a shader to explicitly
+indicate what subset of the fixed pipeline interface will be used.
+This is necessary to establish the interface between multiple programs.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out gl_PerVertex {
+ vec4 gl_Position; <span class="comment">// will use gl_Position</span>
+ <span class="predefined-type">float</span> gl_PointSize; <span class="comment">// will use gl_PointSize</span>
+ vec4 t; <span class="comment">// error, only gl_PerVertex members allowed</span>
+}; <span class="comment">// no other members of gl_PerVertex will be used</span></code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>This establishes the output interface the shader will use with the
+subsequent pipeline stage.
+It must be a subset of the built-in members of <em>gl_PerVertex</em>.
+Such a redeclaration can also add the <strong>invariant</strong> qualifier, interpolation
+qualifiers, and the layout qualifiers <strong>xfb_offset</strong>, <strong>xfb_buffer</strong>, and
+<strong>xfb_stride</strong>.
+It can also add an array size for unsized arrays.
+For example:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">out layout(xfb_buffer = <span class="integer">1</span>, xfb_stride = <span class="integer">16</span>) gl_PerVertex {
+ vec4 gl_Position;
+ layout(xfb_offset = <span class="integer">0</span>) <span class="predefined-type">float</span> gl_ClipDistance[<span class="integer">4</span>];
+};</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>Other layout qualifiers, like <strong>location</strong>, cannot be added to such a
+redeclaration, unless specifically stated.</p>
+</div>
+<div class="paragraph">
+<p>If a built-in interface block is redeclared, it must appear in the shader
+before any use of any member included in the built-in declaration, or a
+compile-time error will result.
+It is also a compile-time error to redeclare the block more than once or to
+redeclare a built-in block and then use a member from that built-in block
+that was not included in the redeclaration.
+Also, if a built-in interface block is redeclared, no member of the built-in
+declaration can be redeclared outside the block redeclaration.
+If multiple shaders using members of a built-in block belonging to the same
+interface are linked together in the same program, they must all redeclare
+the built-in block in the same way, as described in
+“<a href="#interface-blocks">Interface Blocks</a>” for interface block matching, or a
+link-time error will result.
+It will also be a link-time error if some shaders in a program redeclare a
+specific built-in interface block while another shader in that program does
+not redeclare that interface block yet still uses a member of that interface
+block.
+If a built-in block interface is formed across shaders in different
+programs, the shaders must all redeclare the built-in block in the same way
+(as described for a single program), or the values passed along the
+interface are undefined.</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="built-in-functions">8. Built-In Functions</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The OpenGL Shading Language defines an assortment of built-in convenience functions for
+scalar and vector operations.
+Many of these built-in functions can be used in more than one type of
+shader, but some are intended to provide a direct mapping to hardware and so
+are available only for a specific type of shader.</p>
+</div>
+<div class="paragraph">
+<p>The built-in functions basically fall into three categories:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>They expose some necessary hardware functionality in a convenient way
+such as accessing a texture map.
+There is no way in the language for these functions to be emulated by a
+shader.</p>
+</li>
+<li>
+<p>They represent a trivial operation (clamp, mix, etc.) that is very
+simple for the user to write, but they are very common and may have
+direct hardware support.
+It is a very hard problem for the compiler to map expressions to complex
+assembler instructions.</p>
+</li>
+<li>
+<p>They represent an operation graphics hardware is likely to accelerate at
+some point.
+The trigonometry functions fall into this category.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Many of the functions are similar to the same named ones in common C
+libraries, but they support vector input as well as the more traditional
+scalar input.</p>
+</div>
+<div class="paragraph">
+<p>Applications should be encouraged to use the built-in functions rather than
+do the equivalent computations in their own shader code since the built-in
+functions are assumed to be optimal (e.g. perhaps supported directly in
+hardware).</p>
+</div>
+<div class="paragraph">
+<p>User code can replace built-in functions with their own if they choose, by
+simply redeclaring and defining the same name and argument list.
+Because built-in functions are in a more outer scope than user built-in
+functions, doing this will hide all built-in functions with the same name as
+the redeclared function.</p>
+</div>
+<div class="paragraph">
+<p>When the built-in functions are specified below, where the input arguments
+(and corresponding output) can be <strong>float</strong>, <strong>vec2</strong>, <strong>vec3</strong>, or <strong>vec4</strong>,
+<em>genFType</em> is used as the argument.
+Where the input arguments (and corresponding output) can be <strong>int</strong>, <strong>ivec2</strong>,
+<strong>ivec3</strong>, or <strong>ivec4</strong>, <em>genIType</em> is used as the argument.
+Where the input arguments (and corresponding output) can be <strong>uint</strong>, <strong>uvec2</strong>,
+<strong>uvec3</strong>, or <strong>uvec4</strong>, <em>genUType</em> is used as the argument.
+Where the input arguments (or corresponding output) can be <strong>bool</strong>, <strong>bvec2</strong>,
+<strong>bvec3</strong>, or <strong>bvec4</strong>, <em>genBType</em> is used as the argument.
+Where the input arguments (and corresponding output) can be <strong>double</strong>,
+<strong>dvec2</strong>, <strong>dvec3</strong>, <strong>dvec4</strong>, <em>genDType</em> is used as the argument.
+For any specific use of a function, the actual types substituted for
+<em>genFType</em>, <em>genIType</em>, <em>genUType</em>, or <em>genBType</em> have to have the same
+number of components for all arguments and for the return type.
+Similarly, <em>mat</em> is used for any matrix basic
+type with single-precision
+components and <em>dmat</em> is used for any matrix basic type with
+double-precision components.</p>
+</div>
+<div class="sect2">
+<h3 id="angle-and-trigonometry-functions">8.1. Angle and Trigonometry Functions</h3>
+<div class="paragraph">
+<p>Function parameters specified as <em>angle</em> are assumed to be in units of
+radians.
+In no case will any of these functions result in a divide by zero error.
+If the divisor of a ratio is 0, then results will be undefined.</p>
+</div>
+<div class="paragraph">
+<p>These all operate component-wise.
+The description is per component.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>radians</strong>(genFType <em>degrees</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Converts <em>degrees</em> to radians, i.e.,
+ <span class="eq">(π / 180) · degrees</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>degrees</strong>(genFType <em>radians</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Converts <em>radians</em> to degrees, i.e.,
+ <span class="eq">(180 / π) · radians</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>sin</strong>(genFType <em>angle</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The standard trigonometric sine function.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>cos</strong>(genFType <em>angle</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The standard trigonometric cosine function.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>tan</strong>(genFType <em>angle</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">The standard trigonometric tangent.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>asin</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc sine.
+ Returns an angle whose sine is <em>x</em>.
+ The range of values returned by this function is
+ <span class="eq">[-π / 2, π / 2]</span>.
+ Results are undefined if <span class="eq">|x| > 1</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>acos</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc cosine.
+ Returns an angle whose cosine is <em>x</em>.
+ The range of values returned by this function is <span class="eq">[0,π]</span>.
+ Results are undefined if <span class="eq">|x| > 1</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>atan</strong>(genFType <em>y</em>, genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc tangent.
+ Returns an angle whose tangent is <span class="eq">y / x</span>.
+ The signs of <em>x</em> and <em>y</em> are used to determine what quadrant the angle
+ is in.
+ The range of values returned by this function is <span class="eq">[-π, π</span>.
+ Results are undefined if <em>x</em> and <em>y</em> are both 0.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>atan</strong>(genFType <em>y_over_x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc tangent.
+ Returns an angle whose tangent is <em>y_over_x</em>.
+ The range of values returned by this function is
+ <span class="eq">[-π / 2, π / 2]</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>sinh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the hyperbolic sine function <span class="eq">(e<sup>x</sup> - e<sup>-x</sup>) / 2</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>cosh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the hyperbolic cosine function <span class="eq">(e<sup>x</sup> + e<sup>-x</sup>) / 2</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>tanh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the hyperbolic tangent function <span class="eq">sinh(x) / cosh(x)</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>asinh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc hyperbolic sine; returns the inverse of <strong>sinh</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>acosh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc hyperbolic cosine; returns the non-negative inverse of <strong>cosh</strong>.
+ Results are undefined if <span class="eq">x < 1</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>atanh</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Arc hyperbolic tangent; returns the inverse of <strong>tanh</strong>.
+ Results are undefined if <span class="eq">x ≥ 1</span>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="exponential-functions">8.2. Exponential Functions</h3>
+<div class="paragraph">
+<p>These all operate component-wise.
+The description is per component.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>pow</strong>(genFType <em>x</em>, genFType <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> raised to the <em>y</em> power, i.e., <span class="eq">x<sup>y</sup></span>.
+ Results are undefined if <span class="eq">x < 0</span>.
+ Results are undefined if <span class="eq">x = 0</span> and <span class="eq">y ≤ 0</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>exp</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the natural exponentiation of <em>x</em>, i.e., <span class="eq">e<sup>x</sup></span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>log</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the natural logarithm of <em>x</em>, i.e., returns the value <em>y</em>
+ which satisfies the equation <span class="eq">x = e<sup>y</sup></span>.
+ Results are undefined if <span class="eq">x ≤ 0</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>exp2</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns 2 raised to the <em>x</em> power, i.e., <span class="eq">2<sup>x</sup></span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>log2</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the base 2 logarithm of <em>x</em>, i.e., returns the value <em>y</em> which
+ satisfies the equation <span class="eq">x = 2<sup>y</sup></span>.
+ Results are undefined if <span class="eq">x ≤ 0</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>sqrt</strong>(genFType <em>x</em>)<br>
+ genDType <strong>sqrt</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <span class="eq">sqrt(x)</span>.
+ Results are undefined if <span class="eq">x < 0</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>inversesqrt</strong>(genFType <em>x</em>)<br>
+ genDType <strong>inversesqrt</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <span class="eq">1 / sqrt(x)</span>.
+ Results are undefined if <span class="eq">x ≤ 0</span>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="common-functions">8.3. Common Functions</h3>
+<div class="paragraph">
+<p>These all operate component-wise.
+The description is per component.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>abs</strong>(genFType <em>x</em>)<br>
+ genIType <strong>abs</strong>(genIType <em>x</em>)<br>
+ genDType <strong>abs</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> if <span class="eq">x ≥ 0</span>; otherwise it returns -<em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>sign</strong>(genFType <em>x</em>)<br>
+ genIType <strong>sign</strong>(genIType <em>x</em>)<br>
+ genDType <strong>sign</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns 1.0 if <em>x</em> > 0, 0.0 if <em>x</em> = 0, or -1.0 if <em>x</em> < 0.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>floor</strong>(genFType <em>x</em>)<br>
+ genDType <strong>floor</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer that is less than or
+ equal to <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>trunc</strong>(genFType <em>x</em>)<br>
+ genDType <strong>trunc</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer to <em>x</em> whose absolute
+ value is not larger than the absolute value of <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>round</strong>(genFType <em>x</em>)<br>
+ genDType <strong>round</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer to <em>x</em>.
+ The fraction 0.5 will round in a direction chosen by the
+ implementation, presumably the direction that is fastest.
+ This includes the possibility that <strong>round</strong>(<em>x</em>) returns the same value
+ as <strong>roundEven</strong>(<em>x</em>) for all values of <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>roundEven</strong>(genFType <em>x</em>)<br>
+ genDType <strong>roundEven</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer to <em>x</em>.
+ A fractional part of 0.5 will round toward the nearest even integer.
+ (Both 3.5 and 4.5 for x will return 4.0.)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>ceil</strong>(genFType <em>x</em>)<br>
+ genDType <strong>ceil</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a value equal to the nearest integer that is greater than or
+ equal to <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>fract</strong>(genFType <em>x</em>)<br>
+ genDType <strong>fract</strong>(genDType <em>x</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>x</em> - <strong>floor</strong>(<em>x</em>).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>mod</strong>(genFType <em>x</em>, float <em>y</em>)<br>
+ genFType <strong>mod</strong>(genFType <em>x</em>, genFType <em>y</em>)<br>
+ genDType <strong>mod</strong>(genDType <em>x</em>, double <em>y</em>)<br>
+ genDType <strong>mod</strong>(genDType <em>x</em>, genDType <em>y</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Modulus.
+ Returns <span class="eq">x - y · <strong>floor</strong>(x / y)</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>modf</strong>(genFType <em>x</em>, out genFType <em>i</em>)<br>
+ genDType <strong>modf</strong>(genDType <em>x</em>, out genDType <em>i</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the fractional part of <em>x</em> and sets <em>i</em> to the integer part (as
+ a whole number floating-point value).
+ Both the return value and the output parameter will have the same sign
+ as <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>min</strong>(genFType <em>x</em>, genFType <em>y</em>)<br>
+ genFType <strong>min</strong>(genFType <em>x</em>, float <em>y</em>)<br>
+ genDType <strong>min</strong>(genDType <em>x</em>, genDType <em>y</em>)<br>
+ genDType <strong>min</strong>(genDType <em>x</em>, double <em>y</em>)<br>
+ genIType <strong>min</strong>(genIType <em>x</em>, genIType <em>y</em>)<br>
+ genIType <strong>min</strong>(genIType <em>x</em>, int <em>y</em>)<br>
+ genUType <strong>min</strong>(genUType <em>x</em>, genUType <em>y</em>)<br>
+ genUType <strong>min</strong>(genUType <em>x</em>, uint <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>y</em> if <em>y</em> < <em>x;</em> otherwise it returns <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>max</strong>(genFType <em>x</em>, genFType <em>y</em>)<br>
+ genFType <strong>max</strong>(genFType <em>x</em>, float <em>y</em>)<br>
+ genDType <strong>max</strong>(genDType <em>x</em>, genDType <em>y</em>)<br>
+ genDType <strong>max</strong>(genDType <em>x</em>, double <em>y</em>)<br>
+ genIType <strong>max</strong>(genIType <em>x</em>, genIType <em>y</em>)<br>
+ genIType <strong>max</strong>(genIType <em>x</em>, int <em>y</em>)<br>
+ genUType <strong>max</strong>(genUType <em>x</em>, genUType <em>y</em>)<br>
+ genUType <strong>max</strong>(genUType <em>x</em>, uint <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <em>y</em> if <em>x</em> < <em>y;</em> otherwise it returns <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>clamp</strong>(genFType <em>x</em>, genFType <em>minVal</em>, genFType <em>maxVal</em>)<br>
+ genFType <strong>clamp</strong>(genFType <em>x</em>, float <em>minVal</em>, float <em>maxVal</em>)<br>
+ genDType <strong>clamp</strong>(genDType <em>x</em>, genDType <em>minVal</em>, genDType <em>maxVal</em>)<br>
+ genDType <strong>clamp</strong>(genDType <em>x</em>, double <em>minVal</em>, double <em>maxVal</em>)<br>
+ genIType <strong>clamp</strong>(genIType <em>x</em>, genIType <em>minVal</em>, genIType <em>maxVal</em>)<br>
+ genIType <strong>clamp</strong>(genIType <em>x</em>, int <em>minVal</em>, int <em>maxVal</em>)<br>
+ genUType <strong>clamp</strong>(genUType <em>x</em>, genUType <em>minVal</em>, genUType <em>maxVal</em>)<br>
+ genUType <strong>clamp</strong>(genUType <em>x</em>, uint <em>minVal</em>, uint <em>maxVal</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>min</strong>(<strong>max</strong>(<em>x</em>, <em>minVal</em>), <em>maxVal</em>).
+ Results are undefined if <em>minVal</em> > <em>maxVal</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>mix</strong>(genFType <em>x</em>, genFType <em>y</em>, genFType <em>a</em>)<br>
+ genFType <strong>mix</strong>(genFType <em>x</em>, genFType <em>y</em>, float <em>a</em>)<br>
+ genDType <strong>mix</strong>(genDType <em>x</em>, genDType <em>y</em>, genDType <em>a</em>)<br>
+ genDType <strong>mix</strong>(genDType <em>x</em>, genDType <em>y</em>, double <em>a</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the linear blend of <em>x</em> and <em>y</em>, i.e.,
+ <span class="eq">x · (1 - a) + y · a</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>mix</strong>(genFType <em>x</em>, genFType <em>y</em>, genBType <em>a</em>)<br>
+ genDType <strong>mix</strong>(genDType <em>x</em>, genDType <em>y</em>, genBType <em>a</em>)<br>
+ genIType <strong>mix</strong>(genIType <em>x</em>, genIType <em>y</em>, genBType <em>a</em>)<br>
+ genUType <strong>mix</strong>(genUType <em>x</em>, genUType <em>y</em>, genBType <em>a</em>)<br>
+ genBType <strong>mix</strong>(genBType <em>x</em>, genBType <em>y</em>, genBType <em>a</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Selects which vector each returned component comes from.
+ For a component of <em>a</em> that is <strong>false</strong>, the corresponding component of
+ <em>x</em> is returned.
+ For a component of <em>a</em> that is <strong>true</strong>, the corresponding component of
+ <em>y</em> is returned.
+ Components of <em>x</em> and <em>y</em> that are not selected are allowed to be
+ invalid floating-point values and will have no effect on the results.
+ Thus, this provides different functionality than, for example,<br>
+ genFType <strong>mix</strong>(genFType <em>x</em>, genFType <em>y</em>, genFType(<em>a</em>))<br>
+ where <em>a</em> is a Boolean vector.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>step</strong>(genFType <em>edge</em>, genFType <em>x</em>)<br>
+ genFType <strong>step</strong>(float <em>edge</em>, genFType <em>x</em>)<br>
+ genDType <strong>step</strong>(genDType <em>edge</em>, genDType <em>x</em>)<br>
+ genDType <strong>step</strong>(double <em>edge</em>, genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns 0.0 if <em>x</em> < <em>edge;</em> otherwise it returns 1.0.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>smoothstep</strong>(genFType <em>edge0</em>, genFType <em>edge1</em>, genFType <em>x</em>)<br>
+ genFType <strong>smoothstep</strong>(float <em>edge0</em>, float <em>edge1</em>, genFType <em>x</em>)<br>
+ genDType <strong>smoothstep</strong>(genDType <em>edge0</em>, genDType <em>edge1</em>, genDType <em>x</em>)<br>
+ genDType <strong>smoothstep</strong>(double <em>edge0</em>, double <em>edge1</em>, genDType <em>x</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Returns 0.0 if <span class="eq">x ≤ edge0</span> and 1.0 if <span class="eq">x ≥ edge1</span>, and
+ performs smooth Hermite interpolation between 0 and 1 when <span class="eq">edge0
+ < x < edge1</span>.
+ This is useful in cases where you would want a threshold function with
+ a smooth transition.
+ This is equivalent to:</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">genFType t;
+t = clamp ((x - edge0) / (edge1 - edge0), <span class="integer">0</span>, <span class="integer">1</span>);
+<span class="keyword">return</span> t * t * (<span class="integer">3</span> - <span class="integer">2</span> * t);</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>(And similarly for doubles.) Results are undefined if <span class="eq">edge0 ≥
+edge1</span>.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genBType <strong>isnan</strong>(genFType <em>x</em>)<br>
+ genBType <strong>isnan</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if <em>x</em> holds a NaN.
+ Returns <strong>false</strong> otherwise.
+ Always returns <strong>false</strong> if NaNs are not implemented.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genBType <strong>isinf</strong>(genFType <em>x</em>)<br>
+ genBType <strong>isinf</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if <em>x</em> holds a positive infinity or negative infinity.
+ Returns <strong>false</strong> otherwise.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genIType <strong>floatBitsToInt</strong>( genFType <em>value</em>)<br>
+ genUType <strong>floatBitsToUint</strong>( genFType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a signed or unsigned integer value representing the encoding
+ of a floating-point value.
+ The <strong>float</strong> value’s bit-level representation is preserved.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>intBitsToFloat</strong>( genIType <em>value</em>)<br>
+ genFType <strong>uintBitsToFloat</strong>( genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a floating-point value corresponding to a signed or unsigned
+ integer encoding of a floating-point value.
+ If a NaN is passed in, it will not signal, and the resulting value is
+ unspecified.
+ If an Inf is passed in, the resulting value is the corresponding Inf.
+ Otherwise, the bit-level representation is preserved.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>fma</strong>(genFType <em>a</em>, genFType <em>b</em>, genFType <em>c</em>)<br>
+ genDType <strong>fma</strong>(genDType <em>a</em>, genDType <em>b</em>, genDType <em>c</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Computes and returns <span class="eq">a * b + c</span>.
+ In uses where the return value is eventually consumed by a variable
+ declared as <strong>precise</strong>:</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="ulist">
+<ul>
+<li>
+<p><strong>fma</strong>() is considered a single operation, whereas the expression “a * b
++ c” consumed by a variable declared <strong>precise</strong> is considered two
+operations.</p>
+</li>
+<li>
+<p>The precision of <strong>fma</strong>() can differ from the precision of the expression
+“a * b + c”.</p>
+</li>
+<li>
+<p><strong>fma</strong>() will be computed with the same precision as any other <strong>fma</strong>()
+consumed by a precise variable, giving invariant results for the same
+input values of <em>a</em>, <em>b</em>, and <em>c</em>.</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>Otherwise, in the absence of <strong>precise</strong> consumption, there are no special
+constraints on the number of operations or difference in precision between
+<strong>fma</strong>() and the expression “_a_ * <em>b</em> + <em>c</em>”.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>frexp</strong>( genFType <em>x</em>, out genIType <em>exp</em>)
+ genDType <strong>frexp</strong>(genDType <em>x</em>, out genIType <em>exp</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Splits <em>x</em> into a floating-point significand in the range
+ <span class="eq">[0.5,1.0]</span>, and an integral exponent of two, such that</p>
+<p class="tableblock"> <span class="eq">x = significant · 2<sup>exponent</sup></span></p>
+<p class="tableblock"> The significand is returned by the function and the exponent is
+ returned in the parameter <em>exp</em>.
+ For a floating-point value of zero, the significand and exponent are
+ both zero.</p>
+<p class="tableblock"> If an implementation supports signed zero, an input value of minus
+ zero should return a significand of minus zero.
+ For a floating-point value that is an infinity or is not a number, the
+ results are undefined.</p>
+<p class="tableblock"> If the input <em>x</em> is a vector, this operation is performed in a
+ component-wise manner; the value returned by the function and the
+ value written to <em>exp</em> are vectors with the same number of components
+ as <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>ldexp</strong>( genFType <em>x</em>, genIType <em>exp</em>)<br>
+ genDType <strong>ldexp</strong>(genDType <em>x</em>, genIType <em>exp</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Builds a floating-point number from <em>x</em> and the corresponding integral
+ exponent of two in <em>exp</em>, returning:</p>
+<p class="tableblock"> <span class="eq">significand · 2<sup>exponent</sup></span></p>
+<p class="tableblock"> If this product is too large to be represented in the floating-point
+ type, the result is undefined.</p>
+<p class="tableblock"> If <em>exp</em> is greater than +128 (single-precision) or +1024
+ (double-precision), the value returned is undefined.
+ If <em>exp</em> is less than -126 (single-precision) or -1022
+ (double-precision), the value returned may be flushed to zero.
+ Additionally, splitting the value into a significand and exponent
+ using <strong>frexp</strong>() and then reconstructing a floating-point value using
+ <strong>ldexp</strong>() should yield the original input for zero and all finite
+ non-denormalized values.<br>
+ If the input <em>x</em> is a vector, this operation is performed in a
+ component-wise manner; the value passed in <em>exp</em> and returned by the
+ function are vectors with the same number of components as <em>x</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="floating-point-pack-and-unpack-functions">8.4. Floating-Point Pack and Unpack Functions</h3>
+<div class="paragraph">
+<p>These functions do not operate component-wise, rather, as described in each
+case.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>packUnorm2x16</strong>(vec2 <em>v</em>)<br>
+ uint <strong>packSnorm2x16</strong>(vec2 <em>v</em>)<br>
+ uint <strong>packUnorm4x8</strong>(vec4 <em>v</em>)<br>
+ uint <strong>packSnorm4x8</strong>(vec4 <em>v</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">First, converts each component of the normalized floating-point value
+ <em>v</em> into 16-bit (<strong>2x16</strong>) or 8-bit (<strong>4x8</strong>) integer values.
+ Then, the results are packed into the returned 32-bit unsigned
+ integer.</p>
+<p class="tableblock"> The conversion for component <em>c</em> of <em>v</em> to fixed point is done as
+ follows:</p>
+<p class="tableblock"> <strong>packUnorm2x16</strong>: <strong>round</strong>(<strong>clamp</strong>(<em>c</em>, 0, +1) * 65535.0)<br>
+ <strong>packSnorm2x16:</strong> <strong>round</strong>(<strong>clamp</strong>(<em>c</em>, -1, +1) * 32767.0)<br>
+ <strong>packUnorm4x8</strong>: <strong>round</strong>(<strong>clamp</strong>(<em>c</em>, 0, +1) * 255.0)<br>
+ <strong>packSnorm4x8</strong>: <strong>round</strong>(<strong>clamp</strong>(<em>c</em>, -1, +1) * 127.0)</p>
+<p class="tableblock"> The first component of the vector will be written to the least
+ significant bits of the output; the last component will be written to
+ the most significant bits.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> vec2 <strong>unpackUnorm2x16</strong>( uint <em>p</em>)<br>
+ vec2 <strong>unpackSnorm2x16</strong>( uint <em>p</em>)<br>
+ vec4 <strong>unpackUnorm4x8</strong>( uint <em>p</em>)<br>
+ vec4 <strong>unpackSnorm4x8</strong>( uint <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">First, unpacks a single 32-bit unsigned integer <em>p</em> into a pair of
+ 16-bit unsigned integers, a pair of 16-bit signed integers, four 8-bit
+ unsigned integers, or four 8-bit signed integers, respectively.
+ Then, each component is converted to a normalized floating-point value
+ to generate the returned two- or four-component vector.</p>
+<p class="tableblock"> The conversion for unpacked fixed-point value <em>f</em> to floating-point is
+ done as follows:</p>
+<p class="tableblock"> <strong>unpackUnorm2x16</strong>: <em>f</em> / 65535.0<br>
+ <strong>unpackSnorm2x16</strong>: <strong>clamp</strong>(<em>f</em> / 32767.0, -1, +1)<br>
+ <strong>unpackUnorm4x8</strong>: <em>f</em> / 255.0<br>
+ <strong>unpackSnorm4x8</strong>: <strong>clamp</strong>(<em>f</em> / 127.0, -1, +1)</p>
+<p class="tableblock"> The first component of the returned vector will be extracted from the
+ least significant bits of the input; the last component will be
+ extracted from the most significant bits.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>packHalf2x16</strong>(vec2 <em>v</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns an unsigned integer obtained by converting the components of a
+ two-component floating-point vector to the 16-bit floating-point
+ representation found in the <a href="#references">OpenGL Specification</a>, and
+ then packing these two 16-bit integers into a 32-bit unsigned integer.</p>
+<p class="tableblock"> The first vector component specifies the 16 least-significant bits of
+ the result; the second component specifies the 16 most-significant
+ bits.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> vec2 <strong>unpackHalf2x16</strong>( uint <em>v</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a two-component floating-point vector with components obtained
+ by unpacking a 32-bit unsigned integer into a pair of 16-bit values,
+ interpreting those values as 16-bit floating-point numbers according
+ to the <a href="#references">OpenGL Specification</a>, and converting them to
+ 32-bit floating-point values.</p>
+<p class="tableblock"> The first component of the vector is obtained from the 16
+ least-significant bits of <em>v</em>; the second component is obtained from
+ the 16 most-significant bits of <em>v</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">double <strong>packDouble2x32</strong>(uvec2 <em>v</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a double-precision value obtained by packing the components of
+ <em>v</em> into a 64-bit value.
+ If an IEEE 754 Inf or NaN is created, it will not signal, and the
+ resulting floating-point value is unspecified.
+ Otherwise, the bit-level representation of <em>v</em> is preserved.
+ The first vector component specifies the 32 least significant bits;
+ the second component specifies the 32 most significant bits.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uvec2 <strong>unpackDouble2x32</strong>(double <em>v</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a two-component unsigned integer vector representation of <em>v</em>.
+ The bit-level representation of <em>v</em> is preserved.
+ The first component of the vector contains the 32 least significant
+ bits of the double; the second component consists of the 32 most
+ significant bits.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="geometric-functions">8.5. Geometric Functions</h3>
+<div class="paragraph">
+<p>These operate on vectors as vectors, not component-wise.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>length</strong>(genFType <em>x</em>)<br>
+ double <strong>length</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Returns the length of vector <em>x</em>, i.e.,
+ <span class="eq">sqrt( x<sub>0</sub><sup>2</sup> + x<sub>1</sub><sup>2</sup> + …​ )</span>.</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>distance</strong>(genFType <em>p0</em>, genFType <em>p1</em>)<br>
+ double <strong>distance</strong>(genDType <em>p0</em>, genDType <em>p1</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the distance between <em>p0</em> and <em>p1</em>, i.e.,
+ <strong>length</strong>(<em>p0</em> - <em>p1</em>)</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>dot</strong>(genFType <em>x</em>, genFType <em>y</em>)<br>
+ double <strong>dot</strong>(genDType <em>x</em>, genDType <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the dot product of <em>x</em> and <em>y</em>, i.e.,
+ <span class="eq">x<sub>0</sub> · y<sub>0</sub> + x<sub>1</sub> · y<sub>1</sub> + …​</span></p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec3 <strong>cross</strong>(vec3 <em>x</em>, vec3 <em>y</em>)<br>
+ dvec3 <strong>cross</strong>(dvec3 <em>x</em>, dvec3 <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Returns the cross product of <em>x</em> and <em>y</em>, i.e.,
+ <span class="eq">(x<sub>1</sub> · y<sub>2</sub> - y<sub>1</sub> · x<sub>2</sub>,
+ x<sub>2</sub> · y<sub>0</sub> - y<sub>2</sub> · x<sub>0</sub>,
+ x<sub>0</sub> · y<sub>1</sub> - y<sub>0</sub> · x<sub>1</sub>)</span>.</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>normalize</strong>(genFType <em>x</em>)<br>
+ genDType <strong>normalize</strong>(genDType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a vector in the same direction as <em>x</em> but with a length of 1,
+ i.e. <em>x</em> / <strong>length</strong>(x).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">compatibility profile only<br>
+ vec4 <strong>ftransform</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Available only when using the compatibility profile.
+ For core OpenGL, use <strong>invariant</strong>.<br>
+ For vertex shaders only.
+ This function will ensure that the incoming vertex value will be
+ transformed in a way that produces exactly the same result as would be
+ produced by OpenGL’s fixed functionality transform.
+ It is intended to be used to compute <em>gl_Position</em>, e.g.</p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p><em>gl_Position</em> = <strong>ftransform</strong>()</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>This function should be used, for example, when an application is rendering
+the same geometry in separate passes, and one pass uses the fixed
+functionality path to render and another pass uses programmable shaders.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>faceforward</strong>(genFType <em>N</em>, genFType <em>I</em>, genFType <em>Nref</em>)<br>
+ genDType <strong>faceforward</strong>(genDType <em>N</em>, genDType <em>I</em>, genDType <em>Nref</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">If <strong>dot</strong>(<em>Nref</em>, <em>I</em>) < 0 return <em>N</em>, otherwise return -<em>N</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>reflect</strong>(genFType <em>I</em>, genFType <em>N</em>)<br>
+ genDType <strong>reflect</strong>(genDType <em>I</em>, genDType <em>N</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">For the incident vector <em>I</em> and surface orientation <em>N</em>, returns the
+ reflection direction: <span class="eq">I - 2 · <strong>dot</strong>(N, I) · N</span>.
+ <em>N</em> must already be normalized in order to achieve the desired result.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>refract</strong>(genFType <em>I</em>, genFType <em>N</em>, float <em>eta</em>)<br>
+ genDType <strong>refract</strong>(genDType <em>I</em>, genDType <em>N</em>, double <em>eta</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>For the incident vector <em>I</em> and surface normal <em>N</em>, and the ratio of
+ indices of refraction <em>eta</em>, return the refraction vector.
+ The result is computed by the <a href="#refraction-equation">refraction
+ equation</a> shown below.</p>
+</div>
+<div class="literalblock">
+<div class="content">
+<pre>The input parameters for the incident vector _I_ and the surface
+normal _N_ must already be normalized to get the desired results.</pre>
+</div>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+<div class="sect3">
+<h4 id="refraction-equation">8.5.1. Refraction Equation</h4>
+<div class="stemblock">
+<div class="content">
+\[k = 1.0 - eta * eta * (1.0 - \textbf{dot}(N,I) \cdot \textbf{dot}(N,I))\]
+</div>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+result &=
+ \begin{cases}
+ genFType(0.0), & k < 0.0 \\
+ eta * I - (eta * \textbf{dot}(N,I) + \sqrt { k }) * N, & \textbf{otherwise}
+ \end{cases}
+\end{aligned}\]
+</div>
+</div>
+</div>
+</div>
+<div class="sect2">
+<h3 id="matrix-functions">8.6. Matrix Functions</h3>
+<div class="paragraph">
+<p>For each of the following built-in matrix functions, there is both a
+single-precision floating-point version, where all arguments and return
+values are single precision, and a double-precision floating-point version,
+where all arguments and return values are double precision.
+Only the single-precision floating-point version is shown.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mat <strong>matrixCompMult</strong>(mat <em>x</em>, mat <em>y</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Multiply matrix <em>x</em> by matrix <em>y</em> component-wise, i.e., result[i][j]
+ is the scalar product of <em>x</em>[i][j] and <em>y</em>[i][j].<br></p>
+<p class="tableblock"> Note: to get linear algebraic matrix multiplication, use the multiply
+ operator (<strong>*</strong>).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mat2 <strong>outerProduct</strong>(vec2 <em>c</em>, vec2 <em>r</em>)<br>
+ mat3 <strong>outerProduct</strong>(vec3 <em>c</em>, vec3 <em>r</em>)<br>
+ mat4 <strong>outerProduct</strong>(vec4 <em>c</em>, vec4 <em>r</em>)<br>
+ mat2x3 <strong>outerProduct</strong>(vec3 <em>c</em>, vec2 <em>r</em>)<br>
+ mat3x2 <strong>outerProduct</strong>(vec2 <em>c</em>, vec3 <em>r</em>)<br>
+ mat2x4 <strong>outerProduct</strong>(vec4 <em>c</em>, vec2 <em>r</em>)<br>
+ mat4x2 <strong>outerProduct</strong>(vec2 <em>c</em>, vec4 <em>r</em>)<br>
+ mat3x4 <strong>outerProduct</strong>(vec4 <em>c</em>, vec3 <em>r</em>)<br>
+ mat4x3 <strong>outerProduct</strong>(vec3 <em>c</em>, vec4 <em>r</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Treats the first parameter <em>c</em> as a column vector (matrix with one
+ column) and the second parameter <em>r</em> as a row vector (matrix with one
+ row) and does a linear algebraic matrix multiply <em>c</em> * <em>r</em>, yielding a
+ matrix whose number of rows is the number of components in <em>c</em> and
+ whose number of columns is the number of components in <em>r</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mat2 <strong>transpose</strong>(mat2 <em>m</em>)<br>
+ mat3 <strong>transpose</strong>(mat3 <em>m</em>)<br>
+ mat4 <strong>transpose</strong>(mat4 <em>m</em>)<br>
+ mat2x3 <strong>transpose</strong>(mat3x2 <em>m</em>)<br>
+ mat3x2 <strong>transpose</strong>(mat2x3 <em>m</em>)<br>
+ mat2x4 <strong>transpose</strong>(mat4x2 <em>m</em>)<br>
+ mat4x2 <strong>transpose</strong>(mat2x4 <em>m</em>)<br>
+ mat3x4 <strong>transpose</strong>(mat4x3 <em>m</em>)<br>
+ mat4x3 <strong>transpose</strong>(mat3x4 <em>m</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a matrix that is the transpose of <em>m</em>.
+ The input matrix <em>m</em> is not modified.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>determinant</strong>(mat2 <em>m</em>)<br>
+ float <strong>determinant</strong>(mat3 <em>m</em>)<br>
+ float <strong>determinant</strong>(mat4 <em>m</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the determinant of <em>m</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">mat2 <strong>inverse</strong>(mat2 <em>m</em>)<br>
+ mat3 <strong>inverse</strong>(mat3 <em>m</em>)<br>
+ mat4 <strong>inverse</strong>(mat4 <em>m</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a matrix that is the inverse of <em>m</em>.
+ The input matrix <em>m</em> is not modified.
+ The values in the returned matrix are undefined if <em>m</em> is singular or
+ poorly-conditioned (nearly singular).</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="vector-relational-functions">8.7. Vector Relational Functions</h3>
+<div class="paragraph">
+<p>Relational and equality operators (<strong><</strong>, <strong><=</strong>, <strong>></strong>, <strong>>=</strong>, <strong>==</strong>, <strong>!=</strong>) are
+defined to operate on scalars and produce scalar Boolean results.
+For vector results, use the following built-in functions.
+Below, the following placeholders are used for the listed specific types:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Placeholder</th>
+<th class="tableblock halign-left valign-top">Specific Types Allowed</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec2, bvec3, bvec4</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">ivec</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">ivec2, ivec3, ivec4</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uvec</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uvec2, uvec3, uvec4</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec2, vec3, vec4, dvec2, dvec3, dvec4</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>In all cases, the sizes of all the input and return vectors for any
+particular call must match.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>lessThan</strong>(vec x, vec y)<br>
+ bvec <strong>lessThan</strong>(ivec x, ivec y)<br>
+ bvec <strong>lessThan</strong>(uvec x, uvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x < y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>lessThanEqual</strong>(vec x, vec y)<br>
+ bvec <strong>lessThanEqual</strong>(ivec x, ivec y)<br>
+ bvec <strong>lessThanEqual</strong>(uvec x, uvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x ≤ y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>greaterThan</strong>(vec x, vec y)<br>
+ bvec <strong>greaterThan</strong>(ivec x, ivec y)<br>
+ bvec <strong>greaterThan</strong>(uvec x, uvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x > y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>greaterThanEqual</strong>(vec x, vec y)<br>
+ bvec <strong>greaterThanEqual</strong>(ivec x, ivec y)<br>
+ bvec <strong>greaterThanEqual</strong>(uvec x, uvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x ≥ y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>equal</strong>(vec x, vec y)<br>
+ bvec <strong>equal</strong>(ivec x, ivec y)<br>
+ bvec <strong>equal</strong>(uvec x, uvec y)<br>
+ bvec <strong>equal</strong>(bvec x, bvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x == y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>notEqual</strong>(vec x, vec y)<br>
+ bvec <strong>notEqual</strong>(ivec x, ivec y)<br>
+ bvec <strong>notEqual</strong>(uvec x, uvec y)<br>
+ bvec <strong>notEqual</strong>(bvec x, bvec y)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise compare of <span class="eq">x ≠ y</span>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bool <strong>any</strong>(bvec x)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if any component of <em>x</em> is <strong>true</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bool <strong>all</strong>(bvec x)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> only if all components of <em>x</em> are <strong>true</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bvec <strong>not</strong>(bvec x)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the component-wise logical complement of <em>x</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="integer-functions">8.8. Integer Functions</h3>
+<div class="paragraph">
+<p>These all operate component-wise.
+The description is per component.
+The notation [<em>a</em>, <em>b</em>] means the set of bits from bit-number <em>a</em> through
+bit-number <em>b</em>, inclusive.
+The lowest-order bit is bit 0.
+“Bit number” will always refer to counting up from the lowest-order bit as
+bit 0.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genUType <strong>uaddCarry</strong>( genUType <em>x</em>, genUType <em>y</em>, out genUType <em>carry</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Adds 32-bit unsigned integers <em>x</em> and <em>y</em>, returning the sum modulo
+ 2<sup>32</sup>.
+ The value <em>carry</em> is set to zero if the sum was less than 2<sup>32</sup>, or
+ one otherwise.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genUType <strong>usubBorrow</strong>( genUType <em>x</em>, genUType <em>y</em>, out genUType <em>borrow</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Subtracts the 32-bit unsigned integer <em>y</em> from <em>x</em>, returning the
+ difference if non-negative, or 2<sup>32</sup> plus the difference otherwise.
+ The value <em>borrow</em> is set to zero if <span class="eq">x ≥ y</span>, or one
+ otherwise.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>umulExtended</strong>( genUType <em>x</em>, genUType <em>y</em>, out genUType <em>msb</em>, out genUType <em>lsb</em>)<br>
+ void <strong>imulExtended</strong>( genIType <em>x</em>, genIType <em>y</em>, out genIType <em>msb</em>, out genIType <em>lsb</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Multiplies 32-bit unsigned or signed integers <em>x</em> and <em>y</em>, producing a
+ 64-bit result.
+ The 32 least-significant bits are returned in <em>lsb</em>.
+ The 32 most-significant bits are returned in <em>msb</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genIType <strong>bitfieldExtract</strong>(genIType <em>value</em>, int <em>offset</em>, int <em>bits</em>)<br>
+ genUType <strong>bitfieldExtract</strong>(genUType <em>value</em>, int <em>offset</em>, int <em>bits</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Extracts bits <span class="eq">[offset, offset + bits - 1]</span> from <em>value</em>,
+ returning them in the least significant bits of the result.<br></p>
+<p class="tableblock"> For unsigned data types, the most significant bits of the result will
+ be set to zero.
+ For signed data types, the most significant bits will be set to the
+ value of bit <span class="eq">offset + bits - 1</span>.<br></p>
+<p class="tableblock"> If <em>bits</em> is zero, the result will be zero.
+ The result will be undefined if <em>offset</em> or <em>bits</em> is negative, or if
+ the sum of <em>offset</em> and <em>bits</em> is greater than the number of bits used
+ to store the operand.
+ Note that for vector versions of <strong>bitfieldExtract</strong>(), a single pair of
+ <em>offset</em> and <em>bits</em> values is shared for all components.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genIType <strong>bitfieldInsert</strong>(genIType <em>base</em>, genIType <em>insert</em>, int <em>offset</em>, int <em>bits</em>)<br>
+ genUType <strong>bitfieldInsert</strong>(genUType <em>base</em>, genUType <em>insert</em>, int <em>offset</em>, int <em>bits</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Inserts the <em>bits</em> least significant bits of <em>insert</em> into <em>base</em>.</p>
+<p class="tableblock"> The result will have bits <span class="eq">[offset, offset + bits - 1]</span> taken from
+ bits <span class="eq">[0, bits - 1]</span> of <em>insert</em>, and all other bits taken
+ directly from the corresponding bits of <em>base</em>.
+ If <em>bits</em> is zero, the result will simply be <em>base</em>.
+ The result will be undefined if <em>offset</em> or <em>bits</em> is negative, or if
+ the sum of <em>offset</em> and <em>bits</em> is greater than the number of bits used
+ to store the operand.<br>
+ Note that for vector versions of <strong>bitfieldInsert</strong>(), a single pair of
+ <em>offset</em> and <em>bits</em> values is shared for all components.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genIType <strong>bitfieldReverse</strong>( genIType <em>value</em>)<br>
+ genUType <strong>bitfieldReverse</strong>( genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Reverses the bits of <em>value</em>.
+ The bit numbered <em>n</em> of the result will be taken from bit <span class="eq">(bits -
+ 1) - n</span> of <em>value</em>, where <em>bits</em> is the total number of bits used to
+ represent <em>value</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> genIType <strong>bitCount</strong>(genIType <em>value</em>)<br>
+ genIType <strong>bitCount</strong>(genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the number of one bits in the binary representation of
+ <em>value</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> genIType <strong>findLSB</strong>(genIType <em>value</em>)<br>
+ genIType <strong>findLSB</strong>(genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the bit number of the least significant one bit in the binary
+ representation of <em>value</em>.
+ If <em>value</em> is zero, -1 will be returned.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> genIType <strong>findMSB</strong>( genIType <em>value</em>)<br>
+ genIType <strong>findMSB</strong>( genUType <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the bit number of the most significant bit in the binary
+ representation of <em>value</em>.</p>
+<p class="tableblock"> For positive integers, the result will be the bit number of the most
+ significant one bit.
+ For negative integers, the result will be the bit number of the most
+ significant zero bit.
+ For a <em>value</em> of zero or negative one, -1 will be returned.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="texture-functions">8.9. Texture Functions</h3>
+<div class="paragraph">
+<p>Texture lookup functions are available in all shading stages.
+However, level-of-detail is implicitly computed only for fragment shaders.
+Other shaders operate as though the base level-of-detail were computed as
+zero.
+The functions in the table below provide access to textures through
+samplers, as set up through the OpenGL API.
+Texture properties such as size, pixel format, number of dimensions,
+filtering method, number of mipmap levels, depth comparison, and so on are
+also defined by OpenGL API calls.
+Such properties are taken into account as the texture is accessed via the
+built-in functions defined below.</p>
+</div>
+<div class="paragraph">
+<p>Texture data can be stored by the GL as single-precision floating-point,
+unsigned normalized integer, unsigned integer or signed integer data.
+This is determined by the type of the internal format of the texture.</p>
+</div>
+<div class="paragraph">
+<p>Texture lookup functions are provided that can return their result as
+floating-point, unsigned integer or signed integer, depending on the sampler
+type passed to the lookup function.
+Care must be taken to use the right sampler type for texture access.
+The following table lists the supported combinations of sampler types and
+texture internal formats.
+Blank entries are unsupported.
+Doing a texture lookup will return undefined values for unsupported
+combinations.</p>
+</div>
+<div class="paragraph">
+<p>For depth/stencil textures, the internal texture format is determined by the
+component being accessed as set through the OpenGL API.
+When the depth/stencil texture mode is set to DEPTH_COMPONENT, the internal
+format of the depth component should be used.
+When the depth/stencil texture mode is set to STENCIL_INDEX, the internal format
+of the stencil component should be used.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+<col style="width: 25%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Internal Texture Format</th>
+<th class="tableblock halign-left valign-top">Floating-Point Sampler Types</th>
+<th class="tableblock halign-left valign-top">Signed Integer Sampler Types</th>
+<th class="tableblock halign-left valign-top">Unsigned Integer Sampler Types</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Floating-point</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Supported</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Normalized Integer</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Supported</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Signed Integer</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Supported</p></td>
+<td class="tableblock halign-left valign-top"></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Unsigned Integer</p></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Supported</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>If an integer sampler type is used, the result of a texture lookup is an
+<strong>ivec4</strong>.
+If an unsigned integer sampler type is used, the result of a texture lookup
+is a <strong>uvec4</strong>.
+If a floating-point sampler type is used, the result of a texture lookup is
+a <strong>vec4</strong>.</p>
+</div>
+<div class="paragraph">
+<p>In the prototypes below, the “_g_” in the return type “_gvec4_” is used
+as a placeholder for nothing, “_i_”, or “_u_” making a return type of
+<strong>vec4</strong>, <strong>ivec4</strong>, or <strong>uvec4</strong>.
+In these cases, the sampler argument type also starts with “_g_”,
+indicating the same substitution done on the return type; it is either a
+single-precision
+floating-point, signed integer, or unsigned integer sampler, matching the
+basic type of the return type, as described above.</p>
+</div>
+<div class="paragraph">
+<p>For shadow forms (the sampler parameter is a shadow-type), a depth
+comparison lookup on the depth texture bound to <em>sampler</em> is done as
+described in section
+8.23
+“Texture Comparison Modes” of the
+<a href="#references">OpenGL Specification</a>.
+See the table below for which component specifies <em>D<sub>ref</sub></em>.
+The texture bound to <em>sampler</em> must be a depth texture, or results are
+undefined.
+If a non-shadow texture call is made to a sampler that represents a depth
+texture with depth comparisons turned on, then results are undefined.
+If a shadow texture call is made to a sampler that represents a depth
+texture with depth comparisons turned off, then results are undefined.
+If a shadow texture call is made to a sampler that does not represent a
+depth texture, then results are undefined.</p>
+</div>
+<div class="paragraph">
+<p>In all functions below, the <em>bias</em> parameter is optional for fragment
+shaders.
+The <em>bias</em> parameter is not accepted in any other shader stage.
+For a fragment shader, if <em>bias</em> is present, it is added to the implicit
+level-of-detail prior to performing the texture access operation.
+No <em>bias</em> or <em>lod</em> parameters for
+rectangle textures,
+multisample textures, or texture buffers
+are supported because mipmaps are not allowed for these types of textures.</p>
+</div>
+<div class="paragraph">
+<p>The implicit level-of-detail is selected as follows: For a texture that is
+not mipmapped, the texture is used directly.
+If it is mipmapped and running in a fragment shader, the level-of-detail
+computed by the implementation is used to do the texture lookup.
+If it is mipmapped and running in a non-fragment shader, then the base
+texture is used.</p>
+</div>
+<div class="paragraph">
+<p>Some texture functions (non-“<strong>Lod</strong>” and non-“<strong>Grad</strong>” versions) may
+require implicit derivatives.
+Implicit derivatives are undefined within non-uniform control flow and for
+non-fragment shader texture fetches.</p>
+</div>
+<div class="paragraph">
+<p>For <strong>Cube</strong> forms, the direction of <em>P</em> is used to select which face to do a
+2-dimensional texture lookup in, as described in section 8.13 “Cube Map
+Texture Selection” of the <a href="#references">OpenGL Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>For <strong>Array</strong> forms, the array layer used will be</p>
+</div>
+<div class="paragraph">
+<p>\(\max(0,\min(d-1,\left\lfloor layer + 0.5\right\rfloor))\)</p>
+</div>
+<div class="paragraph">
+<p>where <em>d</em> is the depth of the texture array and <em>layer</em> comes from the
+component indicated in the tables below.</p>
+</div>
+<div class="sect3">
+<h4 id="texture-query-functions">8.9.1. Texture Query Functions</h4>
+<div class="paragraph">
+<p>The <strong>textureSize</strong> functions query the dimensions of a specific texture level
+for a sampler.</p>
+</div>
+<div class="paragraph">
+<p>The <strong>textureQueryLod</strong> functions are available only in a fragment shader.
+They take the components of <em>P</em> and compute the level-of-detail information
+that the texture pipe would use to access that texture through a normal
+texture lookup.
+The level-of-detail \(\lambda^{'}\) (equation 3.18 of the
+<a href="#references">OpenGL Specification</a>) is obtained after any level-of-detail bias, but
+prior to clamping to [TEXTURE_MIN_LOD, TEXTURE_MAX_LOD].
+The mipmap array(s) that would be accessed are also computed.
+If a single level-of-detail would be accessed, the level-of-detail number
+relative to the base level is returned.
+If multiple levels-of-detail would be accessed, a floating-point number
+between the two levels is returned, with the fractional part equal to the
+fractional part of the computed and clamped level-of-detail.</p>
+</div>
+<div class="paragraph">
+<p>The algorithm used is given by the following pseudo-code:</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++"><span class="predefined-type">float</span> ComputeAccessedLod(<span class="predefined-type">float</span> computedLod)
+{
+ <span class="comment">// Clamp the computed LOD according to the texture LOD clamps.</span>
+ <span class="keyword">if</span> (computedLod < TEXTURE_MIN_LOD) computedLod = TEXTURE_MIN_LOD;
+ <span class="keyword">if</span> (computedLod > TEXTURE_MAX_LOD) computedLod = TEXTURE_MAX_LOD;
+
+ <span class="comment">// Clamp the computed LOD to the range of accessible levels.</span>
+ <span class="keyword">if</span> (computedLod < <span class="float">0</span><span class="float">.0</span>)
+ computedLod = <span class="float">0</span><span class="float">.0</span>;
+ <span class="keyword">if</span> (computedLod > (<span class="predefined-type">float</span>) maxAccessibleLevel)
+ computedLod = (<span class="predefined-type">float</span>) maxAccessibleLevel;
+
+ <span class="comment">// Return a value according to the min filter.</span>
+ <span class="keyword">if</span> (TEXTURE_MIN_FILTER is LINEAR <span class="keyword">or</span> NEAREST) {
+ <span class="keyword">return</span> <span class="float">0</span><span class="float">.0</span>;
+ } <span class="keyword">else</span> <span class="keyword">if</span> (TEXTURE_MIN_FILTER is NEAREST_MIPMAP_NEAREST
+ <span class="keyword">or</span> LINEAR_MIPMAP_NEAREST) {
+ <span class="keyword">return</span> ceil(computedLod + <span class="float">0</span><span class="float">.5</span>) - <span class="float">1</span><span class="float">.0</span>;
+ } <span class="keyword">else</span> {
+ <span class="keyword">return</span> computedLod;
+ }
+}</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The value <em>maxAccessibleLevel</em> is the level number of the smallest
+accessible level of the mipmap array (the value <em>q</em> in section 8.14.3
+“Mipmapping” of the <a href="#references">OpenGL Specification</a>) minus the base level.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> int <strong>textureSize</strong>(gsampler1D <em>sampler</em>, int <em>lod</em>)<br>
+ ivec2 <strong>textureSize</strong>(gsampler2D <em>sampler</em>, int <em>lod</em>)<br>
+ ivec3 <strong>textureSize</strong>(gsampler3D <em>sampler</em>, int <em>lod</em>)<br>
+ ivec2 <strong>textureSize</strong>(gsamplerCube <em>sampler</em>, int <em>lod</em>)<br>
+ int <strong>textureSize</strong>(sampler1DShadow <em>sampler</em>, int <em>lod</em>)<br>
+ ivec2 <strong>textureSize</strong>(sampler2DShadow <em>sampler</em>, int <em>lod</em>)<br>
+ ivec2 <strong>textureSize</strong>(samplerCubeShadow <em>sampler</em>, int <em>lod</em>)
+ ivec3 <strong>textureSize</strong>(gsamplerCubeArray <em>sampler</em>, int <em>lod</em>)<br>
+ ivec3 <strong>textureSize</strong>(samplerCubeArrayShadow <em>sampler</em>, int <em>lod</em>)<br>
+ ivec2 <strong>textureSize</strong>(gsampler2DRect <em>sampler</em>)<br>
+ ivec2 <strong>textureSize</strong>(sampler2DRectShadow <em>sampler</em>)<br>
+ ivec2 <strong>textureSize</strong>(gsampler1DArray <em>sampler</em>, int <em>lod</em>)<br>
+ ivec2 <strong>textureSize</strong>(sampler1DArrayShadow <em>sampler</em>, int <em>lod</em>)<br>
+ ivec3 <strong>textureSize</strong>(gsampler2DArray <em>sampler</em>, int <em>lod</em>)<br>
+ ivec3 <strong>textureSize</strong>(sampler2DArrayShadow <em>sampler</em>, int <em>lod</em>)<br>
+ int <strong>textureSize</strong>(gsamplerBuffer <em>sampler</em>)<br>
+ ivec2 <strong>textureSize</strong>(gsampler2DMS <em>sampler</em>)<br>
+ ivec3 <strong>textureSize</strong>(gsampler2DMSArray <em>sampler</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the dimensions of level <em>lod</em> (if present) for the texture
+ bound to <em>sampler</em>, as described in section
+ 8.11 “Texture Queries” of the <a href="#references">OpenGL Specification</a>.<br>
+ The components in the return value are filled in, in order, with the
+ width, height, and depth of the texture.</p>
+<p class="tableblock"> For the array forms, the last component of the return value is the
+ number of layers in the texture array, or the number of cubes in the
+ texture cube map array.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec2 <strong>textureQueryLod</strong>(gsampler1D <em>sampler</em>, float <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(gsampler1DArray <em>sampler</em>, float <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(gsampler2DArray <em>sampler</em>, vec2 <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(gsamplerCubeArray <em>sampler</em>, vec3 <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(sampler1DShadow <em>sampler</em>, float <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(sampler2DShadow <em>sampler</em>, vec2 <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(samplerCubeShadow <em>sampler</em>, vec3 <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(sampler1DArrayShadow <em>sampler</em>, float <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(sampler2DArrayShadow <em>sampler</em>, vec2 <em>P</em>)<br>
+ vec2 <strong>textureQueryLod</strong>(samplerCubeArrayShadow <em>sampler</em>, vec3 <em>P</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the mipmap array(s) that would be accessed in the <em>x</em>
+ component of the return value.</p>
+<p class="tableblock"> Returns the computed level-of-detail relative to the base level in the
+ <em>y</em> component of the return value.</p>
+<p class="tableblock"> If called on an incomplete texture, the results are undefined.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">int <strong>textureQueryLevels</strong>(gsampler1D <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(gsampler2D <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(gsampler3D <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(gsamplerCube <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(gsampler1DArray <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(gsampler2DArray <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(gsamplerCubeArray <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(sampler1DShadow <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(sampler2DShadow <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(samplerCubeShadow <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(sampler1DArrayShadow <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(sampler2DArrayShadow <em>sampler</em>)<br>
+ int <strong>textureQueryLevels</strong>(samplerCubeArrayShadow <em>sampler</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the number of mipmap levels accessible in the texture
+ associated with <em>sampler</em>, as defined in the <a href="#references">OpenGL Specification.</a></p>
+<p class="tableblock"> The value zero will be returned if no texture or an incomplete texture
+ is associated with <em>sampler</em>.</p>
+<p class="tableblock"> Available in all shader stages.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">int <strong>textureSamples</strong>(gsampler2DMS <em>sampler</em>)<br>
+ int <strong>textureSamples</strong>(gsampler2DMSArray <em>sampler</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the number of samples of the texture or textures bound to
+ <em>sampler</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="texel-lookup-functions">8.9.2. Texel Lookup Functions</h4>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>texture</strong>(gsampler1D <em>sampler</em>, float <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em>[, float <em>bias</em>] )<br>
+ float <strong>texture</strong>(sampler1DShadow <em>sampler</em>, <em>vec3 _P</em> [, float <em>bias</em>])<br>
+ float <strong>texture</strong>(sampler2DShadow <em>sampler</em>, <em>vec3 _P</em> [, float <em>bias</em>])<br>
+ float <strong>texture</strong>(samplerCubeShadow <em>sampler</em>, <em>vec4 _P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsamplerCubeArray <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>texture</strong>(gsampler1DArray <em>sampler</em>, <em>vec2 _P</em> [, float <em>bias</em>] )<br>
+ float <strong>texture</strong>(sampler1DArrayShadow <em>sampler</em>, <em>P</em> [, float <em>bias</em>] )<br>
+ float <strong>texture</strong>(sampler2DArrayShadow <em>sampler</em>, vec4 <em>P</em>)<br>
+ gvec4 <strong>texture</strong>(gsampler2DRect <em>sampler</em>, vec2 <em>P</em>)<br>
+ float <strong>texture</strong>(sampler2DRectShadow <em>sampler</em>, vec3 <em>P</em>)<br>
+ float <strong>texture</strong>(samplerCubeArrayShadow <em>sampler</em>, vec4 <em>P</em>, float <em>compare</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Use the texture coordinate <em>P</em> to do a texture lookup in the texture
+ currently bound to <em>sampler</em>.</p>
+<p class="tableblock"> For shadow forms: When <em>compare</em> is present, it is used as <em>D<sub>ref</sub></em>
+ and the array layer comes from the last component of <em>P</em>.
+ When <em>compare</em> is not present, the last component of <em>P</em> is used as
+ <em>D<sub>ref</sub></em> and the array layer comes from the second to last component
+ of <em>P</em>.
+ (The second component of <em>P</em> is unused for <strong>1D</strong> shadow lookups.)</p>
+<p class="tableblock"> For non-shadow forms: the array layer comes from the last component of
+ <em>P</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProj</strong>(gsampler1D <em>sampler</em>, vec2 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProj</strong>(gsampler1D <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProj</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProj</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProj</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ float <strong>textureProj</strong>(sampler1DShadow <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ float <strong>textureProj</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProj</strong>(gsampler2DRect <em>sampler</em>, vec3 <em>P</em>)<br>
+ gvec4 <strong>textureProj</strong>(gsampler2DRect <em>sampler</em>, vec4 <em>P</em>)<br>
+ float <strong>textureProj</strong>(sampler2DRectShadow <em>sampler</em>, vec4 <em>P</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup with projection.
+ The texture coordinates consumed from <em>P</em>, not including the last
+ component of <em>P</em>, are divided by the last component of <em>P</em> to
+ form projected coordinates <em>P'</em>.
+ The resulting third component of <em>P</em> in the shadow forms is used as
+ <em>D<sub>ref</sub></em>.
+ The third component of <em>P</em> is ignored when <em>sampler</em> has type
+ <strong>gsampler2D</strong> and <em>P</em> has type <strong>vec4</strong>.
+ After these values are computed, texture lookup proceeds as in
+ <strong>texture</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureLod</strong>(gsampler1D <em>sampler</em>, float <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ float <strong>textureLod</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ float <strong>textureLod</strong>(sampler1DShadow <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsampler1DArray <em>sampler</em>, vec2 <em>P</em>, float <em>lod</em>)<br>
+ float <strong>textureLod</strong>(sampler1DArrayShadow <em>sampler</em>, <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureLod</strong>(gsamplerCubeArray <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Do a texture lookup as in <strong>texture</strong> but with explicit level-of-detail;
+ <em>lod</em> specifies <span class="eq">λ<sub>base</sub>]</span> and sets the partial derivatives
+ as follows:<br>
+ (See section 8.14 “Texture Minification” and equations 8.4-8.6 of
+ the <a href="#references">OpenGL Specification</a>.)<br>
+ <br>
+ <span class="eq">∂u / ∂x =
+ ∂v / ∂x =
+ ∂w / ∂x = 0</span>
+ <br>
+ <span class="eq">∂u / ∂y =
+ ∂v / ∂y =
+ ∂w / ∂y = 0</span></p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureOffset</strong>(gsampler1D <em>sampler</em>, float <em>P</em>, int <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureOffset</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureOffset</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>, ivec3 <em>offset</em> [, float <em>bias</em>] )<br>
+ float <strong>textureOffset</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureOffset</strong>(gsampler2DRect <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureOffset</strong>(sampler2DRectShadow <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureOffset</strong>(sampler1DShadow <em>sampler</em>, vec3 <em>P</em>, int <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureOffset</strong>(gsampler1DArray <em>sampler</em>, vec2 <em>P</em>, int <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureOffset</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ float <strong>textureOffset</strong>(sampler1DArrayShadow <em>sampler</em>, vec3 <em>P</em>, int <em>offset</em> [, float <em>bias</em>] )<br>
+ float <strong>textureOffset</strong>(sampler2DArrayShadow <em>sampler</em>, vec4 P, ivec2 <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup as in <strong>texture</strong> but with <em>offset</em> added to the
+ <span class="eq">(u,v,w)</span> texel coordinates before looking up each texel.
+ The offset value must be a constant expression.
+ A limited range of offset values are supported; the minimum and
+ maximum offset values are implementation-dependent and given by
+ <em>gl_MinProgramTexelOffset</em> and <em>gl_MaxProgramTexelOffset</em>,
+ respectively.</p>
+<p class="tableblock"> Note that <em>offset</em> does not apply to the layer coordinate for texture
+ arrays.
+ This is explained in detail in section 8.14.2 “Coordinate Wrapping
+ and Texel Selection” of the <a href="#references">OpenGL Specification</a>, where <em>offset</em>
+ is <span class="eq">(δ<sub>u</sub>, δ<sub>v</sub>, δ<sub>w</sub>)</span>.<br>
+ Note that texel offsets are also not supported for cube maps.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>texelFetch</strong>(gsampler1D <em>sampler</em>, int <em>P</em>, int <em>lod</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler2D <em>sampler</em>, ivec2 <em>P</em>, int <em>lod</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler3D <em>sampler</em>, ivec3 <em>P</em>, int <em>lod</em>)
+ gvec4 <strong>texelFetch</strong>(gsampler2DRect <em>sampler</em>, ivec2 <em>P</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler1DArray <em>sampler</em>, ivec2 <em>P</em>, int <em>lod</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler2DArray <em>sampler</em>, ivec3 <em>P</em>, int <em>lod</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsamplerBuffer <em>sampler</em>, int <em>P</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler2DMS <em>sampler</em>, ivec2 <em>P</em>, int <em>sample</em>)<br>
+ gvec4 <strong>texelFetch</strong>(gsampler2DMSArray <em>sampler</em>, ivec3 <em>P</em>, int <em>sample</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Use integer texture coordinate <em>P</em> to lookup a single texel from
+ <em>sampler</em>.
+ The array layer comes from the last component of <em>P</em> for the array
+ forms.
+ The level-of-detail <em>lod</em> (if present) is as described in sections
+ 11.1.3.2 “Texel Fetches” and 8.14.1 “Scale Factor and Level of
+ Detail” of the <a href="#references">OpenGL Specification</a>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>texelFetchOffset</strong>(gsampler1D <em>sampler</em>, int <em>P</em>, int <em>lod</em>, int <em>offset</em>)<br>
+ gvec4 <strong>texelFetchOffset</strong>(gsampler2D <em>sampler</em>, ivec2 <em>P</em>, int <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>texelFetchOffset</strong>(gsampler3D <em>sampler</em>, ivec3 <em>P</em>, int <em>lod</em>, ivec3 <em>offset</em>)<br>
+ gvec4 <strong>texelFetchOffset</strong>(gsampler2DRect <em>sampler</em>, ivec2 <em>P</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>texelFetchOffset</strong>(gsampler1DArray <em>sampler</em>, ivec2 <em>P</em>, int <em>lod</em>, int <em>offset</em>)<br>
+ gvec4 <strong>texelFetchOffset</strong>(gsampler2DArray <em>sampler</em>, ivec3 <em>P</em>, int <em>lod</em>, ivec2 <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Fetch a single texel as in <strong>texelFetch</strong>, offset by <em>offset</em> as
+ described in <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjOffset</strong>(gsampler1D <em>sampler</em>, vec2 <em>P</em>, int <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProjOffset</strong>(gsampler1D <em>sampler</em>, vec4 <em>P</em>, int <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProjOffset</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProjOffset</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProjOffset</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, ivec3 <em>offset</em> [, float <em>bias</em>] )<br>
+ gvec4 <strong>textureProjOffset</strong>(gsampler2DRect <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjOffset</strong>(gsampler2DRect <em>sampler</em>, vec4 <em>P</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureProjOffset</strong>(sampler2DRectShadow <em>sampler</em>, vec4 <em>P</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureProjOffset</strong>(sampler1DShadow <em>sampler</em>, vec4 <em>P</em>, int <em>offset</em> [, float <em>bias</em>] )<br>
+ float <strong>textureProjOffset</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, ivec2 <em>offset</em> [, float <em>bias</em>] )</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a projective texture lookup as described in <strong>textureProj</strong>, offset
+ by <em>offset</em> as described in <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureLodOffset</strong>(gsampler1D <em>sampler</em>, float <em>P</em>, float <em>lod</em>, int <em>offset</em>)<br>
+ gvec4 <strong>textureLodOffset</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureLodOffset</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>, ivec3 <em>offset</em>)<br>
+ float <strong>textureLodOffset</strong>(sampler1DShadow <em>sampler</em>, <em>P</em>, float <em>lod</em>, int <em>offset</em>)<br>
+ float <strong>textureLodOffset</strong>(sampler2DShadow <em>sampler</em>, <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureLodOffset</strong>(gsampler1DArray <em>sampler</em>, vec2 <em>P</em>, float <em>lod</em>, int <em>offset</em>)<br>
+ gvec4 <strong>textureLodOffset</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureLodOffset</strong>(sampler1DArrayShadow <em>sampler</em>, <em>P</em>, float <em>lod</em>, int <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do an offset texture lookup with explicit level-of-detail.
+ See <strong>textureLod</strong> and <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjLod</strong>(gsampler1D <em>sampler</em>, vec2 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureProjLod</strong>(gsampler1D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureProjLod</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureProjLod</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)<br>
+ gvec4 <strong>textureProjLod</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)<br>
+ float <strong>textureProjLod</strong>(sampler1DShadow <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)<br>
+ float <strong>textureProjLod</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a projective texture lookup with explicit level-of-detail.
+ See <strong>textureProj</strong> and <strong>textureLod</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjLodOffset</strong>(gsampler1D <em>sampler</em>, vec2 <em>P</em>, float <em>lod</em>, int <em>offset</em>)<br>
+ gvec4 <strong>textureProjLodOffset</strong>(gsampler1D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>, int <em>offset</em>)<br>
+ gvec4 <strong>textureProjLodOffset</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjLodOffset</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjLodOffset</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>, ivec3 <em>offset</em>)<br>
+ float <strong>textureProjLodOffset</strong>(sampler1DShadow <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>, int <em>offset</em>)<br>
+ float <strong>textureProjLodOffset</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, float <em>lod</em>, ivec2 <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do an offset projective texture lookup with explicit level-of-detail.
+ See <strong>textureProj</strong>, <strong>textureLod</strong>, and <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGrad</strong>(gsampler1D <em>sampler</em>, <em>float _P</em>, float <em>dPdx</em>, float <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsampler3D <em>sampler</em>, <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsampler2DRect <em>sampler</em>, vec2 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(sampler2DRectShadow <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(sampler1DShadow <em>sampler</em>, vec3 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsampler1DArray <em>sampler</em>, vec2 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsampler2DArray <em>sampler</em>, <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(sampler1DArrayShadow <em>sampler</em>, vec3 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(samplerCubeShadow <em>sampler</em>, vec4 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br>
+ float <strong>textureGrad</strong>(sampler2DArrayShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureGrad</strong>(gsamplerCubeArray <em>sampler</em>, vec4 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Do a texture lookup as in <strong>texture</strong> but with <a href="#explicit-gradients">explicit gradients</a> as shown below.
+ The partial derivatives of <em>P</em> are with respect to window <em>x</em> and
+ window <em>y</em>.
+ For the cube version, the partial derivatives of <em>P</em> are assumed to be
+ in the coordinate system used before texture coordinates are projected
+ onto the appropriate cube face.</p>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGradOffset</strong>(gsampler1D <em>sampler</em>, float <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>, int <em>offset</em>)<br>
+ gvec4 <strong>textureGradOffset</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureGradOffset</strong>(gsampler3D <em>sampler</em>, vec3 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>, ivec3 <em>offset</em>)<br>
+ gvec4 <strong>textureGradOffset</strong>(gsampler2DRect <em>sampler</em>, vec2 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureGradOffset</strong>(sampler2DRectShadow <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureGradOffset</strong>(sampler1DShadow <em>sampler</em>, vec3 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>, int <em>offset</em>)<br>
+ float <strong>textureGradOffset</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureGradOffset</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureGradOffset</strong>(gsampler1DArray <em>sampler</em>, vec2 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>, int <em>offset</em>)<br>
+ float <strong>textureGradOffset</strong>(sampler1DArrayShadow <em>sampler</em>, vec3 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>, int <em>offset</em>)<br>
+ float <strong>textureGradOffset</strong>(sampler2DArrayShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup with both explicit gradient and offset, as
+ described in <strong>textureGrad</strong> and <strong>textureOffset</strong>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjGrad</strong>(gsampler1D <em>sampler</em>, vec2 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>)<br>
+ gvec4 <strong>textureProjGrad</strong>(gsampler1D <em>sampler</em>, vec4 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>)<br>
+ gvec4 <strong>textureProjGrad</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureProjGrad</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureProjGrad</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>)<br>
+ gvec4 <strong>textureProjGrad</strong>(gsampler2DRect <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ gvec4 <strong>textureProjGrad</strong>(gsampler2DRect <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ float <strong>textureProjGrad</strong>(sampler2DRectShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)<br>
+ float <strong>textureProjGrad</strong>(sampler1DShadow <em>sampler</em>, vec4 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>)<br>
+ float <strong>textureProjGrad</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup both projectively, as described in <strong>textureProj</strong>,
+ and with explicit gradient as described in <strong>textureGrad</strong>.
+ The partial derivatives <em>dPdx</em> and <em>dPdy</em> are assumed to be already
+ projected.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureProjGradOffset</strong>(gsampler1D <em>sampler</em>, vec2 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>, int <em>offset</em>)<br>
+ gvec4 <strong>textureProjGradOffset</strong>(gsampler1D <em>sampler</em>, vec4 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>, int <em>offset</em>)<br>
+ gvec4 <strong>textureProjGradOffset</strong>(gsampler2D <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjGradOffset</strong>(gsampler2D <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjGradOffset</strong>(gsampler3D <em>sampler</em>, vec4 <em>P</em>, vec3 <em>dPdx</em>, vec3 <em>dPdy</em>, ivec3 <em>offset</em>)<br>
+ gvec4 <strong>textureProjGradOffset</strong>(gsampler2DRect <em>sampler</em>, vec3 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureProjGradOffset</strong>(gsampler2DRect <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureProjGradOffset</strong>(sampler2DRectShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)<br>
+ float <strong>textureProjGradOffset</strong>(sampler1DShadow <em>sampler</em>, vec4 <em>P</em>, float <em>dPdx</em>, float <em>dPdy</em>, int <em>offset</em>)<br>
+ float <strong>textureProjGradOffset</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>P</em>, vec2 <em>dPdx</em>, vec2 <em>dPdy</em>, ivec2 <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Do a texture lookup projectively and with explicit gradient as
+ described in <strong>textureProjGrad</strong>, as well as with offset, as described in
+ <strong>textureOffset</strong>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="explicit-gradients">8.9.3. Explicit Gradients</h4>
+<div class="paragraph">
+<p>In the <strong>textureGrad</strong> functions described above, explicit gradients control
+texture lookups as follows:</p>
+</div>
+<div class="stemblock">
+<div class="content">
+\[\begin{aligned}
+ \frac{\partial{s}}{\partial{x}} & =
+ \begin{cases}
+ \frac{\partial{P}}{\partial{x}}, & \text{for a 1D texture} \\[0.8em]
+ \frac{\partial{P.s}}{\partial{x}}, & \text{otherwise}
+ \end{cases} \\[2.5em]
+ \frac{\partial{s}}{\partial{y}} & =
+ \begin{cases}
+ \frac{\partial{P}}{\partial{y}}, & \text{for a 1D texture} \\[0.8em]
+ \frac{\partial{P.s}}{\partial{y}}, & \text{otherwise}
+ \end{cases} \\[2.5em]
+ \frac{\partial{t}}{\partial{x}} & =
+ \begin{cases}
+ 0.0, & \text{for a 1D texture} \\[0.8em]
+ \frac{\partial{P.t}}{\partial{x}}, & \text{otherwise}
+ \end{cases} \\[2.5em]
+ \frac{\partial{t}}{\partial{y}} & =
+ \begin{cases}
+ 0.0, & \text{for a 1D texture} \\[0.8em]
+ \frac{\partial{P.t}}{\partial{y}}, & \text{otherwise}
+ \end{cases} \\[2.5em]
+ \frac{\partial{r}}{\partial{x}} & =
+ \begin{cases}
+ 0.0, & \text{for 1D or 2D} \\[0.8em]
+ \frac{\partial{P.p}}{\partial{x}}, & \text{cube, other}
+ \end{cases} \\[2.5em]
+ \frac{\partial{r}}{\partial{y}} & =
+ \begin{cases}
+ 0.0, & \text{for 1D or 2D} \\[0.8em]
+ \frac{\partial{P.p}}{\partial{y}}, & \text{cube, other}
+ \end{cases}
+\end{aligned}\]
+</div>
+</div>
+</div>
+<div class="sect3">
+<h4 id="texture-gather-functions">8.9.4. Texture Gather Functions</h4>
+<div class="paragraph">
+<p>The texture gather functions take components of a single floating-point
+vector operand as a texture coordinate, determine a set of four texels to
+sample from the base level-of-detail of the specified texture image, and
+return one component from each texel in a four-component result vector.</p>
+</div>
+<div class="paragraph">
+<p>When performing a texture gather operation, the minification and
+magnification filters are ignored, and the rules for LINEAR filtering in the
+<a href="#references">OpenGL Specification</a> are applied to the base level of the texture image
+to identify the four texels <em>i<sub>0</sub> j<sub>1</sub></em>, <em>i<sub>1</sub> j<sub>1</sub></em>, <em>i<sub>1</sub> j<sub>0</sub></em>, and <em>i<sub>0</sub>
+j<sub>0</sub></em>.
+The texels are then converted to texture base colors (<em>R<sub>s</sub></em>, <em>G<sub>s</sub></em>,
+<em>B<sub>s</sub></em>, <em>A<sub>s</sub></em>) according to table 15.1, followed by application of the
+texture swizzle as described in section 15.2.1 “Texture Access” of the
+<a href="#references">OpenGL Specification</a>.
+A four-component vector is assembled by taking the selected component from
+each of the post-swizzled texture source colors in the order (<em>i<sub>0</sub> j<sub>1</sub></em>,
+<em>i<sub>1</sub> j<sub>1</sub></em>, <em>i<sub>1</sub> j<sub>0</sub></em>, <em>i<sub>0</sub> j<sub>0</sub></em>).</p>
+</div>
+<div class="paragraph">
+<p>For texture gather functions using a shadow sampler type, each of the four
+texel lookups perform a depth comparison against the depth reference value
+passed in (<em>refZ</em>), and returns the result of that comparison in the
+appropriate component of the result vector.</p>
+</div>
+<div class="paragraph">
+<p>As with other texture lookup functions, the results of a texture gather are
+undefined for shadow samplers if the texture referenced is not a depth
+texture or has depth comparisons disabled; or for non-shadow samplers if the
+texture referenced is a depth texture with depth comparisons enabled.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGather</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em> [, int <em>comp</em>])<br>
+ gvec4 <strong>textureGather</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em> [, int <em>comp</em>])<br>
+ gvec4 <strong>textureGather</strong>(gsamplerCube <em>sampler</em>, vec3 <em>P</em> [, int <em>comp</em>])<br>
+ gvec4 <strong>textureGather</strong>(gsamplerCubeArray <em>sampler</em>, vec4 <em>P</em>[, int <em>comp</em>])<br>
+ gvec4 <strong>textureGather</strong>(gsampler2DRect <em>sampler</em>, vec2 <em>P</em>[, int <em>comp</em>])<br>
+ vec4 <strong>textureGather</strong>(sampler2DShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>)<br>
+ vec4 <strong>textureGather</strong>(sampler2DArrayShadow <em>sampler</em>, vec3 <em>P</em>, float <em>refZ</em>)<br>
+ vec4 <strong>textureGather</strong>(samplerCubeShadow <em>sampler</em>, vec3 <em>P</em>, float <em>refZ</em>)<br>
+ vec4 <strong>textureGather</strong>(samplerCubeArrayShadow <em>sampler</em>, vec4 <em>P</em>, float <em>refZ</em>)<br>
+ vec4 <strong>textureGather</strong>(sampler2DRectShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Returns the value<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">vec4(Sample_i0_j1(P, base).comp,
+ Sample_i1_j1(P, base).comp,
+ Sample_i1_j0(P, base).comp,
+ Sample_i0_j0(P, base).comp)</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>If specified, the value of <em>comp</em> must be a constant integer expression with
+a value of 0, 1, 2, or 3, identifying the <em>x</em>, <em>y</em>, <em>z</em>, or <em>w</em>
+post-swizzled component of the four-component vector lookup result for each
+texel, respectively.
+If <em>comp</em> is not specified, it is treated as 0, selecting the <em>x</em> component
+of each texel to generate the result.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGatherOffset</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offset</em>, [ int <em>comp</em>])<br>
+ gvec4 <strong>textureGatherOffset</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offset</em> [ int <em>comp</em>])<br>
+ vec4 <strong>textureGatherOffset</strong>(sampler2DShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>, ivec2 <em>offset</em>)<br>
+ vec4 <strong>textureGatherOffset</strong>(sampler2DArrayShadow <em>sampler</em>, vec3 <em>P</em>, float <em>refZ</em>, ivec2 <em>offset</em>)<br>
+ gvec4 <strong>textureGatherOffset</strong>(gsampler2DRect <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offset</em> [ int <em>comp</em>])<br>
+ vec4 <strong>textureGatherOffset</strong>(sampler2DRectShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>, ivec2 <em>offset</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Perform a texture gather operation as in <strong>textureGather</strong> by <em>offset</em>
+ as described in <strong>textureOffset</strong> except that the <em>offset</em> can be
+ variable (non constant) and the implementation-dependent minimum and
+ maximum offset values are given by MIN_PROGRAM_TEXTURE_GATHER_OFFSET
+ and MAX_PROGRAM_TEXTURE_GATHER_OFFSET, respectively.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">gvec4 <strong>textureGatherOffsets</strong>(gsampler2D <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offsets</em>[4] [, int <em>comp</em>])<br>
+ gvec4 <strong>textureGatherOffsets</strong>(gsampler2DArray <em>sampler</em>, vec3 <em>P</em>, ivec2 <em>offsets</em>[4] [, int <em>comp</em>])<br>
+ vec4 <strong>textureGatherOffsets</strong>(sampler2DShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>, ivec2 <em>offsets</em>[4])<br>
+ vec4 <strong>textureGatherOffsets</strong>(sampler2DArrayShadow <em>sampler</em>, vec3 <em>P</em>, float <em>refZ</em>, ivec2 <em>offsets</em>[4])<br>
+ gvec4 <strong>textureGatherOffsets</strong>(gsampler2DRect <em>sampler</em>, vec2 <em>P</em>, ivec2 <em>offsets</em>[4] [, int <em>comp</em>])<br>
+ vec4 <strong>textureGatherOffsets</strong>(sampler2DRectShadow <em>sampler</em>, vec2 <em>P</em>, float <em>refZ</em>, ivec2 <em>offsets</em>[4])</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Operate identically to <strong>textureGatherOffset</strong> except that <em>offsets</em> is
+ used to determine the location of the four texels to sample.
+ Each of the four texels is obtained by applying the corresponding
+ offset in <em>offsets</em> as a (<em>u</em>, <em>v</em>) coordinate offset to <em>P</em>,
+ identifying the four-texel LINEAR footprint, and then selecting the
+ texel <em>i<sub>0</sub> j<sub>0</sub></em> of that footprint.
+ The specified values in <em>offsets</em> must be constant integral
+ expressions.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="compatibility-profile-texture-functions">8.9.5. Compatibility Profile Texture Functions</h4>
+<div class="paragraph">
+<p>The following texture functions are only in the compatibility profile.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec4 <strong>texture1D</strong>(sampler1D <em>sampler</em>, float <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>texture1DProj</strong>(sampler1D <em>sampler</em>, vec2 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>texture1DProj</strong>(sampler1D <em>sampler</em>, vec4 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>texture1DLod</strong>(sampler1D <em>sampler</em>, float <em>coord</em>, float <em>lod</em>)<br>
+ vec4 <strong>texture1DProjLod</strong>(sampler1D <em>sampler</em>, vec2 <em>coord</em>, float <em>lod</em>)<br>
+ vec4 <strong>texture1DProjLod</strong>(sampler1D <em>sampler</em>, vec4 <em>coord</em>, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">See corresponding signature above without “1D” in the name.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec4 <strong>texture2D</strong>(sampler2D <em>sampler</em>, vec2 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>texture2DProj</strong>(sampler2D <em>sampler</em>, vec3 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>texture2DProj</strong>(sampler2D <em>sampler</em>, vec4 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>texture2DLod</strong>(sampler2D <em>sampler</em>, vec2 <em>coord</em>, float <em>lod</em>)<br>
+ vec4 <strong>texture2DProjLod</strong>(sampler2D <em>sampler</em>, vec3 <em>coord</em>, float <em>lod</em>)<br>
+ vec4 <strong>texture2DProjLod</strong>(sampler2D <em>sampler</em>, vec4 <em>coord</em>, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">See corresponding signature above without “2D” in the name.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec4 <strong>texture3D</strong>(sampler3D <em>sampler</em>, vec3 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>texture3DProj</strong>(sampler3D <em>sampler</em>, vec4 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>texture3DLod</strong>(sampler3D <em>sampler</em>, vec3 <em>coord</em>, float <em>lod</em>)<br>
+ vec4 <strong>texture3DProjLod</strong>(sampler3D <em>sampler</em>, vec4 <em>coord</em>, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">See corresponding signature above without “3D” in the name.<br>
+ Use the texture coordinate <em>coord</em> to do a texture lookup in the 3D
+ texture currently bound to <em>sampler</em>.
+ For the projective (“<strong>Proj</strong>”) versions, the texture coordinate is
+ divided by <em>coord.q</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec4 <strong>textureCube</strong>(samplerCube <em>sampler</em>, vec3 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>textureCubeLod</strong>(samplerCube <em>sampler</em>, vec3 <em>coord</em>, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">See corresponding signature above without “Cube” in the name.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec4 <strong>shadow1D</strong>(sampler1DShadow <em>sampler</em>, vec3 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>shadow2D</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>shadow1DProj</strong>(sampler1DShadow <em>sampler</em>, vec4 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>shadow2DProj</strong>(sampler2DShadow <em>sampler</em>, vec4 <em>coord</em> [, float <em>bias</em>] )<br>
+ vec4 <strong>shadow1DLod</strong>(sampler1DShadow <em>sampler</em>, vec3 <em>coord</em>, float <em>lod</em>)<br>
+ vec4 <strong>shadow2DLod</strong>(sampler2DShadow <em>sampler</em>, vec3 <em>coord</em>, float <em>lod</em>)<br>
+ vec4 <strong>shadow1DProjLod</strong>(sampler1DShadow <em>sampler</em>, vec4 <em>coord</em>, float <em>lod</em>)<br>
+ vec4 <strong>shadow2DProjLod</strong>(sampler2DShadow <em>sampler</em>, vec4 coord, float <em>lod</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Same functionality as the “<strong>texture</strong>” based names above with the
+ same signature.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect2">
+<h3 id="atomic-counter-functions">8.10. Atomic Counter Functions</h3>
+<div class="paragraph">
+<p>The atomic-counter operations in this section operate atomically with
+respect to each other.
+They are atomic for any single counter, meaning any of these operations on a
+specific counter in one shader instantiation will be indivisible by any of
+these operations on the same counter from another shader instantiation.
+There is no guarantee that these operations are atomic with respect to other
+forms of access to the counter or that they are serialized when applied to
+separate counters.
+Such cases would require additional use of fences, barriers, or other forms
+of synchronization, if atomicity or serialization is desired.</p>
+</div>
+<div class="paragraph">
+<p>The underlying counter is a 32-bit unsigned integer.
+The result of operations will wrap to [0, 2<sup>32</sup>-1].</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterIncrement</strong>(atomic_uint <em>c</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>increments the counter for <em>c</em>, and</p>
+</li>
+<li>
+<p>returns its value prior to the increment operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterDecrement</strong>(atomic_uint <em>c</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>decrements the counter for <em>c</em>, and</p>
+</li>
+<li>
+<p>returns the value resulting from the decrement operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounter</strong>(atomic_uint <em>c</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the counter value for <em>c</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterAdd</strong>(atomic_uint <em>c</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>adds the value of <em>data</em> to the counter for <em>c</em>, and</p>
+</li>
+<li>
+<p>returns its value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterSubtract</strong>(atomic_uint <em>c</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>subtracts the value of <em>data</em> from the counter for <em>c</em>, and</p>
+</li>
+<li>
+<p>returns its value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterMin</strong>(atomic_uint <em>c</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>sets the counter for <em>c</em> to the minimum of the value of the counter and
+the value of <em>data</em>, and</p>
+</li>
+<li>
+<p>returns the value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterMax</strong>(atomic_uint <em>c</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>sets the counter for <em>c</em> to the maximum of the value of the counter and
+the value of <em>data</em>, and</p>
+</li>
+<li>
+<p>returns the value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterAnd</strong>(atomic_uint <em>c</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>sets the counter for <em>c</em> to the bitwise AND of the value of the counter
+and the value of <em>data</em>, and</p>
+</li>
+<li>
+<p>returns the value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterOr</strong>(atomic_uint <em>c</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>sets the counter for <em>c</em> to the bitwise OR of the value of the counter
+and the value of <em>data</em>, and</p>
+</li>
+<li>
+<p>returns the value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterXor</strong>(atomic_uint <em>c</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>sets the counter for <em>c</em> to the bitwise XOR of the value of the counter
+and the value of <em>data</em>, and</p>
+</li>
+<li>
+<p>returns the value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterExchange</strong>(atomic_uint <em>c</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>sets the counter value for <em>c</em> to the value of <em>data</em>, and</p>
+</li>
+<li>
+<p>returns its value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These two steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCounterCompSwap</strong>(atomic_uint <em>c</em>, uint <em>compare</em>, uint <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><div class="content"><div class="paragraph">
+<p>Atomically<br></p>
+</div>
+<div class="openblock">
+<div class="content">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>compares the value of <em>compare</em> and the counter value for <em>c</em></p>
+</li>
+<li>
+<p>if the values are equal, sets the counter value for <em>c</em> to the value of
+<em>data</em>, and</p>
+</li>
+<li>
+<p>returns its value prior to the operation.</p>
+</li>
+</ol>
+</div>
+<div class="paragraph">
+<p>These three steps are done atomically with respect to the atomic counter
+functions in this table.</p>
+</div>
+</div>
+</div></div></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="atomic-memory-functions">8.11. Atomic Memory Functions</h3>
+<div class="paragraph">
+<p>Atomic memory functions perform atomic operations on an individual signed or
+unsigned integer stored in buffer object or shared variable storage.
+All of the atomic memory operations read a value from memory, compute a new
+value using one of the operations described below, write the new value to
+memory, and return the original value read.
+The contents of the memory being updated by the atomic operation are
+guaranteed not to be modified by any other assignment or atomic memory
+function in any shader invocation between the time the original value is
+read and the time the new value is written.</p>
+</div>
+<div class="paragraph">
+<p>Atomic memory functions are supported only for a limited set of variables.
+A shader will fail to compile if the value passed to the <em>mem</em> argument of
+an atomic memory function does not correspond to a buffer or shared
+variable.
+It is acceptable to pass an element of an array or a single component of a
+vector to the <em>mem</em> argument of an atomic memory function, as long as the
+underlying array or vector is a buffer or shared variable.</p>
+</div>
+<div class="paragraph">
+<p>All the built-in functions in this section accept arguments with
+combinations of <strong>restrict</strong>, <strong>coherent</strong>, and <strong>volatile</strong> memory qualification,
+despite not having them listed in the prototypes.
+The atomic operation will operate as required by the calling argument’s
+memory qualification, not by the built-in function’s formal parameter memory
+qualification.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicAdd</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicAdd</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by adding the value of <em>data</em> to the contents
+ <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicMin</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicMin</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by taking the minimum of the value of <em>data</em> and
+ the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicMax</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicMax</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by taking the maximum of the value of <em>data</em> and
+ the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicAnd</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicAnd</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise AND of the value of
+ <em>data</em> and the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicOr</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicOr</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise OR of the value of
+ <em>data</em> and the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicXor</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicXor</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise EXCLUSIVE OR of the
+ value of <em>data</em> and the contents of <em>mem</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicExchange</strong>(inout uint <em>mem</em>, uint <em>data</em>)<br>
+ int <strong>atomicExchange</strong>(inout int <em>mem</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by simply copying the value of <em>data</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">uint <strong>atomicCompSwap</strong>(inout uint <em>mem</em>, uint <em>compare</em>, uint <em>data</em>)<br>
+ int <strong>atomicCompSwap</strong>(inout int <em>mem</em>, int <em>compare</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Compares the value of <em>compare</em> and the contents of <em>mem</em>.
+ If the values are equal, the new value is given by <em>data</em>; otherwise,
+ it is taken from the original contents of <em>mem</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="image-functions">8.12. Image Functions</h3>
+<div class="paragraph">
+<p>Variables using one of the image basic types may be used by the built-in
+shader image memory functions defined in this section to read and write
+individual texels of a texture.
+Each image variable references an image unit, which has a texture image
+attached.</p>
+</div>
+<div class="paragraph">
+<p>When image memory functions below access memory, an individual texel in the
+image is identified using an (<em>i</em>), (<em>i, j</em>), or (<em>i, j, k</em>) coordinate
+corresponding to the values of <em>P</em>.
+For <strong>image2DMS</strong> and <strong>image2DMSArray</strong> variables (and the corresponding
+int/unsigned int types) corresponding to multisample textures, each texel
+may have multiple samples and an individual sample is identified using the
+integer <em>sample</em> parameter.
+The coordinates
+and sample number
+are used to select an individual texel in the manner described in section
+8.26
+“Texture Image Loads and Stores” of the <a href="#references">OpenGL Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>Loads and stores support float, integer, and unsigned integer types.
+The data types below starting “_gimage_” serve as placeholders meaning
+types starting either “<strong>image</strong>”, “<strong>iimage</strong>”, or “<strong>uimage</strong>” in the same
+way as <em>gvec</em> or <em>gsampler</em> in earlier sections.</p>
+</div>
+<div class="paragraph">
+<p>The <em>IMAGE_PARAMS</em> in the prototypes below is a placeholder representing
+33
+separate functions, each for a different type of image variable.
+The <em>IMAGE_PARAMS</em> placeholder is replaced by one of the following parameter
+lists:</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage2D <em>image</em>, ivec2 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage3D <em>image</em>, ivec3 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimageCube <em>image</em>, ivec3 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimageBuffer <em>image</em>, int <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage2DArray <em>image</em>, ivec3 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimageCubeArray <em>image</em>, ivec3 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage1D <em>image</em>, int <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage1DArray <em>image</em>, ivec2 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage2DRect <em>image</em>, ivec2 <em>P</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage2DMS <em>image</em>, ivec2 <em>P</em>, int <em>sample</em></p>
+</dd>
+<dt class="hdlist1"></dt>
+<dd>
+<p>gimage2DMSArray <em>image</em>, ivec3 <em>P</em>, int <em>sample</em></p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>where each of the lines represents one of three different image variable
+types, and <em>image</em>,
+<em>P</em>, and <em>sample</em>
+specify the individual texel to operate on.
+The method for identifying the individual texel operated on from <em>image</em>,
+<em>P</em>,
+and <em>sample</em>,
+and the method for reading and writing the texel are specified in section
+8.26
+“Texture Image Loads and Stores” of the <a href="#references">OpenGL Specification</a>.</p>
+</div>
+<div class="paragraph">
+<p>The atomic functions perform operations on individual texels or samples of
+an image variable.
+Atomic memory operations read a value from the selected texel, compute a new
+value using one of the operations described below, write the new value to
+the selected texel, and return the original value read.
+The contents of the texel being updated by the atomic operation are
+guaranteed not to be modified by any other image store or atomic function
+between the time the original value is read and the time the new value is
+written.</p>
+</div>
+<div class="paragraph">
+<p>Atomic memory operations are supported on only a subset of all image
+variable types; <em>image</em> must be either:</p>
+</div>
+<div class="ulist">
+<ul>
+<li>
+<p>a signed integer image variable (type starts “<strong>iimage</strong>”) and a format
+qualifier of <strong>r32i</strong>, used with a <em>data</em> argument of type <strong>int</strong>, or</p>
+</li>
+<li>
+<p>an unsigned integer image variable (type starts “<strong>uimage</strong>”) and a
+format qualifier of <strong>r32ui</strong>, used with a <em>data</em> argument of type <strong>uint</strong>,
+or</p>
+</li>
+<li>
+<p>a float image variable (type starts “<strong>image</strong>”) and a format qualifier
+of <strong>r32f</strong>, used with a <em>data</em> argument of type <strong>float</strong>
+(<strong>imageAtomicExchange</strong> only).</p>
+</li>
+</ul>
+</div>
+<div class="paragraph">
+<p>All the built-in functions in this section accept arguments with
+combinations of <strong>restrict</strong>, <strong>coherent</strong>, and <strong>volatile</strong> memory qualification,
+despite not having them listed in the prototypes.
+The image operation will operate as required by the calling argument’s
+memory qualification, not by the built-in function’s formal parameter memory
+qualification.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> int <strong>imageSize</strong>(readonly writeonly gimage1D <em>image</em>)<br>
+ ivec2 <strong>imageSize</strong>(readonly writeonly gimage2D <em>image</em>)<br>
+ ivec3 <strong>imageSize</strong>(readonly writeonly gimage3D <em>image</em>)<br>
+ ivec2 <strong>imageSize</strong>(readonly writeonly gimageCube <em>image</em>)<br>
+ ivec3 <strong>imageSize</strong>(readonly writeonly gimageCubeArray <em>image</em>)<br>
+ ivec3 <strong>imageSize</strong>(readonly writeonly gimage2DArray <em>image</em>)<br>
+ ivec2 <strong>imageSize</strong>(readonly writeonly gimageRect <em>image</em>)<br>
+ ivec2 <strong>imageSize</strong>(readonly writeonly gimage1DArray <em>image</em>)<br>
+ ivec2 <strong>imageSize</strong>(readonly writeonly gimage2DMS <em>image</em>)<br>
+ ivec3 <strong>imageSize</strong>(readonly writeonly gimage2DMSArray <em>image</em>)<br>
+ int <strong>imageSize</strong>(readonly writeonly gimageBuffer <em>image</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the dimensions of the image or images bound to <em>image</em>.
+ For arrayed images, the last component of the return value will hold
+ the size of the array.
+ Cube images only return the dimensions of one face, and the number of
+ cubes in the cube map array, if arrayed.<br>
+ Note: The qualification <strong>readonly writeonly</strong> accepts a variable
+ qualified with <strong>readonly</strong>, <strong>writeonly</strong>, both, or neither.
+ It means the formal argument will be used for neither reading nor
+ writing to the underlying memory.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">int <strong>imageSamples</strong>(readonly writeonly gimage2DMS <em>image</em>)<br>
+ int <strong>imageSamples</strong>(readonly writeonly gimage2DMSArray <em>image</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the number of samples of the image or images bound to <em>image</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> gvec4 <strong>imageLoad</strong>(readonly <em>IMAGE_PARAMS</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Loads the texel at the coordinate <em>P</em> from the image unit <em>image</em> (in
+ <em>IMAGE_PARAMS</em>).
+ For multisample loads, the sample number is given by <em>sample</em>.
+ When <em>image</em>, <em>P</em>, and <em>sample</em>
+ identify a valid texel, the bits used to represent the selected texel in
+ memory are converted to a <strong>vec4</strong>, <strong>ivec4</strong>, or <strong>uvec4</strong> in the manner
+ described in section
+8.26
+ “Texture Image Loads and Stores” of the
+ <a href="#references">OpenGL Specification</a> and returned.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>imageStore</strong>(writeonly <em>IMAGE_PARAMS</em>, gvec4 <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Stores <em>data</em> into the texel at the coordinate <em>P</em> from the image
+ specified by <em>image</em>.
+ For multisample stores, the sample number is given by <em>sample</em>.
+ When <em>image</em>, <em>P</em>, and <em>sample</em>
+ identify a valid texel, the bits used to represent <em>data</em> are converted
+ to the format of the image unit in the manner described in section
+8.26
+ “Texture Image Loads and Stores” of the <a href="#references">OpenGL Specification</a>
+ and stored to the specified texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>imageAtomicAdd</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ int <strong>imageAtomicAdd</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by adding the value of <em>data</em> to the contents of
+ the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>imageAtomicMin</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ int <strong>imageAtomicMin</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by taking the minimum of the value of <em>data</em> and
+ the contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>imageAtomicMax</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ int <strong>imageAtomicMax</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by taking the maximum of the value <em>data</em> and the
+ contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>imageAtomicAnd</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ int <strong>imageAtomicAnd</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise AND of the value of
+ <em>data</em> and the contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>imageAtomicOr</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ int <strong>imageAtomicOr</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise OR of the value of
+ <em>data</em> and the contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>imageAtomicXor</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ int <strong>imageAtomicXor</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by performing a bit-wise EXCLUSIVE OR of the
+ value of <em>data</em> and the contents of the selected texel.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>imageAtomicExchange</strong>(<em>IMAGE_PARAMS</em>, uint <em>data</em>)<br>
+ int <strong>imageAtomicExchange</strong>(<em>IMAGE_PARAMS</em>, int <em>data</em>)<br>
+ float <strong>imageAtomicExchange</strong>(<em>IMAGE_PARAMS</em>, float <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Computes a new value by simply copying the value of <em>data</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock"> uint <strong>imageAtomicCompSwap</strong>(<em>IMAGE_PARAMS</em>, uint <em>compare</em>, uint <em>data</em>)<br>
+ int <strong>imageAtomicCompSwap</strong>(<em>IMAGE_PARAMS</em>, int <em>compare</em>, int <em>data</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Compares the value of <em>compare</em> and the contents of the selected
+ texel.
+ If the values are equal, the new value is given by <em>data</em>; otherwise,
+ it is taken from the original value loaded from the texel.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="geometry-shader-functions">8.13. Geometry Shader Functions</h3>
+<div class="paragraph">
+<p>These functions are only available in geometry shaders.
+They are described in more depth following the table.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>EmitStreamVertex</strong>(int <em>stream</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Emits the current values of output variables to the current output
+ primitive on stream <em>stream</em>.
+ The argument to <em>stream</em> must be a constant integral expression.
+ On return from this call, the values of all output variables are
+ undefined.<br>
+ Can only be used if multiple output streams are supported.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>EndStreamPrimitive</strong>(int <em>stream</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Completes the current output primitive on stream <em>stream</em> and starts a
+ new one.
+ The argument to <em>stream</em> must be a constant integral expression.
+ No vertex is emitted.<br>
+ Can only be used if multiple output streams are supported.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>EmitVertex</strong>()<br></p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Emits the current values of output variables to the current output
+ primitive.
+ When multiple output streams are supported, this is equivalent to
+ calling <strong>EmitStreamVertex</strong>(0).<br>
+ On return from this call, the values of output variables are
+ undefined.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>EndPrimitive</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Completes the current output primitive and starts a new one.
+ When multiple output streams are supported, this is equivalent to
+ calling <strong>EndStreamPrimitive</strong>(0).<br>
+ No vertex is emitted.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The function <strong>EmitStreamVertex</strong>() specifies that a vertex is completed.
+A vertex is added to the current output primitive in vertex stream <em>stream</em>
+using the current values of all built-in and user-defined output variables
+associated with <em>stream</em>.
+The values of all output variables for all output streams are undefined
+after a call to <strong>EmitStreamVertex</strong>().
+If a geometry shader invocation has emitted more vertices than permitted by
+the output layout qualifier <strong>max_vertices</strong>, the results of calling
+<strong>EmitStreamVertex</strong>() are undefined.</p>
+</div>
+<div class="paragraph">
+<p>The function <strong>EndStreamPrimitive</strong>() specifies that the current output
+primitive for vertex stream <em>stream</em> is completed and a new output primitive
+(of the same type) will be started by any subsequent <strong>EmitStreamVertex</strong>().
+This function does not emit a vertex.
+If the output layout is declared to be <strong>points</strong>, calling
+<strong>EndStreamPrimitive</strong>() is optional.</p>
+</div>
+<div class="paragraph">
+<p>A geometry shader starts with an output primitive containing no vertices for
+each stream.
+When a geometry shader terminates, the current output primitive for each
+stream is automatically completed.
+It is not necessary to call <strong>EndStreamPrimitive</strong>() if the geometry shader
+writes only a single primitive.</p>
+</div>
+<div class="paragraph">
+<p>Multiple output streams are supported only if the output primitive type is
+declared to be <strong>points</strong>.
+It is a compile-time or link-time error if a program contains a geometry
+shader calling <strong>EmitStreamVertex</strong>() or <strong>EndStreamPrimitive</strong>() if its output
+primitive type is not <strong>points</strong>.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="fragment-processing-functions">8.14. Fragment Processing Functions</h3>
+<div class="paragraph">
+<p>Fragment processing functions are only available in fragment shaders.</p>
+</div>
+<div class="sect3">
+<h4 id="derivative-functions">8.14.1. Derivative Functions</h4>
+<div class="paragraph">
+<p>Derivatives may be computationally expensive and/or numerically unstable.
+Therefore, an OpenGL implementation may approximate the true derivatives
+by using a fast but not entirely accurate derivative computation.
+Derivatives are undefined within non-uniform control flow.</p>
+</div>
+<div class="paragraph">
+<p>The expected behavior of a derivative is specified using forward/backward
+differencing.</p>
+</div>
+<div class="paragraph">
+<p>Forward differencing:</p>
+</div>
+<div class="paragraph">
+<p>\(F(x+dx) - F(x) \sim dFdx(x) \cdot dx (1a)\)</p>
+</div>
+<div class="paragraph">
+<p>\(dFdx(x) \sim \frac{F(x+dx) - F(x)}{dx} (1b)\)</p>
+</div>
+<div class="paragraph">
+<p>Backward differencing:</p>
+</div>
+<div class="paragraph">
+<p>\(F(x-dx) - F(x) \sim -dFdx(x) \cdot dx (2a)\)</p>
+</div>
+<div class="paragraph">
+<p>\(dFdx(x) \sim \frac{F(x) - F(x-dx)}{dx} (2b)\)</p>
+</div>
+<div class="paragraph">
+<p>With single-sample rasterization, \(dx \leq 1.0\) in equations 1b
+and 2b.
+For multisample rasterization, \(dx < 2.0\) in equations 1b and 2b.</p>
+</div>
+<div class="paragraph">
+<p>\(dFdy\) is approximated similarly, with <em>y</em> replacing <em>x</em>.</p>
+</div>
+<div class="paragraph">
+<p>With multisample rasterization, for any given fragment or sample, either
+neighboring fragments or samples may be considered.</p>
+</div>
+<div class="paragraph">
+<p>It is typical to consider a 2x2 square of fragments or samples, and compute
+independent <strong>dFdxFine</strong> per row and independent <strong>dFdyFine</strong> per column, while
+computing only a single <strong>dFdxCoarse</strong> and a single <strong>dFdyCoarse</strong> for the
+entire 2x2 square.
+Thus, all second-order coarse derivatives, e.g.
+<strong>dFdxCoarse</strong>(<strong>dFdxCoarse</strong>(<em>x</em>)), may be 0, even for non-linear arguments.
+However, second-order fine derivatives, e.g. <strong>dFdxFine</strong>(<strong>dFdyFine</strong>(<em>x</em>))
+will properly reflect the difference between the independent fine
+derivatives computed within the 2x2 square.</p>
+</div>
+<div class="paragraph">
+<p>The method may differ per fragment, subject to the constraint that the
+method may vary by window coordinates, not screen coordinates.
+The invariance requirement described in section 14.2 “Invariance” of the
+<a href="#references">OpenGL Specification</a>, is relaxed for derivative calculations, because
+the method may be a function of fragment location.</p>
+</div>
+<div class="paragraph">
+<p>In some implementations, varying degrees of derivative accuracy for <strong>dFdx</strong>
+and <strong>dFdy</strong> may be obtained by providing GL hints (see section 21.4 “Hints”
+of the <a href="#references">OpenGL Specification</a>), allowing a user to make an image quality
+versus speed trade off.
+These hints have no effect on <strong>dFdxCoarse</strong>, <strong>dFdyCoarse</strong>, <strong>dFdxFine</strong> and
+<strong>dFdyFine</strong>.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>dFdx</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns either <strong>dFdxFine</strong>(<em>p</em>) or <strong>dFdxCoarse</strong>(<em>p</em>), based on
+ implementation choice, presumably whichever is the faster, or by whichever
+ is selected in the API through quality-versus-speed hints.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>dFdy</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns either <strong>dFdyFine</strong>(<em>p</em>) or <strong>dFdyCoarse</strong>(<em>p</em>), based on
+ implementation choice, presumably whichever is the faster, or by
+ whichever is selected in the API through quality-versus-speed hints.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>dFdxFine</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the partial derivative of <em>p</em> with respect to the window x
+ coordinate.
+ Will use local differencing based on the value of <em>p</em> for the current
+ fragment and its immediate neighbor(s).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>dFdyFine</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the partial derivative of <em>p</em> with respect to the window y
+ coordinate.
+ Will use local differencing based on the value of <em>p</em> for the current
+ fragment and its immediate neighbor(s).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>dFdxCoarse</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the partial derivative of <em>p</em> with respect to the window x
+ coordinate.
+ Will use local differencing based on the value of <em>p</em> for the current
+ fragment’s neighbors, and will possibly, but not necessarily, include
+ the value of <em>p</em> for the current fragment.
+ That is, over a given area, the implementation can x compute
+ derivatives in fewer unique locations than would be allowed for
+ <strong>dFdxFine</strong>(<em>p</em>).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>dFdyCoarse</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the partial derivative of <em>p</em> with respect to the window y
+ coordinate.
+ Will use local differencing based on the value of <em>p</em> for the current
+ fragment’s neighbors, and will possibly, but not necessarily, include
+ the value of <em>p</em> for the current fragment.
+ That is, over a given area, the implementation can compute y
+ derivatives in fewer unique locations than would be allowed for
+ <strong>dFdyFine</strong>(<em>p</em>).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>fwidth</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>abs</strong>(<strong>dFdx</strong>(<em>p</em>)) + <strong>abs</strong>(<strong>dFdy</strong>(<em>p</em>)).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>fwidthFine</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>abs</strong>(<strong>dFdxFine</strong>(<em>p</em>)) + <strong>abs</strong>(<strong>dFdyFine</strong>(<em>p</em>)).</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">genFType <strong>fwidthCoarse</strong>(genFType <em>p</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>abs</strong>(<strong>dFdxCoarse</strong>(<em>p</em>)) + <strong>abs</strong>(<strong>dFdyCoarse</strong>(<em>p</em>)).</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect3">
+<h4 id="interpolation-functions">8.14.2. Interpolation Functions</h4>
+<div class="paragraph">
+<p>Built-in interpolation functions are available to compute an interpolated
+value of a fragment shader input variable at a shader-specified (<em>x</em>, <em>y</em>)
+location.
+A separate (<em>x</em>, <em>y</em>) location may be used for each invocation of the
+built-in function, and those locations may differ from the default (<em>x</em>,
+<em>y</em>) location used to produce the default value of the input.</p>
+</div>
+<div class="paragraph">
+<p>For all of the interpolation functions, <em>interpolant</em> must be an l-value
+from an <strong>in</strong> declaration;
+this can include a variable, a block or structure member, an array element, or
+some combination of these.
+Additionally, component selection operators (e.g. <strong>.xy</strong>, <strong>.xxz</strong>) may be applied
+to <em>interpolant</em>, in which case the interpolation function will return the
+result of applying the component selection operator to the interpolated value
+of <em>interpolant</em> (for example, interpolateAt(v.xxz) is defined to return
+interpolateAt(v).xxz).
+Arrayed inputs can be indexed with general (nonuniform) integer expressions.</p>
+</div>
+<div class="paragraph">
+<p>If <em>interpolant</em> is declared with the <strong>flat</strong> qualifier, the interpolated
+value will have the same value everywhere for a single primitive, so the
+location used for interpolation has no effect and the functions just return
+that same value.
+If <em>interpolant</em> is declared with the <strong>centroid</strong> qualifier, the value
+returned by <strong>interpolateAtSample</strong>() and <strong>interpolateAtOffset</strong>() will be
+evaluated at the specified location, ignoring the location normally used
+with the <strong>centroid</strong> qualifier.
+If <em>interpolant</em> is declared with the <strong>noperspective</strong> qualifier, the
+interpolated value will be computed without perspective correction.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>interpolateAtCentroid</strong>(float <em>interpolant</em>)<br>
+ vec2 <strong>interpolateAtCentroid</strong>(vec2 <em>interpolant</em>)<br>
+ vec3 <strong>interpolateAtCentroid</strong>(vec3 <em>interpolant</em>)<br>
+ vec4 <strong>interpolateAtCentroid</strong>(vec4 <em>interpolant</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the value of the input <em>interpolant</em> sampled at a location
+ inside both the pixel and the primitive being processed.
+ The value obtained would be the same value assigned to the input
+ variable if declared with the <strong>centroid</strong> qualifier.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>interpolateAtSample</strong>(float <em>interpolant</em>, int <em>sample</em>)<br>
+ vec2 <strong>interpolateAtSample</strong>(vec2 <em>interpolant</em>, int <em>sample</em>)<br>
+ vec3 <strong>interpolateAtSample</strong>(vec3 <em>interpolant</em>, int <em>sample</em>)<br>
+ vec4 <strong>interpolateAtSample</strong>(vec4 <em>interpolant</em>, int <em>sample</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the value of the input <em>interpolant</em> variable at the location
+ of sample number <em>sample</em>.
+ If multisample buffers are not available, the input variable will be
+ evaluated at the center of the pixel.
+ If sample <em>sample</em> does not exist, the position used to interpolate
+ the input variable is undefined.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>interpolateAtOffset</strong>(float <em>interpolant</em>, vec2 offset)<br>
+ vec2 <strong>interpolateAtOffset</strong>(vec2 <em>interpolant</em>, vec2 offset)<br>
+ vec3 <strong>interpolateAtOffset</strong>(vec3 <em>interpolant</em>, vec2 offset)<br>
+ vec4 <strong>interpolateAtOffset</strong>(vec4 <em>interpolant</em>, vec2 offset)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns the value of the input <em>interpolant</em> variable sampled at an
+ offset from the center of the pixel specified by <em>offset</em>.
+ The two floating-point components of <em>offset</em>, give the offset in
+ pixels in the <em>x</em> and <em>y</em> directions, respectively.<br>
+ An offset of (0, 0) identifies the center of the pixel.
+ The range and granularity of offsets supported by this function is
+ implementation-dependent.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+</div>
+<div class="sect2">
+<h3 id="noise-functions">8.15. Noise Functions</h3>
+<div class="paragraph">
+<p>The noise functions <strong>noise1</strong>, <strong>noise2</strong>, <strong>noise3</strong>, and <strong>noise4</strong> have been
+deprecated starting with version 4.4 of GLSL.
+When not generating SPIR-V they are defined to return the value 0.0 or a vector
+whose components are all 0.0. When generating SPIR-V the noise functions are
+not declared and may not be used.</p>
+</div>
+<div class="paragraph">
+<p>As in previous releases, the noise functions are not semantically considered to
+be compile-time constant expressions.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax (deprecated)</th>
+<th class="tableblock halign-left valign-top">Description (deprecated)</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">float <strong>noise1</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a 1D noise value based on the input value <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec2 <strong>noise2</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a 2D noise value based on the input value <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec3 <strong>noise3</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a 3D noise value based on the input value <em>x</em>.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">vec4 <strong>noise4</strong>(genFType <em>x</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns a 4D noise value based on the input value <em>x</em>.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<div class="sect2">
+<h3 id="shader-invocation-control-functions">8.16. Shader Invocation Control Functions</h3>
+<div class="paragraph">
+<p>The shader invocation control function is only available in tessellation
+control and compute shaders.
+It is used to control the relative execution order of multiple shader
+invocations used to process a patch (in the case of tessellation control
+shaders) or a workgroup (in the case of compute shaders), which are
+otherwise executed with an undefined relative order.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>barrier</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">For any given static instance of <strong>barrier</strong>(), all tessellation control
+ shader invocations for a single input patch must enter it before any
+ will be allowed to continue beyond it, or all compute shader
+ invocations for a single workgroup must enter it before any will
+ continue beyond it.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The function <strong>barrier</strong>() provides a partially defined order of execution
+between shader invocations.
+The ensures that, for some types of memory accesses, values written by one
+invocation prior to a given static instance of <strong>barrier</strong>() can be safely read
+by other invocations after their call to the same static instance <strong>barrier</strong>().
+Because invocations may execute in an undefined order between these barrier
+calls, the values of a per-vertex or per-patch output variable for tessellation
+control shaders, or the values of <strong>shared</strong> variables for compute shaders will be
+undefined in a number of cases enumerated in
+“<a href="#output-variables">Output Variables</a>” (for tessellation control shaders)
+and “<a href="#shared-variables">Shared Variables</a>” (for compute shaders).</p>
+</div>
+<div class="paragraph">
+<p>For tessellation control shaders, the <strong>barrier</strong>() function may only be
+placed inside the function <strong>main</strong>() of the shader and may not be called
+within any control flow.
+Barriers are also disallowed after a return statement in the function
+<strong>main</strong>().
+Any such misplaced barriers result in a compile-time error.</p>
+</div>
+<div class="paragraph">
+<p>A <strong>barrier</strong>() affects control flow but only synchronizes memory accesses
+to <strong>shared</strong> variables and tessellation control output variables.
+For other memory accesses, it does not ensure that values written by one invocation
+prior to a given static instance of <strong>barrier</strong>() can be safely read by other
+invocations after their call to the same static instance of <strong>barrier</strong>().
+To achieve this requires the use of both <strong>barrier</strong>() and a memory barrier.</p>
+</div>
+<div class="paragraph">
+<p>For compute shaders, the <strong>barrier</strong>() function may be placed within control
+flow, but that control flow must be uniform control flow.
+That is, all the controlling expressions that lead to execution of the
+barrier must be dynamically uniform expressions.
+This ensures that if any shader invocation enters a conditional statement,
+then all invocations will enter it.
+While compilers are encouraged to give warnings if they can detect this
+might not happen, compilers cannot completely determine this.
+Hence, it is the author’s responsibility to ensure <strong>barrier</strong>() only exists
+inside uniform control flow.
+Otherwise, some shader invocations will stall indefinitely, waiting for a
+barrier that is never reached by other invocations.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="shader-memory-control-functions">8.17. Shader Memory Control Functions</h3>
+<div class="paragraph">
+<p>Within a single shader invocation, the visibility and order of writes made
+by that invocation are well-defined.
+However, the relative order of reads and writes to a single shared memory
+address from multiple separate shader invocations is largely undefined.
+Additionally, the order of accesses to multiple memory addresses performed
+by a single shader invocation, as observed by other shader invocations, is
+also undefined.</p>
+</div>
+<div class="paragraph">
+<p>The following built-in functions can be used to control the ordering of
+reads and writes:</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrier</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of memory transactions issued by a single shader
+ invocation.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrierAtomicCounter</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of accesses to atomic-counter variables issued by
+ a single shader invocation.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrierBuffer</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of memory transactions to buffer variables issued
+ within a single shader invocation.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrierShared</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of memory transactions to shared variables issued
+ within a single shader invocation, as viewed by other invocations in
+ the same workgroup.<br>
+ Only available in compute shaders.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>memoryBarrierImage</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of memory transactions to images issued within a
+ single shader invocation.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">void <strong>groupMemoryBarrier</strong>()</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Control the ordering of all memory transactions issued within a single
+ shader invocation, as viewed by other invocations in the same workgroup.<br>
+ Only available in compute shaders.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>The memory barrier built-in functions can be used to order reads and writes
+to variables stored in memory accessible to other shader invocations.
+When called, these functions will wait for the completion of all reads and
+writes previously performed by the caller that access selected variable
+types, and then return with no other effect.
+The built-in functions <strong>memoryBarrierAtomicCounter</strong>(),
+<strong>memoryBarrierBuffer</strong>(), <strong>memoryBarrierImage</strong>(), and <strong>memoryBarrierShared</strong>()
+wait for the completion of accesses to atomic counter, buffer, image, and
+shared variables, respectively.
+The built-in functions <strong>memoryBarrier</strong>() and <strong>groupMemoryBarrier</strong>() wait for
+the completion of accesses to all of the above variable types.
+The functions <strong>memoryBarrierShared</strong>() and <strong>groupMemoryBarrier</strong>() are
+available only in compute shaders; the other functions are available in all
+shader types.</p>
+</div>
+<div class="paragraph">
+<p>When these functions return, the effects of any memory stores performed
+using coherent variables prior to the call will be visible to any
+future<sup>1</sup> coherent access to the same memory performed by any other shader
+invocation.
+In particular, the values written this way in one shader stage are
+guaranteed to be visible to coherent memory accesses performed by shader
+invocations in subsequent stages when those invocations were triggered by
+the execution of the original shader invocation (e.g. fragment shader
+invocations for a primitive resulting from a particular geometry
+shader invocation).</p>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1">1</dt>
+<dd>
+<p>An access is only a <em>future</em> access if a <em>happens-before</em> relation can
+be established between the store and the load.</p>
+</dd>
+</dl>
+</div>
+<div class="paragraph">
+<p>Additionally, memory barrier functions order stores performed by the calling
+invocation, as observed by other shader invocations.
+Without memory barriers, if one shader invocation performs two stores to
+coherent variables, a second shader invocation might see the values written
+by the second store prior to seeing those written by the first.
+However, if the first shader invocation calls a memory barrier function
+between the two stores, selected other shader invocations will never see the
+results of the second store before seeing those of the first.
+When using the functions <strong>groupMemoryBarrier</strong>() or <strong>memoryBarrierShared</strong>(),
+this ordering guarantee applies only to other shader invocations in the same
+compute shader workgroup; all other memory barrier functions provide the
+guarantee to all other shader invocations.
+No memory barrier is required to guarantee the order of memory stores as
+observed by the invocation performing the stores; an invocation reading from
+a variable that it previously wrote will always see the most recently
+written value unless another shader invocation also wrote to the same
+memory.</p>
+</div>
+</div>
+<div class="sect2">
+<h3 id="shader-invocation-group-functions">8.18. Shader Invocation Group Functions</h3>
+<div class="paragraph">
+<p>Implementations of the OpenGL Shading Language may optionally group multiple shader
+invocations for a single shader stage into a single SIMD invocation group,
+where invocations are assigned to groups in an undefined,
+implementation-dependent manner.
+Shader algorithms on such implementations may benefit from being able to
+evaluate a composite of Boolean values over all active invocations in a
+group.</p>
+</div>
+<table class="tableblock frame-all grid-all stretch">
+<colgroup>
+<col style="width: 50%;">
+<col style="width: 50%;">
+</colgroup>
+<thead>
+<tr>
+<th class="tableblock halign-left valign-top">Syntax</th>
+<th class="tableblock halign-left valign-top">Description</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bool <strong>anyInvocation</strong>(bool <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if and only if <em>value</em> is <strong>true</strong> for at least one
+ active invocation in the group.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bool <strong>allInvocations</strong>(bool <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if and only if <em>value</em> is <strong>true</strong> for all active
+ invocations in the group.</p></td>
+</tr>
+<tr>
+<td class="tableblock halign-left valign-top"><p class="tableblock">bool <strong>allInvocationsEqual</strong>(bool <em>value</em>)</p></td>
+<td class="tableblock halign-left valign-top"><p class="tableblock">Returns <strong>true</strong> if <em>value</em> is the same for all active invocations in
+ the group.</p></td>
+</tr>
+</tbody>
+</table>
+<div class="paragraph">
+<p>For all of these functions, the same result is returned to all active
+invocations in the group.</p>
+</div>
+<div class="paragraph">
+<p>These functions may be called within conditionally executed code.
+In groups where some invocations do not execute the function call, the value
+returned by the function is not affected by any invocation not calling the
+function, even when value is well-defined for that invocation.</p>
+</div>
+<div class="paragraph">
+<p>Because these functions depend on the values of <em>value</em> in an undefined
+group of invocations, the value returned by these functions is largely
+undefined.
+However, <strong>anyInvocation</strong>() is guaranteed to return <strong>true</strong> if <em>value</em> is
+<strong>true</strong>, and <strong>allInvocations</strong>() is guaranteed to return <strong>false</strong> if <em>value</em> is
+<strong>false</strong>.</p>
+</div>
+<div class="paragraph">
+<p>Because implementations are not required to combine invocations into groups,
+simply returning <em>value</em> for <strong>anyInvocation</strong>() and <strong>allInvocations</strong>() and
+returning true for <strong>allInvocationsEqual</strong>() is a legal implementation of
+these functions.</p>
+</div>
+<div class="paragraph">
+<p>For fragment shaders, invocations in a SIMD invocation group may include
+invocations corresponding to pixels that are covered by a primitive being
+rasterized, as well as invocations corresponding to neighboring pixels not
+covered by the primitive.
+<em>Helper invocations</em> (see “<a href="#built-in-language-variables">Built-In Language
+Variables</a>”) may be created and the value of <em>value</em> for such
+helper-invocation pixels may affect the value returned by <strong>anyInvocation</strong>(),
+<strong>allInvocations</strong>(), and <strong>allInvocationsEqual</strong>().</p>
+</div>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="shading-language-grammar">9. Shading Language Grammar</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>The grammar is fed from the output of lexical analysis.
+The tokens returned from lexical analysis are</p>
+</div>
+<div class="listingblock">
+<div class="content">
+<pre class="CodeRay highlight"><code data-lang="c++">CONST BOOL FLOAT INT UINT
+DOUBLE
+
+BVEC2 BVEC3 BVEC4 IVEC2 IVEC3 IVEC4 UVEC2 UVEC3 UVEC4 VEC2 VEC3 VEC4
+
+MAT2 MAT3 MAT4 CENTROID IN OUT INOUT
+
+UNIFORM PATCH SAMPLE BUFFER SHARED
+
+COHERENT VOLATILE RESTRICT READONLY WRITEONLY
+
+NOPERSPECTIVE
+FLAT SMOOTH LAYOUT
+
+MAT2X2 MAT2X3 MAT2X4
+
+MAT3X2 MAT3X3 MAT3X4
+
+MAT4X2 MAT4X3 MAT4X4
+
+DVEC2 DVEC3 DVEC4 DMAT2 DMAT3 DMAT4
+
+DMAT2X2 DMAT2X3 DMAT2X4
+
+DMAT3X2 DMAT3X3 DMAT3X4
+
+DMAT4X2 DMAT4X3 DMAT4X4
+
+ATOMIC_UINT
+
+SAMPLER2D SAMPLER3D SAMPLERCUBE SAMPLER2DSHADOW
+
+SAMPLERCUBESHADOW SAMPLER2DARRAY SAMPLER2DARRAYSHADOW
+
+ISAMPLER2D ISAMPLER3D ISAMPLERCUBE ISAMPLER2DARRAY
+
+USAMPLER2D USAMPLER3D USAMPLERCUBE USAMPLER2DARRAY
+
+SAMPLER1D SAMPLER1DSHADOW
+
+SAMPLER1DARRAY SAMPLER1DARRAYSHADOW
+
+ISAMPLER1D ISAMPLER1DARRAY
+
+USAMPLER1D USAMPLER1DARRAY
+
+SAMPLER2DRECT SAMPLER2DRECTSHADOW ISAMPLER2DRECT USAMPLER2DRECT
+
+SAMPLERBUFFER ISAMPLERBUFFER USAMPLERBUFFER
+
+SAMPLERCUBEARRAY SAMPLERCUBEARRAYSHADOW
+
+ISAMPLERCUBEARRAY USAMPLERCUBEARRAY
+
+SAMPLER2DMS ISAMPLER2DMS USAMPLER2DMS
+
+SAMPLER2DMSARRAY ISAMPLER2DMSARRAY USAMPLER2DMSARRAY
+
+IMAGE2D IIMAGE2D UIMAGE2D
+
+IMAGE3D IIMAGE3D UIMAGE3D
+
+IMAGECUBE IIMAGECUBE UIMAGECUBE
+
+IMAGEBUFFER IIMAGEBUFFER UIMAGEBUFFER
+
+IMAGE2DARRAY IIMAGE2DARRAY UIMAGE2DARRAY
+
+IMAGECUBEARRAY IIMAGECUBEARRAY UIMAGECUBEARRAY
+
+IMAGE1D IIMAGE1D UIMAGE1D
+
+IMAGE1DARRAY IIMAGE1DARRAY UIMAGE1DARRAY
+
+IMAGE2DRECT IIMAGE2DRECT UIMAGE2DRECT
+
+IMAGE2DMS IIMAGE2DMS UIMAGE2DMS
+
+IMAGE2DMSARRAY IIMAGE2DMSARRAY UIMAGE2DMSARRAY
+
+STRUCT VOID
+
+WHILE BREAK CONTINUE DO ELSE FOR IF DISCARD RETURN SWITCH CASE DEFAULT
+SUBROUTINE
+
+IDENTIFIER TYPE_NAME
+
+FLOATCONSTANT INTCONSTANT UINTCONSTANT BOOLCONSTANT
+DOUBLECONSTANT
+
+FIELD_SELECTION
+
+LEFT_OP RIGHT_OP
+
+INC_OP DEC_OP LE_OP GE_OP EQ_OP NE_OP
+
+AND_OP OR_OP XOR_OP MUL_ASSIGN DIV_ASSIGN ADD_ASSIGN
+
+MOD_ASSIGN LEFT_ASSIGN RIGHT_ASSIGN AND_ASSIGN XOR_ASSIGN OR_ASSIGN
+
+SUB_ASSIGN
+
+LEFT_PAREN RIGHT_PAREN LEFT_BRACKET RIGHT_BRACKET LEFT_BRACE RIGHT_BRACE DOT
+
+COMMA COLON EQUAL SEMICOLON BANG DASH TILDE PLUS STAR SLASH PERCENT
+
+LEFT_ANGLE RIGHT_ANGLE VERTICAL_BAR CARET AMPERSAND QUESTION
+
+INVARIANT PRECISE
+
+HIGH_PRECISION MEDIUM_PRECISION LOW_PRECISION PRECISION</code></pre>
+</div>
+</div>
+<div class="paragraph">
+<p>The following describes the grammar for the OpenGL Shading Language in terms of the above
+tokens.
+The starting rule is <em>translation_unit</em>.
+An empty shader (one having no tokens to parse, after pre-processing) is
+valid, resulting in no compile-time errors, even though the grammar below
+does not have a rule to accept an empty token stream.</p>
+</div>
+<div class="openblock bnf">
+<div class="content">
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>variable_identifier</em> : </dt>
+<dd>
+<p><em>IDENTIFIER</em></p>
+</dd>
+<dt class="hdlist1"><em>primary_expression</em> : </dt>
+<dd>
+<p><em>variable_identifier</em><br>
+<em>INTCONSTANT</em><br>
+<em>UINTCONSTANT</em><br>
+<em>FLOATCONSTANT</em><br>
+<em>BOOLCONSTANT</em><br>
+<em>DOUBLECONSTANT</em><br>
+<em>LEFT_PAREN</em> <em>expression</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>postfix_expression</em> : </dt>
+<dd>
+<p><em>primary_expression</em><br>
+<em>postfix_expression</em> <em>LEFT_BRACKET</em> <em>integer_expression</em> <em>RIGHT_BRACKET</em><br>
+<em>function_call</em><br>
+<em>postfix_expression</em> <em>DOT</em> <em>FIELD_SELECTION</em><br>
+<em>postfix_expression</em> <em>INC_OP</em><br>
+<em>postfix_expression</em> <em>DEC_OP</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>FIELD_SELECTION includes members in structures, component selection for
+vectors and the 'length' identifier for the length() method</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>integer_expression</em> : </dt>
+<dd>
+<p><em>expression</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call</em> : </dt>
+<dd>
+<p><em>function_call_or_method</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_or_method</em> : </dt>
+<dd>
+<p><em>function_call_generic</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_generic</em> : </dt>
+<dd>
+<p><em>function_call_header_with_parameters</em> <em>RIGHT_PAREN</em><br>
+<em>function_call_header_no_parameters</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_header_no_parameters</em> : </dt>
+<dd>
+<p><em>function_call_header</em> <em>VOID</em><br>
+<em>function_call_header</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_header_with_parameters</em> : </dt>
+<dd>
+<p><em>function_call_header</em> <em>assignment_expression</em><br>
+<em>function_call_header_with_parameters</em> <em>COMMA</em> <em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>function_call_header</em> : </dt>
+<dd>
+<p><em>function_identifier</em> <em>LEFT_PAREN</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: Constructors look like functions, but lexical analysis
+recognized most of them as keywords.
+They are now recognized through <em>type_specifier</em>.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Methods (<strong>.length</strong>), subroutine array calls, and identifiers are recognized
+through <em>postfix_expression</em>.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>function_identifier</em> : </dt>
+<dd>
+<p><em>type_specifier</em><br>
+<em>postfix_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>unary_expression</em> : </dt>
+<dd>
+<p><em>postfix_expression</em><br>
+<em>INC_OP</em> <em>unary_expression</em><br>
+<em>DEC_OP</em> <em>unary_expression</em><br>
+<em>unary_operator</em> <em>unary_expression</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: No traditional style type casts.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>unary_operator</em> : </dt>
+<dd>
+<p><em>PLUS</em><br>
+<em>DASH</em><br>
+<em>BANG</em><br>
+<em>TILDE</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: No '*' or '&' unary ops.
+Pointers are not supported.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>multiplicative_expression</em> : </dt>
+<dd>
+<p><em>unary_expression</em><br>
+<em>multiplicative_expression</em> <em>STAR</em> <em>unary_expression</em><br>
+<em>multiplicative_expression</em> <em>SLASH</em> <em>unary_expression</em><br>
+<em>multiplicative_expression</em> <em>PERCENT</em> <em>unary_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>additive_expression</em> : </dt>
+<dd>
+<p><em>multiplicative_expression</em><br>
+<em>additive_expression</em> <em>PLUS</em> <em>multiplicative_expression</em><br>
+<em>additive_expression</em> <em>DASH</em> <em>multiplicative_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>shift_expression</em> : </dt>
+<dd>
+<p><em>additive_expression</em><br>
+<em>shift_expression</em> <em>LEFT_OP</em> <em>additive_expression</em><br>
+<em>shift_expression</em> <em>RIGHT_OP</em> <em>additive_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>relational_expression</em> : </dt>
+<dd>
+<p><em>shift_expression</em><br>
+<em>relational_expression</em> <em>LEFT_ANGLE</em> <em>shift_expression</em><br>
+<em>relational_expression</em> <em>RIGHT_ANGLE</em> <em>shift_expression</em><br>
+<em>relational_expression</em> <em>LE_OP</em> <em>shift_expression</em><br>
+<em>relational_expression</em> <em>GE_OP</em> <em>shift_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>equality_expression</em> : </dt>
+<dd>
+<p><em>relational_expression</em><br>
+<em>equality_expression</em> <em>EQ_OP</em> <em>relational_expression</em><br>
+<em>equality_expression</em> <em>NE_OP</em> <em>relational_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>and_expression</em> : </dt>
+<dd>
+<p><em>equality_expression</em><br>
+<em>and_expression</em> <em>AMPERSAND</em> <em>equality_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>exclusive_or_expression</em> : </dt>
+<dd>
+<p><em>and_expression</em><br>
+<em>exclusive_or_expression</em> <em>CARET</em> <em>and_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>inclusive_or_expression</em> : </dt>
+<dd>
+<p><em>exclusive_or_expression</em><br>
+<em>inclusive_or_expression</em> <em>VERTICAL_BAR</em> <em>exclusive_or_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>logical_and_expression</em> : </dt>
+<dd>
+<p><em>inclusive_or_expression</em><br>
+<em>logical_and_expression</em> <em>AND_OP</em> <em>inclusive_or_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>logical_xor_expression</em> : </dt>
+<dd>
+<p><em>logical_and_expression</em><br>
+<em>logical_xor_expression</em> <em>XOR_OP</em> <em>logical_and_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>logical_or_expression</em> : </dt>
+<dd>
+<p><em>logical_xor_expression</em><br>
+<em>logical_or_expression</em> <em>OR_OP</em> <em>logical_xor_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>conditional_expression</em> : </dt>
+<dd>
+<p><em>logical_or_expression</em><br>
+<em>logical_or_expression</em> <em>QUESTION</em> <em>expression</em> <em>COLON</em>
+<em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>assignment_expression</em> : </dt>
+<dd>
+<p><em>conditional_expression</em><br>
+<em>unary_expression</em> <em>assignment_operator</em> <em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>assignment_operator</em> : </dt>
+<dd>
+<p><em>EQUAL</em><br>
+<em>MUL_ASSIGN</em><br>
+<em>DIV_ASSIGN</em><br>
+<em>MOD_ASSIGN</em><br>
+<em>ADD_ASSIGN</em><br>
+<em>SUB_ASSIGN</em><br>
+<em>LEFT_ASSIGN</em><br>
+<em>RIGHT_ASSIGN</em><br>
+<em>AND_ASSIGN</em><br>
+<em>XOR_ASSIGN</em><br>
+<em>OR_ASSIGN</em></p>
+</dd>
+<dt class="hdlist1"><em>expression</em> : </dt>
+<dd>
+<p><em>assignment_expression</em><br>
+<em>expression</em> <em>COMMA</em> <em>assignment_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>constant_expression</em> : </dt>
+<dd>
+<p><em>conditional_expression</em></p>
+</dd>
+<dt class="hdlist1"><em>declaration</em> : </dt>
+<dd>
+<p><em>function_prototype</em> <em>SEMICOLON</em><br>
+<em>init_declarator_list</em> <em>SEMICOLON</em><br>
+<em>PRECISION</em> <em>precision_qualifier</em> <em>type_specifier</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em>
+<em>RIGHT_BRACE</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em>
+<em>RIGHT_BRACE</em> <em>IDENTIFIER</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em>
+<em>RIGHT_BRACE</em> <em>IDENTIFIER</em> <em>array_specifier</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>IDENTIFIER</em> <em>identifier_list</em> <em>SEMICOLON</em></p>
+</dd>
+<dt class="hdlist1"><em>identifier_list</em> : </dt>
+<dd>
+<p><em>COMMA</em> <em>IDENTIFIER</em><br>
+<em>identifier_list</em> <em>COMMA</em> <em>IDENTIFIER</em></p>
+</dd>
+<dt class="hdlist1"><em>function_prototype</em> : </dt>
+<dd>
+<p><em>function_declarator</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>function_declarator</em> : </dt>
+<dd>
+<p><em>function_header</em><br>
+<em>function_header_with_parameters</em></p>
+</dd>
+<dt class="hdlist1"><em>function_header_with_parameters</em> : </dt>
+<dd>
+<p><em>function_header</em> <em>parameter_declaration</em><br>
+<em>function_header_with_parameters</em> <em>COMMA</em> <em>parameter_declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>function_header</em> : </dt>
+<dd>
+<p><em>fully_specified_type</em> <em>IDENTIFIER</em> <em>LEFT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>parameter_declarator</em> : </dt>
+<dd>
+<p><em>type_specifier</em> <em>IDENTIFIER</em><br>
+<em>type_specifier</em> <em>IDENTIFIER</em> <em>array_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>parameter_declaration</em> : </dt>
+<dd>
+<p><em>type_qualifier</em> <em>parameter_declarator</em><br>
+<em>parameter_declarator</em><br>
+<em>type_qualifier</em> <em>parameter_type_specifier</em><br>
+<em>parameter_type_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>parameter_type_specifier</em> : </dt>
+<dd>
+<p><em>type_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>init_declarator_list</em> : </dt>
+<dd>
+<p><em>single_declaration</em><br>
+<em>init_declarator_list</em> <em>COMMA</em> <em>IDENTIFIER</em><br>
+<em>init_declarator_list</em> <em>COMMA</em> <em>IDENTIFIER</em> <em>array_specifier</em><br>
+<em>init_declarator_list</em> <em>COMMA</em> <em>IDENTIFIER</em> <em>array_specifier</em> <em>EQUAL</em>
+<em>initializer</em><br>
+<em>init_declarator_list</em> <em>COMMA</em> <em>IDENTIFIER</em> <em>EQUAL</em> <em>initializer</em></p>
+</dd>
+<dt class="hdlist1"><em>single_declaration</em> : </dt>
+<dd>
+<p><em>fully_specified_type</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em> <em>array_specifier</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em> <em>array_specifier</em> <em>EQUAL</em>
+<em>initializer</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em> <em>EQUAL</em> <em>initializer</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: No 'enum', or 'typedef'.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>fully_specified_type</em> : </dt>
+<dd>
+<p><em>type_specifier</em><br>
+<em>type_qualifier</em> <em>type_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>invariant_qualifier</em> : </dt>
+<dd>
+<p><em>INVARIANT</em></p>
+</dd>
+<dt class="hdlist1"><em>interpolation_qualifier</em> : </dt>
+<dd>
+<p><em>SMOOTH</em><br>
+<em>FLAT</em><br>
+<em>NOPERSPECTIVE</em></p>
+</dd>
+<dt class="hdlist1"><em>layout_qualifier</em> : </dt>
+<dd>
+<p><em>LAYOUT</em> <em>LEFT_PAREN</em> <em>layout_qualifier_id_list</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>layout_qualifier_id_list</em> : </dt>
+<dd>
+<p><em>layout_qualifier_id</em><br>
+<em>layout_qualifier_id_list</em> <em>COMMA</em> <em>layout_qualifier_id</em></p>
+</dd>
+<dt class="hdlist1"><em>layout_qualifier_id</em> : </dt>
+<dd>
+<p><em>IDENTIFIER</em><br>
+<em>IDENTIFIER</em> <em>EQUAL</em> <em>constant_expression</em><br>
+<em>SHARED</em></p>
+</dd>
+<dt class="hdlist1"><em>precise_qualifier</em> : </dt>
+<dd>
+<p><em>PRECISE</em></p>
+</dd>
+<dt class="hdlist1"><em>type_qualifier</em> : </dt>
+<dd>
+<p><em>single_type_qualifier</em><br>
+<em>type_qualifier</em> <em>single_type_qualifier</em></p>
+</dd>
+<dt class="hdlist1"><em>single_type_qualifier</em> : </dt>
+<dd>
+<p><em>storage_qualifier</em><br>
+<em>layout_qualifier</em><br>
+<em>precision_qualifier</em><br>
+<em>interpolation_qualifier</em><br>
+<em>invariant_qualifier</em><br>
+<em>precise_qualifier</em></p>
+</dd>
+<dt class="hdlist1"><em>storage_qualifier</em> : </dt>
+<dd>
+<p><em>CONST</em><br>
+<em>IN</em><br>
+<em>OUT</em><br>
+<em>INOUT</em><br>
+<em>CENTROID</em><br>
+<em>PATCH</em><br>
+<em>SAMPLE</em><br>
+<em>UNIFORM</em><br>
+<em>BUFFER</em><br>
+<em>SHARED</em><br>
+<em>COHERENT</em><br>
+<em>VOLATILE</em><br>
+<em>RESTRICT</em><br>
+<em>READONLY</em><br>
+<em>WRITEONLY</em><br>
+<em>SUBROUTINE</em><br>
+<em>SUBROUTINE</em> <em>LEFT_PAREN</em> <em>type_name_list</em> <em>RIGHT_PAREN</em></p>
+</dd>
+<dt class="hdlist1"><em>type_name_list</em> : </dt>
+<dd>
+<p><em>TYPE_NAME</em><br>
+<em>type_name_list</em> <em>COMMA</em> <em>TYPE_NAME</em></p>
+</dd>
+<dt class="hdlist1"><em>type_specifier</em> : </dt>
+<dd>
+<p><em>type_specifier_nonarray</em><br>
+<em>type_specifier_nonarray</em> <em>array_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>array_specifier</em> : </dt>
+<dd>
+<p><em>LEFT_BRACKET</em> <em>RIGHT_BRACKET</em><br>
+<em>LEFT_BRACKET</em> <em>constant_expression</em> <em>RIGHT_BRACKET</em><br>
+<em>array_specifier</em> <em>LEFT_BRACKET</em> <em>RIGHT_BRACKET</em><br>
+<em>array_specifier</em> <em>LEFT_BRACKET</em> <em>constant_expression</em> <em>RIGHT_BRACKET</em></p>
+</dd>
+<dt class="hdlist1"><em>type_specifier_nonarray</em> : </dt>
+<dd>
+<p><em>VOID</em><br>
+<em>FLOAT</em><br>
+<em>DOUBLE</em><br>
+<em>INT</em><br>
+<em>UINT</em><br>
+<em>BOOL</em><br>
+<em>VEC2</em><br>
+<em>VEC3</em><br>
+<em>VEC4</em><br>
+<em>DVEC2</em><br>
+<em>DVEC3</em><br>
+<em>DVEC4</em><br>
+<em>BVEC2</em><br>
+<em>BVEC3</em><br>
+<em>BVEC4</em><br>
+<em>IVEC2</em><br>
+<em>IVEC3</em><br>
+<em>IVEC4</em><br>
+<em>UVEC2</em><br>
+<em>UVEC3</em><br>
+<em>UVEC4</em><br>
+<em>MAT2</em><br>
+<em>MAT3</em><br>
+<em>MAT4</em><br>
+<em>MAT2X2</em><br>
+<em>MAT2X3</em><br>
+<em>MAT2X4</em><br>
+<em>MAT3X2</em><br>
+<em>MAT3X3</em><br>
+<em>MAT3X4</em><br>
+<em>MAT4X2</em><br>
+<em>MAT4X3</em><br>
+<em>MAT4X4</em><br>
+<em>DMAT2</em><br>
+<em>DMAT3</em><br>
+<em>DMAT4</em><br>
+<em>DMAT2X2</em><br>
+<em>DMAT2X3</em><br>
+<em>DMAT2X4</em><br>
+<em>DMAT3X2</em><br>
+<em>DMAT3X3</em><br>
+<em>DMAT3X4</em><br>
+<em>DMAT4X2</em><br>
+<em>DMAT4X3</em><br>
+<em>DMAT4X4</em><br>
+<em>ATOMIC_UINT</em><br>
+<em>SAMPLER2D</em><br>
+<em>SAMPLER3D</em><br>
+<em>SAMPLERCUBE</em><br>
+<em>SAMPLER2DSHADOW</em><br>
+<em>SAMPLERCUBESHADOW</em><br>
+<em>SAMPLER2DARRAY</em><br>
+<em>SAMPLER2DARRAYSHADOW</em><br>
+<em>SAMPLERCUBEARRAY</em><br>
+<em>SAMPLERCUBEARRAYSHADOW</em><br>
+<em>ISAMPLER2D</em><br>
+<em>ISAMPLER3D</em><br>
+<em>ISAMPLERCUBE</em><br>
+<em>ISAMPLER2DARRAY</em><br>
+<em>ISAMPLERCUBEARRAY</em><br>
+<em>USAMPLER2D</em><br>
+<em>USAMPLER3D</em><br>
+<em>USAMPLERCUBE</em><br>
+<em>USAMPLER2DARRAY</em><br>
+<em>USAMPLERCUBEARRAY</em><br>
+<em>SAMPLER1D</em><br>
+<em>SAMPLER1DSHADOW</em><br>
+<em>SAMPLER1DARRAY</em><br>
+<em>SAMPLER1DARRAYSHADOW</em><br>
+<em>ISAMPLER1D</em><br>
+<em>ISAMPLER1DARRAY</em><br>
+<em>USAMPLER1D</em><br>
+<em>USAMPLER1DARRAY</em><br>
+<em>SAMPLER2DRECT</em><br>
+<em>SAMPLER2DRECTSHADOW</em><br>
+<em>ISAMPLER2DRECT</em><br>
+<em>USAMPLER2DRECT</em><br>
+<em>SAMPLERBUFFER</em><br>
+<em>ISAMPLERBUFFER</em><br>
+<em>USAMPLERBUFFER</em><br>
+<em>SAMPLER2DMS</em><br>
+<em>ISAMPLER2DMS</em><br>
+<em>USAMPLER2DMS</em><br>
+<em>SAMPLER2DMSARRAY</em><br>
+<em>ISAMPLER2DMSARRAY</em><br>
+<em>USAMPLER2DMSARRAY</em><br>
+<em>IMAGE2D</em><br>
+<em>IIMAGE2D</em><br>
+<em>UIMAGE2D</em><br>
+<em>IMAGE3D</em><br>
+<em>IIMAGE3D</em><br>
+<em>UIMAGE3D</em><br>
+<em>IMAGECUBE</em><br>
+<em>IIMAGECUBE</em><br>
+<em>UIMAGECUBE</em><br>
+<em>IMAGEBUFFER</em><br>
+<em>IIMAGEBUFFER</em><br>
+<em>UIMAGEBUFFER</em><br>
+<em>IMAGE1D</em><br>
+<em>IIMAGE1D</em><br>
+<em>UIMAGE1D</em><br>
+<em>IMAGE1DARRAY</em><br>
+<em>IIMAGE1DARRAY</em><br>
+<em>UIMAGE1DARRAY</em><br>
+<em>IMAGE2DRECT</em><br>
+<em>IIMAGE2DRECT</em><br>
+<em>UIMAGE2DRECT</em><br>
+<em>IMAGE2DARRAY</em><br>
+<em>IIMAGE2DARRAY</em><br>
+<em>UIMAGE2DARRAY</em><br>
+<em>IMAGECUBEARRAY</em><br>
+<em>IIMAGECUBEARRAY</em><br>
+<em>UIMAGECUBEARRAY</em><br>
+<em>IMAGE2DMS</em><br>
+<em>IIMAGE2DMS</em><br>
+<em>UIMAGE2DMS</em><br>
+<em>IMAGE2DMSARRAY</em><br>
+<em>IIMAGE2DMSARRAY</em><br>
+<em>UIMAGE2DMSARRAY</em><br>
+<em>struct_specifier</em><br>
+<em>TYPE_NAME</em></p>
+</dd>
+<dt class="hdlist1"><em>precision_qualifier</em> : </dt>
+<dd>
+<p><em>HIGH_PRECISION</em><br>
+<em>MEDIUM_PRECISION</em><br>
+<em>LOW_PRECISION</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_specifier</em> : </dt>
+<dd>
+<p><em>STRUCT</em> <em>IDENTIFIER</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em>
+<em>RIGHT_BRACE</em><br>
+<em>STRUCT</em> <em>LEFT_BRACE</em> <em>struct_declaration_list</em> <em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_declaration_list</em> : </dt>
+<dd>
+<p><em>struct_declaration</em><br>
+<em>struct_declaration_list</em> <em>struct_declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_declaration</em> : </dt>
+<dd>
+<p><em>type_specifier</em> <em>struct_declarator_list</em> <em>SEMICOLON</em><br>
+<em>type_qualifier</em> <em>type_specifier</em> <em>struct_declarator_list</em> <em>SEMICOLON</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_declarator_list</em> : </dt>
+<dd>
+<p><em>struct_declarator</em><br>
+<em>struct_declarator_list</em> <em>COMMA</em> <em>struct_declarator</em></p>
+</dd>
+<dt class="hdlist1"><em>struct_declarator</em> : </dt>
+<dd>
+<p><em>IDENTIFIER</em><br>
+<em>IDENTIFIER</em> <em>array_specifier</em></p>
+</dd>
+<dt class="hdlist1"><em>initializer</em> : </dt>
+<dd>
+<p><em>assignment_expression</em><br>
+<em>LEFT_BRACE</em> <em>initializer_list</em> <em>RIGHT_BRACE</em><br>
+<em>LEFT_BRACE</em> <em>initializer_list</em> <em>COMMA</em> <em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>initializer_list</em> : </dt>
+<dd>
+<p><em>initializer</em><br>
+<em>initializer_list</em> <em>COMMA</em> <em>initializer</em></p>
+</dd>
+<dt class="hdlist1"><em>declaration_statement</em> : </dt>
+<dd>
+<p><em>declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>statement</em> : </dt>
+<dd>
+<p><em>compound_statement</em><br>
+<em>simple_statement</em></p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: labeled statements for SWITCH only; 'goto' is not supported.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>simple_statement</em> : </dt>
+<dd>
+<p><em>declaration_statement</em><br>
+<em>expression_statement</em><br>
+<em>selection_statement</em><br>
+<em>switch_statement</em><br>
+<em>case_label</em><br>
+<em>iteration_statement</em><br>
+<em>jump_statement</em></p>
+</dd>
+<dt class="hdlist1"><em>compound_statement</em> : </dt>
+<dd>
+<p><em>LEFT_BRACE</em> <em>RIGHT_BRACE</em><br>
+<em>LEFT_BRACE</em> <em>statement_list</em> <em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>statement_no_new_scope</em> : </dt>
+<dd>
+<p><em>compound_statement_no_new_scope</em><br>
+<em>simple_statement</em></p>
+</dd>
+<dt class="hdlist1"><em>compound_statement_no_new_scope</em> : </dt>
+<dd>
+<p><em>LEFT_BRACE</em> <em>RIGHT_BRACE</em><br>
+<em>LEFT_BRACE</em> <em>statement_list</em> <em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>statement_list</em> : </dt>
+<dd>
+<p><em>statement</em><br>
+<em>statement_list</em> <em>statement</em></p>
+</dd>
+<dt class="hdlist1"><em>expression_statement</em> : </dt>
+<dd>
+<p><em>SEMICOLON</em><br>
+<em>expression</em> <em>SEMICOLON</em></p>
+</dd>
+<dt class="hdlist1"><em>selection_statement</em> : </dt>
+<dd>
+<p><em>IF</em> <em>LEFT_PAREN</em> <em>expression</em> <em>RIGHT_PAREN</em> <em>selection_rest_statement</em></p>
+</dd>
+<dt class="hdlist1"><em>selection_rest_statement</em> : </dt>
+<dd>
+<p><em>statement</em> <em>ELSE</em> <em>statement</em><br>
+<em>statement</em></p>
+</dd>
+<dt class="hdlist1"><em>condition</em> : </dt>
+<dd>
+<p><em>expression</em><br>
+<em>fully_specified_type</em> <em>IDENTIFIER</em> <em>EQUAL</em> <em>initializer</em></p>
+</dd>
+<dt class="hdlist1"><em>switch_statement</em> : </dt>
+<dd>
+<p><em>SWITCH</em> <em>LEFT_PAREN</em> <em>expression</em> <em>RIGHT_PAREN</em> <em>LEFT_BRACE</em>
+<em>switch_statement_list</em><br>
+<em>RIGHT_BRACE</em></p>
+</dd>
+<dt class="hdlist1"><em>switch_statement_list</em> : </dt>
+<dd>
+<p>/* <em>nothing</em> */<br>
+<em>statement_list</em></p>
+</dd>
+<dt class="hdlist1"><em>case_label</em> : </dt>
+<dd>
+<p><em>CASE</em> <em>expression</em> <em>COLON</em><br>
+<em>DEFAULT</em> <em>COLON</em></p>
+</dd>
+<dt class="hdlist1"><em>iteration_statement</em> : </dt>
+<dd>
+<p><em>WHILE</em> <em>LEFT_PAREN</em> <em>condition</em> <em>RIGHT_PAREN</em> <em>statement_no_new_scope</em><br>
+<em>DO</em> <em>statement</em> <em>WHILE</em> <em>LEFT_PAREN</em> <em>expression</em> <em>RIGHT_PAREN</em>
+<em>SEMICOLON</em><br>
+<em>FOR</em> <em>LEFT_PAREN</em> <em>for_init_statement</em> <em>for_rest_statement</em>
+<em>RIGHT_PAREN</em> <em>statement_no_new_scope</em></p>
+</dd>
+<dt class="hdlist1"><em>for_init_statement</em> : </dt>
+<dd>
+<p><em>expression_statement</em><br>
+<em>declaration_statement</em></p>
+</dd>
+<dt class="hdlist1"><em>conditionopt</em> : </dt>
+<dd>
+<p><em>condition</em><br>
+/* <em>empty</em> */</p>
+</dd>
+<dt class="hdlist1"><em>for_rest_statement</em> : </dt>
+<dd>
+<p><em>conditionopt</em> <em>SEMICOLON</em><br>
+<em>conditionopt</em> <em>SEMICOLON</em> <em>expression</em></p>
+</dd>
+<dt class="hdlist1"><em>jump_statement</em> : </dt>
+<dd>
+<p><em>CONTINUE</em> <em>SEMICOLON</em><br>
+<em>BREAK</em> <em>SEMICOLON</em><br>
+<em>RETURN</em> <em>SEMICOLON</em><br>
+<em>RETURN</em> <em>expression</em> <em>SEMICOLON</em><br>
+<em>DISCARD</em> <em>SEMICOLON</em> // Fragment shader only.</p>
+</dd>
+</dl>
+</div>
+<div class="admonitionblock note">
+<table>
+<tr>
+<td class="icon">
+<i class="fa icon-note" title="Note"></i>
+</td>
+<td class="content">
+<div class="paragraph">
+<p>Grammar Note: No 'goto'.
+Gotos are not supported.</p>
+</div>
+</td>
+</tr>
+</table>
+</div>
+<div class="dlist">
+<dl>
+<dt class="hdlist1"><em>translation_unit</em> : </dt>
+<dd>
+<p><em>external_declaration</em><br>
+<em>translation_unit</em> <em>external_declaration</em></p>
+</dd>
+<dt class="hdlist1"><em>external_declaration</em> : </dt>
+<dd>
+<p><em>function_definition</em><br>
+<em>declaration</em><br>
+<em>SEMICOLON</em></p>
+</dd>
+<dt class="hdlist1"><em>function_definition</em> : </dt>
+<dd>
+<p><em>function_prototype</em> <em>compound_statement_no_new_scope</em></p>
+</dd>
+</dl>
+</div>
+</div>
+</div>
+<div class="paragraph">
+<p>In general the above grammar describes a super set of the OpenGL Shading Language.
+Certain constructs that are valid purely in terms of the grammar are
+disallowed by statements elsewhere in this specification.</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="acknowledgments">10. Acknowledgments</h2>
+<div class="sectionbody">
+<div class="paragraph">
+<p>This specification is based on the work of those who contributed to past
+versions of the Open GL and Open GL ES Language Specifications and the
+following contributors to this version:</p>
+</div>
+<div class="paragraph">
+<p>Pat Brown, NVIDIA<br>
+Jeff Bolz, NVIDIA<br>
+Frank Chen<br>
+Pierre Boudier, AMD<br>
+Piers Daniell, NVIDIA<br>
+Chris Dodd, NVIDIA<br>
+Nick Haemel, NVIDIA<br>
+Jason Green, Transgaming<br>
+Brent Insko, Intel<br>
+Jon Leech<br>
+Bill Licea-Kane, Qualcomm<br>
+Daniel Koch, NVIDIA<br>
+Graeme Leese, Broadcom<br>
+Barthold Lichtenbelt, NVIDIA<br>
+Bruce Merry, ARM<br>
+Robert Ohannessian<br>
+Tom Olson, ARM<br>
+Brian Paul, VMware<br>
+Acorn Pooley, NVIDIA<br>
+Daniel Rakos, AMD<br>
+Christophe Riccio, AMD<br>
+Kevin Rogovin<br>
+Ian Romanick, Intel<br>
+Greg Roth, NVIDIA<br>
+Graham Sellers, AMD<br>
+Dave Shreiner, ARM<br>
+Jeremy Sandmel, Apple<br>
+Robert Simpson, Qualcomm<br>
+Eric Werness, NVIDIA<br>
+Mark Young, AMD</p>
+</div>
+</div>
+</div>
+<div class="sect1">
+<h2 id="references">11. Normative References</h2>
+<div class="sectionbody">
+<div class="olist arabic">
+<ol class="arabic">
+<li>
+<p>International Standard ISO/IEC 14882:1998(E).
+Programming Languages - C++. Referenced for preprocessor only</p>
+</li>
+<li>
+<p>IEEE 754-2008.
+<em>IEEE Standard for Floating-Point Arithmetic</em></p>
+</li>
+</ol>
+</div>
+</div>
+</div>
+</div>
+<div id="footer">
+<div id="footer-text">
+Version 4.60.6<br>
+Last updated 2018-12-12 16:37:44 MST
+</div>
+</div>
+
+<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-MML-AM_HTMLorMML"></script>
+</body>
+</html>
\ No newline at end of file
diff --git a/specs/gl/GLSLangSpec.4.60.pdf b/specs/gl/GLSLangSpec.4.60.pdf
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+++ b/specs/gl/GLSLangSpec.4.60.pdf
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diff --git a/specs/gl/glspec46.compatibility.pdf b/specs/gl/glspec46.compatibility.pdf
index 9f09b25..467fd68 100644
--- a/specs/gl/glspec46.compatibility.pdf
+++ b/specs/gl/glspec46.compatibility.pdf
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diff --git a/specs/gl/glspec46.compatibility.withchanges.pdf b/specs/gl/glspec46.compatibility.withchanges.pdf
index 5344bbc..b682eab 100644
--- a/specs/gl/glspec46.compatibility.withchanges.pdf
+++ b/specs/gl/glspec46.compatibility.withchanges.pdf
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diff --git a/specs/gl/glspec46.core.pdf b/specs/gl/glspec46.core.pdf
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diff --git a/specs/gl/glspec46.core.withchanges.pdf b/specs/gl/glspec46.core.withchanges.pdf
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diff --git a/specs/gl/katex/README.md b/specs/gl/katex/README.md
new file mode 100644
index 0000000..5f8caa8
--- /dev/null
+++ b/specs/gl/katex/README.md
@@ -0,0 +1,68 @@
+# [<img src="https://khan.github.io/KaTeX/katex-logo.svg" width="130" alt="KaTeX">](https://khan.github.io/KaTeX/) [](https://travis-ci.org/Khan/KaTeX)
+
+[](https://gitter.im/Khan/KaTeX?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge)
+
+KaTeX is a fast, easy-to-use JavaScript library for TeX math rendering on the web.
+
+ * **Fast:** KaTeX renders its math synchronously and doesn't need to reflow the page. See how it compares to a competitor in [this speed test](http://jsperf.com/katex-vs-mathjax/).
+ * **Print quality:** KaTeX’s layout is based on Donald Knuth’s TeX, the gold standard for math typesetting.
+ * **Self contained:** KaTeX has no dependencies and can easily be bundled with your website resources.
+ * **Server side rendering:** KaTeX produces the same output regardless of browser or environment, so you can pre-render expressions using Node.js and send them as plain HTML.
+
+KaTeX supports all major browsers, including Chrome, Safari, Firefox, Opera, and IE 8 - IE 11. A list of supported commands can be on the [wiki](https://github.com/Khan/KaTeX/wiki/Function-Support-in-KaTeX).
+
+## Usage
+
+You can [download KaTeX](https://github.com/khan/katex/releases) and host it on your server or include the `katex.min.js` and `katex.min.css` files on your page directly from a CDN:
+
+```html
+<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.6.0/katex.min.css">
+<script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.6.0/katex.min.js"></script>
+```
+
+#### In-browser rendering
+
+Call `katex.render` with a TeX expression and a DOM element to render into:
+
+```js
+katex.render("c = \\pm\\sqrt{a^2 + b^2}", element);
+```
+
+If KaTeX can't parse the expression, it throws a `katex.ParseError` error.
+
+#### Server side rendering or rendering to a string
+
+To generate HTML on the server or to generate an HTML string of the rendered math, you can use `katex.renderToString`:
+
+```js
+var html = katex.renderToString("c = \\pm\\sqrt{a^2 + b^2}");
+// '<span class="katex">...</span>'
+```
+
+Make sure to include the CSS and font files, but there is no need to include the JavaScript. Like `render`, `renderToString` throws if it can't parse the expression.
+
+#### Rendering options
+
+You can provide an object of options as the last argument to `katex.render` and `katex.renderToString`. Available options are:
+
+- `displayMode`: `boolean`. If `true` the math will be rendered in display mode, which will put the math in display style (so `\int` and `\sum` are large, for example), and will center the math on the page on its own line. If `false` the math will be rendered in inline mode. (default: `false`)
+- `throwOnError`: `boolean`. If `true`, KaTeX will throw a `ParseError` when it encounters an unsupported command. If `false`, KaTeX will render the unsupported command as text in the color given by `errorColor`. (default: `true`)
+- `errorColor`: `string`. A color string given in the format `"#XXX"` or `"#XXXXXX"`. This option determines the color which unsupported commands are rendered in. (default: `#cc0000`)
+
+For example:
+
+```js
+katex.render("c = \\pm\\sqrt{a^2 + b^2}", element, { displayMode: true });
+```
+
+#### Automatic rendering of math on a page
+
+Math on the page can be automatically rendered using the auto-render extension. See [the Auto-render README](contrib/auto-render/README.md) for more information.
+
+## Contributing
+
+See [CONTRIBUTING.md](CONTRIBUTING.md)
+
+## License
+
+KaTeX is licensed under the [MIT License](http://opensource.org/licenses/MIT).
diff --git a/specs/gl/katex/contrib/auto-render.min.js b/specs/gl/katex/contrib/auto-render.min.js
new file mode 100644
index 0000000..30cc312
--- /dev/null
+++ b/specs/gl/katex/contrib/auto-render.min.js
@@ -0,0 +1 @@
+(function(e){if(typeof exports==="object"&&typeof module!=="undefined"){module.exports=e()}else if(typeof define==="function"&&define.amd){define([],e)}else{var t;if(typeof window!=="undefined"){t=window}else if(typeof global!=="undefined"){t=global}else if(typeof self!=="undefined"){t=self}else{t=this}t.renderMathInElement=e()}})(function(){var e,t,r;return function n(e,t,r){function a(o,l){if(!t[o]){if(!e[o]){var f=typeof require=="function"&&require;if(!l&&f)return f(o,!0);if(i)return i(o,!0);var d=new Error("Cannot find module '"+o+"'");throw d.code="MODULE_NOT_FOUND",d}var s=t[o]={exports:{}};e[o][0].call(s.exports,function(t){var r=e[o][1][t];return a(r?r:t)},s,s.exports,n,e,t,r)}return t[o].exports}var i=typeof require=="function"&&require;for(var o=0;o<r.length;o++)a(r[o]);return a}({1:[function(e,t,r){var n=e("./splitAtDelimiters");var a=function(e,t){var r=[{type:"text",data:e}];for(var a=0;a<t.length;a++){var i=t[a];r=n(r,i.left,i.right,i.display||false)}return r};var i=function(e,t){var r=a(e,t);var n=document.createDocumentFragment();for(var i=0;i<r.length;i++){if(r[i].type==="text"){n.appendChild(document.createTextNode(r[i].data))}else{var o=document.createElement("span");var l=r[i].data;try{katex.render(l,o,{displayMode:r[i].display})}catch(f){if(!(f instanceof katex.ParseError)){throw f}console.error("KaTeX auto-render: Failed to parse `"+r[i].data+"` with ",f);n.appendChild(document.createTextNode(r[i].rawData));continue}n.appendChild(o)}}return n};var o=function(e,t,r){for(var n=0;n<e.childNodes.length;n++){var a=e.childNodes[n];if(a.nodeType===3){var l=i(a.textContent,t);n+=l.childNodes.length-1;e.replaceChild(l,a)}else if(a.nodeType===1){var f=r.indexOf(a.nodeName.toLowerCase())===-1;if(f){o(a,t,r)}}}};var l={delimiters:[{left:"$$",right:"$$",display:true},{left:"\\[",right:"\\]",display:true},{left:"\\(",right:"\\)",display:false}],ignoredTags:["script","noscript","style","textarea","pre","code"]};var f=function(e){var t;var r;for(var n=1,a=arguments.length;n<a;n++){t=arguments[n];for(r in t){if(Object.prototype.hasOwnProperty.call(t,r)){e[r]=t[r]}}}return e};var d=function(e,t){if(!e){throw new Error("No element provided to render")}t=f({},l,t);o(e,t.delimiters,t.ignoredTags)};t.exports=d},{"./splitAtDelimiters":2}],2:[function(e,t,r){var n=function(e,t,r){var n=r;var a=0;var i=e.length;while(n<t.length){var o=t[n];if(a<=0&&t.slice(n,n+i)===e){return n}else if(o==="\\"){n++}else if(o==="{"){a++}else if(o==="}"){a--}n++}return-1};var a=function(e,t,r,a){var i=[];for(var o=0;o<e.length;o++){if(e[o].type==="text"){var l=e[o].data;var f=true;var d=0;var s;s=l.indexOf(t);if(s!==-1){d=s;i.push({type:"text",data:l.slice(0,d)});f=false}while(true){if(f){s=l.indexOf(t,d);if(s===-1){break}i.push({type:"text",data:l.slice(d,s)});d=s}else{s=n(r,l,d+t.length);if(s===-1){break}i.push({type:"math",data:l.slice(d+t.length,s),rawData:l.slice(d,s+r.length),display:a});d=s+r.length}f=!f}i.push({type:"text",data:l.slice(d)})}else{i.push(e[o])}}return i};t.exports=a},{}]},{},[1])(1)});
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@@ -0,0 +1,976 @@
+@font-face {
+ font-family: 'KaTeX_AMS';
+ src: url('fonts/KaTeX_AMS-Regular.eot');
+ src: url('fonts/KaTeX_AMS-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_AMS-Regular.woff2') format('woff2'), url('fonts/KaTeX_AMS-Regular.woff') format('woff'), url('fonts/KaTeX_AMS-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Caligraphic';
+ src: url('fonts/KaTeX_Caligraphic-Bold.eot');
+ src: url('fonts/KaTeX_Caligraphic-Bold.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Caligraphic-Bold.woff2') format('woff2'), url('fonts/KaTeX_Caligraphic-Bold.woff') format('woff'), url('fonts/KaTeX_Caligraphic-Bold.ttf') format('truetype');
+ font-weight: bold;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Caligraphic';
+ src: url('fonts/KaTeX_Caligraphic-Regular.eot');
+ src: url('fonts/KaTeX_Caligraphic-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Caligraphic-Regular.woff2') format('woff2'), url('fonts/KaTeX_Caligraphic-Regular.woff') format('woff'), url('fonts/KaTeX_Caligraphic-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Fraktur';
+ src: url('fonts/KaTeX_Fraktur-Bold.eot');
+ src: url('fonts/KaTeX_Fraktur-Bold.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Fraktur-Bold.woff2') format('woff2'), url('fonts/KaTeX_Fraktur-Bold.woff') format('woff'), url('fonts/KaTeX_Fraktur-Bold.ttf') format('truetype');
+ font-weight: bold;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Fraktur';
+ src: url('fonts/KaTeX_Fraktur-Regular.eot');
+ src: url('fonts/KaTeX_Fraktur-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Fraktur-Regular.woff2') format('woff2'), url('fonts/KaTeX_Fraktur-Regular.woff') format('woff'), url('fonts/KaTeX_Fraktur-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Main';
+ src: url('fonts/KaTeX_Main-Bold.eot');
+ src: url('fonts/KaTeX_Main-Bold.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Main-Bold.woff2') format('woff2'), url('fonts/KaTeX_Main-Bold.woff') format('woff'), url('fonts/KaTeX_Main-Bold.ttf') format('truetype');
+ font-weight: bold;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Main';
+ src: url('fonts/KaTeX_Main-Italic.eot');
+ src: url('fonts/KaTeX_Main-Italic.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Main-Italic.woff2') format('woff2'), url('fonts/KaTeX_Main-Italic.woff') format('woff'), url('fonts/KaTeX_Main-Italic.ttf') format('truetype');
+ font-weight: normal;
+ font-style: italic;
+}
+@font-face {
+ font-family: 'KaTeX_Main';
+ src: url('fonts/KaTeX_Main-Regular.eot');
+ src: url('fonts/KaTeX_Main-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Main-Regular.woff2') format('woff2'), url('fonts/KaTeX_Main-Regular.woff') format('woff'), url('fonts/KaTeX_Main-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Math';
+ src: url('fonts/KaTeX_Math-Italic.eot');
+ src: url('fonts/KaTeX_Math-Italic.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Math-Italic.woff2') format('woff2'), url('fonts/KaTeX_Math-Italic.woff') format('woff'), url('fonts/KaTeX_Math-Italic.ttf') format('truetype');
+ font-weight: normal;
+ font-style: italic;
+}
+@font-face {
+ font-family: 'KaTeX_SansSerif';
+ src: url('fonts/KaTeX_SansSerif-Regular.eot');
+ src: url('fonts/KaTeX_SansSerif-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_SansSerif-Regular.woff2') format('woff2'), url('fonts/KaTeX_SansSerif-Regular.woff') format('woff'), url('fonts/KaTeX_SansSerif-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Script';
+ src: url('fonts/KaTeX_Script-Regular.eot');
+ src: url('fonts/KaTeX_Script-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Script-Regular.woff2') format('woff2'), url('fonts/KaTeX_Script-Regular.woff') format('woff'), url('fonts/KaTeX_Script-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Size1';
+ src: url('fonts/KaTeX_Size1-Regular.eot');
+ src: url('fonts/KaTeX_Size1-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Size1-Regular.woff2') format('woff2'), url('fonts/KaTeX_Size1-Regular.woff') format('woff'), url('fonts/KaTeX_Size1-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Size2';
+ src: url('fonts/KaTeX_Size2-Regular.eot');
+ src: url('fonts/KaTeX_Size2-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Size2-Regular.woff2') format('woff2'), url('fonts/KaTeX_Size2-Regular.woff') format('woff'), url('fonts/KaTeX_Size2-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Size3';
+ src: url('fonts/KaTeX_Size3-Regular.eot');
+ src: url('fonts/KaTeX_Size3-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Size3-Regular.woff2') format('woff2'), url('fonts/KaTeX_Size3-Regular.woff') format('woff'), url('fonts/KaTeX_Size3-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Size4';
+ src: url('fonts/KaTeX_Size4-Regular.eot');
+ src: url('fonts/KaTeX_Size4-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Size4-Regular.woff2') format('woff2'), url('fonts/KaTeX_Size4-Regular.woff') format('woff'), url('fonts/KaTeX_Size4-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+@font-face {
+ font-family: 'KaTeX_Typewriter';
+ src: url('fonts/KaTeX_Typewriter-Regular.eot');
+ src: url('fonts/KaTeX_Typewriter-Regular.eot#iefix') format('embedded-opentype'), url('fonts/KaTeX_Typewriter-Regular.woff2') format('woff2'), url('fonts/KaTeX_Typewriter-Regular.woff') format('woff'), url('fonts/KaTeX_Typewriter-Regular.ttf') format('truetype');
+ font-weight: normal;
+ font-style: normal;
+}
+.katex-display {
+ display: block;
+ margin: 1em 0;
+ text-align: center;
+}
+.katex-display > .katex {
+ display: inline-block;
+ text-align: initial;
+}
+.katex {
+ font: normal 1.21em KaTeX_Main, Times New Roman, serif;
+ line-height: 1.2;
+ white-space: nowrap;
+ text-indent: 0;
+}
+.katex .katex-html {
+ display: inline-block;
+}
+.katex .katex-mathml {
+ position: absolute;
+ clip: rect(1px, 1px, 1px, 1px);
+ padding: 0;
+ border: 0;
+ height: 1px;
+ width: 1px;
+ overflow: hidden;
+}
+.katex .base {
+ display: inline-block;
+}
+.katex .strut {
+ display: inline-block;
+}
+.katex .mathrm {
+ font-style: normal;
+}
+.katex .textit {
+ font-style: italic;
+}
+.katex .mathit {
+ font-family: KaTeX_Math;
+ font-style: italic;
+}
+.katex .mathbf {
+ font-family: KaTeX_Main;
+ font-weight: bold;
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+}
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+.katex .fontsize-ensurer.reset-size9.size9 {
+ font-size: 1em;
+}
+.katex .sizing.reset-size9.size10,
+.katex .fontsize-ensurer.reset-size9.size10 {
+ font-size: 1.20289855em;
+}
+.katex .sizing.reset-size10.size1,
+.katex .fontsize-ensurer.reset-size10.size1 {
+ font-size: 0.20080321em;
+}
+.katex .sizing.reset-size10.size2,
+.katex .fontsize-ensurer.reset-size10.size2 {
+ font-size: 0.2811245em;
+}
+.katex .sizing.reset-size10.size3,
+.katex .fontsize-ensurer.reset-size10.size3 {
+ font-size: 0.32128514em;
+}
+.katex .sizing.reset-size10.size4,
+.katex .fontsize-ensurer.reset-size10.size4 {
+ font-size: 0.36144578em;
+}
+.katex .sizing.reset-size10.size5,
+.katex .fontsize-ensurer.reset-size10.size5 {
+ font-size: 0.40160643em;
+}
+.katex .sizing.reset-size10.size6,
+.katex .fontsize-ensurer.reset-size10.size6 {
+ font-size: 0.48192771em;
+}
+.katex .sizing.reset-size10.size7,
+.katex .fontsize-ensurer.reset-size10.size7 {
+ font-size: 0.57831325em;
+}
+.katex .sizing.reset-size10.size8,
+.katex .fontsize-ensurer.reset-size10.size8 {
+ font-size: 0.69477912em;
+}
+.katex .sizing.reset-size10.size9,
+.katex .fontsize-ensurer.reset-size10.size9 {
+ font-size: 0.8313253em;
+}
+.katex .sizing.reset-size10.size10,
+.katex .fontsize-ensurer.reset-size10.size10 {
+ font-size: 1em;
+}
+.katex .delimsizing.size1 {
+ font-family: KaTeX_Size1;
+}
+.katex .delimsizing.size2 {
+ font-family: KaTeX_Size2;
+}
+.katex .delimsizing.size3 {
+ font-family: KaTeX_Size3;
+}
+.katex .delimsizing.size4 {
+ font-family: KaTeX_Size4;
+}
+.katex .delimsizing.mult .delim-size1 > span {
+ font-family: KaTeX_Size1;
+}
+.katex .delimsizing.mult .delim-size4 > span {
+ font-family: KaTeX_Size4;
+}
+.katex .nulldelimiter {
+ display: inline-block;
+ width: 0.12em;
+}
+.katex .op-symbol {
+ position: relative;
+}
+.katex .op-symbol.small-op {
+ font-family: KaTeX_Size1;
+}
+.katex .op-symbol.large-op {
+ font-family: KaTeX_Size2;
+}
+.katex .op-limits > .vlist > span {
+ text-align: center;
+}
+.katex .accent > .vlist > span {
+ text-align: center;
+}
+.katex .accent .accent-body > span {
+ width: 0;
+}
+.katex .accent .accent-body.accent-vec > span {
+ position: relative;
+ left: 0.326em;
+}
+.katex .mtable .vertical-separator {
+ display: inline-block;
+ margin: 0 -0.025em;
+ border-right: 0.05em solid black;
+}
+.katex .mtable .arraycolsep {
+ display: inline-block;
+}
+.katex .mtable .col-align-c > .vlist {
+ text-align: center;
+}
+.katex .mtable .col-align-l > .vlist {
+ text-align: left;
+}
+.katex .mtable .col-align-r > .vlist {
+ text-align: right;
+}
diff --git a/specs/gl/katex/katex.js b/specs/gl/katex/katex.js
new file mode 100644
index 0000000..e104be9
--- /dev/null
+++ b/specs/gl/katex/katex.js
@@ -0,0 +1,9075 @@
+(function(f){if(typeof exports==="object"&&typeof module!=="undefined"){module.exports=f()}else if(typeof define==="function"&&define.amd){define([],f)}else{var g;if(typeof window!=="undefined"){g=window}else if(typeof global!=="undefined"){g=global}else if(typeof self!=="undefined"){g=self}else{g=this}g.katex = f()}})(function(){var define,module,exports;return (function e(t,n,r){function s(o,u){if(!n[o]){if(!t[o]){var a=typeof require=="function"&&require;if(!u&&a)return a(o,!0);if(i)return i(o,!0);var f=new Error("Cannot find module '"+o+"'");throw f.code="MODULE_NOT_FOUND",f}var l=n[o]={exports:{}};t[o][0].call(l.exports,function(e){var n=t[o][1][e];return s(n?n:e)},l,l.exports,e,t,n,r)}return n[o].exports}var i=typeof require=="function"&&require;for(var o=0;o<r.length;o++)s(r[o]);return s})({1:[function(require,module,exports){
+/* eslint no-console:0 */
+/**
+ * This is the main entry point for KaTeX. Here, we expose functions for
+ * rendering expressions either to DOM nodes or to markup strings.
+ *
+ * We also expose the ParseError class to check if errors thrown from KaTeX are
+ * errors in the expression, or errors in javascript handling.
+ */
+
+var ParseError = require("./src/ParseError");
+var Settings = require("./src/Settings");
+
+var buildTree = require("./src/buildTree");
+var parseTree = require("./src/parseTree");
+var utils = require("./src/utils");
+
+/**
+ * Parse and build an expression, and place that expression in the DOM node
+ * given.
+ */
+var render = function(expression, baseNode, options) {
+ utils.clearNode(baseNode);
+
+ var settings = new Settings(options);
+
+ var tree = parseTree(expression, settings);
+ var node = buildTree(tree, expression, settings).toNode();
+
+ baseNode.appendChild(node);
+};
+
+// KaTeX's styles don't work properly in quirks mode. Print out an error, and
+// disable rendering.
+if (typeof document !== "undefined") {
+ if (document.compatMode !== "CSS1Compat") {
+ typeof console !== "undefined" && console.warn(
+ "Warning: KaTeX doesn't work in quirks mode. Make sure your " +
+ "website has a suitable doctype.");
+
+ render = function() {
+ throw new ParseError("KaTeX doesn't work in quirks mode.");
+ };
+ }
+}
+
+/**
+ * Parse and build an expression, and return the markup for that.
+ */
+var renderToString = function(expression, options) {
+ var settings = new Settings(options);
+
+ var tree = parseTree(expression, settings);
+ return buildTree(tree, expression, settings).toMarkup();
+};
+
+/**
+ * Parse an expression and return the parse tree.
+ */
+var generateParseTree = function(expression, options) {
+ var settings = new Settings(options);
+ return parseTree(expression, settings);
+};
+
+module.exports = {
+ render: render,
+ renderToString: renderToString,
+ /**
+ * NOTE: This method is not currently recommended for public use.
+ * The internal tree representation is unstable and is very likely
+ * to change. Use at your own risk.
+ */
+ __parse: generateParseTree,
+ ParseError: ParseError
+};
+
+},{"./src/ParseError":6,"./src/Settings":8,"./src/buildTree":13,"./src/parseTree":22,"./src/utils":25}],2:[function(require,module,exports){
+/** @flow */
+
+"use strict";
+
+function getRelocatable(re) {
+ // In the future, this could use a WeakMap instead of an expando.
+ if (!re.__matchAtRelocatable) {
+ // Disjunctions are the lowest-precedence operator, so we can make any
+ // pattern match the empty string by appending `|()` to it:
+ // https://people.mozilla.org/~jorendorff/es6-draft.html#sec-patterns
+ var source = re.source + "|()";
+
+ // We always make the new regex global.
+ var flags = "g" + (re.ignoreCase ? "i" : "") + (re.multiline ? "m" : "") + (re.unicode ? "u" : "")
+ // sticky (/.../y) doesn't make sense in conjunction with our relocation
+ // logic, so we ignore it here.
+ ;
+
+ re.__matchAtRelocatable = new RegExp(source, flags);
+ }
+ return re.__matchAtRelocatable;
+}
+
+function matchAt(re, str, pos) {
+ if (re.global || re.sticky) {
+ throw new Error("matchAt(...): Only non-global regexes are supported");
+ }
+ var reloc = getRelocatable(re);
+ reloc.lastIndex = pos;
+ var match = reloc.exec(str);
+ // Last capturing group is our sentinel that indicates whether the regex
+ // matched at the given location.
+ if (match[match.length - 1] == null) {
+ // Original regex matched.
+ match.length = match.length - 1;
+ return match;
+ } else {
+ return null;
+ }
+}
+
+module.exports = matchAt;
+},{}],3:[function(require,module,exports){
+/**
+ * The Lexer class handles tokenizing the input in various ways. Since our
+ * parser expects us to be able to backtrack, the lexer allows lexing from any
+ * given starting point.
+ *
+ * Its main exposed function is the `lex` function, which takes a position to
+ * lex from and a type of token to lex. It defers to the appropriate `_innerLex`
+ * function.
+ *
+ * The various `_innerLex` functions perform the actual lexing of different
+ * kinds.
+ */
+
+var matchAt = require("match-at");
+
+var ParseError = require("./ParseError");
+
+// The main lexer class
+function Lexer(input) {
+ this.input = input;
+ this.pos = 0;
+}
+
+/**
+ * The resulting token returned from `lex`.
+ *
+ * It consists of the token text plus some position information.
+ * The position information is essentially a range in an input string,
+ * but instead of referencing the bare input string, we refer to the lexer.
+ * That way it is possible to attach extra metadata to the input string,
+ * like for example a file name or similar.
+ *
+ * The position information (all three parameters) is optional,
+ * so it is OK to construct synthetic tokens if appropriate.
+ * Not providing available position information may lead to
+ * degraded error reporting, though.
+ *
+ * @param {string} text the text of this token
+ * @param {number=} start the start offset, zero-based inclusive
+ * @param {number=} end the end offset, zero-based exclusive
+ * @param {Lexer=} lexer the lexer which in turn holds the input string
+ */
+function Token(text, start, end, lexer) {
+ this.text = text;
+ this.start = start;
+ this.end = end;
+ this.lexer = lexer;
+}
+
+/**
+ * Given a pair of tokens (this and endToken), compute a “Token” encompassing
+ * the whole input range enclosed by these two.
+ *
+ * @param {Token} endToken last token of the range, inclusive
+ * @param {string} text the text of the newly constructed token
+ */
+Token.prototype.range = function(endToken, text) {
+ if (endToken.lexer !== this.lexer) {
+ return new Token(text); // sorry, no position information available
+ }
+ return new Token(text, this.start, endToken.end, this.lexer);
+};
+
+/* The following tokenRegex
+ * - matches typical whitespace (but not NBSP etc.) using its first group
+ * - does not match any control character \x00-\x1f except whitespace
+ * - does not match a bare backslash
+ * - matches any ASCII character except those just mentioned
+ * - does not match the BMP private use area \uE000-\uF8FF
+ * - does not match bare surrogate code units
+ * - matches any BMP character except for those just described
+ * - matches any valid Unicode surrogate pair
+ * - matches a backslash followed by one or more letters
+ * - matches a backslash followed by any BMP character, including newline
+ * Just because the Lexer matches something doesn't mean it's valid input:
+ * If there is no matching function or symbol definition, the Parser will
+ * still reject the input.
+ */
+var tokenRegex = new RegExp(
+ "([ \r\n\t]+)|" + // whitespace
+ "([!-\\[\\]-\u2027\u202A-\uD7FF\uF900-\uFFFF]" + // single codepoint
+ "|[\uD800-\uDBFF][\uDC00-\uDFFF]" + // surrogate pair
+ "|\\\\(?:[a-zA-Z]+|[^\uD800-\uDFFF])" + // function name
+ ")"
+);
+
+/**
+ * This function lexes a single token.
+ */
+Lexer.prototype.lex = function() {
+ var input = this.input;
+ var pos = this.pos;
+ if (pos === input.length) {
+ return new Token("EOF", pos, pos, this);
+ }
+ var match = matchAt(tokenRegex, input, pos);
+ if (match === null) {
+ throw new ParseError(
+ "Unexpected character: '" + input[pos] + "'",
+ new Token(input[pos], pos, pos + 1, this));
+ }
+ var text = match[2] || " ";
+ var start = this.pos;
+ this.pos += match[0].length;
+ var end = this.pos;
+ return new Token(text, start, end, this);
+};
+
+module.exports = Lexer;
+
+},{"./ParseError":6,"match-at":2}],4:[function(require,module,exports){
+/**
+ * This file contains the “gullet” where macros are expanded
+ * until only non-macro tokens remain.
+ */
+
+var Lexer = require("./Lexer");
+
+function MacroExpander(input, macros) {
+ this.lexer = new Lexer(input);
+ this.macros = macros;
+ this.stack = []; // contains tokens in REVERSE order
+ this.discardedWhiteSpace = [];
+}
+
+/**
+ * Recursively expand first token, then return first non-expandable token.
+ */
+MacroExpander.prototype.nextToken = function() {
+ for (;;) {
+ if (this.stack.length === 0) {
+ this.stack.push(this.lexer.lex());
+ }
+ var topToken = this.stack.pop();
+ var name = topToken.text;
+ if (!(name.charAt(0) === "\\" && this.macros.hasOwnProperty(name))) {
+ return topToken;
+ }
+ var expansion = this.macros[name];
+ if (typeof expansion === "string") {
+ var bodyLexer = new Lexer(expansion);
+ expansion = [];
+ var tok = bodyLexer.lex();
+ while (tok.text !== "EOF") {
+ expansion.push(tok);
+ tok = bodyLexer.lex();
+ }
+ expansion.reverse(); // to fit in with stack using push and pop
+ this.macros[name] = expansion;
+ }
+ this.stack = this.stack.concat(expansion);
+ }
+};
+
+MacroExpander.prototype.get = function(ignoreSpace) {
+ this.discardedWhiteSpace = [];
+ var token = this.nextToken();
+ if (ignoreSpace) {
+ while (token.text === " ") {
+ this.discardedWhiteSpace.push(token);
+ token = this.nextToken();
+ }
+ }
+ return token;
+};
+
+/**
+ * Undo the effect of the preceding call to the get method.
+ * A call to this method MUST be immediately preceded and immediately followed
+ * by a call to get. Only used during mode switching, i.e. after one token
+ * was got in the old mode but should get got again in a new mode
+ * with possibly different whitespace handling.
+ */
+MacroExpander.prototype.unget = function(token) {
+ this.stack.push(token);
+ while (this.discardedWhiteSpace.length !== 0) {
+ this.stack.push(this.discardedWhiteSpace.pop());
+ }
+};
+
+module.exports = MacroExpander;
+
+},{"./Lexer":3}],5:[function(require,module,exports){
+/**
+ * This file contains information about the options that the Parser carries
+ * around with it while parsing. Data is held in an `Options` object, and when
+ * recursing, a new `Options` object can be created with the `.with*` and
+ * `.reset` functions.
+ */
+
+/**
+ * This is the main options class. It contains the style, size, color, and font
+ * of the current parse level. It also contains the style and size of the parent
+ * parse level, so size changes can be handled efficiently.
+ *
+ * Each of the `.with*` and `.reset` functions passes its current style and size
+ * as the parentStyle and parentSize of the new options class, so parent
+ * handling is taken care of automatically.
+ */
+function Options(data) {
+ this.style = data.style;
+ this.color = data.color;
+ this.size = data.size;
+ this.phantom = data.phantom;
+ this.font = data.font;
+
+ if (data.parentStyle === undefined) {
+ this.parentStyle = data.style;
+ } else {
+ this.parentStyle = data.parentStyle;
+ }
+
+ if (data.parentSize === undefined) {
+ this.parentSize = data.size;
+ } else {
+ this.parentSize = data.parentSize;
+ }
+}
+
+/**
+ * Returns a new options object with the same properties as "this". Properties
+ * from "extension" will be copied to the new options object.
+ */
+Options.prototype.extend = function(extension) {
+ var data = {
+ style: this.style,
+ size: this.size,
+ color: this.color,
+ parentStyle: this.style,
+ parentSize: this.size,
+ phantom: this.phantom,
+ font: this.font
+ };
+
+ for (var key in extension) {
+ if (extension.hasOwnProperty(key)) {
+ data[key] = extension[key];
+ }
+ }
+
+ return new Options(data);
+};
+
+/**
+ * Create a new options object with the given style.
+ */
+Options.prototype.withStyle = function(style) {
+ return this.extend({
+ style: style
+ });
+};
+
+/**
+ * Create a new options object with the given size.
+ */
+Options.prototype.withSize = function(size) {
+ return this.extend({
+ size: size
+ });
+};
+
+/**
+ * Create a new options object with the given color.
+ */
+Options.prototype.withColor = function(color) {
+ return this.extend({
+ color: color
+ });
+};
+
+/**
+ * Create a new options object with "phantom" set to true.
+ */
+Options.prototype.withPhantom = function() {
+ return this.extend({
+ phantom: true
+ });
+};
+
+/**
+ * Create a new options objects with the give font.
+ */
+Options.prototype.withFont = function(font) {
+ return this.extend({
+ font: font || this.font
+ });
+};
+
+/**
+ * Create a new options object with the same style, size, and color. This is
+ * used so that parent style and size changes are handled correctly.
+ */
+Options.prototype.reset = function() {
+ return this.extend({});
+};
+
+/**
+ * A map of color names to CSS colors.
+ * TODO(emily): Remove this when we have real macros
+ */
+var colorMap = {
+ "katex-blue": "#6495ed",
+ "katex-orange": "#ffa500",
+ "katex-pink": "#ff00af",
+ "katex-red": "#df0030",
+ "katex-green": "#28ae7b",
+ "katex-gray": "gray",
+ "katex-purple": "#9d38bd",
+ "katex-blueA": "#ccfaff",
+ "katex-blueB": "#80f6ff",
+ "katex-blueC": "#63d9ea",
+ "katex-blueD": "#11accd",
+ "katex-blueE": "#0c7f99",
+ "katex-tealA": "#94fff5",
+ "katex-tealB": "#26edd5",
+ "katex-tealC": "#01d1c1",
+ "katex-tealD": "#01a995",
+ "katex-tealE": "#208170",
+ "katex-greenA": "#b6ffb0",
+ "katex-greenB": "#8af281",
+ "katex-greenC": "#74cf70",
+ "katex-greenD": "#1fab54",
+ "katex-greenE": "#0d923f",
+ "katex-goldA": "#ffd0a9",
+ "katex-goldB": "#ffbb71",
+ "katex-goldC": "#ff9c39",
+ "katex-goldD": "#e07d10",
+ "katex-goldE": "#a75a05",
+ "katex-redA": "#fca9a9",
+ "katex-redB": "#ff8482",
+ "katex-redC": "#f9685d",
+ "katex-redD": "#e84d39",
+ "katex-redE": "#bc2612",
+ "katex-maroonA": "#ffbde0",
+ "katex-maroonB": "#ff92c6",
+ "katex-maroonC": "#ed5fa6",
+ "katex-maroonD": "#ca337c",
+ "katex-maroonE": "#9e034e",
+ "katex-purpleA": "#ddd7ff",
+ "katex-purpleB": "#c6b9fc",
+ "katex-purpleC": "#aa87ff",
+ "katex-purpleD": "#7854ab",
+ "katex-purpleE": "#543b78",
+ "katex-mintA": "#f5f9e8",
+ "katex-mintB": "#edf2df",
+ "katex-mintC": "#e0e5cc",
+ "katex-grayA": "#f6f7f7",
+ "katex-grayB": "#f0f1f2",
+ "katex-grayC": "#e3e5e6",
+ "katex-grayD": "#d6d8da",
+ "katex-grayE": "#babec2",
+ "katex-grayF": "#888d93",
+ "katex-grayG": "#626569",
+ "katex-grayH": "#3b3e40",
+ "katex-grayI": "#21242c",
+ "katex-kaBlue": "#314453",
+ "katex-kaGreen": "#71B307"
+};
+
+/**
+ * Gets the CSS color of the current options object, accounting for the
+ * `colorMap`.
+ */
+Options.prototype.getColor = function() {
+ if (this.phantom) {
+ return "transparent";
+ } else {
+ return colorMap[this.color] || this.color;
+ }
+};
+
+module.exports = Options;
+
+},{}],6:[function(require,module,exports){
+/**
+ * This is the ParseError class, which is the main error thrown by KaTeX
+ * functions when something has gone wrong. This is used to distinguish internal
+ * errors from errors in the expression that the user provided.
+ *
+ * If possible, a caller should provide a Token or ParseNode with information
+ * about where in the source string the problem occurred.
+ *
+ * @param {string} message The error message
+ * @param {(Token|ParseNode)=} token An object providing position information
+ */
+function ParseError(message, token) {
+ var error = "KaTeX parse error: " + message;
+ var start;
+ var end;
+
+ if (token && token.lexer && token.start <= token.end) {
+ // If we have the input and a position, make the error a bit fancier
+
+ // Get the input
+ var input = token.lexer.input;
+
+ // Prepend some information
+ start = token.start;
+ end = token.end;
+ if (start === input.length) {
+ error += " at end of input: ";
+ } else {
+ error += " at position " + (start + 1) + ": ";
+ }
+
+ // Underline token in question using combining underscores
+ var underlined = input.slice(start, end).replace(/[^]/g, "$&\u0332");
+
+ // Extract some context from the input and add it to the error
+ var left;
+ if (start > 15) {
+ left = "…" + input.slice(start - 15, start);
+ } else {
+ left = input.slice(0, start);
+ }
+ var right;
+ if (end + 15 < input.length) {
+ right = input.slice(end, end + 15) + "…";
+ } else {
+ right = input.slice(end);
+ }
+ error += left + underlined + right;
+ }
+
+ // Some hackery to make ParseError a prototype of Error
+ // See http://stackoverflow.com/a/8460753
+ var self = new Error(error);
+ self.name = "ParseError";
+ self.__proto__ = ParseError.prototype;
+
+ self.position = start;
+ return self;
+}
+
+// More hackery
+ParseError.prototype.__proto__ = Error.prototype;
+
+module.exports = ParseError;
+
+},{}],7:[function(require,module,exports){
+/* eslint no-constant-condition:0 */
+var functions = require("./functions");
+var environments = require("./environments");
+var MacroExpander = require("./MacroExpander");
+var symbols = require("./symbols");
+var utils = require("./utils");
+var cjkRegex = require("./unicodeRegexes").cjkRegex;
+
+var parseData = require("./parseData");
+var ParseError = require("./ParseError");
+
+/**
+ * This file contains the parser used to parse out a TeX expression from the
+ * input. Since TeX isn't context-free, standard parsers don't work particularly
+ * well.
+ *
+ * The strategy of this parser is as such:
+ *
+ * The main functions (the `.parse...` ones) take a position in the current
+ * parse string to parse tokens from. The lexer (found in Lexer.js, stored at
+ * this.lexer) also supports pulling out tokens at arbitrary places. When
+ * individual tokens are needed at a position, the lexer is called to pull out a
+ * token, which is then used.
+ *
+ * The parser has a property called "mode" indicating the mode that
+ * the parser is currently in. Currently it has to be one of "math" or
+ * "text", which denotes whether the current environment is a math-y
+ * one or a text-y one (e.g. inside \text). Currently, this serves to
+ * limit the functions which can be used in text mode.
+ *
+ * The main functions then return an object which contains the useful data that
+ * was parsed at its given point, and a new position at the end of the parsed
+ * data. The main functions can call each other and continue the parsing by
+ * using the returned position as a new starting point.
+ *
+ * There are also extra `.handle...` functions, which pull out some reused
+ * functionality into self-contained functions.
+ *
+ * The earlier functions return ParseNodes.
+ * The later functions (which are called deeper in the parse) sometimes return
+ * ParseFuncOrArgument, which contain a ParseNode as well as some data about
+ * whether the parsed object is a function which is missing some arguments, or a
+ * standalone object which can be used as an argument to another function.
+ */
+
+/**
+ * Main Parser class
+ */
+function Parser(input, settings) {
+ // Create a new macro expander (gullet) and (indirectly via that) also a
+ // new lexer (mouth) for this parser (stomach, in the language of TeX)
+ this.gullet = new MacroExpander(input, settings.macros);
+ // Store the settings for use in parsing
+ this.settings = settings;
+ // Count leftright depth (for \middle errors)
+ this.leftrightDepth = 0;
+}
+
+var ParseNode = parseData.ParseNode;
+
+/**
+ * An initial function (without its arguments), or an argument to a function.
+ * The `result` argument should be a ParseNode.
+ */
+function ParseFuncOrArgument(result, isFunction, token) {
+ this.result = result;
+ // Is this a function (i.e. is it something defined in functions.js)?
+ this.isFunction = isFunction;
+ this.token = token;
+}
+
+/**
+ * Checks a result to make sure it has the right type, and throws an
+ * appropriate error otherwise.
+ *
+ * @param {boolean=} consume whether to consume the expected token,
+ * defaults to true
+ */
+Parser.prototype.expect = function(text, consume) {
+ if (this.nextToken.text !== text) {
+ throw new ParseError(
+ "Expected '" + text + "', got '" + this.nextToken.text + "'",
+ this.nextToken
+ );
+ }
+ if (consume !== false) {
+ this.consume();
+ }
+};
+
+/**
+ * Considers the current look ahead token as consumed,
+ * and fetches the one after that as the new look ahead.
+ */
+Parser.prototype.consume = function() {
+ this.nextToken = this.gullet.get(this.mode === "math");
+};
+
+Parser.prototype.switchMode = function(newMode) {
+ this.gullet.unget(this.nextToken);
+ this.mode = newMode;
+ this.consume();
+};
+
+/**
+ * Main parsing function, which parses an entire input.
+ *
+ * @return {?Array.<ParseNode>}
+ */
+Parser.prototype.parse = function() {
+ // Try to parse the input
+ this.mode = "math";
+ this.consume();
+ var parse = this.parseInput();
+ return parse;
+};
+
+/**
+ * Parses an entire input tree.
+ */
+Parser.prototype.parseInput = function() {
+ // Parse an expression
+ var expression = this.parseExpression(false);
+ // If we succeeded, make sure there's an EOF at the end
+ this.expect("EOF", false);
+ return expression;
+};
+
+var endOfExpression = ["}", "\\end", "\\right", "&", "\\\\", "\\cr"];
+
+/**
+ * Parses an "expression", which is a list of atoms.
+ *
+ * @param {boolean} breakOnInfix Should the parsing stop when we hit infix
+ * nodes? This happens when functions have higher precendence
+ * than infix nodes in implicit parses.
+ *
+ * @param {?string} breakOnTokenText The text of the token that the expression
+ * should end with, or `null` if something else should end the
+ * expression.
+ *
+ * @return {ParseNode}
+ */
+Parser.prototype.parseExpression = function(breakOnInfix, breakOnTokenText) {
+ var body = [];
+ // Keep adding atoms to the body until we can't parse any more atoms (either
+ // we reached the end, a }, or a \right)
+ while (true) {
+ var lex = this.nextToken;
+ if (endOfExpression.indexOf(lex.text) !== -1) {
+ break;
+ }
+ if (breakOnTokenText && lex.text === breakOnTokenText) {
+ break;
+ }
+ if (breakOnInfix && functions[lex.text] && functions[lex.text].infix) {
+ break;
+ }
+ var atom = this.parseAtom();
+ if (!atom) {
+ if (!this.settings.throwOnError && lex.text[0] === "\\") {
+ var errorNode = this.handleUnsupportedCmd();
+ body.push(errorNode);
+ continue;
+ }
+
+ break;
+ }
+ body.push(atom);
+ }
+ return this.handleInfixNodes(body);
+};
+
+/**
+ * Rewrites infix operators such as \over with corresponding commands such
+ * as \frac.
+ *
+ * There can only be one infix operator per group. If there's more than one
+ * then the expression is ambiguous. This can be resolved by adding {}.
+ *
+ * @returns {Array}
+ */
+Parser.prototype.handleInfixNodes = function(body) {
+ var overIndex = -1;
+ var funcName;
+
+ for (var i = 0; i < body.length; i++) {
+ var node = body[i];
+ if (node.type === "infix") {
+ if (overIndex !== -1) {
+ throw new ParseError(
+ "only one infix operator per group",
+ node.value.token);
+ }
+ overIndex = i;
+ funcName = node.value.replaceWith;
+ }
+ }
+
+ if (overIndex !== -1) {
+ var numerNode;
+ var denomNode;
+
+ var numerBody = body.slice(0, overIndex);
+ var denomBody = body.slice(overIndex + 1);
+
+ if (numerBody.length === 1 && numerBody[0].type === "ordgroup") {
+ numerNode = numerBody[0];
+ } else {
+ numerNode = new ParseNode("ordgroup", numerBody, this.mode);
+ }
+
+ if (denomBody.length === 1 && denomBody[0].type === "ordgroup") {
+ denomNode = denomBody[0];
+ } else {
+ denomNode = new ParseNode("ordgroup", denomBody, this.mode);
+ }
+
+ var value = this.callFunction(
+ funcName, [numerNode, denomNode], null);
+ return [new ParseNode(value.type, value, this.mode)];
+ } else {
+ return body;
+ }
+};
+
+// The greediness of a superscript or subscript
+var SUPSUB_GREEDINESS = 1;
+
+/**
+ * Handle a subscript or superscript with nice errors.
+ */
+Parser.prototype.handleSupSubscript = function(name) {
+ var symbolToken = this.nextToken;
+ var symbol = symbolToken.text;
+ this.consume();
+ var group = this.parseGroup();
+
+ if (!group) {
+ if (!this.settings.throwOnError && this.nextToken.text[0] === "\\") {
+ return this.handleUnsupportedCmd();
+ } else {
+ throw new ParseError(
+ "Expected group after '" + symbol + "'",
+ symbolToken
+ );
+ }
+ } else if (group.isFunction) {
+ // ^ and _ have a greediness, so handle interactions with functions'
+ // greediness
+ var funcGreediness = functions[group.result].greediness;
+ if (funcGreediness > SUPSUB_GREEDINESS) {
+ return this.parseFunction(group);
+ } else {
+ throw new ParseError(
+ "Got function '" + group.result + "' with no arguments " +
+ "as " + name, symbolToken);
+ }
+ } else {
+ return group.result;
+ }
+};
+
+/**
+ * Converts the textual input of an unsupported command into a text node
+ * contained within a color node whose color is determined by errorColor
+ */
+Parser.prototype.handleUnsupportedCmd = function() {
+ var text = this.nextToken.text;
+ var textordArray = [];
+
+ for (var i = 0; i < text.length; i++) {
+ textordArray.push(new ParseNode("textord", text[i], "text"));
+ }
+
+ var textNode = new ParseNode(
+ "text",
+ {
+ body: textordArray,
+ type: "text"
+ },
+ this.mode);
+
+ var colorNode = new ParseNode(
+ "color",
+ {
+ color: this.settings.errorColor,
+ value: [textNode],
+ type: "color"
+ },
+ this.mode);
+
+ this.consume();
+ return colorNode;
+};
+
+/**
+ * Parses a group with optional super/subscripts.
+ *
+ * @return {?ParseNode}
+ */
+Parser.prototype.parseAtom = function() {
+ // The body of an atom is an implicit group, so that things like
+ // \left(x\right)^2 work correctly.
+ var base = this.parseImplicitGroup();
+
+ // In text mode, we don't have superscripts or subscripts
+ if (this.mode === "text") {
+ return base;
+ }
+
+ // Note that base may be empty (i.e. null) at this point.
+
+ var superscript;
+ var subscript;
+ while (true) {
+ // Lex the first token
+ var lex = this.nextToken;
+
+ if (lex.text === "\\limits" || lex.text === "\\nolimits") {
+ // We got a limit control
+ if (!base || base.type !== "op") {
+ throw new ParseError(
+ "Limit controls must follow a math operator",
+ lex);
+ } else {
+ var limits = lex.text === "\\limits";
+ base.value.limits = limits;
+ base.value.alwaysHandleSupSub = true;
+ }
+ this.consume();
+ } else if (lex.text === "^") {
+ // We got a superscript start
+ if (superscript) {
+ throw new ParseError("Double superscript", lex);
+ }
+ superscript = this.handleSupSubscript("superscript");
+ } else if (lex.text === "_") {
+ // We got a subscript start
+ if (subscript) {
+ throw new ParseError("Double subscript", lex);
+ }
+ subscript = this.handleSupSubscript("subscript");
+ } else if (lex.text === "'") {
+ // We got a prime
+ var prime = new ParseNode("textord", "\\prime", this.mode);
+
+ // Many primes can be grouped together, so we handle this here
+ var primes = [prime];
+ this.consume();
+ // Keep lexing tokens until we get something that's not a prime
+ while (this.nextToken.text === "'") {
+ // For each one, add another prime to the list
+ primes.push(prime);
+ this.consume();
+ }
+ // Put them into an ordgroup as the superscript
+ superscript = new ParseNode("ordgroup", primes, this.mode);
+ } else {
+ // If it wasn't ^, _, or ', stop parsing super/subscripts
+ break;
+ }
+ }
+
+ if (superscript || subscript) {
+ // If we got either a superscript or subscript, create a supsub
+ return new ParseNode("supsub", {
+ base: base,
+ sup: superscript,
+ sub: subscript
+ }, this.mode);
+ } else {
+ // Otherwise return the original body
+ return base;
+ }
+};
+
+// A list of the size-changing functions, for use in parseImplicitGroup
+var sizeFuncs = [
+ "\\tiny", "\\scriptsize", "\\footnotesize", "\\small", "\\normalsize",
+ "\\large", "\\Large", "\\LARGE", "\\huge", "\\Huge"
+];
+
+// A list of the style-changing functions, for use in parseImplicitGroup
+var styleFuncs = [
+ "\\displaystyle", "\\textstyle", "\\scriptstyle", "\\scriptscriptstyle"
+];
+
+/**
+ * Parses an implicit group, which is a group that starts at the end of a
+ * specified, and ends right before a higher explicit group ends, or at EOL. It
+ * is used for functions that appear to affect the current style, like \Large or
+ * \textrm, where instead of keeping a style we just pretend that there is an
+ * implicit grouping after it until the end of the group. E.g.
+ * small text {\Large large text} small text again
+ * It is also used for \left and \right to get the correct grouping.
+ *
+ * @return {?ParseNode}
+ */
+Parser.prototype.parseImplicitGroup = function() {
+ var start = this.parseSymbol();
+
+ if (start == null) {
+ // If we didn't get anything we handle, fall back to parseFunction
+ return this.parseFunction();
+ }
+
+ var func = start.result;
+ var body;
+
+ if (func === "\\left") {
+ // If we see a left:
+ // Parse the entire left function (including the delimiter)
+ var left = this.parseFunction(start);
+ // Parse out the implicit body
+ ++this.leftrightDepth;
+ body = this.parseExpression(false);
+ --this.leftrightDepth;
+ // Check the next token
+ this.expect("\\right", false);
+ var right = this.parseFunction();
+ return new ParseNode("leftright", {
+ body: body,
+ left: left.value.value,
+ right: right.value.value
+ }, this.mode);
+ } else if (func === "\\begin") {
+ // begin...end is similar to left...right
+ var begin = this.parseFunction(start);
+ var envName = begin.value.name;
+ if (!environments.hasOwnProperty(envName)) {
+ throw new ParseError(
+ "No such environment: " + envName, begin.value.nameGroup);
+ }
+ // Build the environment object. Arguments and other information will
+ // be made available to the begin and end methods using properties.
+ var env = environments[envName];
+ var args = this.parseArguments("\\begin{" + envName + "}", env);
+ var context = {
+ mode: this.mode,
+ envName: envName,
+ parser: this,
+ positions: args.pop()
+ };
+ var result = env.handler(context, args);
+ this.expect("\\end", false);
+ var endNameToken = this.nextToken;
+ var end = this.parseFunction();
+ if (end.value.name !== envName) {
+ throw new ParseError(
+ "Mismatch: \\begin{" + envName + "} matched " +
+ "by \\end{" + end.value.name + "}",
+ endNameToken);
+ }
+ result.position = end.position;
+ return result;
+ } else if (utils.contains(sizeFuncs, func)) {
+ // If we see a sizing function, parse out the implict body
+ body = this.parseExpression(false);
+ return new ParseNode("sizing", {
+ // Figure out what size to use based on the list of functions above
+ size: "size" + (utils.indexOf(sizeFuncs, func) + 1),
+ value: body
+ }, this.mode);
+ } else if (utils.contains(styleFuncs, func)) {
+ // If we see a styling function, parse out the implict body
+ body = this.parseExpression(true);
+ return new ParseNode("styling", {
+ // Figure out what style to use by pulling out the style from
+ // the function name
+ style: func.slice(1, func.length - 5),
+ value: body
+ }, this.mode);
+ } else {
+ // Defer to parseFunction if it's not a function we handle
+ return this.parseFunction(start);
+ }
+};
+
+/**
+ * Parses an entire function, including its base and all of its arguments.
+ * The base might either have been parsed already, in which case
+ * it is provided as an argument, or it's the next group in the input.
+ *
+ * @param {ParseFuncOrArgument=} baseGroup optional as described above
+ * @return {?ParseNode}
+ */
+Parser.prototype.parseFunction = function(baseGroup) {
+ if (!baseGroup) {
+ baseGroup = this.parseGroup();
+ }
+
+ if (baseGroup) {
+ if (baseGroup.isFunction) {
+ var func = baseGroup.result;
+ var funcData = functions[func];
+ if (this.mode === "text" && !funcData.allowedInText) {
+ throw new ParseError(
+ "Can't use function '" + func + "' in text mode",
+ baseGroup.token);
+ }
+
+ var args = this.parseArguments(func, funcData);
+ var token = baseGroup.token;
+ var result = this.callFunction(func, args, args.pop(), token);
+ return new ParseNode(result.type, result, this.mode);
+ } else {
+ return baseGroup.result;
+ }
+ } else {
+ return null;
+ }
+};
+
+/**
+ * Call a function handler with a suitable context and arguments.
+ */
+Parser.prototype.callFunction = function(name, args, positions, token) {
+ var context = {
+ funcName: name,
+ parser: this,
+ positions: positions,
+ token: token
+ };
+ return functions[name].handler(context, args);
+};
+
+/**
+ * Parses the arguments of a function or environment
+ *
+ * @param {string} func "\name" or "\begin{name}"
+ * @param {{numArgs:number,numOptionalArgs:number|undefined}} funcData
+ * @return the array of arguments, with the list of positions as last element
+ */
+Parser.prototype.parseArguments = function(func, funcData) {
+ var totalArgs = funcData.numArgs + funcData.numOptionalArgs;
+ if (totalArgs === 0) {
+ return [[this.pos]];
+ }
+
+ var baseGreediness = funcData.greediness;
+ var positions = [this.pos];
+ var args = [];
+
+ for (var i = 0; i < totalArgs; i++) {
+ var nextToken = this.nextToken;
+ var argType = funcData.argTypes && funcData.argTypes[i];
+ var arg;
+ if (i < funcData.numOptionalArgs) {
+ if (argType) {
+ arg = this.parseGroupOfType(argType, true);
+ } else {
+ arg = this.parseGroup(true);
+ }
+ if (!arg) {
+ args.push(null);
+ positions.push(this.pos);
+ continue;
+ }
+ } else {
+ if (argType) {
+ arg = this.parseGroupOfType(argType);
+ } else {
+ arg = this.parseGroup();
+ }
+ if (!arg) {
+ if (!this.settings.throwOnError &&
+ this.nextToken.text[0] === "\\") {
+ arg = new ParseFuncOrArgument(
+ this.handleUnsupportedCmd(this.nextToken.text),
+ false);
+ } else {
+ throw new ParseError(
+ "Expected group after '" + func + "'", nextToken);
+ }
+ }
+ }
+ var argNode;
+ if (arg.isFunction) {
+ var argGreediness =
+ functions[arg.result].greediness;
+ if (argGreediness > baseGreediness) {
+ argNode = this.parseFunction(arg);
+ } else {
+ throw new ParseError(
+ "Got function '" + arg.result + "' as " +
+ "argument to '" + func + "'", nextToken);
+ }
+ } else {
+ argNode = arg.result;
+ }
+ args.push(argNode);
+ positions.push(this.pos);
+ }
+
+ args.push(positions);
+
+ return args;
+};
+
+
+/**
+ * Parses a group when the mode is changing.
+ *
+ * @return {?ParseFuncOrArgument}
+ */
+Parser.prototype.parseGroupOfType = function(innerMode, optional) {
+ var outerMode = this.mode;
+ // Handle `original` argTypes
+ if (innerMode === "original") {
+ innerMode = outerMode;
+ }
+
+ if (innerMode === "color") {
+ return this.parseColorGroup(optional);
+ }
+ if (innerMode === "size") {
+ return this.parseSizeGroup(optional);
+ }
+
+ this.switchMode(innerMode);
+ if (innerMode === "text") {
+ // text mode is special because it should ignore the whitespace before
+ // it
+ while (this.nextToken.text === " ") {
+ this.consume();
+ }
+ }
+ // By the time we get here, innerMode is one of "text" or "math".
+ // We switch the mode of the parser, recurse, then restore the old mode.
+ var res = this.parseGroup(optional);
+ this.switchMode(outerMode);
+ return res;
+};
+
+/**
+ * Parses a group, essentially returning the string formed by the
+ * brace-enclosed tokens plus some position information.
+ *
+ * @param {string} modeName Used to describe the mode in error messages
+ * @param {boolean=} optional Whether the group is optional or required
+ */
+Parser.prototype.parseStringGroup = function(modeName, optional) {
+ if (optional && this.nextToken.text !== "[") {
+ return null;
+ }
+ var outerMode = this.mode;
+ this.mode = "text";
+ this.expect(optional ? "[" : "{");
+ var str = "";
+ var firstToken = this.nextToken;
+ var lastToken = firstToken;
+ while (this.nextToken.text !== (optional ? "]" : "}")) {
+ if (this.nextToken.text === "EOF") {
+ throw new ParseError(
+ "Unexpected end of input in " + modeName,
+ firstToken.range(this.nextToken, str));
+ }
+ lastToken = this.nextToken;
+ str += lastToken.text;
+ this.consume();
+ }
+ this.mode = outerMode;
+ this.expect(optional ? "]" : "}");
+ return firstToken.range(lastToken, str);
+};
+
+/**
+ * Parses a regex-delimited group: the largest sequence of tokens
+ * whose concatenated strings match `regex`. Returns the string
+ * formed by the tokens plus some position information.
+ *
+ * @param {RegExp} regex
+ * @param {string} modeName Used to describe the mode in error messages
+ */
+Parser.prototype.parseRegexGroup = function(regex, modeName) {
+ var outerMode = this.mode;
+ this.mode = "text";
+ var firstToken = this.nextToken;
+ var lastToken = firstToken;
+ var str = "";
+ while (this.nextToken.text !== "EOF"
+ && regex.test(str + this.nextToken.text)) {
+ lastToken = this.nextToken;
+ str += lastToken.text;
+ this.consume();
+ }
+ if (str === "") {
+ throw new ParseError(
+ "Invalid " + modeName + ": '" + firstToken.text + "'",
+ firstToken);
+ }
+ this.mode = outerMode;
+ return firstToken.range(lastToken, str);
+};
+
+/**
+ * Parses a color description.
+ */
+Parser.prototype.parseColorGroup = function(optional) {
+ var res = this.parseStringGroup("color", optional);
+ if (!res) {
+ return null;
+ }
+ var match = (/^(#[a-z0-9]+|[a-z]+)$/i).exec(res.text);
+ if (!match) {
+ throw new ParseError("Invalid color: '" + res.text + "'", res);
+ }
+ return new ParseFuncOrArgument(
+ new ParseNode("color", match[0], this.mode),
+ false);
+};
+
+/**
+ * Parses a size specification, consisting of magnitude and unit.
+ */
+Parser.prototype.parseSizeGroup = function(optional) {
+ var res;
+ if (!optional && this.nextToken.text !== "{") {
+ res = this.parseRegexGroup(
+ /^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2}$/, "size");
+ } else {
+ res = this.parseStringGroup("size", optional);
+ }
+ if (!res) {
+ return null;
+ }
+ var match = (/([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/).exec(res.text);
+ if (!match) {
+ throw new ParseError("Invalid size: '" + res.text + "'", res);
+ }
+ var data = {
+ number: +(match[1] + match[2]), // sign + magnitude, cast to number
+ unit: match[3]
+ };
+ if (data.unit !== "em" && data.unit !== "ex" && data.unit !== "mu") {
+ throw new ParseError("Invalid unit: '" + data.unit + "'", res);
+ }
+ return new ParseFuncOrArgument(
+ new ParseNode("color", data, this.mode),
+ false);
+};
+
+/**
+ * If the argument is false or absent, this parses an ordinary group,
+ * which is either a single nucleus (like "x") or an expression
+ * in braces (like "{x+y}").
+ * If the argument is true, it parses either a bracket-delimited expression
+ * (like "[x+y]") or returns null to indicate the absence of a
+ * bracket-enclosed group.
+ *
+ * @param {boolean=} optional Whether the group is optional or required
+ * @return {?ParseFuncOrArgument}
+ */
+Parser.prototype.parseGroup = function(optional) {
+ var firstToken = this.nextToken;
+ // Try to parse an open brace
+ if (this.nextToken.text === (optional ? "[" : "{")) {
+ // If we get a brace, parse an expression
+ this.consume();
+ var expression = this.parseExpression(false, optional ? "]" : null);
+ var lastToken = this.nextToken;
+ // Make sure we get a close brace
+ this.expect(optional ? "]" : "}");
+ if (this.mode === "text") {
+ this.formLigatures(expression);
+ }
+ return new ParseFuncOrArgument(
+ new ParseNode("ordgroup", expression, this.mode,
+ firstToken, lastToken),
+ false);
+ } else {
+ // Otherwise, just return a nucleus, or nothing for an optional group
+ return optional ? null : this.parseSymbol();
+ }
+};
+
+/**
+ * Form ligature-like combinations of characters for text mode.
+ * This includes inputs like "--", "---", "``" and "''".
+ * The result will simply replace multiple textord nodes with a single
+ * character in each value by a single textord node having multiple
+ * characters in its value. The representation is still ASCII source.
+ *
+ * @param {Array.<ParseNode>} group the nodes of this group,
+ * list will be moified in place
+ */
+Parser.prototype.formLigatures = function(group) {
+ var i;
+ var n = group.length - 1;
+ for (i = 0; i < n; ++i) {
+ var a = group[i];
+ var v = a.value;
+ if (v === "-" && group[i + 1].value === "-") {
+ if (i + 1 < n && group[i + 2].value === "-") {
+ group.splice(i, 3, new ParseNode(
+ "textord", "---", "text", a, group[i + 2]));
+ n -= 2;
+ } else {
+ group.splice(i, 2, new ParseNode(
+ "textord", "--", "text", a, group[i + 1]));
+ n -= 1;
+ }
+ }
+ if ((v === "'" || v === "`") && group[i + 1].value === v) {
+ group.splice(i, 2, new ParseNode(
+ "textord", v + v, "text", a, group[i + 1]));
+ n -= 1;
+ }
+ }
+};
+
+/**
+ * Parse a single symbol out of the string. Here, we handle both the functions
+ * we have defined, as well as the single character symbols
+ *
+ * @return {?ParseFuncOrArgument}
+ */
+Parser.prototype.parseSymbol = function() {
+ var nucleus = this.nextToken;
+
+ if (functions[nucleus.text]) {
+ this.consume();
+ // If there exists a function with this name, we return the function and
+ // say that it is a function.
+ return new ParseFuncOrArgument(
+ nucleus.text,
+ true, nucleus);
+ } else if (symbols[this.mode][nucleus.text]) {
+ this.consume();
+ // Otherwise if this is a no-argument function, find the type it
+ // corresponds to in the symbols map
+ return new ParseFuncOrArgument(
+ new ParseNode(symbols[this.mode][nucleus.text].group,
+ nucleus.text, this.mode, nucleus),
+ false, nucleus);
+ } else if (this.mode === "text" && cjkRegex.test(nucleus.text)) {
+ this.consume();
+ return new ParseFuncOrArgument(
+ new ParseNode("textord", nucleus.text, this.mode, nucleus),
+ false, nucleus);
+ } else {
+ return null;
+ }
+};
+
+Parser.prototype.ParseNode = ParseNode;
+
+module.exports = Parser;
+
+},{"./MacroExpander":4,"./ParseError":6,"./environments":16,"./functions":19,"./parseData":21,"./symbols":23,"./unicodeRegexes":24,"./utils":25}],8:[function(require,module,exports){
+/**
+ * This is a module for storing settings passed into KaTeX. It correctly handles
+ * default settings.
+ */
+
+/**
+ * Helper function for getting a default value if the value is undefined
+ */
+function get(option, defaultValue) {
+ return option === undefined ? defaultValue : option;
+}
+
+/**
+ * The main Settings object
+ *
+ * The current options stored are:
+ * - displayMode: Whether the expression should be typeset by default in
+ * textstyle or displaystyle (default false)
+ */
+function Settings(options) {
+ // allow null options
+ options = options || {};
+ this.displayMode = get(options.displayMode, false);
+ this.throwOnError = get(options.throwOnError, true);
+ this.errorColor = get(options.errorColor, "#cc0000");
+ this.macros = options.macros || {};
+}
+
+module.exports = Settings;
+
+},{}],9:[function(require,module,exports){
+/**
+ * This file contains information and classes for the various kinds of styles
+ * used in TeX. It provides a generic `Style` class, which holds information
+ * about a specific style. It then provides instances of all the different kinds
+ * of styles possible, and provides functions to move between them and get
+ * information about them.
+ */
+
+var sigmas = require("./fontMetrics.js").sigmas;
+
+var metrics = [{}, {}, {}];
+var i;
+for (var key in sigmas) {
+ if (sigmas.hasOwnProperty(key)) {
+ for (i = 0; i < 3; i++) {
+ metrics[i][key] = sigmas[key][i];
+ }
+ }
+}
+for (i = 0; i < 3; i++) {
+ metrics[i].emPerEx = sigmas.xHeight[i] / sigmas.quad[i];
+}
+
+/**
+ * The main style class. Contains a unique id for the style, a size (which is
+ * the same for cramped and uncramped version of a style), a cramped flag, and a
+ * size multiplier, which gives the size difference between a style and
+ * textstyle.
+ */
+function Style(id, size, multiplier, cramped) {
+ this.id = id;
+ this.size = size;
+ this.cramped = cramped;
+ this.sizeMultiplier = multiplier;
+ this.metrics = metrics[size > 0 ? size - 1 : 0];
+}
+
+/**
+ * Get the style of a superscript given a base in the current style.
+ */
+Style.prototype.sup = function() {
+ return styles[sup[this.id]];
+};
+
+/**
+ * Get the style of a subscript given a base in the current style.
+ */
+Style.prototype.sub = function() {
+ return styles[sub[this.id]];
+};
+
+/**
+ * Get the style of a fraction numerator given the fraction in the current
+ * style.
+ */
+Style.prototype.fracNum = function() {
+ return styles[fracNum[this.id]];
+};
+
+/**
+ * Get the style of a fraction denominator given the fraction in the current
+ * style.
+ */
+Style.prototype.fracDen = function() {
+ return styles[fracDen[this.id]];
+};
+
+/**
+ * Get the cramped version of a style (in particular, cramping a cramped style
+ * doesn't change the style).
+ */
+Style.prototype.cramp = function() {
+ return styles[cramp[this.id]];
+};
+
+/**
+ * HTML class name, like "displaystyle cramped"
+ */
+Style.prototype.cls = function() {
+ return sizeNames[this.size] + (this.cramped ? " cramped" : " uncramped");
+};
+
+/**
+ * HTML Reset class name, like "reset-textstyle"
+ */
+Style.prototype.reset = function() {
+ return resetNames[this.size];
+};
+
+/**
+ * Return if this style is tightly spaced (scriptstyle/scriptscriptstyle)
+ */
+Style.prototype.isTight = function() {
+ return this.size >= 2;
+};
+
+// IDs of the different styles
+var D = 0;
+var Dc = 1;
+var T = 2;
+var Tc = 3;
+var S = 4;
+var Sc = 5;
+var SS = 6;
+var SSc = 7;
+
+// String names for the different sizes
+var sizeNames = [
+ "displaystyle textstyle",
+ "textstyle",
+ "scriptstyle",
+ "scriptscriptstyle"
+];
+
+// Reset names for the different sizes
+var resetNames = [
+ "reset-textstyle",
+ "reset-textstyle",
+ "reset-scriptstyle",
+ "reset-scriptscriptstyle"
+];
+
+// Instances of the different styles
+var styles = [
+ new Style(D, 0, 1.0, false),
+ new Style(Dc, 0, 1.0, true),
+ new Style(T, 1, 1.0, false),
+ new Style(Tc, 1, 1.0, true),
+ new Style(S, 2, 0.7, false),
+ new Style(Sc, 2, 0.7, true),
+ new Style(SS, 3, 0.5, false),
+ new Style(SSc, 3, 0.5, true)
+];
+
+// Lookup tables for switching from one style to another
+var sup = [S, Sc, S, Sc, SS, SSc, SS, SSc];
+var sub = [Sc, Sc, Sc, Sc, SSc, SSc, SSc, SSc];
+var fracNum = [T, Tc, S, Sc, SS, SSc, SS, SSc];
+var fracDen = [Tc, Tc, Sc, Sc, SSc, SSc, SSc, SSc];
+var cramp = [Dc, Dc, Tc, Tc, Sc, Sc, SSc, SSc];
+
+// We only export some of the styles. Also, we don't export the `Style` class so
+// no more styles can be generated.
+module.exports = {
+ DISPLAY: styles[D],
+ TEXT: styles[T],
+ SCRIPT: styles[S],
+ SCRIPTSCRIPT: styles[SS]
+};
+
+},{"./fontMetrics.js":17}],10:[function(require,module,exports){
+/* eslint no-console:0 */
+/**
+ * This module contains general functions that can be used for building
+ * different kinds of domTree nodes in a consistent manner.
+ */
+
+var domTree = require("./domTree");
+var fontMetrics = require("./fontMetrics");
+var symbols = require("./symbols");
+var utils = require("./utils");
+
+var greekCapitals = [
+ "\\Gamma",
+ "\\Delta",
+ "\\Theta",
+ "\\Lambda",
+ "\\Xi",
+ "\\Pi",
+ "\\Sigma",
+ "\\Upsilon",
+ "\\Phi",
+ "\\Psi",
+ "\\Omega"
+];
+
+// The following have to be loaded from Main-Italic font, using class mainit
+var mainitLetters = [
+ "\u0131", // dotless i, \imath
+ "\u0237", // dotless j, \jmath
+ "\u00a3" // \pounds
+];
+
+/**
+ * Makes a symbolNode after translation via the list of symbols in symbols.js.
+ * Correctly pulls out metrics for the character, and optionally takes a list of
+ * classes to be attached to the node.
+ *
+ * TODO: make argument order closer to makeSpan
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ */
+var makeSymbol = function(value, fontFamily, mode, options, classes) {
+ // Replace the value with its replaced value from symbol.js
+ if (symbols[mode][value] && symbols[mode][value].replace) {
+ value = symbols[mode][value].replace;
+ }
+
+ var metrics = fontMetrics.getCharacterMetrics(value, fontFamily);
+
+ var symbolNode;
+ if (metrics) {
+ var italic = metrics.italic;
+ if (mode === "text") {
+ italic = 0;
+ }
+ symbolNode = new domTree.symbolNode(
+ value, metrics.height, metrics.depth, italic, metrics.skew,
+ classes);
+ } else {
+ // TODO(emily): Figure out a good way to only print this in development
+ typeof console !== "undefined" && console.warn(
+ "No character metrics for '" + value + "' in style '" +
+ fontFamily + "'");
+ symbolNode = new domTree.symbolNode(value, 0, 0, 0, 0, classes);
+ }
+
+ if (options) {
+ if (options.style.isTight()) {
+ symbolNode.classes.push("mtight");
+ }
+ if (options.getColor()) {
+ symbolNode.style.color = options.getColor();
+ }
+ }
+
+ return symbolNode;
+};
+
+/**
+ * Makes a symbol in Main-Regular or AMS-Regular.
+ * Used for rel, bin, open, close, inner, and punct.
+ */
+var mathsym = function(value, mode, options, classes) {
+ // Decide what font to render the symbol in by its entry in the symbols
+ // table.
+ // Have a special case for when the value = \ because the \ is used as a
+ // textord in unsupported command errors but cannot be parsed as a regular
+ // text ordinal and is therefore not present as a symbol in the symbols
+ // table for text
+ if (value === "\\" || symbols[mode][value].font === "main") {
+ return makeSymbol(value, "Main-Regular", mode, options, classes);
+ } else {
+ return makeSymbol(
+ value, "AMS-Regular", mode, options, classes.concat(["amsrm"]));
+ }
+};
+
+/**
+ * Makes a symbol in the default font for mathords and textords.
+ */
+var mathDefault = function(value, mode, options, classes, type) {
+ if (type === "mathord") {
+ return mathit(value, mode, options, classes);
+ } else if (type === "textord") {
+ return makeSymbol(
+ value, "Main-Regular", mode, options, classes.concat(["mathrm"]));
+ } else {
+ throw new Error("unexpected type: " + type + " in mathDefault");
+ }
+};
+
+/**
+ * Makes a symbol in the italic math font.
+ */
+var mathit = function(value, mode, options, classes) {
+ if (/[0-9]/.test(value.charAt(0)) ||
+ // glyphs for \imath and \jmath do not exist in Math-Italic so we
+ // need to use Main-Italic instead
+ utils.contains(mainitLetters, value) ||
+ utils.contains(greekCapitals, value)) {
+ return makeSymbol(
+ value, "Main-Italic", mode, options, classes.concat(["mainit"]));
+ } else {
+ return makeSymbol(
+ value, "Math-Italic", mode, options, classes.concat(["mathit"]));
+ }
+};
+
+/**
+ * Makes either a mathord or textord in the correct font and color.
+ */
+var makeOrd = function(group, options, type) {
+ var mode = group.mode;
+ var value = group.value;
+ if (symbols[mode][value] && symbols[mode][value].replace) {
+ value = symbols[mode][value].replace;
+ }
+
+ var classes = ["mord"];
+
+ var font = options.font;
+ if (font) {
+ if (font === "mathit" || utils.contains(mainitLetters, value)) {
+ return mathit(value, mode, options, classes);
+ } else {
+ var fontName = fontMap[font].fontName;
+ if (fontMetrics.getCharacterMetrics(value, fontName)) {
+ return makeSymbol(
+ value, fontName, mode, options, classes.concat([font]));
+ } else {
+ return mathDefault(value, mode, options, classes, type);
+ }
+ }
+ } else {
+ return mathDefault(value, mode, options, classes, type);
+ }
+};
+
+/**
+ * Calculate the height, depth, and maxFontSize of an element based on its
+ * children.
+ */
+var sizeElementFromChildren = function(elem) {
+ var height = 0;
+ var depth = 0;
+ var maxFontSize = 0;
+
+ if (elem.children) {
+ for (var i = 0; i < elem.children.length; i++) {
+ if (elem.children[i].height > height) {
+ height = elem.children[i].height;
+ }
+ if (elem.children[i].depth > depth) {
+ depth = elem.children[i].depth;
+ }
+ if (elem.children[i].maxFontSize > maxFontSize) {
+ maxFontSize = elem.children[i].maxFontSize;
+ }
+ }
+ }
+
+ elem.height = height;
+ elem.depth = depth;
+ elem.maxFontSize = maxFontSize;
+};
+
+/**
+ * Makes a span with the given list of classes, list of children, and options.
+ *
+ * TODO: Ensure that `options` is always provided (currently some call sites
+ * don't pass it).
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ */
+var makeSpan = function(classes, children, options) {
+ var span = new domTree.span(classes, children, options);
+
+ sizeElementFromChildren(span);
+
+ return span;
+};
+
+/**
+ * Prepends the given children to the given span, updating height, depth, and
+ * maxFontSize.
+ */
+var prependChildren = function(span, children) {
+ span.children = children.concat(span.children);
+
+ sizeElementFromChildren(span);
+};
+
+/**
+ * Makes a document fragment with the given list of children.
+ */
+var makeFragment = function(children) {
+ var fragment = new domTree.documentFragment(children);
+
+ sizeElementFromChildren(fragment);
+
+ return fragment;
+};
+
+/**
+ * Makes an element placed in each of the vlist elements to ensure that each
+ * element has the same max font size. To do this, we create a zero-width space
+ * with the correct font size.
+ */
+var makeFontSizer = function(options, fontSize) {
+ var fontSizeInner = makeSpan([], [new domTree.symbolNode("\u200b")]);
+ fontSizeInner.style.fontSize =
+ (fontSize / options.style.sizeMultiplier) + "em";
+
+ var fontSizer = makeSpan(
+ ["fontsize-ensurer", "reset-" + options.size, "size5"],
+ [fontSizeInner]);
+
+ return fontSizer;
+};
+
+/**
+ * Makes a vertical list by stacking elements and kerns on top of each other.
+ * Allows for many different ways of specifying the positioning method.
+ *
+ * Arguments:
+ * - children: A list of child or kern nodes to be stacked on top of each other
+ * (i.e. the first element will be at the bottom, and the last at
+ * the top). Element nodes are specified as
+ * {type: "elem", elem: node}
+ * while kern nodes are specified as
+ * {type: "kern", size: size}
+ * - positionType: The method by which the vlist should be positioned. Valid
+ * values are:
+ * - "individualShift": The children list only contains elem
+ * nodes, and each node contains an extra
+ * "shift" value of how much it should be
+ * shifted (note that shifting is always
+ * moving downwards). positionData is
+ * ignored.
+ * - "top": The positionData specifies the topmost point of
+ * the vlist (note this is expected to be a height,
+ * so positive values move up)
+ * - "bottom": The positionData specifies the bottommost point
+ * of the vlist (note this is expected to be a
+ * depth, so positive values move down
+ * - "shift": The vlist will be positioned such that its
+ * baseline is positionData away from the baseline
+ * of the first child. Positive values move
+ * downwards.
+ * - "firstBaseline": The vlist will be positioned such that
+ * its baseline is aligned with the
+ * baseline of the first child.
+ * positionData is ignored. (this is
+ * equivalent to "shift" with
+ * positionData=0)
+ * - positionData: Data used in different ways depending on positionType
+ * - options: An Options object
+ *
+ */
+var makeVList = function(children, positionType, positionData, options) {
+ var depth;
+ var currPos;
+ var i;
+ if (positionType === "individualShift") {
+ var oldChildren = children;
+ children = [oldChildren[0]];
+
+ // Add in kerns to the list of children to get each element to be
+ // shifted to the correct specified shift
+ depth = -oldChildren[0].shift - oldChildren[0].elem.depth;
+ currPos = depth;
+ for (i = 1; i < oldChildren.length; i++) {
+ var diff = -oldChildren[i].shift - currPos -
+ oldChildren[i].elem.depth;
+ var size = diff -
+ (oldChildren[i - 1].elem.height +
+ oldChildren[i - 1].elem.depth);
+
+ currPos = currPos + diff;
+
+ children.push({type: "kern", size: size});
+ children.push(oldChildren[i]);
+ }
+ } else if (positionType === "top") {
+ // We always start at the bottom, so calculate the bottom by adding up
+ // all the sizes
+ var bottom = positionData;
+ for (i = 0; i < children.length; i++) {
+ if (children[i].type === "kern") {
+ bottom -= children[i].size;
+ } else {
+ bottom -= children[i].elem.height + children[i].elem.depth;
+ }
+ }
+ depth = bottom;
+ } else if (positionType === "bottom") {
+ depth = -positionData;
+ } else if (positionType === "shift") {
+ depth = -children[0].elem.depth - positionData;
+ } else if (positionType === "firstBaseline") {
+ depth = -children[0].elem.depth;
+ } else {
+ depth = 0;
+ }
+
+ // Make the fontSizer
+ var maxFontSize = 0;
+ for (i = 0; i < children.length; i++) {
+ if (children[i].type === "elem") {
+ maxFontSize = Math.max(maxFontSize, children[i].elem.maxFontSize);
+ }
+ }
+ var fontSizer = makeFontSizer(options, maxFontSize);
+
+ // Create a new list of actual children at the correct offsets
+ var realChildren = [];
+ currPos = depth;
+ for (i = 0; i < children.length; i++) {
+ if (children[i].type === "kern") {
+ currPos += children[i].size;
+ } else {
+ var child = children[i].elem;
+
+ var shift = -child.depth - currPos;
+ currPos += child.height + child.depth;
+
+ var childWrap = makeSpan([], [fontSizer, child]);
+ childWrap.height -= shift;
+ childWrap.depth += shift;
+ childWrap.style.top = shift + "em";
+
+ realChildren.push(childWrap);
+ }
+ }
+
+ // Add in an element at the end with no offset to fix the calculation of
+ // baselines in some browsers (namely IE, sometimes safari)
+ var baselineFix = makeSpan(
+ ["baseline-fix"], [fontSizer, new domTree.symbolNode("\u200b")]);
+ realChildren.push(baselineFix);
+
+ var vlist = makeSpan(["vlist"], realChildren);
+ // Fix the final height and depth, in case there were kerns at the ends
+ // since the makeSpan calculation won't take that in to account.
+ vlist.height = Math.max(currPos, vlist.height);
+ vlist.depth = Math.max(-depth, vlist.depth);
+ return vlist;
+};
+
+// A table of size -> font size for the different sizing functions
+var sizingMultiplier = {
+ size1: 0.5,
+ size2: 0.7,
+ size3: 0.8,
+ size4: 0.9,
+ size5: 1.0,
+ size6: 1.2,
+ size7: 1.44,
+ size8: 1.73,
+ size9: 2.07,
+ size10: 2.49
+};
+
+// A map of spacing functions to their attributes, like size and corresponding
+// CSS class
+var spacingFunctions = {
+ "\\qquad": {
+ size: "2em",
+ className: "qquad"
+ },
+ "\\quad": {
+ size: "1em",
+ className: "quad"
+ },
+ "\\enspace": {
+ size: "0.5em",
+ className: "enspace"
+ },
+ "\\;": {
+ size: "0.277778em",
+ className: "thickspace"
+ },
+ "\\:": {
+ size: "0.22222em",
+ className: "mediumspace"
+ },
+ "\\,": {
+ size: "0.16667em",
+ className: "thinspace"
+ },
+ "\\!": {
+ size: "-0.16667em",
+ className: "negativethinspace"
+ }
+};
+
+/**
+ * Maps TeX font commands to objects containing:
+ * - variant: string used for "mathvariant" attribute in buildMathML.js
+ * - fontName: the "style" parameter to fontMetrics.getCharacterMetrics
+ */
+// A map between tex font commands an MathML mathvariant attribute values
+var fontMap = {
+ // styles
+ "mathbf": {
+ variant: "bold",
+ fontName: "Main-Bold"
+ },
+ "mathrm": {
+ variant: "normal",
+ fontName: "Main-Regular"
+ },
+ "textit": {
+ variant: "italic",
+ fontName: "Main-Italic"
+ },
+
+ // "mathit" is missing because it requires the use of two fonts: Main-Italic
+ // and Math-Italic. This is handled by a special case in makeOrd which ends
+ // up calling mathit.
+
+ // families
+ "mathbb": {
+ variant: "double-struck",
+ fontName: "AMS-Regular"
+ },
+ "mathcal": {
+ variant: "script",
+ fontName: "Caligraphic-Regular"
+ },
+ "mathfrak": {
+ variant: "fraktur",
+ fontName: "Fraktur-Regular"
+ },
+ "mathscr": {
+ variant: "script",
+ fontName: "Script-Regular"
+ },
+ "mathsf": {
+ variant: "sans-serif",
+ fontName: "SansSerif-Regular"
+ },
+ "mathtt": {
+ variant: "monospace",
+ fontName: "Typewriter-Regular"
+ }
+};
+
+module.exports = {
+ fontMap: fontMap,
+ makeSymbol: makeSymbol,
+ mathsym: mathsym,
+ makeSpan: makeSpan,
+ makeFragment: makeFragment,
+ makeVList: makeVList,
+ makeOrd: makeOrd,
+ prependChildren: prependChildren,
+ sizingMultiplier: sizingMultiplier,
+ spacingFunctions: spacingFunctions
+};
+
+},{"./domTree":15,"./fontMetrics":17,"./symbols":23,"./utils":25}],11:[function(require,module,exports){
+/* eslint no-console:0 */
+/**
+ * This file does the main work of building a domTree structure from a parse
+ * tree. The entry point is the `buildHTML` function, which takes a parse tree.
+ * Then, the buildExpression, buildGroup, and various groupTypes functions are
+ * called, to produce a final HTML tree.
+ */
+
+var ParseError = require("./ParseError");
+var Style = require("./Style");
+
+var buildCommon = require("./buildCommon");
+var delimiter = require("./delimiter");
+var domTree = require("./domTree");
+var fontMetrics = require("./fontMetrics");
+var utils = require("./utils");
+
+var makeSpan = buildCommon.makeSpan;
+
+var isSpace = function(node) {
+ return node instanceof domTree.span && node.classes[0] === "mspace";
+};
+
+// Binary atoms (first class `mbin`) change into ordinary atoms (`mord`)
+// depending on their surroundings. See TeXbook pg. 442-446, Rules 5 and 6,
+// and the text before Rule 19.
+
+var isBin = function(node) {
+ return node && node.classes[0] === "mbin";
+};
+
+var isBinLeftCanceller = function(node, isRealGroup) {
+ // TODO: This code assumes that a node's math class is the first element
+ // of its `classes` array. A later cleanup should ensure this, for
+ // instance by changing the signature of `makeSpan`.
+ if (node) {
+ return utils.contains(["mbin", "mopen", "mrel", "mop", "mpunct"],
+ node.classes[0]);
+ } else {
+ return isRealGroup;
+ }
+};
+
+var isBinRightCanceller = function(node, isRealGroup) {
+ if (node) {
+ return utils.contains(["mrel", "mclose", "mpunct"], node.classes[0]);
+ } else {
+ return isRealGroup;
+ }
+};
+
+/**
+ * Take a list of nodes, build them in order, and return a list of the built
+ * nodes. documentFragments are flattened into their contents, so the
+ * returned list contains no fragments. `isRealGroup` is true if `expression`
+ * is a real group (no atoms will be added on either side), as opposed to
+ * a partial group (e.g. one created by \color).
+ */
+var buildExpression = function(expression, options, isRealGroup) {
+ // Parse expressions into `groups`.
+ var groups = [];
+ for (var i = 0; i < expression.length; i++) {
+ var group = expression[i];
+ var output = buildGroup(group, options);
+ if (output instanceof domTree.documentFragment) {
+ Array.prototype.push.apply(groups, output.children);
+ } else {
+ groups.push(output);
+ }
+ }
+ // At this point `groups` consists entirely of `symbolNode`s and `span`s.
+
+ // Explicit spaces (e.g., \;, \,) should be ignored with respect to atom
+ // spacing (e.g., "add thick space between mord and mrel"). Since CSS
+ // adjacency rules implement atom spacing, spaces should be invisible to
+ // CSS. So we splice them out of `groups` and into the atoms themselves.
+ var spaces = null;
+ for (i = 0; i < groups.length; i++) {
+ if (isSpace(groups[i])) {
+ spaces = spaces || [];
+ spaces.push(groups[i]);
+ groups.splice(i, 1);
+ i--;
+ } else if (spaces) {
+ if (groups[i] instanceof domTree.symbolNode) {
+ groups[i] = makeSpan([].concat(groups[i].classes), [groups[i]]);
+ }
+ buildCommon.prependChildren(groups[i], spaces);
+ spaces = null;
+ }
+ }
+ if (spaces) {
+ Array.prototype.push.apply(groups, spaces);
+ }
+
+ // Binary operators change to ordinary symbols in some contexts.
+ for (i = 0; i < groups.length; i++) {
+ if (isBin(groups[i])
+ && (isBinLeftCanceller(groups[i - 1], isRealGroup)
+ || isBinRightCanceller(groups[i + 1], isRealGroup))) {
+ groups[i].classes[0] = "mord";
+ }
+ }
+
+ return groups;
+};
+
+// Return math atom class (mclass) of a domTree.
+var getTypeOfDomTree = function(node) {
+ if (node instanceof domTree.documentFragment) {
+ if (node.children.length) {
+ return getTypeOfDomTree(
+ node.children[node.children.length - 1]);
+ }
+ } else {
+ if (utils.contains(["mord", "mop", "mbin", "mrel", "mopen", "mclose",
+ "mpunct", "minner"], node.classes[0])) {
+ return node.classes[0];
+ }
+ }
+ return null;
+};
+
+/**
+ * Sometimes, groups perform special rules when they have superscripts or
+ * subscripts attached to them. This function lets the `supsub` group know that
+ * its inner element should handle the superscripts and subscripts instead of
+ * handling them itself.
+ */
+var shouldHandleSupSub = function(group, options) {
+ if (!group) {
+ return false;
+ } else if (group.type === "op") {
+ // Operators handle supsubs differently when they have limits
+ // (e.g. `\displaystyle\sum_2^3`)
+ return group.value.limits &&
+ (options.style.size === Style.DISPLAY.size ||
+ group.value.alwaysHandleSupSub);
+ } else if (group.type === "accent") {
+ return isCharacterBox(group.value.base);
+ } else {
+ return null;
+ }
+};
+
+/**
+ * Sometimes we want to pull out the innermost element of a group. In most
+ * cases, this will just be the group itself, but when ordgroups and colors have
+ * a single element, we want to pull that out.
+ */
+var getBaseElem = function(group) {
+ if (!group) {
+ return false;
+ } else if (group.type === "ordgroup") {
+ if (group.value.length === 1) {
+ return getBaseElem(group.value[0]);
+ } else {
+ return group;
+ }
+ } else if (group.type === "color") {
+ if (group.value.value.length === 1) {
+ return getBaseElem(group.value.value[0]);
+ } else {
+ return group;
+ }
+ } else if (group.type === "font") {
+ return getBaseElem(group.value.body);
+ } else {
+ return group;
+ }
+};
+
+/**
+ * TeXbook algorithms often reference "character boxes", which are simply groups
+ * with a single character in them. To decide if something is a character box,
+ * we find its innermost group, and see if it is a single character.
+ */
+var isCharacterBox = function(group) {
+ var baseElem = getBaseElem(group);
+
+ // These are all they types of groups which hold single characters
+ return baseElem.type === "mathord" ||
+ baseElem.type === "textord" ||
+ baseElem.type === "bin" ||
+ baseElem.type === "rel" ||
+ baseElem.type === "inner" ||
+ baseElem.type === "open" ||
+ baseElem.type === "close" ||
+ baseElem.type === "punct";
+};
+
+var makeNullDelimiter = function(options, classes) {
+ return makeSpan(classes.concat([
+ "sizing", "reset-" + options.size, "size5",
+ options.style.reset(), Style.TEXT.cls(),
+ "nulldelimiter"]));
+};
+
+/**
+ * This is a map of group types to the function used to handle that type.
+ * Simpler types come at the beginning, while complicated types come afterwards.
+ */
+var groupTypes = {};
+
+groupTypes.mathord = function(group, options) {
+ return buildCommon.makeOrd(group, options, "mathord");
+};
+
+groupTypes.textord = function(group, options) {
+ return buildCommon.makeOrd(group, options, "textord");
+};
+
+groupTypes.bin = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mbin"]);
+};
+
+groupTypes.rel = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mrel"]);
+};
+
+groupTypes.open = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mopen"]);
+};
+
+groupTypes.close = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mclose"]);
+};
+
+groupTypes.inner = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["minner"]);
+};
+
+groupTypes.punct = function(group, options) {
+ return buildCommon.mathsym(
+ group.value, group.mode, options, ["mpunct"]);
+};
+
+groupTypes.ordgroup = function(group, options) {
+ return makeSpan(
+ ["mord", options.style.cls()],
+ buildExpression(group.value, options.reset(), true),
+ options
+ );
+};
+
+groupTypes.text = function(group, options) {
+ var newOptions = options.withFont(group.value.style);
+ var inner = buildExpression(group.value.body, newOptions, true);
+ for (var i = 0; i < inner.length - 1; i++) {
+ if (inner[i].tryCombine(inner[i + 1])) {
+ inner.splice(i + 1, 1);
+ i--;
+ }
+ }
+ return makeSpan(["mord", "text", newOptions.style.cls()],
+ inner, newOptions);
+};
+
+groupTypes.color = function(group, options) {
+ var elements = buildExpression(
+ group.value.value,
+ options.withColor(group.value.color),
+ false
+ );
+
+ // \color isn't supposed to affect the type of the elements it contains.
+ // To accomplish this, we wrap the results in a fragment, so the inner
+ // elements will be able to directly interact with their neighbors. For
+ // example, `\color{red}{2 +} 3` has the same spacing as `2 + 3`
+ return new buildCommon.makeFragment(elements);
+};
+
+groupTypes.supsub = function(group, options) {
+ // Superscript and subscripts are handled in the TeXbook on page
+ // 445-446, rules 18(a-f).
+
+ // Here is where we defer to the inner group if it should handle
+ // superscripts and subscripts itself.
+ if (shouldHandleSupSub(group.value.base, options)) {
+ return groupTypes[group.value.base.type](group, options);
+ }
+
+ var base = buildGroup(group.value.base, options.reset());
+ var supmid;
+ var submid;
+ var sup;
+ var sub;
+
+ var style = options.style;
+ var newOptions;
+
+ if (group.value.sup) {
+ newOptions = options.withStyle(style.sup());
+ sup = buildGroup(group.value.sup, newOptions);
+ supmid = makeSpan([style.reset(), style.sup().cls()],
+ [sup], newOptions);
+ }
+
+ if (group.value.sub) {
+ newOptions = options.withStyle(style.sub());
+ sub = buildGroup(group.value.sub, newOptions);
+ submid = makeSpan([style.reset(), style.sub().cls()],
+ [sub], newOptions);
+ }
+
+ // Rule 18a
+ var supShift;
+ var subShift;
+ if (isCharacterBox(group.value.base)) {
+ supShift = 0;
+ subShift = 0;
+ } else {
+ supShift = base.height - style.metrics.supDrop;
+ subShift = base.depth + style.metrics.subDrop;
+ }
+
+ // Rule 18c
+ var minSupShift;
+ if (style === Style.DISPLAY) {
+ minSupShift = style.metrics.sup1;
+ } else if (style.cramped) {
+ minSupShift = style.metrics.sup3;
+ } else {
+ minSupShift = style.metrics.sup2;
+ }
+
+ // scriptspace is a font-size-independent size, so scale it
+ // appropriately
+ var multiplier = Style.TEXT.sizeMultiplier *
+ style.sizeMultiplier;
+ var scriptspace =
+ (0.5 / fontMetrics.metrics.ptPerEm) / multiplier + "em";
+
+ var supsub;
+ if (!group.value.sup) {
+ // Rule 18b
+ subShift = Math.max(
+ subShift, style.metrics.sub1,
+ sub.height - 0.8 * style.metrics.xHeight);
+
+ supsub = buildCommon.makeVList([
+ {type: "elem", elem: submid}
+ ], "shift", subShift, options);
+
+ supsub.children[0].style.marginRight = scriptspace;
+
+ // Subscripts shouldn't be shifted by the base's italic correction.
+ // Account for that by shifting the subscript back the appropriate
+ // amount. Note we only do this when the base is a single symbol.
+ if (base instanceof domTree.symbolNode) {
+ supsub.children[0].style.marginLeft = -base.italic + "em";
+ }
+ } else if (!group.value.sub) {
+ // Rule 18c, d
+ supShift = Math.max(supShift, minSupShift,
+ sup.depth + 0.25 * style.metrics.xHeight);
+
+ supsub = buildCommon.makeVList([
+ {type: "elem", elem: supmid}
+ ], "shift", -supShift, options);
+
+ supsub.children[0].style.marginRight = scriptspace;
+ } else {
+ supShift = Math.max(
+ supShift, minSupShift, sup.depth + 0.25 * style.metrics.xHeight);
+ subShift = Math.max(subShift, style.metrics.sub2);
+
+ var ruleWidth = fontMetrics.metrics.defaultRuleThickness;
+
+ // Rule 18e
+ if ((supShift - sup.depth) - (sub.height - subShift) <
+ 4 * ruleWidth) {
+ subShift = 4 * ruleWidth - (supShift - sup.depth) + sub.height;
+ var psi = 0.8 * style.metrics.xHeight - (supShift - sup.depth);
+ if (psi > 0) {
+ supShift += psi;
+ subShift -= psi;
+ }
+ }
+
+ supsub = buildCommon.makeVList([
+ {type: "elem", elem: submid, shift: subShift},
+ {type: "elem", elem: supmid, shift: -supShift}
+ ], "individualShift", null, options);
+
+ // See comment above about subscripts not being shifted
+ if (base instanceof domTree.symbolNode) {
+ supsub.children[0].style.marginLeft = -base.italic + "em";
+ }
+
+ supsub.children[0].style.marginRight = scriptspace;
+ supsub.children[1].style.marginRight = scriptspace;
+ }
+
+ // We ensure to wrap the supsub vlist in a span.msupsub to reset text-align
+ var mclass = getTypeOfDomTree(base) || "mord";
+ return makeSpan([mclass],
+ [base, makeSpan(["msupsub"], [supsub])],
+ options);
+};
+
+groupTypes.genfrac = function(group, options) {
+ // Fractions are handled in the TeXbook on pages 444-445, rules 15(a-e).
+ // Figure out what style this fraction should be in based on the
+ // function used
+ var style = options.style;
+ if (group.value.size === "display") {
+ style = Style.DISPLAY;
+ } else if (group.value.size === "text") {
+ style = Style.TEXT;
+ }
+
+ var nstyle = style.fracNum();
+ var dstyle = style.fracDen();
+ var newOptions;
+
+ newOptions = options.withStyle(nstyle);
+ var numer = buildGroup(group.value.numer, newOptions);
+ var numerreset = makeSpan([style.reset(), nstyle.cls()],
+ [numer], newOptions);
+
+ newOptions = options.withStyle(dstyle);
+ var denom = buildGroup(group.value.denom, newOptions);
+ var denomreset = makeSpan([style.reset(), dstyle.cls()],
+ [denom], newOptions);
+
+ var ruleWidth;
+ if (group.value.hasBarLine) {
+ ruleWidth = fontMetrics.metrics.defaultRuleThickness /
+ options.style.sizeMultiplier;
+ } else {
+ ruleWidth = 0;
+ }
+
+ // Rule 15b
+ var numShift;
+ var clearance;
+ var denomShift;
+ if (style.size === Style.DISPLAY.size) {
+ numShift = style.metrics.num1;
+ if (ruleWidth > 0) {
+ clearance = 3 * ruleWidth;
+ } else {
+ clearance = 7 * fontMetrics.metrics.defaultRuleThickness;
+ }
+ denomShift = style.metrics.denom1;
+ } else {
+ if (ruleWidth > 0) {
+ numShift = style.metrics.num2;
+ clearance = ruleWidth;
+ } else {
+ numShift = style.metrics.num3;
+ clearance = 3 * fontMetrics.metrics.defaultRuleThickness;
+ }
+ denomShift = style.metrics.denom2;
+ }
+
+ var frac;
+ if (ruleWidth === 0) {
+ // Rule 15c
+ var candidateClearance =
+ (numShift - numer.depth) - (denom.height - denomShift);
+ if (candidateClearance < clearance) {
+ numShift += 0.5 * (clearance - candidateClearance);
+ denomShift += 0.5 * (clearance - candidateClearance);
+ }
+
+ frac = buildCommon.makeVList([
+ {type: "elem", elem: denomreset, shift: denomShift},
+ {type: "elem", elem: numerreset, shift: -numShift}
+ ], "individualShift", null, options);
+ } else {
+ // Rule 15d
+ var axisHeight = style.metrics.axisHeight;
+
+ if ((numShift - numer.depth) - (axisHeight + 0.5 * ruleWidth) <
+ clearance) {
+ numShift +=
+ clearance - ((numShift - numer.depth) -
+ (axisHeight + 0.5 * ruleWidth));
+ }
+
+ if ((axisHeight - 0.5 * ruleWidth) - (denom.height - denomShift) <
+ clearance) {
+ denomShift +=
+ clearance - ((axisHeight - 0.5 * ruleWidth) -
+ (denom.height - denomShift));
+ }
+
+ var mid = makeSpan(
+ [options.style.reset(), Style.TEXT.cls(), "frac-line"]);
+ // Manually set the height of the line because its height is
+ // created in CSS
+ mid.height = ruleWidth;
+
+ var midShift = -(axisHeight - 0.5 * ruleWidth);
+
+ frac = buildCommon.makeVList([
+ {type: "elem", elem: denomreset, shift: denomShift},
+ {type: "elem", elem: mid, shift: midShift},
+ {type: "elem", elem: numerreset, shift: -numShift}
+ ], "individualShift", null, options);
+ }
+
+ // Since we manually change the style sometimes (with \dfrac or \tfrac),
+ // account for the possible size change here.
+ frac.height *= style.sizeMultiplier / options.style.sizeMultiplier;
+ frac.depth *= style.sizeMultiplier / options.style.sizeMultiplier;
+
+ // Rule 15e
+ var delimSize;
+ if (style.size === Style.DISPLAY.size) {
+ delimSize = style.metrics.delim1;
+ } else {
+ delimSize = style.metrics.delim2;
+ }
+
+ var leftDelim;
+ var rightDelim;
+ if (group.value.leftDelim == null) {
+ leftDelim = makeNullDelimiter(options, ["mopen"]);
+ } else {
+ leftDelim = delimiter.customSizedDelim(
+ group.value.leftDelim, delimSize, true,
+ options.withStyle(style), group.mode, ["mopen"]);
+ }
+ if (group.value.rightDelim == null) {
+ rightDelim = makeNullDelimiter(options, ["mclose"]);
+ } else {
+ rightDelim = delimiter.customSizedDelim(
+ group.value.rightDelim, delimSize, true,
+ options.withStyle(style), group.mode, ["mclose"]);
+ }
+
+ return makeSpan(
+ ["mord", options.style.reset(), style.cls()],
+ [leftDelim, makeSpan(["mfrac"], [frac]), rightDelim],
+ options);
+};
+
+var calculateSize = function(sizeValue, style) {
+ var x = sizeValue.number;
+ if (sizeValue.unit === "ex") {
+ x *= style.metrics.emPerEx;
+ } else if (sizeValue.unit === "mu") {
+ x /= 18;
+ }
+ return x;
+};
+
+groupTypes.array = function(group, options) {
+ var r;
+ var c;
+ var nr = group.value.body.length;
+ var nc = 0;
+ var body = new Array(nr);
+
+ var style = options.style;
+
+ // Horizontal spacing
+ var pt = 1 / fontMetrics.metrics.ptPerEm;
+ var arraycolsep = 5 * pt; // \arraycolsep in article.cls
+
+ // Vertical spacing
+ var baselineskip = 12 * pt; // see size10.clo
+ // Default \arraystretch from lttab.dtx
+ // TODO(gagern): may get redefined once we have user-defined macros
+ var arraystretch = utils.deflt(group.value.arraystretch, 1);
+ var arrayskip = arraystretch * baselineskip;
+ var arstrutHeight = 0.7 * arrayskip; // \strutbox in ltfsstrc.dtx and
+ var arstrutDepth = 0.3 * arrayskip; // \@arstrutbox in lttab.dtx
+
+ var totalHeight = 0;
+ for (r = 0; r < group.value.body.length; ++r) {
+ var inrow = group.value.body[r];
+ var height = arstrutHeight; // \@array adds an \@arstrut
+ var depth = arstrutDepth; // to each tow (via the template)
+
+ if (nc < inrow.length) {
+ nc = inrow.length;
+ }
+
+ var outrow = new Array(inrow.length);
+ for (c = 0; c < inrow.length; ++c) {
+ var elt = buildGroup(inrow[c], options);
+ if (depth < elt.depth) {
+ depth = elt.depth;
+ }
+ if (height < elt.height) {
+ height = elt.height;
+ }
+ outrow[c] = elt;
+ }
+
+ var gap = 0;
+ if (group.value.rowGaps[r]) {
+ gap = calculateSize(group.value.rowGaps[r].value, style);
+ if (gap > 0) { // \@argarraycr
+ gap += arstrutDepth;
+ if (depth < gap) {
+ depth = gap; // \@xargarraycr
+ }
+ gap = 0;
+ }
+ }
+
+ outrow.height = height;
+ outrow.depth = depth;
+ totalHeight += height;
+ outrow.pos = totalHeight;
+ totalHeight += depth + gap; // \@yargarraycr
+ body[r] = outrow;
+ }
+
+ var offset = totalHeight / 2 + style.metrics.axisHeight;
+ var colDescriptions = group.value.cols || [];
+ var cols = [];
+ var colSep;
+ var colDescrNum;
+ for (c = 0, colDescrNum = 0;
+ // Continue while either there are more columns or more column
+ // descriptions, so trailing separators don't get lost.
+ c < nc || colDescrNum < colDescriptions.length;
+ ++c, ++colDescrNum) {
+
+ var colDescr = colDescriptions[colDescrNum] || {};
+
+ var firstSeparator = true;
+ while (colDescr.type === "separator") {
+ // If there is more than one separator in a row, add a space
+ // between them.
+ if (!firstSeparator) {
+ colSep = makeSpan(["arraycolsep"], []);
+ colSep.style.width =
+ fontMetrics.metrics.doubleRuleSep + "em";
+ cols.push(colSep);
+ }
+
+ if (colDescr.separator === "|") {
+ var separator = makeSpan(
+ ["vertical-separator"],
+ []);
+ separator.style.height = totalHeight + "em";
+ separator.style.verticalAlign =
+ -(totalHeight - offset) + "em";
+
+ cols.push(separator);
+ } else {
+ throw new ParseError(
+ "Invalid separator type: " + colDescr.separator);
+ }
+
+ colDescrNum++;
+ colDescr = colDescriptions[colDescrNum] || {};
+ firstSeparator = false;
+ }
+
+ if (c >= nc) {
+ continue;
+ }
+
+ var sepwidth;
+ if (c > 0 || group.value.hskipBeforeAndAfter) {
+ sepwidth = utils.deflt(colDescr.pregap, arraycolsep);
+ if (sepwidth !== 0) {
+ colSep = makeSpan(["arraycolsep"], []);
+ colSep.style.width = sepwidth + "em";
+ cols.push(colSep);
+ }
+ }
+
+ var col = [];
+ for (r = 0; r < nr; ++r) {
+ var row = body[r];
+ var elem = row[c];
+ if (!elem) {
+ continue;
+ }
+ var shift = row.pos - offset;
+ elem.depth = row.depth;
+ elem.height = row.height;
+ col.push({type: "elem", elem: elem, shift: shift});
+ }
+
+ col = buildCommon.makeVList(col, "individualShift", null, options);
+ col = makeSpan(
+ ["col-align-" + (colDescr.align || "c")],
+ [col]);
+ cols.push(col);
+
+ if (c < nc - 1 || group.value.hskipBeforeAndAfter) {
+ sepwidth = utils.deflt(colDescr.postgap, arraycolsep);
+ if (sepwidth !== 0) {
+ colSep = makeSpan(["arraycolsep"], []);
+ colSep.style.width = sepwidth + "em";
+ cols.push(colSep);
+ }
+ }
+ }
+ body = makeSpan(["mtable"], cols);
+ return makeSpan(["mord"], [body], options);
+};
+
+groupTypes.spacing = function(group, options) {
+ if (group.value === "\\ " || group.value === "\\space" ||
+ group.value === " " || group.value === "~") {
+ // Spaces are generated by adding an actual space. Each of these
+ // things has an entry in the symbols table, so these will be turned
+ // into appropriate outputs.
+ if (group.mode === "text") {
+ return buildCommon.makeOrd(group, options, "textord");
+ } else {
+ return makeSpan(["mspace"],
+ [buildCommon.mathsym(group.value, group.mode, options)],
+ options);
+ }
+ } else {
+ // Other kinds of spaces are of arbitrary width. We use CSS to
+ // generate these.
+ return makeSpan(
+ ["mspace",
+ buildCommon.spacingFunctions[group.value].className],
+ [], options);
+ }
+};
+
+groupTypes.llap = function(group, options) {
+ var inner = makeSpan(
+ ["inner"], [buildGroup(group.value.body, options.reset())]);
+ var fix = makeSpan(["fix"], []);
+ return makeSpan(
+ ["mord", "llap", options.style.cls()], [inner, fix], options);
+};
+
+groupTypes.rlap = function(group, options) {
+ var inner = makeSpan(
+ ["inner"], [buildGroup(group.value.body, options.reset())]);
+ var fix = makeSpan(["fix"], []);
+ return makeSpan(
+ ["mord", "rlap", options.style.cls()], [inner, fix], options);
+};
+
+groupTypes.op = function(group, options) {
+ // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+ var supGroup;
+ var subGroup;
+ var hasLimits = false;
+ if (group.type === "supsub") {
+ // If we have limits, supsub will pass us its group to handle. Pull
+ // out the superscript and subscript and set the group to the op in
+ // its base.
+ supGroup = group.value.sup;
+ subGroup = group.value.sub;
+ group = group.value.base;
+ hasLimits = true;
+ }
+
+ var style = options.style;
+
+ // Most operators have a large successor symbol, but these don't.
+ var noSuccessor = [
+ "\\smallint"
+ ];
+
+ var large = false;
+ if (style.size === Style.DISPLAY.size &&
+ group.value.symbol &&
+ !utils.contains(noSuccessor, group.value.body)) {
+
+ // Most symbol operators get larger in displaystyle (rule 13)
+ large = true;
+ }
+
+ var base;
+ var baseShift = 0;
+ var slant = 0;
+ if (group.value.symbol) {
+ // If this is a symbol, create the symbol.
+ var fontName = large ? "Size2-Regular" : "Size1-Regular";
+ base = buildCommon.makeSymbol(
+ group.value.body, fontName, "math", options,
+ ["mop", "op-symbol", large ? "large-op" : "small-op"]);
+
+ // Shift the symbol so its center lies on the axis (rule 13). It
+ // appears that our fonts have the centers of the symbols already
+ // almost on the axis, so these numbers are very small. Note we
+ // don't actually apply this here, but instead it is used either in
+ // the vlist creation or separately when there are no limits.
+ baseShift = (base.height - base.depth) / 2 -
+ style.metrics.axisHeight * style.sizeMultiplier;
+
+ // The slant of the symbol is just its italic correction.
+ slant = base.italic;
+ } else if (group.value.value) {
+ // If this is a list, compose that list.
+ var inner = buildExpression(group.value.value, options, true);
+
+ base = makeSpan(["mop"], inner, options);
+ } else {
+ // Otherwise, this is a text operator. Build the text from the
+ // operator's name.
+ // TODO(emily): Add a space in the middle of some of these
+ // operators, like \limsup
+ var output = [];
+ for (var i = 1; i < group.value.body.length; i++) {
+ output.push(buildCommon.mathsym(group.value.body[i], group.mode));
+ }
+ base = makeSpan(["mop"], output, options);
+ }
+
+ if (hasLimits) {
+ // IE 8 clips \int if it is in a display: inline-block. We wrap it
+ // in a new span so it is an inline, and works.
+ base = makeSpan([], [base]);
+
+ var supmid;
+ var supKern;
+ var submid;
+ var subKern;
+ var newOptions;
+ // We manually have to handle the superscripts and subscripts. This,
+ // aside from the kern calculations, is copied from supsub.
+ if (supGroup) {
+ newOptions = options.withStyle(style.sup());
+ var sup = buildGroup(supGroup, newOptions);
+ supmid = makeSpan([style.reset(), style.sup().cls()],
+ [sup], newOptions);
+
+ supKern = Math.max(
+ fontMetrics.metrics.bigOpSpacing1,
+ fontMetrics.metrics.bigOpSpacing3 - sup.depth);
+ }
+
+ if (subGroup) {
+ newOptions = options.withStyle(style.sub());
+ var sub = buildGroup(subGroup, newOptions);
+ submid = makeSpan([style.reset(), style.sub().cls()],
+ [sub], newOptions);
+
+ subKern = Math.max(
+ fontMetrics.metrics.bigOpSpacing2,
+ fontMetrics.metrics.bigOpSpacing4 - sub.height);
+ }
+
+ // Build the final group as a vlist of the possible subscript, base,
+ // and possible superscript.
+ var finalGroup;
+ var top;
+ var bottom;
+ if (!supGroup) {
+ top = base.height - baseShift;
+
+ finalGroup = buildCommon.makeVList([
+ {type: "kern", size: fontMetrics.metrics.bigOpSpacing5},
+ {type: "elem", elem: submid},
+ {type: "kern", size: subKern},
+ {type: "elem", elem: base}
+ ], "top", top, options);
+
+ // Here, we shift the limits by the slant of the symbol. Note
+ // that we are supposed to shift the limits by 1/2 of the slant,
+ // but since we are centering the limits adding a full slant of
+ // margin will shift by 1/2 that.
+ finalGroup.children[0].style.marginLeft = -slant + "em";
+ } else if (!subGroup) {
+ bottom = base.depth + baseShift;
+
+ finalGroup = buildCommon.makeVList([
+ {type: "elem", elem: base},
+ {type: "kern", size: supKern},
+ {type: "elem", elem: supmid},
+ {type: "kern", size: fontMetrics.metrics.bigOpSpacing5}
+ ], "bottom", bottom, options);
+
+ // See comment above about slants
+ finalGroup.children[1].style.marginLeft = slant + "em";
+ } else if (!supGroup && !subGroup) {
+ // This case probably shouldn't occur (this would mean the
+ // supsub was sending us a group with no superscript or
+ // subscript) but be safe.
+ return base;
+ } else {
+ bottom = fontMetrics.metrics.bigOpSpacing5 +
+ submid.height + submid.depth +
+ subKern +
+ base.depth + baseShift;
+
+ finalGroup = buildCommon.makeVList([
+ {type: "kern", size: fontMetrics.metrics.bigOpSpacing5},
+ {type: "elem", elem: submid},
+ {type: "kern", size: subKern},
+ {type: "elem", elem: base},
+ {type: "kern", size: supKern},
+ {type: "elem", elem: supmid},
+ {type: "kern", size: fontMetrics.metrics.bigOpSpacing5}
+ ], "bottom", bottom, options);
+
+ // See comment above about slants
+ finalGroup.children[0].style.marginLeft = -slant + "em";
+ finalGroup.children[2].style.marginLeft = slant + "em";
+ }
+
+ return makeSpan(["mop", "op-limits"], [finalGroup], options);
+ } else {
+ if (group.value.symbol) {
+ base.style.top = baseShift + "em";
+ }
+
+ return base;
+ }
+};
+
+groupTypes.mod = function(group, options) {
+ var inner = [];
+
+ if (group.value.modType === "bmod") {
+ // “\nonscript\mskip-\medmuskip\mkern5mu”
+ if (!options.style.isTight()) {
+ inner.push(makeSpan(
+ ["mspace", "negativemediumspace"], [], options));
+ }
+ inner.push(makeSpan(["mspace", "thickspace"], [], options));
+ } else if (options.style.size === Style.DISPLAY.size) {
+ inner.push(makeSpan(["mspace", "quad"], [], options));
+ } else if (group.value.modType === "mod") {
+ inner.push(makeSpan(["mspace", "twelvemuspace"], [], options));
+ } else {
+ inner.push(makeSpan(["mspace", "eightmuspace"], [], options));
+ }
+
+ if (group.value.modType === "pod" || group.value.modType === "pmod") {
+ inner.push(buildCommon.mathsym("(", group.mode));
+ }
+
+ if (group.value.modType !== "pod") {
+ var modInner = [
+ buildCommon.mathsym("m", group.mode),
+ buildCommon.mathsym("o", group.mode),
+ buildCommon.mathsym("d", group.mode)];
+ if (group.value.modType === "bmod") {
+ inner.push(makeSpan(["mbin"], modInner, options));
+ // “\mkern5mu\nonscript\mskip-\medmuskip”
+ inner.push(makeSpan(["mspace", "thickspace"], [], options));
+ if (!options.style.isTight()) {
+ inner.push(makeSpan(
+ ["mspace", "negativemediumspace"], [], options));
+ }
+ } else {
+ Array.prototype.push.apply(inner, modInner);
+ inner.push(makeSpan(["mspace", "sixmuspace"], [], options));
+ }
+ }
+
+ if (group.value.value) {
+ Array.prototype.push.apply(inner,
+ buildExpression(group.value.value, options, false));
+ }
+
+ if (group.value.modType === "pod" || group.value.modType === "pmod") {
+ inner.push(buildCommon.mathsym(")", group.mode));
+ }
+
+ return buildCommon.makeFragment(inner);
+};
+
+groupTypes.katex = function(group, options) {
+ // The KaTeX logo. The offsets for the K and a were chosen to look
+ // good, but the offsets for the T, E, and X were taken from the
+ // definition of \TeX in TeX (see TeXbook pg. 356)
+ var k = makeSpan(
+ ["k"], [buildCommon.mathsym("K", group.mode)], options);
+ var a = makeSpan(
+ ["a"], [buildCommon.mathsym("A", group.mode)], options);
+
+ a.height = (a.height + 0.2) * 0.75;
+ a.depth = (a.height - 0.2) * 0.75;
+
+ var t = makeSpan(
+ ["t"], [buildCommon.mathsym("T", group.mode)], options);
+ var e = makeSpan(
+ ["e"], [buildCommon.mathsym("E", group.mode)], options);
+
+ e.height = (e.height - 0.2155);
+ e.depth = (e.depth + 0.2155);
+
+ var x = makeSpan(
+ ["x"], [buildCommon.mathsym("X", group.mode)], options);
+
+ return makeSpan(
+ ["mord", "katex-logo"], [k, a, t, e, x], options);
+};
+
+groupTypes.overline = function(group, options) {
+ // Overlines are handled in the TeXbook pg 443, Rule 9.
+ var style = options.style;
+
+ // Build the inner group in the cramped style.
+ var innerGroup = buildGroup(group.value.body,
+ options.withStyle(style.cramp()));
+
+ var ruleWidth = fontMetrics.metrics.defaultRuleThickness /
+ style.sizeMultiplier;
+
+ // Create the line above the body
+ var line = makeSpan(
+ [style.reset(), Style.TEXT.cls(), "overline-line"]);
+ line.height = ruleWidth;
+ line.maxFontSize = 1.0;
+
+ // Generate the vlist, with the appropriate kerns
+ var vlist = buildCommon.makeVList([
+ {type: "elem", elem: innerGroup},
+ {type: "kern", size: 3 * ruleWidth},
+ {type: "elem", elem: line},
+ {type: "kern", size: ruleWidth}
+ ], "firstBaseline", null, options);
+
+ return makeSpan(["mord", "overline"], [vlist], options);
+};
+
+groupTypes.underline = function(group, options) {
+ // Underlines are handled in the TeXbook pg 443, Rule 10.
+ var style = options.style;
+
+ // Build the inner group.
+ var innerGroup = buildGroup(group.value.body, options);
+
+ var ruleWidth = fontMetrics.metrics.defaultRuleThickness /
+ style.sizeMultiplier;
+
+ // Create the line above the body
+ var line = makeSpan([style.reset(), Style.TEXT.cls(), "underline-line"]);
+ line.height = ruleWidth;
+ line.maxFontSize = 1.0;
+
+ // Generate the vlist, with the appropriate kerns
+ var vlist = buildCommon.makeVList([
+ {type: "kern", size: ruleWidth},
+ {type: "elem", elem: line},
+ {type: "kern", size: 3 * ruleWidth},
+ {type: "elem", elem: innerGroup}
+ ], "top", innerGroup.height, options);
+
+ return makeSpan(["mord", "underline"], [vlist], options);
+};
+
+groupTypes.sqrt = function(group, options) {
+ // Square roots are handled in the TeXbook pg. 443, Rule 11.
+ var style = options.style;
+
+ // First, we do the same steps as in overline to build the inner group
+ // and line
+ var inner = buildGroup(group.value.body, options.withStyle(style.cramp()));
+
+ var ruleWidth = fontMetrics.metrics.defaultRuleThickness /
+ style.sizeMultiplier;
+
+ var line = makeSpan(
+ [style.reset(), Style.TEXT.cls(), "sqrt-line"], [],
+ options);
+ line.height = ruleWidth;
+ line.maxFontSize = 1.0;
+
+ var phi = ruleWidth;
+ if (style.id < Style.TEXT.id) {
+ phi = style.metrics.xHeight;
+ }
+
+ // Calculate the clearance between the body and line
+ var lineClearance = ruleWidth + phi / 4;
+
+ var innerHeight = (inner.height + inner.depth) * style.sizeMultiplier;
+ var minDelimiterHeight = innerHeight + lineClearance + ruleWidth;
+
+ // Create a \surd delimiter of the required minimum size
+ var delim = makeSpan(["sqrt-sign"], [
+ delimiter.customSizedDelim("\\surd", minDelimiterHeight,
+ false, options, group.mode)],
+ options);
+
+ var delimDepth = (delim.height + delim.depth) - ruleWidth;
+
+ // Adjust the clearance based on the delimiter size
+ if (delimDepth > inner.height + inner.depth + lineClearance) {
+ lineClearance =
+ (lineClearance + delimDepth - inner.height - inner.depth) / 2;
+ }
+
+ // Shift the delimiter so that its top lines up with the top of the line
+ var delimShift = -(inner.height + lineClearance + ruleWidth) + delim.height;
+ delim.style.top = delimShift + "em";
+ delim.height -= delimShift;
+ delim.depth += delimShift;
+
+ // We add a special case here, because even when `inner` is empty, we
+ // still get a line. So, we use a simple heuristic to decide if we
+ // should omit the body entirely. (note this doesn't work for something
+ // like `\sqrt{\rlap{x}}`, but if someone is doing that they deserve for
+ // it not to work.
+ var body;
+ if (inner.height === 0 && inner.depth === 0) {
+ body = makeSpan();
+ } else {
+ body = buildCommon.makeVList([
+ {type: "elem", elem: inner},
+ {type: "kern", size: lineClearance},
+ {type: "elem", elem: line},
+ {type: "kern", size: ruleWidth}
+ ], "firstBaseline", null, options);
+ }
+
+ if (!group.value.index) {
+ return makeSpan(["mord", "sqrt"], [delim, body], options);
+ } else {
+ // Handle the optional root index
+
+ // The index is always in scriptscript style
+ var newOptions = options.withStyle(Style.SCRIPTSCRIPT);
+ var root = buildGroup(group.value.index, newOptions);
+ var rootWrap = makeSpan(
+ [style.reset(), Style.SCRIPTSCRIPT.cls()],
+ [root],
+ newOptions);
+
+ // Figure out the height and depth of the inner part
+ var innerRootHeight = Math.max(delim.height, body.height);
+ var innerRootDepth = Math.max(delim.depth, body.depth);
+
+ // The amount the index is shifted by. This is taken from the TeX
+ // source, in the definition of `\r@@t`.
+ var toShift = 0.6 * (innerRootHeight - innerRootDepth);
+
+ // Build a VList with the superscript shifted up correctly
+ var rootVList = buildCommon.makeVList(
+ [{type: "elem", elem: rootWrap}],
+ "shift", -toShift, options);
+ // Add a class surrounding it so we can add on the appropriate
+ // kerning
+ var rootVListWrap = makeSpan(["root"], [rootVList]);
+
+ return makeSpan(["mord", "sqrt"],
+ [rootVListWrap, delim, body], options);
+ }
+};
+
+groupTypes.sizing = function(group, options) {
+ // Handle sizing operators like \Huge. Real TeX doesn't actually allow
+ // these functions inside of math expressions, so we do some special
+ // handling.
+ var inner = buildExpression(group.value.value,
+ options.withSize(group.value.size), false);
+
+ // Compute the correct maxFontSize.
+ var style = options.style;
+ var fontSize = buildCommon.sizingMultiplier[group.value.size];
+ fontSize = fontSize * style.sizeMultiplier;
+
+ // Add size-resetting classes to the inner list and set maxFontSize
+ // manually. Handle nested size changes.
+ for (var i = 0; i < inner.length; i++) {
+ var pos = utils.indexOf(inner[i].classes, "sizing");
+ if (pos < 0) {
+ inner[i].classes.push("sizing", "reset-" + options.size,
+ group.value.size, style.cls());
+ inner[i].maxFontSize = fontSize;
+ } else if (inner[i].classes[pos + 1] === "reset-" + group.value.size) {
+ // This is a nested size change: e.g., inner[i] is the "b" in
+ // `\Huge a \small b`. Override the old size (the `reset-` class)
+ // but not the new size.
+ inner[i].classes[pos + 1] = "reset-" + options.size;
+ }
+ }
+
+ return buildCommon.makeFragment(inner);
+};
+
+groupTypes.styling = function(group, options) {
+ // Style changes are handled in the TeXbook on pg. 442, Rule 3.
+
+ // Figure out what style we're changing to.
+ var styleMap = {
+ "display": Style.DISPLAY,
+ "text": Style.TEXT,
+ "script": Style.SCRIPT,
+ "scriptscript": Style.SCRIPTSCRIPT
+ };
+
+ var newStyle = styleMap[group.value.style];
+ var newOptions = options.withStyle(newStyle);
+
+ // Build the inner expression in the new style.
+ var inner = buildExpression(
+ group.value.value, newOptions, false);
+
+ // Add style-resetting classes to the inner list. Handle nested changes.
+ for (var i = 0; i < inner.length; i++) {
+ var pos = utils.indexOf(inner[i].classes, newStyle.reset());
+ if (pos < 0) {
+ inner[i].classes.push(options.style.reset(), newStyle.cls());
+ } else {
+ // This is a nested style change, as `\textstyle a\scriptstyle b`.
+ // Only override the old style (the reset class).
+ inner[i].classes[pos] = options.style.reset();
+ }
+ }
+
+ return new buildCommon.makeFragment(inner);
+};
+
+groupTypes.font = function(group, options) {
+ var font = group.value.font;
+ return buildGroup(group.value.body, options.withFont(font));
+};
+
+groupTypes.delimsizing = function(group, options) {
+ var delim = group.value.value;
+
+ if (delim === ".") {
+ // Empty delimiters still count as elements, even though they don't
+ // show anything.
+ return makeSpan([group.value.mclass]);
+ }
+
+ // Use delimiter.sizedDelim to generate the delimiter.
+ return delimiter.sizedDelim(
+ delim, group.value.size, options, group.mode,
+ [group.value.mclass]);
+};
+
+groupTypes.leftright = function(group, options) {
+ // Build the inner expression
+ var inner = buildExpression(group.value.body, options.reset(), true);
+
+ var innerHeight = 0;
+ var innerDepth = 0;
+ var hadMiddle = false;
+
+ // Calculate its height and depth
+ for (var i = 0; i < inner.length; i++) {
+ if (inner[i].isMiddle) {
+ hadMiddle = true;
+ } else {
+ innerHeight = Math.max(inner[i].height, innerHeight);
+ innerDepth = Math.max(inner[i].depth, innerDepth);
+ }
+ }
+
+ var style = options.style;
+
+ // The size of delimiters is the same, regardless of what style we are
+ // in. Thus, to correctly calculate the size of delimiter we need around
+ // a group, we scale down the inner size based on the size.
+ innerHeight *= style.sizeMultiplier;
+ innerDepth *= style.sizeMultiplier;
+
+ var leftDelim;
+ if (group.value.left === ".") {
+ // Empty delimiters in \left and \right make null delimiter spaces.
+ leftDelim = makeNullDelimiter(options, ["mopen"]);
+ } else {
+ // Otherwise, use leftRightDelim to generate the correct sized
+ // delimiter.
+ leftDelim = delimiter.leftRightDelim(
+ group.value.left, innerHeight, innerDepth, options,
+ group.mode, ["mopen"]);
+ }
+ // Add it to the beginning of the expression
+ inner.unshift(leftDelim);
+
+ // Handle middle delimiters
+ if (hadMiddle) {
+ for (i = 1; i < inner.length; i++) {
+ if (inner[i].isMiddle) {
+ // Apply the options that were active when \middle was called
+ inner[i] = delimiter.leftRightDelim(
+ inner[i].isMiddle.value, innerHeight, innerDepth,
+ inner[i].isMiddle.options, group.mode, []);
+ }
+ }
+ }
+
+ var rightDelim;
+ // Same for the right delimiter
+ if (group.value.right === ".") {
+ rightDelim = makeNullDelimiter(options, ["mclose"]);
+ } else {
+ rightDelim = delimiter.leftRightDelim(
+ group.value.right, innerHeight, innerDepth, options,
+ group.mode, ["mclose"]);
+ }
+ // Add it to the end of the expression.
+ inner.push(rightDelim);
+
+ return makeSpan(
+ ["minner", style.cls()], inner, options);
+};
+
+groupTypes.middle = function(group, options) {
+ var middleDelim;
+ if (group.value.value === ".") {
+ middleDelim = makeNullDelimiter(options, []);
+ } else {
+ middleDelim = delimiter.sizedDelim(
+ group.value.value, 1, options,
+ group.mode, []);
+ middleDelim.isMiddle = {value: group.value.value, options: options};
+ }
+ return middleDelim;
+};
+
+groupTypes.rule = function(group, options) {
+ // Make an empty span for the rule
+ var rule = makeSpan(["mord", "rule"], [], options);
+ var style = options.style;
+
+ // Calculate the shift, width, and height of the rule, and account for units
+ var shift = 0;
+ if (group.value.shift) {
+ shift = calculateSize(group.value.shift, style);
+ }
+
+ var width = calculateSize(group.value.width, style);
+ var height = calculateSize(group.value.height, style);
+
+ // The sizes of rules are absolute, so make it larger if we are in a
+ // smaller style.
+ shift /= style.sizeMultiplier;
+ width /= style.sizeMultiplier;
+ height /= style.sizeMultiplier;
+
+ // Style the rule to the right size
+ rule.style.borderRightWidth = width + "em";
+ rule.style.borderTopWidth = height + "em";
+ rule.style.bottom = shift + "em";
+
+ // Record the height and width
+ rule.width = width;
+ rule.height = height + shift;
+ rule.depth = -shift;
+
+ return rule;
+};
+
+groupTypes.kern = function(group, options) {
+ // Make an empty span for the rule
+ var rule = makeSpan(["mord", "rule"], [], options);
+ var style = options.style;
+
+ var dimension = 0;
+ if (group.value.dimension) {
+ dimension = calculateSize(group.value.dimension, style);
+ }
+
+ dimension /= style.sizeMultiplier;
+
+ rule.style.marginLeft = dimension + "em";
+
+ return rule;
+};
+
+groupTypes.accent = function(group, options) {
+ // Accents are handled in the TeXbook pg. 443, rule 12.
+ var base = group.value.base;
+ var style = options.style;
+
+ var supsubGroup;
+ if (group.type === "supsub") {
+ // If our base is a character box, and we have superscripts and
+ // subscripts, the supsub will defer to us. In particular, we want
+ // to attach the superscripts and subscripts to the inner body (so
+ // that the position of the superscripts and subscripts won't be
+ // affected by the height of the accent). We accomplish this by
+ // sticking the base of the accent into the base of the supsub, and
+ // rendering that, while keeping track of where the accent is.
+
+ // The supsub group is the group that was passed in
+ var supsub = group;
+ // The real accent group is the base of the supsub group
+ group = supsub.value.base;
+ // The character box is the base of the accent group
+ base = group.value.base;
+ // Stick the character box into the base of the supsub group
+ supsub.value.base = base;
+
+ // Rerender the supsub group with its new base, and store that
+ // result.
+ supsubGroup = buildGroup(
+ supsub, options.reset());
+ }
+
+ // Build the base group
+ var body = buildGroup(
+ base, options.withStyle(style.cramp()));
+
+ // Calculate the skew of the accent. This is based on the line "If the
+ // nucleus is not a single character, let s = 0; otherwise set s to the
+ // kern amount for the nucleus followed by the \skewchar of its font."
+ // Note that our skew metrics are just the kern between each character
+ // and the skewchar.
+ var skew;
+ if (isCharacterBox(base)) {
+ // If the base is a character box, then we want the skew of the
+ // innermost character. To do that, we find the innermost character:
+ var baseChar = getBaseElem(base);
+ // Then, we render its group to get the symbol inside it
+ var baseGroup = buildGroup(
+ baseChar, options.withStyle(style.cramp()));
+ // Finally, we pull the skew off of the symbol.
+ skew = baseGroup.skew;
+ // Note that we now throw away baseGroup, because the layers we
+ // removed with getBaseElem might contain things like \color which
+ // we can't get rid of.
+ // TODO(emily): Find a better way to get the skew
+ } else {
+ skew = 0;
+ }
+
+ // calculate the amount of space between the body and the accent
+ var clearance = Math.min(
+ body.height,
+ style.metrics.xHeight);
+
+ // Build the accent
+ var accent = buildCommon.makeSymbol(
+ group.value.accent, "Main-Regular", "math", options);
+ // Remove the italic correction of the accent, because it only serves to
+ // shift the accent over to a place we don't want.
+ accent.italic = 0;
+
+ // The \vec character that the fonts use is a combining character, and
+ // thus shows up much too far to the left. To account for this, we add a
+ // specific class which shifts the accent over to where we want it.
+ // TODO(emily): Fix this in a better way, like by changing the font
+ var vecClass = group.value.accent === "\\vec" ? "accent-vec" : null;
+
+ var accentBody = makeSpan(["accent-body", vecClass], [
+ makeSpan([], [accent])]);
+
+ accentBody = buildCommon.makeVList([
+ {type: "elem", elem: body},
+ {type: "kern", size: -clearance},
+ {type: "elem", elem: accentBody}
+ ], "firstBaseline", null, options);
+
+ // Shift the accent over by the skew. Note we shift by twice the skew
+ // because we are centering the accent, so by adding 2*skew to the left,
+ // we shift it to the right by 1*skew.
+ accentBody.children[1].style.marginLeft = 2 * skew + "em";
+
+ var accentWrap = makeSpan(["mord", "accent"], [accentBody], options);
+
+ if (supsubGroup) {
+ // Here, we replace the "base" child of the supsub with our newly
+ // generated accent.
+ supsubGroup.children[0] = accentWrap;
+
+ // Since we don't rerun the height calculation after replacing the
+ // accent, we manually recalculate height.
+ supsubGroup.height = Math.max(accentWrap.height, supsubGroup.height);
+
+ // Accents should always be ords, even when their innards are not.
+ supsubGroup.classes[0] = "mord";
+
+ return supsubGroup;
+ } else {
+ return accentWrap;
+ }
+};
+
+groupTypes.phantom = function(group, options) {
+ var elements = buildExpression(
+ group.value.value,
+ options.withPhantom(),
+ false
+ );
+
+ // \phantom isn't supposed to affect the elements it contains.
+ // See "color" for more details.
+ return new buildCommon.makeFragment(elements);
+};
+
+groupTypes.mclass = function(group, options) {
+ var elements = buildExpression(group.value.value, options, true);
+
+ return makeSpan([group.value.mclass], elements, options);
+};
+
+/**
+ * buildGroup is the function that takes a group and calls the correct groupType
+ * function for it. It also handles the interaction of size and style changes
+ * between parents and children.
+ */
+var buildGroup = function(group, options) {
+ if (!group) {
+ return makeSpan();
+ }
+
+ if (groupTypes[group.type]) {
+ // Call the groupTypes function
+ var groupNode = groupTypes[group.type](group, options);
+ var multiplier;
+
+ // If the style changed between the parent and the current group,
+ // account for the size difference
+ if (options.style !== options.parentStyle) {
+ multiplier = options.style.sizeMultiplier /
+ options.parentStyle.sizeMultiplier;
+
+ groupNode.height *= multiplier;
+ groupNode.depth *= multiplier;
+ }
+
+ // If the size changed between the parent and the current group, account
+ // for that size difference.
+ if (options.size !== options.parentSize) {
+ multiplier = buildCommon.sizingMultiplier[options.size] /
+ buildCommon.sizingMultiplier[options.parentSize];
+
+ groupNode.height *= multiplier;
+ groupNode.depth *= multiplier;
+ }
+
+ return groupNode;
+ } else {
+ throw new ParseError(
+ "Got group of unknown type: '" + group.type + "'");
+ }
+};
+
+/**
+ * Take an entire parse tree, and build it into an appropriate set of HTML
+ * nodes.
+ */
+var buildHTML = function(tree, options) {
+ // buildExpression is destructive, so we need to make a clone
+ // of the incoming tree so that it isn't accidentally changed
+ tree = JSON.parse(JSON.stringify(tree));
+
+ // Build the expression contained in the tree
+ var expression = buildExpression(tree, options, true);
+ var body = makeSpan(["base", options.style.cls()], expression, options);
+
+ // Add struts, which ensure that the top of the HTML element falls at the
+ // height of the expression, and the bottom of the HTML element falls at the
+ // depth of the expression.
+ var topStrut = makeSpan(["strut"]);
+ var bottomStrut = makeSpan(["strut", "bottom"]);
+
+ topStrut.style.height = body.height + "em";
+ bottomStrut.style.height = (body.height + body.depth) + "em";
+ // We'd like to use `vertical-align: top` but in IE 9 this lowers the
+ // baseline of the box to the bottom of this strut (instead staying in the
+ // normal place) so we use an absolute value for vertical-align instead
+ bottomStrut.style.verticalAlign = -body.depth + "em";
+
+ // Wrap the struts and body together
+ var htmlNode = makeSpan(["katex-html"], [topStrut, bottomStrut, body]);
+
+ htmlNode.setAttribute("aria-hidden", "true");
+
+ return htmlNode;
+};
+
+module.exports = buildHTML;
+
+},{"./ParseError":6,"./Style":9,"./buildCommon":10,"./delimiter":14,"./domTree":15,"./fontMetrics":17,"./utils":25}],12:[function(require,module,exports){
+/**
+ * This file converts a parse tree into a cooresponding MathML tree. The main
+ * entry point is the `buildMathML` function, which takes a parse tree from the
+ * parser.
+ */
+
+var buildCommon = require("./buildCommon");
+var fontMetrics = require("./fontMetrics");
+var mathMLTree = require("./mathMLTree");
+var ParseError = require("./ParseError");
+var symbols = require("./symbols");
+var utils = require("./utils");
+
+var makeSpan = buildCommon.makeSpan;
+var fontMap = buildCommon.fontMap;
+
+/**
+ * Takes a symbol and converts it into a MathML text node after performing
+ * optional replacement from symbols.js.
+ */
+var makeText = function(text, mode) {
+ if (symbols[mode][text] && symbols[mode][text].replace) {
+ text = symbols[mode][text].replace;
+ }
+
+ return new mathMLTree.TextNode(text);
+};
+
+/**
+ * Returns the math variant as a string or null if none is required.
+ */
+var getVariant = function(group, options) {
+ var font = options.font;
+ if (!font) {
+ return null;
+ }
+
+ var mode = group.mode;
+ if (font === "mathit") {
+ return "italic";
+ }
+
+ var value = group.value;
+ if (utils.contains(["\\imath", "\\jmath"], value)) {
+ return null;
+ }
+
+ if (symbols[mode][value] && symbols[mode][value].replace) {
+ value = symbols[mode][value].replace;
+ }
+
+ var fontName = fontMap[font].fontName;
+ if (fontMetrics.getCharacterMetrics(value, fontName)) {
+ return fontMap[options.font].variant;
+ }
+
+ return null;
+};
+
+/**
+ * Functions for handling the different types of groups found in the parse
+ * tree. Each function should take a parse group and return a MathML node.
+ */
+var groupTypes = {};
+
+groupTypes.mathord = function(group, options) {
+ var node = new mathMLTree.MathNode(
+ "mi",
+ [makeText(group.value, group.mode)]);
+
+ var variant = getVariant(group, options);
+ if (variant) {
+ node.setAttribute("mathvariant", variant);
+ }
+ return node;
+};
+
+groupTypes.textord = function(group, options) {
+ var text = makeText(group.value, group.mode);
+
+ var variant = getVariant(group, options) || "normal";
+
+ var node;
+ if (/[0-9]/.test(group.value)) {
+ // TODO(kevinb) merge adjacent <mn> nodes
+ // do it as a post processing step
+ node = new mathMLTree.MathNode("mn", [text]);
+ if (options.font) {
+ node.setAttribute("mathvariant", variant);
+ }
+ } else {
+ node = new mathMLTree.MathNode("mi", [text]);
+ node.setAttribute("mathvariant", variant);
+ }
+
+ return node;
+};
+
+groupTypes.bin = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.rel = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.open = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.close = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.inner = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ return node;
+};
+
+groupTypes.punct = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value, group.mode)]);
+
+ node.setAttribute("separator", "true");
+
+ return node;
+};
+
+groupTypes.ordgroup = function(group, options) {
+ var inner = buildExpression(group.value, options);
+
+ var node = new mathMLTree.MathNode("mrow", inner);
+
+ return node;
+};
+
+groupTypes.text = function(group, options) {
+ var inner = buildExpression(group.value.body, options);
+
+ var node = new mathMLTree.MathNode("mtext", inner);
+
+ return node;
+};
+
+groupTypes.color = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+
+ var node = new mathMLTree.MathNode("mstyle", inner);
+
+ node.setAttribute("mathcolor", group.value.color);
+
+ return node;
+};
+
+groupTypes.supsub = function(group, options) {
+ var children = [buildGroup(group.value.base, options)];
+
+ if (group.value.sub) {
+ children.push(buildGroup(group.value.sub, options));
+ }
+
+ if (group.value.sup) {
+ children.push(buildGroup(group.value.sup, options));
+ }
+
+ var nodeType;
+ if (!group.value.sub) {
+ nodeType = "msup";
+ } else if (!group.value.sup) {
+ nodeType = "msub";
+ } else {
+ nodeType = "msubsup";
+ }
+
+ var node = new mathMLTree.MathNode(nodeType, children);
+
+ return node;
+};
+
+groupTypes.genfrac = function(group, options) {
+ var node = new mathMLTree.MathNode(
+ "mfrac",
+ [buildGroup(group.value.numer, options),
+ buildGroup(group.value.denom, options)]);
+
+ if (!group.value.hasBarLine) {
+ node.setAttribute("linethickness", "0px");
+ }
+
+ if (group.value.leftDelim != null || group.value.rightDelim != null) {
+ var withDelims = [];
+
+ if (group.value.leftDelim != null) {
+ var leftOp = new mathMLTree.MathNode(
+ "mo", [new mathMLTree.TextNode(group.value.leftDelim)]);
+
+ leftOp.setAttribute("fence", "true");
+
+ withDelims.push(leftOp);
+ }
+
+ withDelims.push(node);
+
+ if (group.value.rightDelim != null) {
+ var rightOp = new mathMLTree.MathNode(
+ "mo", [new mathMLTree.TextNode(group.value.rightDelim)]);
+
+ rightOp.setAttribute("fence", "true");
+
+ withDelims.push(rightOp);
+ }
+
+ var outerNode = new mathMLTree.MathNode("mrow", withDelims);
+
+ return outerNode;
+ }
+
+ return node;
+};
+
+groupTypes.array = function(group, options) {
+ return new mathMLTree.MathNode(
+ "mtable", group.value.body.map(function(row) {
+ return new mathMLTree.MathNode(
+ "mtr", row.map(function(cell) {
+ return new mathMLTree.MathNode(
+ "mtd", [buildGroup(cell, options)]);
+ }));
+ }));
+};
+
+groupTypes.sqrt = function(group, options) {
+ var node;
+ if (group.value.index) {
+ node = new mathMLTree.MathNode(
+ "mroot", [
+ buildGroup(group.value.body, options),
+ buildGroup(group.value.index, options)
+ ]);
+ } else {
+ node = new mathMLTree.MathNode(
+ "msqrt", [buildGroup(group.value.body, options)]);
+ }
+
+ return node;
+};
+
+groupTypes.leftright = function(group, options) {
+ var inner = buildExpression(group.value.body, options);
+
+ if (group.value.left !== ".") {
+ var leftNode = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.left, group.mode)]);
+
+ leftNode.setAttribute("fence", "true");
+
+ inner.unshift(leftNode);
+ }
+
+ if (group.value.right !== ".") {
+ var rightNode = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.right, group.mode)]);
+
+ rightNode.setAttribute("fence", "true");
+
+ inner.push(rightNode);
+ }
+
+ var outerNode = new mathMLTree.MathNode("mrow", inner);
+
+ return outerNode;
+};
+
+groupTypes.middle = function(group, options) {
+ var middleNode = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.middle, group.mode)]);
+ middleNode.setAttribute("fence", "true");
+ return middleNode;
+};
+
+groupTypes.accent = function(group, options) {
+ var accentNode = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.accent, group.mode)]);
+
+ var node = new mathMLTree.MathNode(
+ "mover",
+ [buildGroup(group.value.base, options),
+ accentNode]);
+
+ node.setAttribute("accent", "true");
+
+ return node;
+};
+
+groupTypes.spacing = function(group) {
+ var node;
+
+ if (group.value === "\\ " || group.value === "\\space" ||
+ group.value === " " || group.value === "~") {
+ node = new mathMLTree.MathNode(
+ "mtext", [new mathMLTree.TextNode("\u00a0")]);
+ } else {
+ node = new mathMLTree.MathNode("mspace");
+
+ node.setAttribute(
+ "width", buildCommon.spacingFunctions[group.value].size);
+ }
+
+ return node;
+};
+
+groupTypes.op = function(group, options) {
+ var node;
+
+ // TODO(emily): handle big operators using the `largeop` attribute
+
+ if (group.value.symbol) {
+ // This is a symbol. Just add the symbol.
+ node = new mathMLTree.MathNode(
+ "mo", [makeText(group.value.body, group.mode)]);
+ } else if (group.value.value) {
+ // This is an operator with children. Add them.
+ node = new mathMLTree.MathNode(
+ "mo", buildExpression(group.value.value, options));
+ } else {
+ // This is a text operator. Add all of the characters from the
+ // operator's name.
+ // TODO(emily): Add a space in the middle of some of these
+ // operators, like \limsup.
+ node = new mathMLTree.MathNode(
+ "mi", [new mathMLTree.TextNode(group.value.body.slice(1))]);
+ }
+
+ return node;
+};
+
+groupTypes.mod = function(group, options) {
+ var inner = [];
+
+ if (group.value.modType === "pod" || group.value.modType === "pmod") {
+ inner.push(new mathMLTree.MathNode(
+ "mo", [makeText("(", group.mode)]));
+ }
+ if (group.value.modType !== "pod") {
+ inner.push(new mathMLTree.MathNode(
+ "mo", [makeText("mod", group.mode)]));
+ }
+ if (group.value.value) {
+ var space = new mathMLTree.MathNode("mspace");
+ space.setAttribute("width", "0.333333em");
+ inner.push(space);
+ inner = inner.concat(buildExpression(group.value.value, options));
+ }
+ if (group.value.modType === "pod" || group.value.modType === "pmod") {
+ inner.push(new mathMLTree.MathNode(
+ "mo", [makeText(")", group.mode)]));
+ }
+
+ return new mathMLTree.MathNode("mo", inner);
+};
+
+groupTypes.katex = function(group) {
+ var node = new mathMLTree.MathNode(
+ "mtext", [new mathMLTree.TextNode("KaTeX")]);
+
+ return node;
+};
+
+groupTypes.font = function(group, options) {
+ var font = group.value.font;
+ return buildGroup(group.value.body, options.withFont(font));
+};
+
+groupTypes.delimsizing = function(group) {
+ var children = [];
+
+ if (group.value.value !== ".") {
+ children.push(makeText(group.value.value, group.mode));
+ }
+
+ var node = new mathMLTree.MathNode("mo", children);
+
+ if (group.value.mclass === "mopen" ||
+ group.value.mclass === "mclose") {
+ // Only some of the delimsizing functions act as fences, and they
+ // return "mopen" or "mclose" mclass.
+ node.setAttribute("fence", "true");
+ } else {
+ // Explicitly disable fencing if it's not a fence, to override the
+ // defaults.
+ node.setAttribute("fence", "false");
+ }
+
+ return node;
+};
+
+groupTypes.styling = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+
+ var node = new mathMLTree.MathNode("mstyle", inner);
+
+ var styleAttributes = {
+ "display": ["0", "true"],
+ "text": ["0", "false"],
+ "script": ["1", "false"],
+ "scriptscript": ["2", "false"]
+ };
+
+ var attr = styleAttributes[group.value.style];
+
+ node.setAttribute("scriptlevel", attr[0]);
+ node.setAttribute("displaystyle", attr[1]);
+
+ return node;
+};
+
+groupTypes.sizing = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+
+ var node = new mathMLTree.MathNode("mstyle", inner);
+
+ // TODO(emily): This doesn't produce the correct size for nested size
+ // changes, because we don't keep state of what style we're currently
+ // in, so we can't reset the size to normal before changing it. Now
+ // that we're passing an options parameter we should be able to fix
+ // this.
+ node.setAttribute(
+ "mathsize", buildCommon.sizingMultiplier[group.value.size] + "em");
+
+ return node;
+};
+
+groupTypes.overline = function(group, options) {
+ var operator = new mathMLTree.MathNode(
+ "mo", [new mathMLTree.TextNode("\u203e")]);
+ operator.setAttribute("stretchy", "true");
+
+ var node = new mathMLTree.MathNode(
+ "mover",
+ [buildGroup(group.value.body, options),
+ operator]);
+ node.setAttribute("accent", "true");
+
+ return node;
+};
+
+groupTypes.underline = function(group, options) {
+ var operator = new mathMLTree.MathNode(
+ "mo", [new mathMLTree.TextNode("\u203e")]);
+ operator.setAttribute("stretchy", "true");
+
+ var node = new mathMLTree.MathNode(
+ "munder",
+ [buildGroup(group.value.body, options),
+ operator]);
+ node.setAttribute("accentunder", "true");
+
+ return node;
+};
+
+groupTypes.rule = function(group) {
+ // TODO(emily): Figure out if there's an actual way to draw black boxes
+ // in MathML.
+ var node = new mathMLTree.MathNode("mrow");
+
+ return node;
+};
+
+groupTypes.kern = function(group) {
+ // TODO(kevin): Figure out if there's a way to add space in MathML
+ var node = new mathMLTree.MathNode("mrow");
+
+ return node;
+};
+
+groupTypes.llap = function(group, options) {
+ var node = new mathMLTree.MathNode(
+ "mpadded", [buildGroup(group.value.body, options)]);
+
+ node.setAttribute("lspace", "-1width");
+ node.setAttribute("width", "0px");
+
+ return node;
+};
+
+groupTypes.rlap = function(group, options) {
+ var node = new mathMLTree.MathNode(
+ "mpadded", [buildGroup(group.value.body, options)]);
+
+ node.setAttribute("width", "0px");
+
+ return node;
+};
+
+groupTypes.phantom = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+ return new mathMLTree.MathNode("mphantom", inner);
+};
+
+groupTypes.mclass = function(group, options) {
+ var inner = buildExpression(group.value.value, options);
+ return new mathMLTree.MathNode("mstyle", inner);
+};
+
+/**
+ * Takes a list of nodes, builds them, and returns a list of the generated
+ * MathML nodes. A little simpler than the HTML version because we don't do any
+ * previous-node handling.
+ */
+var buildExpression = function(expression, options) {
+ var groups = [];
+ for (var i = 0; i < expression.length; i++) {
+ var group = expression[i];
+ groups.push(buildGroup(group, options));
+ }
+ return groups;
+};
+
+/**
+ * Takes a group from the parser and calls the appropriate groupTypes function
+ * on it to produce a MathML node.
+ */
+var buildGroup = function(group, options) {
+ if (!group) {
+ return new mathMLTree.MathNode("mrow");
+ }
+
+ if (groupTypes[group.type]) {
+ // Call the groupTypes function
+ return groupTypes[group.type](group, options);
+ } else {
+ throw new ParseError(
+ "Got group of unknown type: '" + group.type + "'");
+ }
+};
+
+/**
+ * Takes a full parse tree and settings and builds a MathML representation of
+ * it. In particular, we put the elements from building the parse tree into a
+ * <semantics> tag so we can also include that TeX source as an annotation.
+ *
+ * Note that we actually return a domTree element with a `<math>` inside it so
+ * we can do appropriate styling.
+ */
+var buildMathML = function(tree, texExpression, options) {
+ var expression = buildExpression(tree, options);
+
+ // Wrap up the expression in an mrow so it is presented in the semantics
+ // tag correctly.
+ var wrapper = new mathMLTree.MathNode("mrow", expression);
+
+ // Build a TeX annotation of the source
+ var annotation = new mathMLTree.MathNode(
+ "annotation", [new mathMLTree.TextNode(texExpression)]);
+
+ annotation.setAttribute("encoding", "application/x-tex");
+
+ var semantics = new mathMLTree.MathNode(
+ "semantics", [wrapper, annotation]);
+
+ var math = new mathMLTree.MathNode("math", [semantics]);
+
+ // You can't style <math> nodes, so we wrap the node in a span.
+ return makeSpan(["katex-mathml"], [math]);
+};
+
+module.exports = buildMathML;
+
+},{"./ParseError":6,"./buildCommon":10,"./fontMetrics":17,"./mathMLTree":20,"./symbols":23,"./utils":25}],13:[function(require,module,exports){
+var buildHTML = require("./buildHTML");
+var buildMathML = require("./buildMathML");
+var buildCommon = require("./buildCommon");
+var Options = require("./Options");
+var Settings = require("./Settings");
+var Style = require("./Style");
+
+var makeSpan = buildCommon.makeSpan;
+
+var buildTree = function(tree, expression, settings) {
+ settings = settings || new Settings({});
+
+ var startStyle = Style.TEXT;
+ if (settings.displayMode) {
+ startStyle = Style.DISPLAY;
+ }
+
+ // Setup the default options
+ var options = new Options({
+ style: startStyle,
+ size: "size5"
+ });
+
+ // `buildHTML` sometimes messes with the parse tree (like turning bins ->
+ // ords), so we build the MathML version first.
+ var mathMLNode = buildMathML(tree, expression, options);
+ var htmlNode = buildHTML(tree, options);
+
+ var katexNode = makeSpan(["katex"], [
+ mathMLNode, htmlNode
+ ]);
+
+ if (settings.displayMode) {
+ return makeSpan(["katex-display"], [katexNode]);
+ } else {
+ return katexNode;
+ }
+};
+
+module.exports = buildTree;
+
+},{"./Options":5,"./Settings":8,"./Style":9,"./buildCommon":10,"./buildHTML":11,"./buildMathML":12}],14:[function(require,module,exports){
+/**
+ * This file deals with creating delimiters of various sizes. The TeXbook
+ * discusses these routines on page 441-442, in the "Another subroutine sets box
+ * x to a specified variable delimiter" paragraph.
+ *
+ * There are three main routines here. `makeSmallDelim` makes a delimiter in the
+ * normal font, but in either text, script, or scriptscript style.
+ * `makeLargeDelim` makes a delimiter in textstyle, but in one of the Size1,
+ * Size2, Size3, or Size4 fonts. `makeStackedDelim` makes a delimiter out of
+ * smaller pieces that are stacked on top of one another.
+ *
+ * The functions take a parameter `center`, which determines if the delimiter
+ * should be centered around the axis.
+ *
+ * Then, there are three exposed functions. `sizedDelim` makes a delimiter in
+ * one of the given sizes. This is used for things like `\bigl`.
+ * `customSizedDelim` makes a delimiter with a given total height+depth. It is
+ * called in places like `\sqrt`. `leftRightDelim` makes an appropriate
+ * delimiter which surrounds an expression of a given height an depth. It is
+ * used in `\left` and `\right`.
+ */
+
+var ParseError = require("./ParseError");
+var Style = require("./Style");
+
+var buildCommon = require("./buildCommon");
+var fontMetrics = require("./fontMetrics");
+var symbols = require("./symbols");
+var utils = require("./utils");
+
+var makeSpan = buildCommon.makeSpan;
+
+/**
+ * Get the metrics for a given symbol and font, after transformation (i.e.
+ * after following replacement from symbols.js)
+ */
+var getMetrics = function(symbol, font) {
+ if (symbols.math[symbol] && symbols.math[symbol].replace) {
+ return fontMetrics.getCharacterMetrics(
+ symbols.math[symbol].replace, font);
+ } else {
+ return fontMetrics.getCharacterMetrics(
+ symbol, font);
+ }
+};
+
+/**
+ * Builds a symbol in the given font size (note size is an integer)
+ */
+var mathrmSize = function(value, size, mode, options) {
+ return buildCommon.makeSymbol(value, "Size" + size + "-Regular",
+ mode, options);
+};
+
+/**
+ * Puts a delimiter span in a given style, and adds appropriate height, depth,
+ * and maxFontSizes.
+ */
+var styleWrap = function(delim, toStyle, options, classes) {
+ classes = classes || [];
+ var span = makeSpan(
+ classes.concat(["style-wrap", options.style.reset(), toStyle.cls()]),
+ [delim], options);
+
+ var multiplier = toStyle.sizeMultiplier / options.style.sizeMultiplier;
+
+ span.height *= multiplier;
+ span.depth *= multiplier;
+ span.maxFontSize = toStyle.sizeMultiplier;
+
+ return span;
+};
+
+/**
+ * Makes a small delimiter. This is a delimiter that comes in the Main-Regular
+ * font, but is restyled to either be in textstyle, scriptstyle, or
+ * scriptscriptstyle.
+ */
+var makeSmallDelim = function(delim, style, center, options, mode, classes) {
+ var text = buildCommon.makeSymbol(delim, "Main-Regular", mode, options);
+
+ var span = styleWrap(text, style, options, classes);
+
+ if (center) {
+ var shift =
+ (1 - options.style.sizeMultiplier / style.sizeMultiplier) *
+ options.style.metrics.axisHeight;
+
+ span.style.top = shift + "em";
+ span.height -= shift;
+ span.depth += shift;
+ }
+
+ return span;
+};
+
+/**
+ * Makes a large delimiter. This is a delimiter that comes in the Size1, Size2,
+ * Size3, or Size4 fonts. It is always rendered in textstyle.
+ */
+var makeLargeDelim = function(delim, size, center, options, mode, classes) {
+ var inner = mathrmSize(delim, size, mode, options);
+
+ var span = styleWrap(
+ makeSpan(["delimsizing", "size" + size], [inner], options),
+ Style.TEXT, options, classes);
+
+ if (center) {
+ var shift = (1 - options.style.sizeMultiplier) *
+ options.style.metrics.axisHeight;
+
+ span.style.top = shift + "em";
+ span.height -= shift;
+ span.depth += shift;
+ }
+
+ return span;
+};
+
+/**
+ * Make an inner span with the given offset and in the given font. This is used
+ * in `makeStackedDelim` to make the stacking pieces for the delimiter.
+ */
+var makeInner = function(symbol, font, mode) {
+ var sizeClass;
+ // Apply the correct CSS class to choose the right font.
+ if (font === "Size1-Regular") {
+ sizeClass = "delim-size1";
+ } else if (font === "Size4-Regular") {
+ sizeClass = "delim-size4";
+ }
+
+ var inner = makeSpan(
+ ["delimsizinginner", sizeClass],
+ [makeSpan([], [buildCommon.makeSymbol(symbol, font, mode)])]);
+
+ // Since this will be passed into `makeVList` in the end, wrap the element
+ // in the appropriate tag that VList uses.
+ return {type: "elem", elem: inner};
+};
+
+/**
+ * Make a stacked delimiter out of a given delimiter, with the total height at
+ * least `heightTotal`. This routine is mentioned on page 442 of the TeXbook.
+ */
+var makeStackedDelim = function(delim, heightTotal, center, options, mode,
+ classes) {
+ // There are four parts, the top, an optional middle, a repeated part, and a
+ // bottom.
+ var top;
+ var middle;
+ var repeat;
+ var bottom;
+ top = repeat = bottom = delim;
+ middle = null;
+ // Also keep track of what font the delimiters are in
+ var font = "Size1-Regular";
+
+ // We set the parts and font based on the symbol. Note that we use
+ // '\u23d0' instead of '|' and '\u2016' instead of '\\|' for the
+ // repeats of the arrows
+ if (delim === "\\uparrow") {
+ repeat = bottom = "\u23d0";
+ } else if (delim === "\\Uparrow") {
+ repeat = bottom = "\u2016";
+ } else if (delim === "\\downarrow") {
+ top = repeat = "\u23d0";
+ } else if (delim === "\\Downarrow") {
+ top = repeat = "\u2016";
+ } else if (delim === "\\updownarrow") {
+ top = "\\uparrow";
+ repeat = "\u23d0";
+ bottom = "\\downarrow";
+ } else if (delim === "\\Updownarrow") {
+ top = "\\Uparrow";
+ repeat = "\u2016";
+ bottom = "\\Downarrow";
+ } else if (delim === "[" || delim === "\\lbrack") {
+ top = "\u23a1";
+ repeat = "\u23a2";
+ bottom = "\u23a3";
+ font = "Size4-Regular";
+ } else if (delim === "]" || delim === "\\rbrack") {
+ top = "\u23a4";
+ repeat = "\u23a5";
+ bottom = "\u23a6";
+ font = "Size4-Regular";
+ } else if (delim === "\\lfloor") {
+ repeat = top = "\u23a2";
+ bottom = "\u23a3";
+ font = "Size4-Regular";
+ } else if (delim === "\\lceil") {
+ top = "\u23a1";
+ repeat = bottom = "\u23a2";
+ font = "Size4-Regular";
+ } else if (delim === "\\rfloor") {
+ repeat = top = "\u23a5";
+ bottom = "\u23a6";
+ font = "Size4-Regular";
+ } else if (delim === "\\rceil") {
+ top = "\u23a4";
+ repeat = bottom = "\u23a5";
+ font = "Size4-Regular";
+ } else if (delim === "(") {
+ top = "\u239b";
+ repeat = "\u239c";
+ bottom = "\u239d";
+ font = "Size4-Regular";
+ } else if (delim === ")") {
+ top = "\u239e";
+ repeat = "\u239f";
+ bottom = "\u23a0";
+ font = "Size4-Regular";
+ } else if (delim === "\\{" || delim === "\\lbrace") {
+ top = "\u23a7";
+ middle = "\u23a8";
+ bottom = "\u23a9";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\}" || delim === "\\rbrace") {
+ top = "\u23ab";
+ middle = "\u23ac";
+ bottom = "\u23ad";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\lgroup") {
+ top = "\u23a7";
+ bottom = "\u23a9";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\rgroup") {
+ top = "\u23ab";
+ bottom = "\u23ad";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\lmoustache") {
+ top = "\u23a7";
+ bottom = "\u23ad";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\rmoustache") {
+ top = "\u23ab";
+ bottom = "\u23a9";
+ repeat = "\u23aa";
+ font = "Size4-Regular";
+ } else if (delim === "\\surd") {
+ top = "\ue001";
+ bottom = "\u23b7";
+ repeat = "\ue000";
+ font = "Size4-Regular";
+ }
+
+ // Get the metrics of the four sections
+ var topMetrics = getMetrics(top, font);
+ var topHeightTotal = topMetrics.height + topMetrics.depth;
+ var repeatMetrics = getMetrics(repeat, font);
+ var repeatHeightTotal = repeatMetrics.height + repeatMetrics.depth;
+ var bottomMetrics = getMetrics(bottom, font);
+ var bottomHeightTotal = bottomMetrics.height + bottomMetrics.depth;
+ var middleHeightTotal = 0;
+ var middleFactor = 1;
+ if (middle !== null) {
+ var middleMetrics = getMetrics(middle, font);
+ middleHeightTotal = middleMetrics.height + middleMetrics.depth;
+ middleFactor = 2; // repeat symmetrically above and below middle
+ }
+
+ // Calcuate the minimal height that the delimiter can have.
+ // It is at least the size of the top, bottom, and optional middle combined.
+ var minHeight = topHeightTotal + bottomHeightTotal + middleHeightTotal;
+
+ // Compute the number of copies of the repeat symbol we will need
+ var repeatCount = Math.ceil(
+ (heightTotal - minHeight) / (middleFactor * repeatHeightTotal));
+
+ // Compute the total height of the delimiter including all the symbols
+ var realHeightTotal =
+ minHeight + repeatCount * middleFactor * repeatHeightTotal;
+
+ // The center of the delimiter is placed at the center of the axis. Note
+ // that in this context, "center" means that the delimiter should be
+ // centered around the axis in the current style, while normally it is
+ // centered around the axis in textstyle.
+ var axisHeight = options.style.metrics.axisHeight;
+ if (center) {
+ axisHeight *= options.style.sizeMultiplier;
+ }
+ // Calculate the depth
+ var depth = realHeightTotal / 2 - axisHeight;
+
+ // Now, we start building the pieces that will go into the vlist
+
+ // Keep a list of the inner pieces
+ var inners = [];
+
+ // Add the bottom symbol
+ inners.push(makeInner(bottom, font, mode));
+
+ var i;
+ if (middle === null) {
+ // Add that many symbols
+ for (i = 0; i < repeatCount; i++) {
+ inners.push(makeInner(repeat, font, mode));
+ }
+ } else {
+ // When there is a middle bit, we need the middle part and two repeated
+ // sections
+ for (i = 0; i < repeatCount; i++) {
+ inners.push(makeInner(repeat, font, mode));
+ }
+ inners.push(makeInner(middle, font, mode));
+ for (i = 0; i < repeatCount; i++) {
+ inners.push(makeInner(repeat, font, mode));
+ }
+ }
+
+ // Add the top symbol
+ inners.push(makeInner(top, font, mode));
+
+ // Finally, build the vlist
+ var inner = buildCommon.makeVList(inners, "bottom", depth, options);
+
+ return styleWrap(
+ makeSpan(["delimsizing", "mult"], [inner], options),
+ Style.TEXT, options, classes);
+};
+
+// There are three kinds of delimiters, delimiters that stack when they become
+// too large
+var stackLargeDelimiters = [
+ "(", ")", "[", "\\lbrack", "]", "\\rbrack",
+ "\\{", "\\lbrace", "\\}", "\\rbrace",
+ "\\lfloor", "\\rfloor", "\\lceil", "\\rceil",
+ "\\surd"
+];
+
+// delimiters that always stack
+var stackAlwaysDelimiters = [
+ "\\uparrow", "\\downarrow", "\\updownarrow",
+ "\\Uparrow", "\\Downarrow", "\\Updownarrow",
+ "|", "\\|", "\\vert", "\\Vert",
+ "\\lvert", "\\rvert", "\\lVert", "\\rVert",
+ "\\lgroup", "\\rgroup", "\\lmoustache", "\\rmoustache"
+];
+
+// and delimiters that never stack
+var stackNeverDelimiters = [
+ "<", ">", "\\langle", "\\rangle", "/", "\\backslash", "\\lt", "\\gt"
+];
+
+// Metrics of the different sizes. Found by looking at TeX's output of
+// $\bigl| // \Bigl| \biggl| \Biggl| \showlists$
+// Used to create stacked delimiters of appropriate sizes in makeSizedDelim.
+var sizeToMaxHeight = [0, 1.2, 1.8, 2.4, 3.0];
+
+/**
+ * Used to create a delimiter of a specific size, where `size` is 1, 2, 3, or 4.
+ */
+var makeSizedDelim = function(delim, size, options, mode, classes) {
+ // < and > turn into \langle and \rangle in delimiters
+ if (delim === "<" || delim === "\\lt") {
+ delim = "\\langle";
+ } else if (delim === ">" || delim === "\\gt") {
+ delim = "\\rangle";
+ }
+
+ // Sized delimiters are never centered.
+ if (utils.contains(stackLargeDelimiters, delim) ||
+ utils.contains(stackNeverDelimiters, delim)) {
+ return makeLargeDelim(delim, size, false, options, mode, classes);
+ } else if (utils.contains(stackAlwaysDelimiters, delim)) {
+ return makeStackedDelim(
+ delim, sizeToMaxHeight[size], false, options, mode, classes);
+ } else {
+ throw new ParseError("Illegal delimiter: '" + delim + "'");
+ }
+};
+
+/**
+ * There are three different sequences of delimiter sizes that the delimiters
+ * follow depending on the kind of delimiter. This is used when creating custom
+ * sized delimiters to decide whether to create a small, large, or stacked
+ * delimiter.
+ *
+ * In real TeX, these sequences aren't explicitly defined, but are instead
+ * defined inside the font metrics. Since there are only three sequences that
+ * are possible for the delimiters that TeX defines, it is easier to just encode
+ * them explicitly here.
+ */
+
+// Delimiters that never stack try small delimiters and large delimiters only
+var stackNeverDelimiterSequence = [
+ {type: "small", style: Style.SCRIPTSCRIPT},
+ {type: "small", style: Style.SCRIPT},
+ {type: "small", style: Style.TEXT},
+ {type: "large", size: 1},
+ {type: "large", size: 2},
+ {type: "large", size: 3},
+ {type: "large", size: 4}
+];
+
+// Delimiters that always stack try the small delimiters first, then stack
+var stackAlwaysDelimiterSequence = [
+ {type: "small", style: Style.SCRIPTSCRIPT},
+ {type: "small", style: Style.SCRIPT},
+ {type: "small", style: Style.TEXT},
+ {type: "stack"}
+];
+
+// Delimiters that stack when large try the small and then large delimiters, and
+// stack afterwards
+var stackLargeDelimiterSequence = [
+ {type: "small", style: Style.SCRIPTSCRIPT},
+ {type: "small", style: Style.SCRIPT},
+ {type: "small", style: Style.TEXT},
+ {type: "large", size: 1},
+ {type: "large", size: 2},
+ {type: "large", size: 3},
+ {type: "large", size: 4},
+ {type: "stack"}
+];
+
+/**
+ * Get the font used in a delimiter based on what kind of delimiter it is.
+ */
+var delimTypeToFont = function(type) {
+ if (type.type === "small") {
+ return "Main-Regular";
+ } else if (type.type === "large") {
+ return "Size" + type.size + "-Regular";
+ } else if (type.type === "stack") {
+ return "Size4-Regular";
+ }
+};
+
+/**
+ * Traverse a sequence of types of delimiters to decide what kind of delimiter
+ * should be used to create a delimiter of the given height+depth.
+ */
+var traverseSequence = function(delim, height, sequence, options) {
+ // Here, we choose the index we should start at in the sequences. In smaller
+ // sizes (which correspond to larger numbers in style.size) we start earlier
+ // in the sequence. Thus, scriptscript starts at index 3-3=0, script starts
+ // at index 3-2=1, text starts at 3-1=2, and display starts at min(2,3-0)=2
+ var start = Math.min(2, 3 - options.style.size);
+ for (var i = start; i < sequence.length; i++) {
+ if (sequence[i].type === "stack") {
+ // This is always the last delimiter, so we just break the loop now.
+ break;
+ }
+
+ var metrics = getMetrics(delim, delimTypeToFont(sequence[i]));
+ var heightDepth = metrics.height + metrics.depth;
+
+ // Small delimiters are scaled down versions of the same font, so we
+ // account for the style change size.
+
+ if (sequence[i].type === "small") {
+ heightDepth *= sequence[i].style.sizeMultiplier;
+ }
+
+ // Check if the delimiter at this size works for the given height.
+ if (heightDepth > height) {
+ return sequence[i];
+ }
+ }
+
+ // If we reached the end of the sequence, return the last sequence element.
+ return sequence[sequence.length - 1];
+};
+
+/**
+ * Make a delimiter of a given height+depth, with optional centering. Here, we
+ * traverse the sequences, and create a delimiter that the sequence tells us to.
+ */
+var makeCustomSizedDelim = function(delim, height, center, options, mode,
+ classes) {
+ if (delim === "<" || delim === "\\lt") {
+ delim = "\\langle";
+ } else if (delim === ">" || delim === "\\gt") {
+ delim = "\\rangle";
+ }
+
+ // Decide what sequence to use
+ var sequence;
+ if (utils.contains(stackNeverDelimiters, delim)) {
+ sequence = stackNeverDelimiterSequence;
+ } else if (utils.contains(stackLargeDelimiters, delim)) {
+ sequence = stackLargeDelimiterSequence;
+ } else {
+ sequence = stackAlwaysDelimiterSequence;
+ }
+
+ // Look through the sequence
+ var delimType = traverseSequence(delim, height, sequence, options);
+
+ // Depending on the sequence element we decided on, call the appropriate
+ // function.
+ if (delimType.type === "small") {
+ return makeSmallDelim(delim, delimType.style, center, options, mode,
+ classes);
+ } else if (delimType.type === "large") {
+ return makeLargeDelim(delim, delimType.size, center, options, mode,
+ classes);
+ } else if (delimType.type === "stack") {
+ return makeStackedDelim(delim, height, center, options, mode, classes);
+ }
+};
+
+/**
+ * Make a delimiter for use with `\left` and `\right`, given a height and depth
+ * of an expression that the delimiters surround.
+ */
+var makeLeftRightDelim = function(delim, height, depth, options, mode,
+ classes) {
+ // We always center \left/\right delimiters, so the axis is always shifted
+ var axisHeight =
+ options.style.metrics.axisHeight * options.style.sizeMultiplier;
+
+ // Taken from TeX source, tex.web, function make_left_right
+ var delimiterFactor = 901;
+ var delimiterExtend = 5.0 / fontMetrics.metrics.ptPerEm;
+
+ var maxDistFromAxis = Math.max(
+ height - axisHeight, depth + axisHeight);
+
+ var totalHeight = Math.max(
+ // In real TeX, calculations are done using integral values which are
+ // 65536 per pt, or 655360 per em. So, the division here truncates in
+ // TeX but doesn't here, producing different results. If we wanted to
+ // exactly match TeX's calculation, we could do
+ // Math.floor(655360 * maxDistFromAxis / 500) *
+ // delimiterFactor / 655360
+ // (To see the difference, compare
+ // x^{x^{\left(\rule{0.1em}{0.68em}\right)}}
+ // in TeX and KaTeX)
+ maxDistFromAxis / 500 * delimiterFactor,
+ 2 * maxDistFromAxis - delimiterExtend);
+
+ // Finally, we defer to `makeCustomSizedDelim` with our calculated total
+ // height
+ return makeCustomSizedDelim(delim, totalHeight, true, options, mode,
+ classes);
+};
+
+module.exports = {
+ sizedDelim: makeSizedDelim,
+ customSizedDelim: makeCustomSizedDelim,
+ leftRightDelim: makeLeftRightDelim
+};
+
+},{"./ParseError":6,"./Style":9,"./buildCommon":10,"./fontMetrics":17,"./symbols":23,"./utils":25}],15:[function(require,module,exports){
+/**
+ * These objects store the data about the DOM nodes we create, as well as some
+ * extra data. They can then be transformed into real DOM nodes with the
+ * `toNode` function or HTML markup using `toMarkup`. They are useful for both
+ * storing extra properties on the nodes, as well as providing a way to easily
+ * work with the DOM.
+ *
+ * Similar functions for working with MathML nodes exist in mathMLTree.js.
+ */
+var unicodeRegexes = require("./unicodeRegexes");
+var utils = require("./utils");
+
+/**
+ * Create an HTML className based on a list of classes. In addition to joining
+ * with spaces, we also remove null or empty classes.
+ */
+var createClass = function(classes) {
+ classes = classes.slice();
+ for (var i = classes.length - 1; i >= 0; i--) {
+ if (!classes[i]) {
+ classes.splice(i, 1);
+ }
+ }
+
+ return classes.join(" ");
+};
+
+/**
+ * This node represents a span node, with a className, a list of children, and
+ * an inline style. It also contains information about its height, depth, and
+ * maxFontSize.
+ */
+function span(classes, children, options) {
+ this.classes = classes || [];
+ this.children = children || [];
+ this.height = 0;
+ this.depth = 0;
+ this.maxFontSize = 0;
+ this.style = {};
+ this.attributes = {};
+ if (options) {
+ if (options.style.isTight()) {
+ this.classes.push("mtight");
+ }
+ if (options.getColor()) {
+ this.style.color = options.getColor();
+ }
+ }
+}
+
+/**
+ * Sets an arbitrary attribute on the span. Warning: use this wisely. Not all
+ * browsers support attributes the same, and having too many custom attributes
+ * is probably bad.
+ */
+span.prototype.setAttribute = function(attribute, value) {
+ this.attributes[attribute] = value;
+};
+
+span.prototype.tryCombine = function(sibling) {
+ return false;
+};
+
+/**
+ * Convert the span into an HTML node
+ */
+span.prototype.toNode = function() {
+ var span = document.createElement("span");
+
+ // Apply the class
+ span.className = createClass(this.classes);
+
+ // Apply inline styles
+ for (var style in this.style) {
+ if (Object.prototype.hasOwnProperty.call(this.style, style)) {
+ span.style[style] = this.style[style];
+ }
+ }
+
+ // Apply attributes
+ for (var attr in this.attributes) {
+ if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+ span.setAttribute(attr, this.attributes[attr]);
+ }
+ }
+
+ // Append the children, also as HTML nodes
+ for (var i = 0; i < this.children.length; i++) {
+ span.appendChild(this.children[i].toNode());
+ }
+
+ return span;
+};
+
+/**
+ * Convert the span into an HTML markup string
+ */
+span.prototype.toMarkup = function() {
+ var markup = "<span";
+
+ // Add the class
+ if (this.classes.length) {
+ markup += " class=\"";
+ markup += utils.escape(createClass(this.classes));
+ markup += "\"";
+ }
+
+ var styles = "";
+
+ // Add the styles, after hyphenation
+ for (var style in this.style) {
+ if (this.style.hasOwnProperty(style)) {
+ styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+ }
+ }
+
+ if (styles) {
+ markup += " style=\"" + utils.escape(styles) + "\"";
+ }
+
+ // Add the attributes
+ for (var attr in this.attributes) {
+ if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+ markup += " " + attr + "=\"";
+ markup += utils.escape(this.attributes[attr]);
+ markup += "\"";
+ }
+ }
+
+ markup += ">";
+
+ // Add the markup of the children, also as markup
+ for (var i = 0; i < this.children.length; i++) {
+ markup += this.children[i].toMarkup();
+ }
+
+ markup += "</span>";
+
+ return markup;
+};
+
+/**
+ * This node represents a document fragment, which contains elements, but when
+ * placed into the DOM doesn't have any representation itself. Thus, it only
+ * contains children and doesn't have any HTML properties. It also keeps track
+ * of a height, depth, and maxFontSize.
+ */
+function documentFragment(children) {
+ this.children = children || [];
+ this.height = 0;
+ this.depth = 0;
+ this.maxFontSize = 0;
+}
+
+/**
+ * Convert the fragment into a node
+ */
+documentFragment.prototype.toNode = function() {
+ // Create a fragment
+ var frag = document.createDocumentFragment();
+
+ // Append the children
+ for (var i = 0; i < this.children.length; i++) {
+ frag.appendChild(this.children[i].toNode());
+ }
+
+ return frag;
+};
+
+/**
+ * Convert the fragment into HTML markup
+ */
+documentFragment.prototype.toMarkup = function() {
+ var markup = "";
+
+ // Simply concatenate the markup for the children together
+ for (var i = 0; i < this.children.length; i++) {
+ markup += this.children[i].toMarkup();
+ }
+
+ return markup;
+};
+
+var iCombinations = {
+ 'î': '\u0131\u0302',
+ 'ï': '\u0131\u0308',
+ 'í': '\u0131\u0301',
+ // 'ī': '\u0131\u0304', // enable when we add Extended Latin
+ 'ì': '\u0131\u0300'
+};
+
+/**
+ * A symbol node contains information about a single symbol. It either renders
+ * to a single text node, or a span with a single text node in it, depending on
+ * whether it has CSS classes, styles, or needs italic correction.
+ */
+function symbolNode(value, height, depth, italic, skew, classes, style) {
+ this.value = value || "";
+ this.height = height || 0;
+ this.depth = depth || 0;
+ this.italic = italic || 0;
+ this.skew = skew || 0;
+ this.classes = classes || [];
+ this.style = style || {};
+ this.maxFontSize = 0;
+
+ // Mark CJK characters with specific classes so that we can specify which
+ // fonts to use. This allows us to render these characters with a serif
+ // font in situations where the browser would either default to a sans serif
+ // or render a placeholder character.
+ if (unicodeRegexes.cjkRegex.test(value)) {
+ // I couldn't find any fonts that contained Hangul as well as all of
+ // the other characters we wanted to test there for it gets its own
+ // CSS class.
+ if (unicodeRegexes.hangulRegex.test(value)) {
+ this.classes.push('hangul_fallback');
+ } else {
+ this.classes.push('cjk_fallback');
+ }
+ }
+
+ if (/[îïíì]/.test(this.value)) { // add ī when we add Extended Latin
+ this.value = iCombinations[this.value];
+ }
+}
+
+symbolNode.prototype.tryCombine = function(sibling) {
+ if (!sibling
+ || !(sibling instanceof symbolNode)
+ || this.italic > 0
+ || createClass(this.classes) !== createClass(sibling.classes)
+ || this.skew !== sibling.skew
+ || this.maxFontSize !== sibling.maxFontSize) {
+ return false;
+ }
+ for (var style in this.style) {
+ if (this.style.hasOwnProperty(style)
+ && this.style[style] !== sibling.style[style]) {
+ return false;
+ }
+ }
+ for (style in sibling.style) {
+ if (sibling.style.hasOwnProperty(style)
+ && this.style[style] !== sibling.style[style]) {
+ return false;
+ }
+ }
+ this.value += sibling.value;
+ this.height = Math.max(this.height, sibling.height);
+ this.depth = Math.max(this.depth, sibling.depth);
+ this.italic = sibling.italic;
+ return true;
+};
+
+/**
+ * Creates a text node or span from a symbol node. Note that a span is only
+ * created if it is needed.
+ */
+symbolNode.prototype.toNode = function() {
+ var node = document.createTextNode(this.value);
+ var span = null;
+
+ if (this.italic > 0) {
+ span = document.createElement("span");
+ span.style.marginRight = this.italic + "em";
+ }
+
+ if (this.classes.length > 0) {
+ span = span || document.createElement("span");
+ span.className = createClass(this.classes);
+ }
+
+ for (var style in this.style) {
+ if (this.style.hasOwnProperty(style)) {
+ span = span || document.createElement("span");
+ span.style[style] = this.style[style];
+ }
+ }
+
+ if (span) {
+ span.appendChild(node);
+ return span;
+ } else {
+ return node;
+ }
+};
+
+/**
+ * Creates markup for a symbol node.
+ */
+symbolNode.prototype.toMarkup = function() {
+ // TODO(alpert): More duplication than I'd like from
+ // span.prototype.toMarkup and symbolNode.prototype.toNode...
+ var needsSpan = false;
+
+ var markup = "<span";
+
+ if (this.classes.length) {
+ needsSpan = true;
+ markup += " class=\"";
+ markup += utils.escape(createClass(this.classes));
+ markup += "\"";
+ }
+
+ var styles = "";
+
+ if (this.italic > 0) {
+ styles += "margin-right:" + this.italic + "em;";
+ }
+ for (var style in this.style) {
+ if (this.style.hasOwnProperty(style)) {
+ styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+ }
+ }
+
+ if (styles) {
+ needsSpan = true;
+ markup += " style=\"" + utils.escape(styles) + "\"";
+ }
+
+ var escaped = utils.escape(this.value);
+ if (needsSpan) {
+ markup += ">";
+ markup += escaped;
+ markup += "</span>";
+ return markup;
+ } else {
+ return escaped;
+ }
+};
+
+module.exports = {
+ span: span,
+ documentFragment: documentFragment,
+ symbolNode: symbolNode
+};
+
+},{"./unicodeRegexes":24,"./utils":25}],16:[function(require,module,exports){
+/* eslint no-constant-condition:0 */
+var parseData = require("./parseData");
+var ParseError = require("./ParseError");
+var Style = require("./Style");
+
+var ParseNode = parseData.ParseNode;
+
+/**
+ * Parse the body of the environment, with rows delimited by \\ and
+ * columns delimited by &, and create a nested list in row-major order
+ * with one group per cell.
+ */
+function parseArray(parser, result) {
+ var row = [];
+ var body = [row];
+ var rowGaps = [];
+ while (true) {
+ var cell = parser.parseExpression(false, null);
+ row.push(new ParseNode("ordgroup", cell, parser.mode));
+ var next = parser.nextToken.text;
+ if (next === "&") {
+ parser.consume();
+ } else if (next === "\\end") {
+ break;
+ } else if (next === "\\\\" || next === "\\cr") {
+ var cr = parser.parseFunction();
+ rowGaps.push(cr.value.size);
+ row = [];
+ body.push(row);
+ } else {
+ throw new ParseError("Expected & or \\\\ or \\end",
+ parser.nextToken);
+ }
+ }
+ result.body = body;
+ result.rowGaps = rowGaps;
+ return new ParseNode(result.type, result, parser.mode);
+}
+
+/*
+ * An environment definition is very similar to a function definition:
+ * it is declared with a name or a list of names, a set of properties
+ * and a handler containing the actual implementation.
+ *
+ * The properties include:
+ * - numArgs: The number of arguments after the \begin{name} function.
+ * - argTypes: (optional) Just like for a function
+ * - allowedInText: (optional) Whether or not the environment is allowed inside
+ * text mode (default false) (not enforced yet)
+ * - numOptionalArgs: (optional) Just like for a function
+ * A bare number instead of that object indicates the numArgs value.
+ *
+ * The handler function will receive two arguments
+ * - context: information and references provided by the parser
+ * - args: an array of arguments passed to \begin{name}
+ * The context contains the following properties:
+ * - envName: the name of the environment, one of the listed names.
+ * - parser: the parser object
+ * - lexer: the lexer object
+ * - positions: the positions associated with these arguments from args.
+ * The handler must return a ParseResult.
+ */
+
+function defineEnvironment(names, props, handler) {
+ if (typeof names === "string") {
+ names = [names];
+ }
+ if (typeof props === "number") {
+ props = { numArgs: props };
+ }
+ // Set default values of environments
+ var data = {
+ numArgs: props.numArgs || 0,
+ argTypes: props.argTypes,
+ greediness: 1,
+ allowedInText: !!props.allowedInText,
+ numOptionalArgs: props.numOptionalArgs || 0,
+ handler: handler
+ };
+ for (var i = 0; i < names.length; ++i) {
+ module.exports[names[i]] = data;
+ }
+}
+
+// Arrays are part of LaTeX, defined in lttab.dtx so its documentation
+// is part of the source2e.pdf file of LaTeX2e source documentation.
+defineEnvironment("array", {
+ numArgs: 1
+}, function(context, args) {
+ var colalign = args[0];
+ colalign = colalign.value.map ? colalign.value : [colalign];
+ var cols = colalign.map(function(node) {
+ var ca = node.value;
+ if ("lcr".indexOf(ca) !== -1) {
+ return {
+ type: "align",
+ align: ca
+ };
+ } else if (ca === "|") {
+ return {
+ type: "separator",
+ separator: "|"
+ };
+ }
+ throw new ParseError(
+ "Unknown column alignment: " + node.value,
+ node);
+ });
+ var res = {
+ type: "array",
+ cols: cols,
+ hskipBeforeAndAfter: true // \@preamble in lttab.dtx
+ };
+ res = parseArray(context.parser, res);
+ return res;
+});
+
+// The matrix environments of amsmath builds on the array environment
+// of LaTeX, which is discussed above.
+defineEnvironment([
+ "matrix",
+ "pmatrix",
+ "bmatrix",
+ "Bmatrix",
+ "vmatrix",
+ "Vmatrix"
+], {
+}, function(context) {
+ var delimiters = {
+ "matrix": null,
+ "pmatrix": ["(", ")"],
+ "bmatrix": ["[", "]"],
+ "Bmatrix": ["\\{", "\\}"],
+ "vmatrix": ["|", "|"],
+ "Vmatrix": ["\\Vert", "\\Vert"]
+ }[context.envName];
+ var res = {
+ type: "array",
+ hskipBeforeAndAfter: false // \hskip -\arraycolsep in amsmath
+ };
+ res = parseArray(context.parser, res);
+ if (delimiters) {
+ res = new ParseNode("leftright", {
+ body: [res],
+ left: delimiters[0],
+ right: delimiters[1]
+ }, context.mode);
+ }
+ return res;
+});
+
+// A cases environment (in amsmath.sty) is almost equivalent to
+// \def\arraystretch{1.2}%
+// \left\{\begin{array}{@{}l@{\quad}l@{}} … \end{array}\right.
+defineEnvironment("cases", {
+}, function(context) {
+ var res = {
+ type: "array",
+ arraystretch: 1.2,
+ cols: [{
+ type: "align",
+ align: "l",
+ pregap: 0,
+ // TODO(kevinb) get the current style.
+ // For now we use the metrics for TEXT style which is what we were
+ // doing before. Before attempting to get the current style we
+ // should look at TeX's behavior especially for \over and matrices.
+ postgap: Style.TEXT.metrics.quad
+ }, {
+ type: "align",
+ align: "l",
+ pregap: 0,
+ postgap: 0
+ }]
+ };
+ res = parseArray(context.parser, res);
+ res = new ParseNode("leftright", {
+ body: [res],
+ left: "\\{",
+ right: "."
+ }, context.mode);
+ return res;
+});
+
+// An aligned environment is like the align* environment
+// except it operates within math mode.
+// Note that we assume \nomallineskiplimit to be zero,
+// so that \strut@ is the same as \strut.
+defineEnvironment("aligned", {
+}, function(context) {
+ var res = {
+ type: "array",
+ cols: []
+ };
+ res = parseArray(context.parser, res);
+ var emptyGroup = new ParseNode("ordgroup", [], context.mode);
+ var numCols = 0;
+ res.value.body.forEach(function(row) {
+ var i;
+ for (i = 1; i < row.length; i += 2) {
+ row[i].value.unshift(emptyGroup);
+ }
+ if (numCols < row.length) {
+ numCols = row.length;
+ }
+ });
+ for (var i = 0; i < numCols; ++i) {
+ var align = "r";
+ var pregap = 0;
+ if (i % 2 === 1) {
+ align = "l";
+ } else if (i > 0) {
+ pregap = 2; // one \qquad between columns
+ }
+ res.value.cols[i] = {
+ type: "align",
+ align: align,
+ pregap: pregap,
+ postgap: 0
+ };
+ }
+ return res;
+});
+
+},{"./ParseError":6,"./Style":9,"./parseData":21}],17:[function(require,module,exports){
+/* eslint no-unused-vars:0 */
+
+var Style = require("./Style");
+var cjkRegex = require("./unicodeRegexes").cjkRegex;
+
+/**
+ * This file contains metrics regarding fonts and individual symbols. The sigma
+ * and xi variables, as well as the metricMap map contain data extracted from
+ * TeX, TeX font metrics, and the TTF files. These data are then exposed via the
+ * `metrics` variable and the getCharacterMetrics function.
+ */
+
+// In TeX, there are actually three sets of dimensions, one for each of
+// textstyle, scriptstyle, and scriptscriptstyle. These are provided in the
+// the arrays below, in that order.
+//
+// The font metrics are stored in fonts cmsy10, cmsy7, and cmsy5 respsectively.
+// This was determined by running the folllowing script:
+//
+// latex -interaction=nonstopmode \
+// '\documentclass{article}\usepackage{amsmath}\begin{document}' \
+// '$a$ \expandafter\show\the\textfont2' \
+// '\expandafter\show\the\scriptfont2' \
+// '\expandafter\show\the\scriptscriptfont2' \
+// '\stop'
+//
+// The metrics themselves were retreived using the following commands:
+//
+// tftopl cmsy10
+// tftopl cmsy7
+// tftopl cmsy5
+//
+// The output of each of these commands is quite lengthy. The only part we
+// care about is the FONTDIMEN section. Each value is measured in EMs.
+var sigmas = {
+ slant: [0.250, 0.250, 0.250], // sigma1
+ space: [0.000, 0.000, 0.000], // sigma2
+ stretch: [0.000, 0.000, 0.000], // sigma3
+ shrink: [0.000, 0.000, 0.000], // sigma4
+ xHeight: [0.431, 0.431, 0.431], // sigma5
+ quad: [1.000, 1.171, 1.472], // sigma6
+ extraSpace: [0.000, 0.000, 0.000], // sigma7
+ num1: [0.677, 0.732, 0.925], // sigma8
+ num2: [0.394, 0.384, 0.387], // sigma9
+ num3: [0.444, 0.471, 0.504], // sigma10
+ denom1: [0.686, 0.752, 1.025], // sigma11
+ denom2: [0.345, 0.344, 0.532], // sigma12
+ sup1: [0.413, 0.503, 0.504], // sigma13
+ sup2: [0.363, 0.431, 0.404], // sigma14
+ sup3: [0.289, 0.286, 0.294], // sigma15
+ sub1: [0.150, 0.143, 0.200], // sigma16
+ sub2: [0.247, 0.286, 0.400], // sigma17
+ supDrop: [0.386, 0.353, 0.494], // sigma18
+ subDrop: [0.050, 0.071, 0.100], // sigma19
+ delim1: [2.390, 1.700, 1.980], // sigma20
+ delim2: [1.010, 1.157, 1.420], // sigma21
+ axisHeight: [0.250, 0.250, 0.250] // sigma22
+};
+
+// These font metrics are extracted from TeX by using
+// \font\a=cmex10
+// \showthe\fontdimenX\a
+// where X is the corresponding variable number. These correspond to the font
+// parameters of the extension fonts (family 3). See the TeXbook, page 441.
+var xi1 = 0;
+var xi2 = 0;
+var xi3 = 0;
+var xi4 = 0;
+var xi5 = 0.431;
+var xi6 = 1;
+var xi7 = 0;
+var xi8 = 0.04;
+var xi9 = 0.111;
+var xi10 = 0.166;
+var xi11 = 0.2;
+var xi12 = 0.6;
+var xi13 = 0.1;
+
+// This value determines how large a pt is, for metrics which are defined in
+// terms of pts.
+// This value is also used in katex.less; if you change it make sure the values
+// match.
+var ptPerEm = 10.0;
+
+// The space between adjacent `|` columns in an array definition. From
+// `\showthe\doublerulesep` in LaTeX.
+var doubleRuleSep = 2.0 / ptPerEm;
+
+/**
+ * This is just a mapping from common names to real metrics
+ */
+var metrics = {
+ defaultRuleThickness: xi8,
+ bigOpSpacing1: xi9,
+ bigOpSpacing2: xi10,
+ bigOpSpacing3: xi11,
+ bigOpSpacing4: xi12,
+ bigOpSpacing5: xi13,
+ ptPerEm: ptPerEm,
+ doubleRuleSep: doubleRuleSep
+};
+
+// This map contains a mapping from font name and character code to character
+// metrics, including height, depth, italic correction, and skew (kern from the
+// character to the corresponding \skewchar)
+// This map is generated via `make metrics`. It should not be changed manually.
+var metricMap = require("./fontMetricsData");
+
+// These are very rough approximations. We default to Times New Roman which
+// should have Latin-1 and Cyrillic characters, but may not depending on the
+// operating system. The metrics do not account for extra height from the
+// accents. In the case of Cyrillic characters which have both ascenders and
+// descenders we prefer approximations with ascenders, primarily to prevent
+// the fraction bar or root line from intersecting the glyph.
+// TODO(kevinb) allow union of multiple glyph metrics for better accuracy.
+var extraCharacterMap = {
+ // Latin-1
+ 'À': 'A',
+ 'Á': 'A',
+ 'Â': 'A',
+ 'Ã': 'A',
+ 'Ä': 'A',
+ 'Å': 'A',
+ 'Æ': 'A',
+ 'Ç': 'C',
+ 'È': 'E',
+ 'É': 'E',
+ 'Ê': 'E',
+ 'Ë': 'E',
+ 'Ì': 'I',
+ 'Í': 'I',
+ 'Î': 'I',
+ 'Ï': 'I',
+ 'Ð': 'D',
+ 'Ñ': 'N',
+ 'Ò': 'O',
+ 'Ó': 'O',
+ 'Ô': 'O',
+ 'Õ': 'O',
+ 'Ö': 'O',
+ 'Ø': 'O',
+ 'Ù': 'U',
+ 'Ú': 'U',
+ 'Û': 'U',
+ 'Ü': 'U',
+ 'Ý': 'Y',
+ 'Þ': 'o',
+ 'ß': 'B',
+ 'à': 'a',
+ 'á': 'a',
+ 'â': 'a',
+ 'ã': 'a',
+ 'ä': 'a',
+ 'å': 'a',
+ 'æ': 'a',
+ 'ç': 'c',
+ 'è': 'e',
+ 'é': 'e',
+ 'ê': 'e',
+ 'ë': 'e',
+ 'ì': 'i',
+ 'í': 'i',
+ 'î': 'i',
+ 'ï': 'i',
+ 'ð': 'd',
+ 'ñ': 'n',
+ 'ò': 'o',
+ 'ó': 'o',
+ 'ô': 'o',
+ 'õ': 'o',
+ 'ö': 'o',
+ 'ø': 'o',
+ 'ù': 'u',
+ 'ú': 'u',
+ 'û': 'u',
+ 'ü': 'u',
+ 'ý': 'y',
+ 'þ': 'o',
+ 'ÿ': 'y',
+
+ // Cyrillic
+ 'А': 'A',
+ 'Б': 'B',
+ 'В': 'B',
+ 'Г': 'F',
+ 'Д': 'A',
+ 'Е': 'E',
+ 'Ж': 'K',
+ 'З': '3',
+ 'И': 'N',
+ 'Й': 'N',
+ 'К': 'K',
+ 'Л': 'N',
+ 'М': 'M',
+ 'Н': 'H',
+ 'О': 'O',
+ 'П': 'N',
+ 'Р': 'P',
+ 'С': 'C',
+ 'Т': 'T',
+ 'У': 'y',
+ 'Ф': 'O',
+ 'Х': 'X',
+ 'Ц': 'U',
+ 'Ч': 'h',
+ 'Ш': 'W',
+ 'Щ': 'W',
+ 'Ъ': 'B',
+ 'Ы': 'X',
+ 'Ь': 'B',
+ 'Э': '3',
+ 'Ю': 'X',
+ 'Я': 'R',
+ 'а': 'a',
+ 'б': 'b',
+ 'в': 'a',
+ 'г': 'r',
+ 'д': 'y',
+ 'е': 'e',
+ 'ж': 'm',
+ 'з': 'e',
+ 'и': 'n',
+ 'й': 'n',
+ 'к': 'n',
+ 'л': 'n',
+ 'м': 'm',
+ 'н': 'n',
+ 'о': 'o',
+ 'п': 'n',
+ 'р': 'p',
+ 'с': 'c',
+ 'т': 'o',
+ 'у': 'y',
+ 'ф': 'b',
+ 'х': 'x',
+ 'ц': 'n',
+ 'ч': 'n',
+ 'ш': 'w',
+ 'щ': 'w',
+ 'ъ': 'a',
+ 'ы': 'm',
+ 'ь': 'a',
+ 'э': 'e',
+ 'ю': 'm',
+ 'я': 'r'
+};
+
+/**
+ * This function is a convenience function for looking up information in the
+ * metricMap table. It takes a character as a string, and a style.
+ *
+ * Note: the `width` property may be undefined if fontMetricsData.js wasn't
+ * built using `Make extended_metrics`.
+ */
+var getCharacterMetrics = function(character, style) {
+ var ch = character.charCodeAt(0);
+ if (character[0] in extraCharacterMap) {
+ ch = extraCharacterMap[character[0]].charCodeAt(0);
+ } else if (cjkRegex.test(character[0])) {
+ ch = 'M'.charCodeAt(0);
+ }
+ var metrics = metricMap[style][ch];
+ if (metrics) {
+ return {
+ depth: metrics[0],
+ height: metrics[1],
+ italic: metrics[2],
+ skew: metrics[3],
+ width: metrics[4]
+ };
+ }
+};
+
+module.exports = {
+ metrics: metrics,
+ sigmas: sigmas,
+ getCharacterMetrics: getCharacterMetrics
+};
+
+},{"./Style":9,"./fontMetricsData":18,"./unicodeRegexes":24}],18:[function(require,module,exports){
+module.exports = {
+ "AMS-Regular": {
+ "65": [0, 0.68889, 0, 0],
+ "66": [0, 0.68889, 0, 0],
+ "67": [0, 0.68889, 0, 0],
+ "68": [0, 0.68889, 0, 0],
+ "69": [0, 0.68889, 0, 0],
+ "70": [0, 0.68889, 0, 0],
+ "71": [0, 0.68889, 0, 0],
+ "72": [0, 0.68889, 0, 0],
+ "73": [0, 0.68889, 0, 0],
+ "74": [0.16667, 0.68889, 0, 0],
+ "75": [0, 0.68889, 0, 0],
+ "76": [0, 0.68889, 0, 0],
+ "77": [0, 0.68889, 0, 0],
+ "78": [0, 0.68889, 0, 0],
+ "79": [0.16667, 0.68889, 0, 0],
+ "80": [0, 0.68889, 0, 0],
+ "81": [0.16667, 0.68889, 0, 0],
+ "82": [0, 0.68889, 0, 0],
+ "83": [0, 0.68889, 0, 0],
+ "84": [0, 0.68889, 0, 0],
+ "85": [0, 0.68889, 0, 0],
+ "86": [0, 0.68889, 0, 0],
+ "87": [0, 0.68889, 0, 0],
+ "88": [0, 0.68889, 0, 0],
+ "89": [0, 0.68889, 0, 0],
+ "90": [0, 0.68889, 0, 0],
+ "107": [0, 0.68889, 0, 0],
+ "165": [0, 0.675, 0.025, 0],
+ "174": [0.15559, 0.69224, 0, 0],
+ "240": [0, 0.68889, 0, 0],
+ "295": [0, 0.68889, 0, 0],
+ "710": [0, 0.825, 0, 0],
+ "732": [0, 0.9, 0, 0],
+ "770": [0, 0.825, 0, 0],
+ "771": [0, 0.9, 0, 0],
+ "989": [0.08167, 0.58167, 0, 0],
+ "1008": [0, 0.43056, 0.04028, 0],
+ "8245": [0, 0.54986, 0, 0],
+ "8463": [0, 0.68889, 0, 0],
+ "8487": [0, 0.68889, 0, 0],
+ "8498": [0, 0.68889, 0, 0],
+ "8502": [0, 0.68889, 0, 0],
+ "8503": [0, 0.68889, 0, 0],
+ "8504": [0, 0.68889, 0, 0],
+ "8513": [0, 0.68889, 0, 0],
+ "8592": [-0.03598, 0.46402, 0, 0],
+ "8594": [-0.03598, 0.46402, 0, 0],
+ "8602": [-0.13313, 0.36687, 0, 0],
+ "8603": [-0.13313, 0.36687, 0, 0],
+ "8606": [0.01354, 0.52239, 0, 0],
+ "8608": [0.01354, 0.52239, 0, 0],
+ "8610": [0.01354, 0.52239, 0, 0],
+ "8611": [0.01354, 0.52239, 0, 0],
+ "8619": [0, 0.54986, 0, 0],
+ "8620": [0, 0.54986, 0, 0],
+ "8621": [-0.13313, 0.37788, 0, 0],
+ "8622": [-0.13313, 0.36687, 0, 0],
+ "8624": [0, 0.69224, 0, 0],
+ "8625": [0, 0.69224, 0, 0],
+ "8630": [0, 0.43056, 0, 0],
+ "8631": [0, 0.43056, 0, 0],
+ "8634": [0.08198, 0.58198, 0, 0],
+ "8635": [0.08198, 0.58198, 0, 0],
+ "8638": [0.19444, 0.69224, 0, 0],
+ "8639": [0.19444, 0.69224, 0, 0],
+ "8642": [0.19444, 0.69224, 0, 0],
+ "8643": [0.19444, 0.69224, 0, 0],
+ "8644": [0.1808, 0.675, 0, 0],
+ "8646": [0.1808, 0.675, 0, 0],
+ "8647": [0.1808, 0.675, 0, 0],
+ "8648": [0.19444, 0.69224, 0, 0],
+ "8649": [0.1808, 0.675, 0, 0],
+ "8650": [0.19444, 0.69224, 0, 0],
+ "8651": [0.01354, 0.52239, 0, 0],
+ "8652": [0.01354, 0.52239, 0, 0],
+ "8653": [-0.13313, 0.36687, 0, 0],
+ "8654": [-0.13313, 0.36687, 0, 0],
+ "8655": [-0.13313, 0.36687, 0, 0],
+ "8666": [0.13667, 0.63667, 0, 0],
+ "8667": [0.13667, 0.63667, 0, 0],
+ "8669": [-0.13313, 0.37788, 0, 0],
+ "8672": [-0.064, 0.437, 0, 0],
+ "8674": [-0.064, 0.437, 0, 0],
+ "8705": [0, 0.825, 0, 0],
+ "8708": [0, 0.68889, 0, 0],
+ "8709": [0.08167, 0.58167, 0, 0],
+ "8717": [0, 0.43056, 0, 0],
+ "8722": [-0.03598, 0.46402, 0, 0],
+ "8724": [0.08198, 0.69224, 0, 0],
+ "8726": [0.08167, 0.58167, 0, 0],
+ "8733": [0, 0.69224, 0, 0],
+ "8736": [0, 0.69224, 0, 0],
+ "8737": [0, 0.69224, 0, 0],
+ "8738": [0.03517, 0.52239, 0, 0],
+ "8739": [0.08167, 0.58167, 0, 0],
+ "8740": [0.25142, 0.74111, 0, 0],
+ "8741": [0.08167, 0.58167, 0, 0],
+ "8742": [0.25142, 0.74111, 0, 0],
+ "8756": [0, 0.69224, 0, 0],
+ "8757": [0, 0.69224, 0, 0],
+ "8764": [-0.13313, 0.36687, 0, 0],
+ "8765": [-0.13313, 0.37788, 0, 0],
+ "8769": [-0.13313, 0.36687, 0, 0],
+ "8770": [-0.03625, 0.46375, 0, 0],
+ "8774": [0.30274, 0.79383, 0, 0],
+ "8776": [-0.01688, 0.48312, 0, 0],
+ "8778": [0.08167, 0.58167, 0, 0],
+ "8782": [0.06062, 0.54986, 0, 0],
+ "8783": [0.06062, 0.54986, 0, 0],
+ "8785": [0.08198, 0.58198, 0, 0],
+ "8786": [0.08198, 0.58198, 0, 0],
+ "8787": [0.08198, 0.58198, 0, 0],
+ "8790": [0, 0.69224, 0, 0],
+ "8791": [0.22958, 0.72958, 0, 0],
+ "8796": [0.08198, 0.91667, 0, 0],
+ "8806": [0.25583, 0.75583, 0, 0],
+ "8807": [0.25583, 0.75583, 0, 0],
+ "8808": [0.25142, 0.75726, 0, 0],
+ "8809": [0.25142, 0.75726, 0, 0],
+ "8812": [0.25583, 0.75583, 0, 0],
+ "8814": [0.20576, 0.70576, 0, 0],
+ "8815": [0.20576, 0.70576, 0, 0],
+ "8816": [0.30274, 0.79383, 0, 0],
+ "8817": [0.30274, 0.79383, 0, 0],
+ "8818": [0.22958, 0.72958, 0, 0],
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+ "933": [0, 0.61111, 0, 0],
+ "934": [0, 0.61111, 0, 0],
+ "936": [0, 0.61111, 0, 0],
+ "937": [0, 0.61111, 0, 0],
+ "2018": [0, 0.61111, 0, 0],
+ "2019": [0, 0.61111, 0, 0],
+ "8242": [0, 0.61111, 0, 0]
+ }
+};
+
+},{}],19:[function(require,module,exports){
+var utils = require("./utils");
+var ParseError = require("./ParseError");
+var parseData = require("./parseData");
+var ParseNode = parseData.ParseNode;
+
+/* This file contains a list of functions that we parse, identified by
+ * the calls to defineFunction.
+ *
+ * The first argument to defineFunction is a single name or a list of names.
+ * All functions named in such a list will share a single implementation.
+ *
+ * Each declared function can have associated properties, which
+ * include the following:
+ *
+ * - numArgs: The number of arguments the function takes.
+ * If this is the only property, it can be passed as a number
+ * instead of an element of a properties object.
+ * - argTypes: (optional) An array corresponding to each argument of the
+ * function, giving the type of argument that should be parsed. Its
+ * length should be equal to `numArgs + numOptionalArgs`. Valid
+ * types:
+ * - "size": A size-like thing, such as "1em" or "5ex"
+ * - "color": An html color, like "#abc" or "blue"
+ * - "original": The same type as the environment that the
+ * function being parsed is in (e.g. used for the
+ * bodies of functions like \color where the first
+ * argument is special and the second argument is
+ * parsed normally)
+ * Other possible types (probably shouldn't be used)
+ * - "text": Text-like (e.g. \text)
+ * - "math": Normal math
+ * If undefined, this will be treated as an appropriate length
+ * array of "original" strings
+ * - greediness: (optional) The greediness of the function to use ungrouped
+ * arguments.
+ *
+ * E.g. if you have an expression
+ * \sqrt \frac 1 2
+ * since \frac has greediness=2 vs \sqrt's greediness=1, \frac
+ * will use the two arguments '1' and '2' as its two arguments,
+ * then that whole function will be used as the argument to
+ * \sqrt. On the other hand, the expressions
+ * \frac \frac 1 2 3
+ * and
+ * \frac \sqrt 1 2
+ * will fail because \frac and \frac have equal greediness
+ * and \sqrt has a lower greediness than \frac respectively. To
+ * make these parse, we would have to change them to:
+ * \frac {\frac 1 2} 3
+ * and
+ * \frac {\sqrt 1} 2
+ *
+ * The default value is `1`
+ * - allowedInText: (optional) Whether or not the function is allowed inside
+ * text mode (default false)
+ * - numOptionalArgs: (optional) The number of optional arguments the function
+ * should parse. If the optional arguments aren't found,
+ * `null` will be passed to the handler in their place.
+ * (default 0)
+ * - infix: (optional) Must be true if the function is an infix operator.
+ *
+ * The last argument is that implementation, the handler for the function(s).
+ * It is called to handle these functions and their arguments.
+ * It receives two arguments:
+ * - context contains information and references provided by the parser
+ * - args is an array of arguments obtained from TeX input
+ * The context contains the following properties:
+ * - funcName: the text (i.e. name) of the function, including \
+ * - parser: the parser object
+ * - lexer: the lexer object
+ * - positions: the positions in the overall string of the function
+ * and the arguments.
+ * The latter three should only be used to produce error messages.
+ *
+ * The function should return an object with the following keys:
+ * - type: The type of element that this is. This is then used in
+ * buildHTML/buildMathML to determine which function
+ * should be called to build this node into a DOM node
+ * Any other data can be added to the object, which will be passed
+ * in to the function in buildHTML/buildMathML as `group.value`.
+ */
+
+function defineFunction(names, props, handler) {
+ if (typeof names === "string") {
+ names = [names];
+ }
+ if (typeof props === "number") {
+ props = { numArgs: props };
+ }
+ // Set default values of functions
+ var data = {
+ numArgs: props.numArgs,
+ argTypes: props.argTypes,
+ greediness: (props.greediness === undefined) ? 1 : props.greediness,
+ allowedInText: !!props.allowedInText,
+ numOptionalArgs: props.numOptionalArgs || 0,
+ infix: !!props.infix,
+ handler: handler
+ };
+ for (var i = 0; i < names.length; ++i) {
+ module.exports[names[i]] = data;
+ }
+}
+
+// Since the corresponding buildHTML/buildMathML function expects a
+// list of elements, we normalize for different kinds of arguments
+var ordargument = function(arg) {
+ if (arg.type === "ordgroup") {
+ return arg.value;
+ } else {
+ return [arg];
+ }
+};
+
+// A normal square root
+defineFunction("\\sqrt", {
+ numArgs: 1,
+ numOptionalArgs: 1
+}, function(context, args) {
+ var index = args[0];
+ var body = args[1];
+ return {
+ type: "sqrt",
+ body: body,
+ index: index
+ };
+});
+
+// Non-mathy text, possibly in a font
+var textFunctionStyles = {
+ "\\text": undefined, "\\textrm": "mathrm", "\\textsf": "mathsf",
+ "\\texttt": "mathtt", "\\textnormal": "mathrm", "\\textbf": "mathbf",
+ "\\textit": "textit"
+};
+
+defineFunction([
+ "\\text", "\\textrm", "\\textsf", "\\texttt", "\\textnormal",
+ "\\textbf", "\\textit"
+], {
+ numArgs: 1,
+ argTypes: ["text"],
+ greediness: 2,
+ allowedInText: true
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "text",
+ body: ordargument(body),
+ style: textFunctionStyles[context.funcName]
+ };
+});
+
+// A two-argument custom color
+defineFunction("\\color", {
+ numArgs: 2,
+ allowedInText: true,
+ greediness: 3,
+ argTypes: ["color", "original"]
+}, function(context, args) {
+ var color = args[0];
+ var body = args[1];
+ return {
+ type: "color",
+ color: color.value,
+ value: ordargument(body)
+ };
+});
+
+// An overline
+defineFunction("\\overline", {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "overline",
+ body: body
+ };
+});
+
+// An underline
+defineFunction("\\underline", {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "underline",
+ body: body
+ };
+});
+
+// A box of the width and height
+defineFunction("\\rule", {
+ numArgs: 2,
+ numOptionalArgs: 1,
+ argTypes: ["size", "size", "size"]
+}, function(context, args) {
+ var shift = args[0];
+ var width = args[1];
+ var height = args[2];
+ return {
+ type: "rule",
+ shift: shift && shift.value,
+ width: width.value,
+ height: height.value
+ };
+});
+
+// TODO: In TeX, \mkern only accepts mu-units, and \kern does not accept
+// mu-units. In current KaTeX we relax this; both commands accept any unit.
+defineFunction(["\\kern", "\\mkern"], {
+ numArgs: 1,
+ argTypes: ["size"]
+}, function(context, args) {
+ return {
+ type: "kern",
+ dimension: args[0].value
+ };
+});
+
+// A KaTeX logo
+defineFunction("\\KaTeX", {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "katex"
+ };
+});
+
+defineFunction("\\phantom", {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "phantom",
+ value: ordargument(body)
+ };
+});
+
+// Math class commands except \mathop
+defineFunction([
+ "\\mathord", "\\mathbin", "\\mathrel", "\\mathopen",
+ "\\mathclose", "\\mathpunct", "\\mathinner"
+], {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "mclass",
+ mclass: "m" + context.funcName.substr(5),
+ value: ordargument(body)
+ };
+});
+
+// Build a relation by placing one symbol on top of another
+defineFunction("\\stackrel", {
+ numArgs: 2
+}, function(context, args) {
+ var top = args[0];
+ var bottom = args[1];
+
+ var bottomop = new ParseNode("op", {
+ type: "op",
+ limits: true,
+ alwaysHandleSupSub: true,
+ symbol: false,
+ value: ordargument(bottom)
+ }, bottom.mode);
+
+ var supsub = new ParseNode("supsub", {
+ base: bottomop,
+ sup: top,
+ sub: null
+ }, top.mode);
+
+ return {
+ type: "mclass",
+ mclass: "mrel",
+ value: [supsub]
+ };
+});
+
+// \mod-type functions
+defineFunction("\\bmod", {
+ numArgs: 0
+}, function(context, args) {
+ return {
+ type: "mod",
+ modType: "bmod",
+ value: null
+ };
+});
+
+defineFunction(["\\pod", "\\pmod", "\\mod"], {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "mod",
+ modType: context.funcName.substr(1),
+ value: ordargument(body)
+ };
+});
+
+// Extra data needed for the delimiter handler down below
+var delimiterSizes = {
+ "\\bigl" : {mclass: "mopen", size: 1},
+ "\\Bigl" : {mclass: "mopen", size: 2},
+ "\\biggl": {mclass: "mopen", size: 3},
+ "\\Biggl": {mclass: "mopen", size: 4},
+ "\\bigr" : {mclass: "mclose", size: 1},
+ "\\Bigr" : {mclass: "mclose", size: 2},
+ "\\biggr": {mclass: "mclose", size: 3},
+ "\\Biggr": {mclass: "mclose", size: 4},
+ "\\bigm" : {mclass: "mrel", size: 1},
+ "\\Bigm" : {mclass: "mrel", size: 2},
+ "\\biggm": {mclass: "mrel", size: 3},
+ "\\Biggm": {mclass: "mrel", size: 4},
+ "\\big" : {mclass: "mord", size: 1},
+ "\\Big" : {mclass: "mord", size: 2},
+ "\\bigg" : {mclass: "mord", size: 3},
+ "\\Bigg" : {mclass: "mord", size: 4}
+};
+
+var delimiters = [
+ "(", ")", "[", "\\lbrack", "]", "\\rbrack",
+ "\\{", "\\lbrace", "\\}", "\\rbrace",
+ "\\lfloor", "\\rfloor", "\\lceil", "\\rceil",
+ "<", ">", "\\langle", "\\rangle", "\\lt", "\\gt",
+ "\\lvert", "\\rvert", "\\lVert", "\\rVert",
+ "\\lgroup", "\\rgroup", "\\lmoustache", "\\rmoustache",
+ "/", "\\backslash",
+ "|", "\\vert", "\\|", "\\Vert",
+ "\\uparrow", "\\Uparrow",
+ "\\downarrow", "\\Downarrow",
+ "\\updownarrow", "\\Updownarrow",
+ "."
+];
+
+var fontAliases = {
+ "\\Bbb": "\\mathbb",
+ "\\bold": "\\mathbf",
+ "\\frak": "\\mathfrak"
+};
+
+// Single-argument color functions
+defineFunction([
+ "\\blue", "\\orange", "\\pink", "\\red",
+ "\\green", "\\gray", "\\purple",
+ "\\blueA", "\\blueB", "\\blueC", "\\blueD", "\\blueE",
+ "\\tealA", "\\tealB", "\\tealC", "\\tealD", "\\tealE",
+ "\\greenA", "\\greenB", "\\greenC", "\\greenD", "\\greenE",
+ "\\goldA", "\\goldB", "\\goldC", "\\goldD", "\\goldE",
+ "\\redA", "\\redB", "\\redC", "\\redD", "\\redE",
+ "\\maroonA", "\\maroonB", "\\maroonC", "\\maroonD", "\\maroonE",
+ "\\purpleA", "\\purpleB", "\\purpleC", "\\purpleD", "\\purpleE",
+ "\\mintA", "\\mintB", "\\mintC",
+ "\\grayA", "\\grayB", "\\grayC", "\\grayD", "\\grayE",
+ "\\grayF", "\\grayG", "\\grayH", "\\grayI",
+ "\\kaBlue", "\\kaGreen"
+], {
+ numArgs: 1,
+ allowedInText: true,
+ greediness: 3
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "color",
+ color: "katex-" + context.funcName.slice(1),
+ value: ordargument(body)
+ };
+});
+
+// There are 2 flags for operators; whether they produce limits in
+// displaystyle, and whether they are symbols and should grow in
+// displaystyle. These four groups cover the four possible choices.
+
+// No limits, not symbols
+defineFunction([
+ "\\arcsin", "\\arccos", "\\arctan", "\\arg", "\\cos", "\\cosh",
+ "\\cot", "\\coth", "\\csc", "\\deg", "\\dim", "\\exp", "\\hom",
+ "\\ker", "\\lg", "\\ln", "\\log", "\\sec", "\\sin", "\\sinh",
+ "\\tan", "\\tanh"
+], {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "op",
+ limits: false,
+ symbol: false,
+ body: context.funcName
+ };
+});
+
+// Limits, not symbols
+defineFunction([
+ "\\det", "\\gcd", "\\inf", "\\lim", "\\liminf", "\\limsup", "\\max",
+ "\\min", "\\Pr", "\\sup"
+], {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "op",
+ limits: true,
+ symbol: false,
+ body: context.funcName
+ };
+});
+
+// No limits, symbols
+defineFunction([
+ "\\int", "\\iint", "\\iiint", "\\oint"
+], {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "op",
+ limits: false,
+ symbol: true,
+ body: context.funcName
+ };
+});
+
+// Limits, symbols
+defineFunction([
+ "\\coprod", "\\bigvee", "\\bigwedge", "\\biguplus", "\\bigcap",
+ "\\bigcup", "\\intop", "\\prod", "\\sum", "\\bigotimes",
+ "\\bigoplus", "\\bigodot", "\\bigsqcup", "\\smallint"
+], {
+ numArgs: 0
+}, function(context) {
+ return {
+ type: "op",
+ limits: true,
+ symbol: true,
+ body: context.funcName
+ };
+});
+
+// \mathop class command
+defineFunction("\\mathop", {
+ numArgs: 1
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: "op",
+ limits: false,
+ symbol: false,
+ value: ordargument(body)
+ };
+});
+
+// Fractions
+defineFunction([
+ "\\dfrac", "\\frac", "\\tfrac",
+ "\\dbinom", "\\binom", "\\tbinom",
+ "\\\\atopfrac" // can’t be entered directly
+], {
+ numArgs: 2,
+ greediness: 2
+}, function(context, args) {
+ var numer = args[0];
+ var denom = args[1];
+ var hasBarLine;
+ var leftDelim = null;
+ var rightDelim = null;
+ var size = "auto";
+
+ switch (context.funcName) {
+ case "\\dfrac":
+ case "\\frac":
+ case "\\tfrac":
+ hasBarLine = true;
+ break;
+ case "\\\\atopfrac":
+ hasBarLine = false;
+ break;
+ case "\\dbinom":
+ case "\\binom":
+ case "\\tbinom":
+ hasBarLine = false;
+ leftDelim = "(";
+ rightDelim = ")";
+ break;
+ default:
+ throw new Error("Unrecognized genfrac command");
+ }
+
+ switch (context.funcName) {
+ case "\\dfrac":
+ case "\\dbinom":
+ size = "display";
+ break;
+ case "\\tfrac":
+ case "\\tbinom":
+ size = "text";
+ break;
+ }
+
+ return {
+ type: "genfrac",
+ numer: numer,
+ denom: denom,
+ hasBarLine: hasBarLine,
+ leftDelim: leftDelim,
+ rightDelim: rightDelim,
+ size: size
+ };
+});
+
+// Left and right overlap functions
+defineFunction(["\\llap", "\\rlap"], {
+ numArgs: 1,
+ allowedInText: true
+}, function(context, args) {
+ var body = args[0];
+ return {
+ type: context.funcName.slice(1),
+ body: body
+ };
+});
+
+// Delimiter functions
+var checkDelimiter = function(delim, context) {
+ if (utils.contains(delimiters, delim.value)) {
+ return delim;
+ } else {
+ throw new ParseError(
+ "Invalid delimiter: '" + delim.value + "' after '" +
+ context.funcName + "'", delim);
+ }
+};
+
+defineFunction([
+ "\\bigl", "\\Bigl", "\\biggl", "\\Biggl",
+ "\\bigr", "\\Bigr", "\\biggr", "\\Biggr",
+ "\\bigm", "\\Bigm", "\\biggm", "\\Biggm",
+ "\\big", "\\Big", "\\bigg", "\\Bigg"
+], {
+ numArgs: 1
+}, function(context, args) {
+ var delim = checkDelimiter(args[0], context);
+
+ return {
+ type: "delimsizing",
+ size: delimiterSizes[context.funcName].size,
+ mclass: delimiterSizes[context.funcName].mclass,
+ value: delim.value
+ };
+});
+
+defineFunction([
+ "\\left", "\\right"
+], {
+ numArgs: 1
+}, function(context, args) {
+ var delim = checkDelimiter(args[0], context);
+
+ // \left and \right are caught somewhere in Parser.js, which is
+ // why this data doesn't match what is in buildHTML.
+ return {
+ type: "leftright",
+ value: delim.value
+ };
+});
+
+defineFunction("\\middle", {
+ numArgs: 1
+}, function(context, args) {
+ var delim = checkDelimiter(args[0], context);
+ if (!context.parser.leftrightDepth) {
+ throw new ParseError("\\middle without preceding \\left", delim);
+ }
+
+ return {
+ type: "middle",
+ value: delim.value
+ };
+});
+
+// Sizing functions (handled in Parser.js explicitly, hence no handler)
+defineFunction([
+ "\\tiny", "\\scriptsize", "\\footnotesize", "\\small",
+ "\\normalsize", "\\large", "\\Large", "\\LARGE", "\\huge", "\\Huge"
+], 0, null);
+
+// Style changing functions (handled in Parser.js explicitly, hence no
+// handler)
+defineFunction([
+ "\\displaystyle", "\\textstyle", "\\scriptstyle",
+ "\\scriptscriptstyle"
+], 0, null);
+
+defineFunction([
+ // styles
+ "\\mathrm", "\\mathit", "\\mathbf",
+
+ // families
+ "\\mathbb", "\\mathcal", "\\mathfrak", "\\mathscr", "\\mathsf",
+ "\\mathtt",
+
+ // aliases
+ "\\Bbb", "\\bold", "\\frak"
+], {
+ numArgs: 1,
+ greediness: 2
+}, function(context, args) {
+ var body = args[0];
+ var func = context.funcName;
+ if (func in fontAliases) {
+ func = fontAliases[func];
+ }
+ return {
+ type: "font",
+ font: func.slice(1),
+ body: body
+ };
+});
+
+// Accents
+defineFunction([
+ "\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve",
+ "\\check", "\\hat", "\\vec", "\\dot"
+ // We don't support expanding accents yet
+ // "\\widetilde", "\\widehat"
+], {
+ numArgs: 1
+}, function(context, args) {
+ var base = args[0];
+ return {
+ type: "accent",
+ accent: context.funcName,
+ base: base
+ };
+});
+
+// Infix generalized fractions
+defineFunction(["\\over", "\\choose", "\\atop"], {
+ numArgs: 0,
+ infix: true
+}, function(context) {
+ var replaceWith;
+ switch (context.funcName) {
+ case "\\over":
+ replaceWith = "\\frac";
+ break;
+ case "\\choose":
+ replaceWith = "\\binom";
+ break;
+ case "\\atop":
+ replaceWith = "\\\\atopfrac";
+ break;
+ default:
+ throw new Error("Unrecognized infix genfrac command");
+ }
+ return {
+ type: "infix",
+ replaceWith: replaceWith,
+ token: context.token
+ };
+});
+
+// Row breaks for aligned data
+defineFunction(["\\\\", "\\cr"], {
+ numArgs: 0,
+ numOptionalArgs: 1,
+ argTypes: ["size"]
+}, function(context, args) {
+ var size = args[0];
+ return {
+ type: "cr",
+ size: size
+ };
+});
+
+// Environment delimiters
+defineFunction(["\\begin", "\\end"], {
+ numArgs: 1,
+ argTypes: ["text"]
+}, function(context, args) {
+ var nameGroup = args[0];
+ if (nameGroup.type !== "ordgroup") {
+ throw new ParseError("Invalid environment name", nameGroup);
+ }
+ var name = "";
+ for (var i = 0; i < nameGroup.value.length; ++i) {
+ name += nameGroup.value[i].value;
+ }
+ return {
+ type: "environment",
+ name: name,
+ nameGroup: nameGroup
+ };
+});
+
+},{"./ParseError":6,"./parseData":21,"./utils":25}],20:[function(require,module,exports){
+/**
+ * These objects store data about MathML nodes. This is the MathML equivalent
+ * of the types in domTree.js. Since MathML handles its own rendering, and
+ * since we're mainly using MathML to improve accessibility, we don't manage
+ * any of the styling state that the plain DOM nodes do.
+ *
+ * The `toNode` and `toMarkup` functions work simlarly to how they do in
+ * domTree.js, creating namespaced DOM nodes and HTML text markup respectively.
+ */
+
+var utils = require("./utils");
+
+/**
+ * This node represents a general purpose MathML node of any type. The
+ * constructor requires the type of node to create (for example, `"mo"` or
+ * `"mspace"`, corresponding to `<mo>` and `<mspace>` tags).
+ */
+function MathNode(type, children) {
+ this.type = type;
+ this.attributes = {};
+ this.children = children || [];
+}
+
+/**
+ * Sets an attribute on a MathML node. MathML depends on attributes to convey a
+ * semantic content, so this is used heavily.
+ */
+MathNode.prototype.setAttribute = function(name, value) {
+ this.attributes[name] = value;
+};
+
+/**
+ * Converts the math node into a MathML-namespaced DOM element.
+ */
+MathNode.prototype.toNode = function() {
+ var node = document.createElementNS(
+ "http://www.w3.org/1998/Math/MathML", this.type);
+
+ for (var attr in this.attributes) {
+ if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+ node.setAttribute(attr, this.attributes[attr]);
+ }
+ }
+
+ for (var i = 0; i < this.children.length; i++) {
+ node.appendChild(this.children[i].toNode());
+ }
+
+ return node;
+};
+
+/**
+ * Converts the math node into an HTML markup string.
+ */
+MathNode.prototype.toMarkup = function() {
+ var markup = "<" + this.type;
+
+ // Add the attributes
+ for (var attr in this.attributes) {
+ if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+ markup += " " + attr + "=\"";
+ markup += utils.escape(this.attributes[attr]);
+ markup += "\"";
+ }
+ }
+
+ markup += ">";
+
+ for (var i = 0; i < this.children.length; i++) {
+ markup += this.children[i].toMarkup();
+ }
+
+ markup += "</" + this.type + ">";
+
+ return markup;
+};
+
+/**
+ * This node represents a piece of text.
+ */
+function TextNode(text) {
+ this.text = text;
+}
+
+/**
+ * Converts the text node into a DOM text node.
+ */
+TextNode.prototype.toNode = function() {
+ return document.createTextNode(this.text);
+};
+
+/**
+ * Converts the text node into HTML markup (which is just the text itself).
+ */
+TextNode.prototype.toMarkup = function() {
+ return utils.escape(this.text);
+};
+
+module.exports = {
+ MathNode: MathNode,
+ TextNode: TextNode
+};
+
+},{"./utils":25}],21:[function(require,module,exports){
+/**
+ * The resulting parse tree nodes of the parse tree.
+ *
+ * It is possible to provide position information, so that a ParseNode can
+ * fulfil a role similar to a Token in error reporting.
+ * For details on the corresponding properties see Token constructor.
+ * Providing such information can lead to better error reporting.
+ *
+ * @param {string} type type of node, like e.g. "ordgroup"
+ * @param {?object} value type-specific representation of the node
+ * @param {string} mode parse mode in action for this node,
+ * "math" or "text"
+ * @param {Token=} firstToken first token of the input for this node,
+ * will omit position information if unset
+ * @param {Token=} lastToken last token of the input for this node,
+ * will default to firstToken if unset
+ */
+function ParseNode(type, value, mode, firstToken, lastToken) {
+ this.type = type;
+ this.value = value;
+ this.mode = mode;
+ if (firstToken && (!lastToken || lastToken.lexer === firstToken.lexer)) {
+ this.lexer = firstToken.lexer;
+ this.start = firstToken.start;
+ this.end = (lastToken || firstToken).end;
+ }
+}
+
+module.exports = {
+ ParseNode: ParseNode
+};
+
+
+},{}],22:[function(require,module,exports){
+/**
+ * Provides a single function for parsing an expression using a Parser
+ * TODO(emily): Remove this
+ */
+
+var Parser = require("./Parser");
+
+/**
+ * Parses an expression using a Parser, then returns the parsed result.
+ */
+var parseTree = function(toParse, settings) {
+ if (!(typeof toParse === 'string' || toParse instanceof String)) {
+ throw new TypeError('KaTeX can only parse string typed expression');
+ }
+ var parser = new Parser(toParse, settings);
+
+ return parser.parse();
+};
+
+module.exports = parseTree;
+
+},{"./Parser":7}],23:[function(require,module,exports){
+/**
+ * This file holds a list of all no-argument functions and single-character
+ * symbols (like 'a' or ';').
+ *
+ * For each of the symbols, there are three properties they can have:
+ * - font (required): the font to be used for this symbol. Either "main" (the
+ normal font), or "ams" (the ams fonts).
+ * - group (required): the ParseNode group type the symbol should have (i.e.
+ "textord", "mathord", etc).
+ See https://github.com/Khan/KaTeX/wiki/Examining-TeX#group-types
+ * - replace: the character that this symbol or function should be
+ * replaced with (i.e. "\phi" has a replace value of "\u03d5", the phi
+ * character in the main font).
+ *
+ * The outermost map in the table indicates what mode the symbols should be
+ * accepted in (e.g. "math" or "text").
+ */
+
+module.exports = {
+ math: {},
+ text: {}
+};
+
+function defineSymbol(mode, font, group, replace, name) {
+ module.exports[mode][name] = {
+ font: font,
+ group: group,
+ replace: replace
+ };
+}
+
+// Some abbreviations for commonly used strings.
+// This helps minify the code, and also spotting typos using jshint.
+
+// modes:
+var math = "math";
+var text = "text";
+
+// fonts:
+var main = "main";
+var ams = "ams";
+
+// groups:
+var accent = "accent";
+var bin = "bin";
+var close = "close";
+var inner = "inner";
+var mathord = "mathord";
+var op = "op";
+var open = "open";
+var punct = "punct";
+var rel = "rel";
+var spacing = "spacing";
+var textord = "textord";
+
+// Now comes the symbol table
+
+// Relation Symbols
+defineSymbol(math, main, rel, "\u2261", "\\equiv");
+defineSymbol(math, main, rel, "\u227a", "\\prec");
+defineSymbol(math, main, rel, "\u227b", "\\succ");
+defineSymbol(math, main, rel, "\u223c", "\\sim");
+defineSymbol(math, main, rel, "\u22a5", "\\perp");
+defineSymbol(math, main, rel, "\u2aaf", "\\preceq");
+defineSymbol(math, main, rel, "\u2ab0", "\\succeq");
+defineSymbol(math, main, rel, "\u2243", "\\simeq");
+defineSymbol(math, main, rel, "\u2223", "\\mid");
+defineSymbol(math, main, rel, "\u226a", "\\ll");
+defineSymbol(math, main, rel, "\u226b", "\\gg");
+defineSymbol(math, main, rel, "\u224d", "\\asymp");
+defineSymbol(math, main, rel, "\u2225", "\\parallel");
+defineSymbol(math, main, rel, "\u22c8", "\\bowtie");
+defineSymbol(math, main, rel, "\u2323", "\\smile");
+defineSymbol(math, main, rel, "\u2291", "\\sqsubseteq");
+defineSymbol(math, main, rel, "\u2292", "\\sqsupseteq");
+defineSymbol(math, main, rel, "\u2250", "\\doteq");
+defineSymbol(math, main, rel, "\u2322", "\\frown");
+defineSymbol(math, main, rel, "\u220b", "\\ni");
+defineSymbol(math, main, rel, "\u221d", "\\propto");
+defineSymbol(math, main, rel, "\u22a2", "\\vdash");
+defineSymbol(math, main, rel, "\u22a3", "\\dashv");
+defineSymbol(math, main, rel, "\u220b", "\\owns");
+
+// Punctuation
+defineSymbol(math, main, punct, "\u002e", "\\ldotp");
+defineSymbol(math, main, punct, "\u22c5", "\\cdotp");
+
+// Misc Symbols
+defineSymbol(math, main, textord, "\u0023", "\\#");
+defineSymbol(text, main, textord, "\u0023", "\\#");
+defineSymbol(math, main, textord, "\u0026", "\\&");
+defineSymbol(text, main, textord, "\u0026", "\\&");
+defineSymbol(math, main, textord, "\u2135", "\\aleph");
+defineSymbol(math, main, textord, "\u2200", "\\forall");
+defineSymbol(math, main, textord, "\u210f", "\\hbar");
+defineSymbol(math, main, textord, "\u2203", "\\exists");
+defineSymbol(math, main, textord, "\u2207", "\\nabla");
+defineSymbol(math, main, textord, "\u266d", "\\flat");
+defineSymbol(math, main, textord, "\u2113", "\\ell");
+defineSymbol(math, main, textord, "\u266e", "\\natural");
+defineSymbol(math, main, textord, "\u2663", "\\clubsuit");
+defineSymbol(math, main, textord, "\u2118", "\\wp");
+defineSymbol(math, main, textord, "\u266f", "\\sharp");
+defineSymbol(math, main, textord, "\u2662", "\\diamondsuit");
+defineSymbol(math, main, textord, "\u211c", "\\Re");
+defineSymbol(math, main, textord, "\u2661", "\\heartsuit");
+defineSymbol(math, main, textord, "\u2111", "\\Im");
+defineSymbol(math, main, textord, "\u2660", "\\spadesuit");
+
+// Math and Text
+defineSymbol(math, main, textord, "\u2020", "\\dag");
+defineSymbol(math, main, textord, "\u2021", "\\ddag");
+
+// Large Delimiters
+defineSymbol(math, main, close, "\u23b1", "\\rmoustache");
+defineSymbol(math, main, open, "\u23b0", "\\lmoustache");
+defineSymbol(math, main, close, "\u27ef", "\\rgroup");
+defineSymbol(math, main, open, "\u27ee", "\\lgroup");
+
+// Binary Operators
+defineSymbol(math, main, bin, "\u2213", "\\mp");
+defineSymbol(math, main, bin, "\u2296", "\\ominus");
+defineSymbol(math, main, bin, "\u228e", "\\uplus");
+defineSymbol(math, main, bin, "\u2293", "\\sqcap");
+defineSymbol(math, main, bin, "\u2217", "\\ast");
+defineSymbol(math, main, bin, "\u2294", "\\sqcup");
+defineSymbol(math, main, bin, "\u25ef", "\\bigcirc");
+defineSymbol(math, main, bin, "\u2219", "\\bullet");
+defineSymbol(math, main, bin, "\u2021", "\\ddagger");
+defineSymbol(math, main, bin, "\u2240", "\\wr");
+defineSymbol(math, main, bin, "\u2a3f", "\\amalg");
+
+// Arrow Symbols
+defineSymbol(math, main, rel, "\u27f5", "\\longleftarrow");
+defineSymbol(math, main, rel, "\u21d0", "\\Leftarrow");
+defineSymbol(math, main, rel, "\u27f8", "\\Longleftarrow");
+defineSymbol(math, main, rel, "\u27f6", "\\longrightarrow");
+defineSymbol(math, main, rel, "\u21d2", "\\Rightarrow");
+defineSymbol(math, main, rel, "\u27f9", "\\Longrightarrow");
+defineSymbol(math, main, rel, "\u2194", "\\leftrightarrow");
+defineSymbol(math, main, rel, "\u27f7", "\\longleftrightarrow");
+defineSymbol(math, main, rel, "\u21d4", "\\Leftrightarrow");
+defineSymbol(math, main, rel, "\u27fa", "\\Longleftrightarrow");
+defineSymbol(math, main, rel, "\u21a6", "\\mapsto");
+defineSymbol(math, main, rel, "\u27fc", "\\longmapsto");
+defineSymbol(math, main, rel, "\u2197", "\\nearrow");
+defineSymbol(math, main, rel, "\u21a9", "\\hookleftarrow");
+defineSymbol(math, main, rel, "\u21aa", "\\hookrightarrow");
+defineSymbol(math, main, rel, "\u2198", "\\searrow");
+defineSymbol(math, main, rel, "\u21bc", "\\leftharpoonup");
+defineSymbol(math, main, rel, "\u21c0", "\\rightharpoonup");
+defineSymbol(math, main, rel, "\u2199", "\\swarrow");
+defineSymbol(math, main, rel, "\u21bd", "\\leftharpoondown");
+defineSymbol(math, main, rel, "\u21c1", "\\rightharpoondown");
+defineSymbol(math, main, rel, "\u2196", "\\nwarrow");
+defineSymbol(math, main, rel, "\u21cc", "\\rightleftharpoons");
+
+// AMS Negated Binary Relations
+defineSymbol(math, ams, rel, "\u226e", "\\nless");
+defineSymbol(math, ams, rel, "\ue010", "\\nleqslant");
+defineSymbol(math, ams, rel, "\ue011", "\\nleqq");
+defineSymbol(math, ams, rel, "\u2a87", "\\lneq");
+defineSymbol(math, ams, rel, "\u2268", "\\lneqq");
+defineSymbol(math, ams, rel, "\ue00c", "\\lvertneqq");
+defineSymbol(math, ams, rel, "\u22e6", "\\lnsim");
+defineSymbol(math, ams, rel, "\u2a89", "\\lnapprox");
+defineSymbol(math, ams, rel, "\u2280", "\\nprec");
+defineSymbol(math, ams, rel, "\u22e0", "\\npreceq");
+defineSymbol(math, ams, rel, "\u22e8", "\\precnsim");
+defineSymbol(math, ams, rel, "\u2ab9", "\\precnapprox");
+defineSymbol(math, ams, rel, "\u2241", "\\nsim");
+defineSymbol(math, ams, rel, "\ue006", "\\nshortmid");
+defineSymbol(math, ams, rel, "\u2224", "\\nmid");
+defineSymbol(math, ams, rel, "\u22ac", "\\nvdash");
+defineSymbol(math, ams, rel, "\u22ad", "\\nvDash");
+defineSymbol(math, ams, rel, "\u22ea", "\\ntriangleleft");
+defineSymbol(math, ams, rel, "\u22ec", "\\ntrianglelefteq");
+defineSymbol(math, ams, rel, "\u228a", "\\subsetneq");
+defineSymbol(math, ams, rel, "\ue01a", "\\varsubsetneq");
+defineSymbol(math, ams, rel, "\u2acb", "\\subsetneqq");
+defineSymbol(math, ams, rel, "\ue017", "\\varsubsetneqq");
+defineSymbol(math, ams, rel, "\u226f", "\\ngtr");
+defineSymbol(math, ams, rel, "\ue00f", "\\ngeqslant");
+defineSymbol(math, ams, rel, "\ue00e", "\\ngeqq");
+defineSymbol(math, ams, rel, "\u2a88", "\\gneq");
+defineSymbol(math, ams, rel, "\u2269", "\\gneqq");
+defineSymbol(math, ams, rel, "\ue00d", "\\gvertneqq");
+defineSymbol(math, ams, rel, "\u22e7", "\\gnsim");
+defineSymbol(math, ams, rel, "\u2a8a", "\\gnapprox");
+defineSymbol(math, ams, rel, "\u2281", "\\nsucc");
+defineSymbol(math, ams, rel, "\u22e1", "\\nsucceq");
+defineSymbol(math, ams, rel, "\u22e9", "\\succnsim");
+defineSymbol(math, ams, rel, "\u2aba", "\\succnapprox");
+defineSymbol(math, ams, rel, "\u2246", "\\ncong");
+defineSymbol(math, ams, rel, "\ue007", "\\nshortparallel");
+defineSymbol(math, ams, rel, "\u2226", "\\nparallel");
+defineSymbol(math, ams, rel, "\u22af", "\\nVDash");
+defineSymbol(math, ams, rel, "\u22eb", "\\ntriangleright");
+defineSymbol(math, ams, rel, "\u22ed", "\\ntrianglerighteq");
+defineSymbol(math, ams, rel, "\ue018", "\\nsupseteqq");
+defineSymbol(math, ams, rel, "\u228b", "\\supsetneq");
+defineSymbol(math, ams, rel, "\ue01b", "\\varsupsetneq");
+defineSymbol(math, ams, rel, "\u2acc", "\\supsetneqq");
+defineSymbol(math, ams, rel, "\ue019", "\\varsupsetneqq");
+defineSymbol(math, ams, rel, "\u22ae", "\\nVdash");
+defineSymbol(math, ams, rel, "\u2ab5", "\\precneqq");
+defineSymbol(math, ams, rel, "\u2ab6", "\\succneqq");
+defineSymbol(math, ams, rel, "\ue016", "\\nsubseteqq");
+defineSymbol(math, ams, bin, "\u22b4", "\\unlhd");
+defineSymbol(math, ams, bin, "\u22b5", "\\unrhd");
+
+// AMS Negated Arrows
+defineSymbol(math, ams, rel, "\u219a", "\\nleftarrow");
+defineSymbol(math, ams, rel, "\u219b", "\\nrightarrow");
+defineSymbol(math, ams, rel, "\u21cd", "\\nLeftarrow");
+defineSymbol(math, ams, rel, "\u21cf", "\\nRightarrow");
+defineSymbol(math, ams, rel, "\u21ae", "\\nleftrightarrow");
+defineSymbol(math, ams, rel, "\u21ce", "\\nLeftrightarrow");
+
+// AMS Misc
+defineSymbol(math, ams, rel, "\u25b3", "\\vartriangle");
+defineSymbol(math, ams, textord, "\u210f", "\\hslash");
+defineSymbol(math, ams, textord, "\u25bd", "\\triangledown");
+defineSymbol(math, ams, textord, "\u25ca", "\\lozenge");
+defineSymbol(math, ams, textord, "\u24c8", "\\circledS");
+defineSymbol(math, ams, textord, "\u00ae", "\\circledR");
+defineSymbol(math, ams, textord, "\u2221", "\\measuredangle");
+defineSymbol(math, ams, textord, "\u2204", "\\nexists");
+defineSymbol(math, ams, textord, "\u2127", "\\mho");
+defineSymbol(math, ams, textord, "\u2132", "\\Finv");
+defineSymbol(math, ams, textord, "\u2141", "\\Game");
+defineSymbol(math, ams, textord, "\u006b", "\\Bbbk");
+defineSymbol(math, ams, textord, "\u2035", "\\backprime");
+defineSymbol(math, ams, textord, "\u25b2", "\\blacktriangle");
+defineSymbol(math, ams, textord, "\u25bc", "\\blacktriangledown");
+defineSymbol(math, ams, textord, "\u25a0", "\\blacksquare");
+defineSymbol(math, ams, textord, "\u29eb", "\\blacklozenge");
+defineSymbol(math, ams, textord, "\u2605", "\\bigstar");
+defineSymbol(math, ams, textord, "\u2222", "\\sphericalangle");
+defineSymbol(math, ams, textord, "\u2201", "\\complement");
+defineSymbol(math, ams, textord, "\u00f0", "\\eth");
+defineSymbol(math, ams, textord, "\u2571", "\\diagup");
+defineSymbol(math, ams, textord, "\u2572", "\\diagdown");
+defineSymbol(math, ams, textord, "\u25a1", "\\square");
+defineSymbol(math, ams, textord, "\u25a1", "\\Box");
+defineSymbol(math, ams, textord, "\u25ca", "\\Diamond");
+defineSymbol(math, ams, textord, "\u00a5", "\\yen");
+defineSymbol(math, ams, textord, "\u2713", "\\checkmark");
+
+// AMS Hebrew
+defineSymbol(math, ams, textord, "\u2136", "\\beth");
+defineSymbol(math, ams, textord, "\u2138", "\\daleth");
+defineSymbol(math, ams, textord, "\u2137", "\\gimel");
+
+// AMS Greek
+defineSymbol(math, ams, textord, "\u03dd", "\\digamma");
+defineSymbol(math, ams, textord, "\u03f0", "\\varkappa");
+
+// AMS Delimiters
+defineSymbol(math, ams, open, "\u250c", "\\ulcorner");
+defineSymbol(math, ams, close, "\u2510", "\\urcorner");
+defineSymbol(math, ams, open, "\u2514", "\\llcorner");
+defineSymbol(math, ams, close, "\u2518", "\\lrcorner");
+
+// AMS Binary Relations
+defineSymbol(math, ams, rel, "\u2266", "\\leqq");
+defineSymbol(math, ams, rel, "\u2a7d", "\\leqslant");
+defineSymbol(math, ams, rel, "\u2a95", "\\eqslantless");
+defineSymbol(math, ams, rel, "\u2272", "\\lesssim");
+defineSymbol(math, ams, rel, "\u2a85", "\\lessapprox");
+defineSymbol(math, ams, rel, "\u224a", "\\approxeq");
+defineSymbol(math, ams, bin, "\u22d6", "\\lessdot");
+defineSymbol(math, ams, rel, "\u22d8", "\\lll");
+defineSymbol(math, ams, rel, "\u2276", "\\lessgtr");
+defineSymbol(math, ams, rel, "\u22da", "\\lesseqgtr");
+defineSymbol(math, ams, rel, "\u2a8b", "\\lesseqqgtr");
+defineSymbol(math, ams, rel, "\u2251", "\\doteqdot");
+defineSymbol(math, ams, rel, "\u2253", "\\risingdotseq");
+defineSymbol(math, ams, rel, "\u2252", "\\fallingdotseq");
+defineSymbol(math, ams, rel, "\u223d", "\\backsim");
+defineSymbol(math, ams, rel, "\u22cd", "\\backsimeq");
+defineSymbol(math, ams, rel, "\u2ac5", "\\subseteqq");
+defineSymbol(math, ams, rel, "\u22d0", "\\Subset");
+defineSymbol(math, ams, rel, "\u228f", "\\sqsubset");
+defineSymbol(math, ams, rel, "\u227c", "\\preccurlyeq");
+defineSymbol(math, ams, rel, "\u22de", "\\curlyeqprec");
+defineSymbol(math, ams, rel, "\u227e", "\\precsim");
+defineSymbol(math, ams, rel, "\u2ab7", "\\precapprox");
+defineSymbol(math, ams, rel, "\u22b2", "\\vartriangleleft");
+defineSymbol(math, ams, rel, "\u22b4", "\\trianglelefteq");
+defineSymbol(math, ams, rel, "\u22a8", "\\vDash");
+defineSymbol(math, ams, rel, "\u22aa", "\\Vvdash");
+defineSymbol(math, ams, rel, "\u2323", "\\smallsmile");
+defineSymbol(math, ams, rel, "\u2322", "\\smallfrown");
+defineSymbol(math, ams, rel, "\u224f", "\\bumpeq");
+defineSymbol(math, ams, rel, "\u224e", "\\Bumpeq");
+defineSymbol(math, ams, rel, "\u2267", "\\geqq");
+defineSymbol(math, ams, rel, "\u2a7e", "\\geqslant");
+defineSymbol(math, ams, rel, "\u2a96", "\\eqslantgtr");
+defineSymbol(math, ams, rel, "\u2273", "\\gtrsim");
+defineSymbol(math, ams, rel, "\u2a86", "\\gtrapprox");
+defineSymbol(math, ams, bin, "\u22d7", "\\gtrdot");
+defineSymbol(math, ams, rel, "\u22d9", "\\ggg");
+defineSymbol(math, ams, rel, "\u2277", "\\gtrless");
+defineSymbol(math, ams, rel, "\u22db", "\\gtreqless");
+defineSymbol(math, ams, rel, "\u2a8c", "\\gtreqqless");
+defineSymbol(math, ams, rel, "\u2256", "\\eqcirc");
+defineSymbol(math, ams, rel, "\u2257", "\\circeq");
+defineSymbol(math, ams, rel, "\u225c", "\\triangleq");
+defineSymbol(math, ams, rel, "\u223c", "\\thicksim");
+defineSymbol(math, ams, rel, "\u2248", "\\thickapprox");
+defineSymbol(math, ams, rel, "\u2ac6", "\\supseteqq");
+defineSymbol(math, ams, rel, "\u22d1", "\\Supset");
+defineSymbol(math, ams, rel, "\u2290", "\\sqsupset");
+defineSymbol(math, ams, rel, "\u227d", "\\succcurlyeq");
+defineSymbol(math, ams, rel, "\u22df", "\\curlyeqsucc");
+defineSymbol(math, ams, rel, "\u227f", "\\succsim");
+defineSymbol(math, ams, rel, "\u2ab8", "\\succapprox");
+defineSymbol(math, ams, rel, "\u22b3", "\\vartriangleright");
+defineSymbol(math, ams, rel, "\u22b5", "\\trianglerighteq");
+defineSymbol(math, ams, rel, "\u22a9", "\\Vdash");
+defineSymbol(math, ams, rel, "\u2223", "\\shortmid");
+defineSymbol(math, ams, rel, "\u2225", "\\shortparallel");
+defineSymbol(math, ams, rel, "\u226c", "\\between");
+defineSymbol(math, ams, rel, "\u22d4", "\\pitchfork");
+defineSymbol(math, ams, rel, "\u221d", "\\varpropto");
+defineSymbol(math, ams, rel, "\u25c0", "\\blacktriangleleft");
+defineSymbol(math, ams, rel, "\u2234", "\\therefore");
+defineSymbol(math, ams, rel, "\u220d", "\\backepsilon");
+defineSymbol(math, ams, rel, "\u25b6", "\\blacktriangleright");
+defineSymbol(math, ams, rel, "\u2235", "\\because");
+defineSymbol(math, ams, rel, "\u22d8", "\\llless");
+defineSymbol(math, ams, rel, "\u22d9", "\\gggtr");
+defineSymbol(math, ams, bin, "\u22b2", "\\lhd");
+defineSymbol(math, ams, bin, "\u22b3", "\\rhd");
+defineSymbol(math, ams, rel, "\u2242", "\\eqsim");
+defineSymbol(math, main, rel, "\u22c8", "\\Join");
+defineSymbol(math, ams, rel, "\u2251", "\\Doteq");
+
+// AMS Binary Operators
+defineSymbol(math, ams, bin, "\u2214", "\\dotplus");
+defineSymbol(math, ams, bin, "\u2216", "\\smallsetminus");
+defineSymbol(math, ams, bin, "\u22d2", "\\Cap");
+defineSymbol(math, ams, bin, "\u22d3", "\\Cup");
+defineSymbol(math, ams, bin, "\u2a5e", "\\doublebarwedge");
+defineSymbol(math, ams, bin, "\u229f", "\\boxminus");
+defineSymbol(math, ams, bin, "\u229e", "\\boxplus");
+defineSymbol(math, ams, bin, "\u22c7", "\\divideontimes");
+defineSymbol(math, ams, bin, "\u22c9", "\\ltimes");
+defineSymbol(math, ams, bin, "\u22ca", "\\rtimes");
+defineSymbol(math, ams, bin, "\u22cb", "\\leftthreetimes");
+defineSymbol(math, ams, bin, "\u22cc", "\\rightthreetimes");
+defineSymbol(math, ams, bin, "\u22cf", "\\curlywedge");
+defineSymbol(math, ams, bin, "\u22ce", "\\curlyvee");
+defineSymbol(math, ams, bin, "\u229d", "\\circleddash");
+defineSymbol(math, ams, bin, "\u229b", "\\circledast");
+defineSymbol(math, ams, bin, "\u22c5", "\\centerdot");
+defineSymbol(math, ams, bin, "\u22ba", "\\intercal");
+defineSymbol(math, ams, bin, "\u22d2", "\\doublecap");
+defineSymbol(math, ams, bin, "\u22d3", "\\doublecup");
+defineSymbol(math, ams, bin, "\u22a0", "\\boxtimes");
+
+// AMS Arrows
+defineSymbol(math, ams, rel, "\u21e2", "\\dashrightarrow");
+defineSymbol(math, ams, rel, "\u21e0", "\\dashleftarrow");
+defineSymbol(math, ams, rel, "\u21c7", "\\leftleftarrows");
+defineSymbol(math, ams, rel, "\u21c6", "\\leftrightarrows");
+defineSymbol(math, ams, rel, "\u21da", "\\Lleftarrow");
+defineSymbol(math, ams, rel, "\u219e", "\\twoheadleftarrow");
+defineSymbol(math, ams, rel, "\u21a2", "\\leftarrowtail");
+defineSymbol(math, ams, rel, "\u21ab", "\\looparrowleft");
+defineSymbol(math, ams, rel, "\u21cb", "\\leftrightharpoons");
+defineSymbol(math, ams, rel, "\u21b6", "\\curvearrowleft");
+defineSymbol(math, ams, rel, "\u21ba", "\\circlearrowleft");
+defineSymbol(math, ams, rel, "\u21b0", "\\Lsh");
+defineSymbol(math, ams, rel, "\u21c8", "\\upuparrows");
+defineSymbol(math, ams, rel, "\u21bf", "\\upharpoonleft");
+defineSymbol(math, ams, rel, "\u21c3", "\\downharpoonleft");
+defineSymbol(math, ams, rel, "\u22b8", "\\multimap");
+defineSymbol(math, ams, rel, "\u21ad", "\\leftrightsquigarrow");
+defineSymbol(math, ams, rel, "\u21c9", "\\rightrightarrows");
+defineSymbol(math, ams, rel, "\u21c4", "\\rightleftarrows");
+defineSymbol(math, ams, rel, "\u21a0", "\\twoheadrightarrow");
+defineSymbol(math, ams, rel, "\u21a3", "\\rightarrowtail");
+defineSymbol(math, ams, rel, "\u21ac", "\\looparrowright");
+defineSymbol(math, ams, rel, "\u21b7", "\\curvearrowright");
+defineSymbol(math, ams, rel, "\u21bb", "\\circlearrowright");
+defineSymbol(math, ams, rel, "\u21b1", "\\Rsh");
+defineSymbol(math, ams, rel, "\u21ca", "\\downdownarrows");
+defineSymbol(math, ams, rel, "\u21be", "\\upharpoonright");
+defineSymbol(math, ams, rel, "\u21c2", "\\downharpoonright");
+defineSymbol(math, ams, rel, "\u21dd", "\\rightsquigarrow");
+defineSymbol(math, ams, rel, "\u21dd", "\\leadsto");
+defineSymbol(math, ams, rel, "\u21db", "\\Rrightarrow");
+defineSymbol(math, ams, rel, "\u21be", "\\restriction");
+
+defineSymbol(math, main, textord, "\u2018", "`");
+defineSymbol(math, main, textord, "$", "\\$");
+defineSymbol(text, main, textord, "$", "\\$");
+defineSymbol(math, main, textord, "%", "\\%");
+defineSymbol(text, main, textord, "%", "\\%");
+defineSymbol(math, main, textord, "_", "\\_");
+defineSymbol(text, main, textord, "_", "\\_");
+defineSymbol(math, main, textord, "\u2220", "\\angle");
+defineSymbol(math, main, textord, "\u221e", "\\infty");
+defineSymbol(math, main, textord, "\u2032", "\\prime");
+defineSymbol(math, main, textord, "\u25b3", "\\triangle");
+defineSymbol(math, main, textord, "\u0393", "\\Gamma");
+defineSymbol(math, main, textord, "\u0394", "\\Delta");
+defineSymbol(math, main, textord, "\u0398", "\\Theta");
+defineSymbol(math, main, textord, "\u039b", "\\Lambda");
+defineSymbol(math, main, textord, "\u039e", "\\Xi");
+defineSymbol(math, main, textord, "\u03a0", "\\Pi");
+defineSymbol(math, main, textord, "\u03a3", "\\Sigma");
+defineSymbol(math, main, textord, "\u03a5", "\\Upsilon");
+defineSymbol(math, main, textord, "\u03a6", "\\Phi");
+defineSymbol(math, main, textord, "\u03a8", "\\Psi");
+defineSymbol(math, main, textord, "\u03a9", "\\Omega");
+defineSymbol(math, main, textord, "\u00ac", "\\neg");
+defineSymbol(math, main, textord, "\u00ac", "\\lnot");
+defineSymbol(math, main, textord, "\u22a4", "\\top");
+defineSymbol(math, main, textord, "\u22a5", "\\bot");
+defineSymbol(math, main, textord, "\u2205", "\\emptyset");
+defineSymbol(math, ams, textord, "\u2205", "\\varnothing");
+defineSymbol(math, main, mathord, "\u03b1", "\\alpha");
+defineSymbol(math, main, mathord, "\u03b2", "\\beta");
+defineSymbol(math, main, mathord, "\u03b3", "\\gamma");
+defineSymbol(math, main, mathord, "\u03b4", "\\delta");
+defineSymbol(math, main, mathord, "\u03f5", "\\epsilon");
+defineSymbol(math, main, mathord, "\u03b6", "\\zeta");
+defineSymbol(math, main, mathord, "\u03b7", "\\eta");
+defineSymbol(math, main, mathord, "\u03b8", "\\theta");
+defineSymbol(math, main, mathord, "\u03b9", "\\iota");
+defineSymbol(math, main, mathord, "\u03ba", "\\kappa");
+defineSymbol(math, main, mathord, "\u03bb", "\\lambda");
+defineSymbol(math, main, mathord, "\u03bc", "\\mu");
+defineSymbol(math, main, mathord, "\u03bd", "\\nu");
+defineSymbol(math, main, mathord, "\u03be", "\\xi");
+defineSymbol(math, main, mathord, "o", "\\omicron");
+defineSymbol(math, main, mathord, "\u03c0", "\\pi");
+defineSymbol(math, main, mathord, "\u03c1", "\\rho");
+defineSymbol(math, main, mathord, "\u03c3", "\\sigma");
+defineSymbol(math, main, mathord, "\u03c4", "\\tau");
+defineSymbol(math, main, mathord, "\u03c5", "\\upsilon");
+defineSymbol(math, main, mathord, "\u03d5", "\\phi");
+defineSymbol(math, main, mathord, "\u03c7", "\\chi");
+defineSymbol(math, main, mathord, "\u03c8", "\\psi");
+defineSymbol(math, main, mathord, "\u03c9", "\\omega");
+defineSymbol(math, main, mathord, "\u03b5", "\\varepsilon");
+defineSymbol(math, main, mathord, "\u03d1", "\\vartheta");
+defineSymbol(math, main, mathord, "\u03d6", "\\varpi");
+defineSymbol(math, main, mathord, "\u03f1", "\\varrho");
+defineSymbol(math, main, mathord, "\u03c2", "\\varsigma");
+defineSymbol(math, main, mathord, "\u03c6", "\\varphi");
+defineSymbol(math, main, bin, "\u2217", "*");
+defineSymbol(math, main, bin, "+", "+");
+defineSymbol(math, main, bin, "\u2212", "-");
+defineSymbol(math, main, bin, "\u22c5", "\\cdot");
+defineSymbol(math, main, bin, "\u2218", "\\circ");
+defineSymbol(math, main, bin, "\u00f7", "\\div");
+defineSymbol(math, main, bin, "\u00b1", "\\pm");
+defineSymbol(math, main, bin, "\u00d7", "\\times");
+defineSymbol(math, main, bin, "\u2229", "\\cap");
+defineSymbol(math, main, bin, "\u222a", "\\cup");
+defineSymbol(math, main, bin, "\u2216", "\\setminus");
+defineSymbol(math, main, bin, "\u2227", "\\land");
+defineSymbol(math, main, bin, "\u2228", "\\lor");
+defineSymbol(math, main, bin, "\u2227", "\\wedge");
+defineSymbol(math, main, bin, "\u2228", "\\vee");
+defineSymbol(math, main, textord, "\u221a", "\\surd");
+defineSymbol(math, main, open, "(", "(");
+defineSymbol(math, main, open, "[", "[");
+defineSymbol(math, main, open, "\u27e8", "\\langle");
+defineSymbol(math, main, open, "\u2223", "\\lvert");
+defineSymbol(math, main, open, "\u2225", "\\lVert");
+defineSymbol(math, main, close, ")", ")");
+defineSymbol(math, main, close, "]", "]");
+defineSymbol(math, main, close, "?", "?");
+defineSymbol(math, main, close, "!", "!");
+defineSymbol(math, main, close, "\u27e9", "\\rangle");
+defineSymbol(math, main, close, "\u2223", "\\rvert");
+defineSymbol(math, main, close, "\u2225", "\\rVert");
+defineSymbol(math, main, rel, "=", "=");
+defineSymbol(math, main, rel, "<", "<");
+defineSymbol(math, main, rel, ">", ">");
+defineSymbol(math, main, rel, ":", ":");
+defineSymbol(math, main, rel, "\u2248", "\\approx");
+defineSymbol(math, main, rel, "\u2245", "\\cong");
+defineSymbol(math, main, rel, "\u2265", "\\ge");
+defineSymbol(math, main, rel, "\u2265", "\\geq");
+defineSymbol(math, main, rel, "\u2190", "\\gets");
+defineSymbol(math, main, rel, ">", "\\gt");
+defineSymbol(math, main, rel, "\u2208", "\\in");
+defineSymbol(math, main, rel, "\u2209", "\\notin");
+defineSymbol(math, main, rel, "\u2282", "\\subset");
+defineSymbol(math, main, rel, "\u2283", "\\supset");
+defineSymbol(math, main, rel, "\u2286", "\\subseteq");
+defineSymbol(math, main, rel, "\u2287", "\\supseteq");
+defineSymbol(math, ams, rel, "\u2288", "\\nsubseteq");
+defineSymbol(math, ams, rel, "\u2289", "\\nsupseteq");
+defineSymbol(math, main, rel, "\u22a8", "\\models");
+defineSymbol(math, main, rel, "\u2190", "\\leftarrow");
+defineSymbol(math, main, rel, "\u2264", "\\le");
+defineSymbol(math, main, rel, "\u2264", "\\leq");
+defineSymbol(math, main, rel, "<", "\\lt");
+defineSymbol(math, main, rel, "\u2260", "\\ne");
+defineSymbol(math, main, rel, "\u2260", "\\neq");
+defineSymbol(math, main, rel, "\u2192", "\\rightarrow");
+defineSymbol(math, main, rel, "\u2192", "\\to");
+defineSymbol(math, ams, rel, "\u2271", "\\ngeq");
+defineSymbol(math, ams, rel, "\u2270", "\\nleq");
+defineSymbol(math, main, spacing, null, "\\!");
+defineSymbol(math, main, spacing, "\u00a0", "\\ ");
+defineSymbol(math, main, spacing, "\u00a0", "~");
+defineSymbol(math, main, spacing, null, "\\,");
+defineSymbol(math, main, spacing, null, "\\:");
+defineSymbol(math, main, spacing, null, "\\;");
+defineSymbol(math, main, spacing, null, "\\enspace");
+defineSymbol(math, main, spacing, null, "\\qquad");
+defineSymbol(math, main, spacing, null, "\\quad");
+defineSymbol(math, main, spacing, "\u00a0", "\\space");
+defineSymbol(math, main, punct, ",", ",");
+defineSymbol(math, main, punct, ";", ";");
+defineSymbol(math, main, punct, ":", "\\colon");
+defineSymbol(math, ams, bin, "\u22bc", "\\barwedge");
+defineSymbol(math, ams, bin, "\u22bb", "\\veebar");
+defineSymbol(math, main, bin, "\u2299", "\\odot");
+defineSymbol(math, main, bin, "\u2295", "\\oplus");
+defineSymbol(math, main, bin, "\u2297", "\\otimes");
+defineSymbol(math, main, textord, "\u2202", "\\partial");
+defineSymbol(math, main, bin, "\u2298", "\\oslash");
+defineSymbol(math, ams, bin, "\u229a", "\\circledcirc");
+defineSymbol(math, ams, bin, "\u22a1", "\\boxdot");
+defineSymbol(math, main, bin, "\u25b3", "\\bigtriangleup");
+defineSymbol(math, main, bin, "\u25bd", "\\bigtriangledown");
+defineSymbol(math, main, bin, "\u2020", "\\dagger");
+defineSymbol(math, main, bin, "\u22c4", "\\diamond");
+defineSymbol(math, main, bin, "\u22c6", "\\star");
+defineSymbol(math, main, bin, "\u25c3", "\\triangleleft");
+defineSymbol(math, main, bin, "\u25b9", "\\triangleright");
+defineSymbol(math, main, open, "{", "\\{");
+defineSymbol(text, main, textord, "{", "\\{");
+defineSymbol(math, main, close, "}", "\\}");
+defineSymbol(text, main, textord, "}", "\\}");
+defineSymbol(math, main, open, "{", "\\lbrace");
+defineSymbol(math, main, close, "}", "\\rbrace");
+defineSymbol(math, main, open, "[", "\\lbrack");
+defineSymbol(math, main, close, "]", "\\rbrack");
+defineSymbol(math, main, open, "\u230a", "\\lfloor");
+defineSymbol(math, main, close, "\u230b", "\\rfloor");
+defineSymbol(math, main, open, "\u2308", "\\lceil");
+defineSymbol(math, main, close, "\u2309", "\\rceil");
+defineSymbol(math, main, textord, "\\", "\\backslash");
+defineSymbol(math, main, textord, "\u2223", "|");
+defineSymbol(math, main, textord, "\u2223", "\\vert");
+defineSymbol(math, main, textord, "\u2225", "\\|");
+defineSymbol(math, main, textord, "\u2225", "\\Vert");
+defineSymbol(math, main, rel, "\u2191", "\\uparrow");
+defineSymbol(math, main, rel, "\u21d1", "\\Uparrow");
+defineSymbol(math, main, rel, "\u2193", "\\downarrow");
+defineSymbol(math, main, rel, "\u21d3", "\\Downarrow");
+defineSymbol(math, main, rel, "\u2195", "\\updownarrow");
+defineSymbol(math, main, rel, "\u21d5", "\\Updownarrow");
+defineSymbol(math, math, op, "\u2210", "\\coprod");
+defineSymbol(math, math, op, "\u22c1", "\\bigvee");
+defineSymbol(math, math, op, "\u22c0", "\\bigwedge");
+defineSymbol(math, math, op, "\u2a04", "\\biguplus");
+defineSymbol(math, math, op, "\u22c2", "\\bigcap");
+defineSymbol(math, math, op, "\u22c3", "\\bigcup");
+defineSymbol(math, math, op, "\u222b", "\\int");
+defineSymbol(math, math, op, "\u222b", "\\intop");
+defineSymbol(math, math, op, "\u222c", "\\iint");
+defineSymbol(math, math, op, "\u222d", "\\iiint");
+defineSymbol(math, math, op, "\u220f", "\\prod");
+defineSymbol(math, math, op, "\u2211", "\\sum");
+defineSymbol(math, math, op, "\u2a02", "\\bigotimes");
+defineSymbol(math, math, op, "\u2a01", "\\bigoplus");
+defineSymbol(math, math, op, "\u2a00", "\\bigodot");
+defineSymbol(math, math, op, "\u222e", "\\oint");
+defineSymbol(math, math, op, "\u2a06", "\\bigsqcup");
+defineSymbol(math, math, op, "\u222b", "\\smallint");
+defineSymbol(text, main, inner, "\u2026", "\\textellipsis");
+defineSymbol(math, main, inner, "\u2026", "\\mathellipsis");
+defineSymbol(text, main, inner, "\u2026", "\\ldots");
+defineSymbol(math, main, inner, "\u2026", "\\ldots");
+defineSymbol(math, main, inner, "\u22ef", "\\cdots");
+defineSymbol(math, main, inner, "\u22f1", "\\ddots");
+defineSymbol(math, main, textord, "\u22ee", "\\vdots");
+defineSymbol(math, main, accent, "\u00b4", "\\acute");
+defineSymbol(math, main, accent, "\u0060", "\\grave");
+defineSymbol(math, main, accent, "\u00a8", "\\ddot");
+defineSymbol(math, main, accent, "\u007e", "\\tilde");
+defineSymbol(math, main, accent, "\u00af", "\\bar");
+defineSymbol(math, main, accent, "\u02d8", "\\breve");
+defineSymbol(math, main, accent, "\u02c7", "\\check");
+defineSymbol(math, main, accent, "\u005e", "\\hat");
+defineSymbol(math, main, accent, "\u20d7", "\\vec");
+defineSymbol(math, main, accent, "\u02d9", "\\dot");
+defineSymbol(math, main, mathord, "\u0131", "\\imath");
+defineSymbol(math, main, mathord, "\u0237", "\\jmath");
+
+defineSymbol(text, main, textord, "\u2013", "--");
+defineSymbol(text, main, textord, "\u2014", "---");
+defineSymbol(text, main, textord, "\u2018", "`");
+defineSymbol(text, main, textord, "\u2019", "'");
+defineSymbol(text, main, textord, "\u201c", "``");
+defineSymbol(text, main, textord, "\u201d", "''");
+defineSymbol(math, main, textord, "\u00b0", "\\degree");
+defineSymbol(text, main, textord, "\u00b0", "\\degree");
+defineSymbol(math, main, mathord, "\u00a3", "\\pounds");
+defineSymbol(math, ams, textord, "\u2720", "\\maltese");
+defineSymbol(text, ams, textord, "\u2720", "\\maltese");
+
+defineSymbol(text, main, spacing, "\u00a0", "\\ ");
+defineSymbol(text, main, spacing, "\u00a0", " ");
+defineSymbol(text, main, spacing, "\u00a0", "~");
+
+// There are lots of symbols which are the same, so we add them in afterwards.
+var i;
+var ch;
+
+// All of these are textords in math mode
+var mathTextSymbols = "0123456789/@.\"";
+for (i = 0; i < mathTextSymbols.length; i++) {
+ ch = mathTextSymbols.charAt(i);
+ defineSymbol(math, main, textord, ch, ch);
+}
+
+// All of these are textords in text mode
+var textSymbols = "0123456789!@*()-=+[]\";:?/.,";
+for (i = 0; i < textSymbols.length; i++) {
+ ch = textSymbols.charAt(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+// All of these are textords in text mode, and mathords in math mode
+var letters = "abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ";
+for (i = 0; i < letters.length; i++) {
+ ch = letters.charAt(i);
+ defineSymbol(math, main, mathord, ch, ch);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+// Latin-1 letters
+for (i = 0x00C0; i <= 0x00D6; i++) {
+ ch = String.fromCharCode(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+for (i = 0x00D8; i <= 0x00F6; i++) {
+ ch = String.fromCharCode(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+for (i = 0x00F8; i <= 0x00FF; i++) {
+ ch = String.fromCharCode(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+// Cyrillic
+for (i = 0x0410; i <= 0x044F; i++) {
+ ch = String.fromCharCode(i);
+ defineSymbol(text, main, textord, ch, ch);
+}
+
+// Unicode versions of existing characters
+defineSymbol(text, main, textord, "\u2013", "–");
+defineSymbol(text, main, textord, "\u2014", "—");
+defineSymbol(text, main, textord, "\u2018", "‘");
+defineSymbol(text, main, textord, "\u2019", "’");
+defineSymbol(text, main, textord, "\u201c", "“");
+defineSymbol(text, main, textord, "\u201d", "”");
+
+},{}],24:[function(require,module,exports){
+var hangulRegex = /[\uAC00-\uD7AF]/;
+
+// This regex combines
+// - Hiragana: [\u3040-\u309F]
+// - Katakana: [\u30A0-\u30FF]
+// - CJK ideograms: [\u4E00-\u9FAF]
+// - Hangul syllables: [\uAC00-\uD7AF]
+// Notably missing are halfwidth Katakana and Romanji glyphs.
+var cjkRegex =
+ /[\u3040-\u309F]|[\u30A0-\u30FF]|[\u4E00-\u9FAF]|[\uAC00-\uD7AF]/;
+
+module.exports = {
+ cjkRegex: cjkRegex,
+ hangulRegex: hangulRegex
+};
+
+},{}],25:[function(require,module,exports){
+/**
+ * This file contains a list of utility functions which are useful in other
+ * files.
+ */
+
+/**
+ * Provide an `indexOf` function which works in IE8, but defers to native if
+ * possible.
+ */
+var nativeIndexOf = Array.prototype.indexOf;
+var indexOf = function(list, elem) {
+ if (list == null) {
+ return -1;
+ }
+ if (nativeIndexOf && list.indexOf === nativeIndexOf) {
+ return list.indexOf(elem);
+ }
+ var i = 0;
+ var l = list.length;
+ for (; i < l; i++) {
+ if (list[i] === elem) {
+ return i;
+ }
+ }
+ return -1;
+};
+
+/**
+ * Return whether an element is contained in a list
+ */
+var contains = function(list, elem) {
+ return indexOf(list, elem) !== -1;
+};
+
+/**
+ * Provide a default value if a setting is undefined
+ */
+var deflt = function(setting, defaultIfUndefined) {
+ return setting === undefined ? defaultIfUndefined : setting;
+};
+
+// hyphenate and escape adapted from Facebook's React under Apache 2 license
+
+var uppercase = /([A-Z])/g;
+var hyphenate = function(str) {
+ return str.replace(uppercase, "-$1").toLowerCase();
+};
+
+var ESCAPE_LOOKUP = {
+ "&": "&",
+ ">": ">",
+ "<": "<",
+ "\"": """,
+ "'": "'"
+};
+
+var ESCAPE_REGEX = /[&><"']/g;
+
+function escaper(match) {
+ return ESCAPE_LOOKUP[match];
+}
+
+/**
+ * Escapes text to prevent scripting attacks.
+ *
+ * @param {*} text Text value to escape.
+ * @return {string} An escaped string.
+ */
+function escape(text) {
+ return ("" + text).replace(ESCAPE_REGEX, escaper);
+}
+
+/**
+ * A function to set the text content of a DOM element in all supported
+ * browsers. Note that we don't define this if there is no document.
+ */
+var setTextContent;
+if (typeof document !== "undefined") {
+ var testNode = document.createElement("span");
+ if ("textContent" in testNode) {
+ setTextContent = function(node, text) {
+ node.textContent = text;
+ };
+ } else {
+ setTextContent = function(node, text) {
+ node.innerText = text;
+ };
+ }
+}
+
+/**
+ * A function to clear a node.
+ */
+function clearNode(node) {
+ setTextContent(node, "");
+}
+
+module.exports = {
+ contains: contains,
+ deflt: deflt,
+ escape: escape,
+ hyphenate: hyphenate,
+ indexOf: indexOf,
+ setTextContent: setTextContent,
+ clearNode: clearNode
+};
+
+},{}]},{},[1])(1)
+});
\ No newline at end of file
diff --git a/specs/gl/katex/katex.min.css b/specs/gl/katex/katex.min.css
new file mode 100644
index 0000000..d6fb837
--- /dev/null
+++ b/specs/gl/katex/katex.min.css
@@ -0,0 +1 @@
+@font-face{font-family:KaTeX_AMS;src:url(fonts/KaTeX_AMS-Regular.eot);src:url(fonts/KaTeX_AMS-Regular.eot#iefix) format('embedded-opentype'),url(fonts/KaTeX_AMS-Regular.woff2) format('woff2'),url(fonts/KaTeX_AMS-Regular.woff) format('woff'),url(fonts/KaTeX_AMS-Regular.ttf) format('truetype');font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Caligraphic;src:url(fonts/KaTeX_Caligraphic-Bold.eot);src:url(fonts/KaTeX_Caligraphic-Bold.eot#iefix) format('embedded-opentype'),url(fonts/KaTeX_Caligraphic-Bold.woff2) format('woff2'),url(fonts/KaTeX_Caligraphic-Bold.woff) format('woff'),url(fonts/KaTeX_Caligraphic-Bold.ttf) format('truetype');font-weight:700;font-style:normal}@font-face{font-family:KaTeX_Caligraphic;src:url(fonts/KaTeX_Caligraphic-Regular.eot);src:url(fonts/KaTeX_Caligraphic-Regular.eot#iefix) format('embedded-opentype'),url(fonts/KaTeX_Caligraphic-Regular.woff2) format('woff2'),url(fonts/KaTeX_Caligraphic-Regular.woff) format('woff'),url(fonts/KaTeX_Caligraphic-Regular.ttf) format('truetype');font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Fraktur;src:url(fonts/KaTeX_Fraktur-Bold.eot);src:url(fonts/KaTeX_Fraktur-Bold.eot#iefix) format('embedded-opentype'),url(fonts/KaTeX_Fraktur-Bold.woff2) format('woff2'),url(fonts/KaTeX_Fraktur-Bold.woff) format('woff'),url(fonts/KaTeX_Fraktur-Bold.ttf) format('truetype');font-weight:700;font-style:normal}@font-face{font-family:KaTeX_Fraktur;src:url(fonts/KaTeX_Fraktur-Regular.eot);src:url(fonts/KaTeX_Fraktur-Regular.eot#iefix) format('embedded-opentype'),url(fonts/KaTeX_Fraktur-Regular.woff2) format('woff2'),url(fonts/KaTeX_Fraktur-Regular.woff) format('woff'),url(fonts/KaTeX_Fraktur-Regular.ttf) format('truetype');font-weight:400;font-style:normal}@font-face{font-family:KaTeX_Main;src:url(fonts/KaTeX_Main-Bold.eot);src:url(fonts/KaTeX_Main-Bold.eot#iefix) format('embedded-opentype'),url(fonts/KaTeX_Main-Bold.woff2) 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diff --git a/specs/gl/katex/katex.min.js b/specs/gl/katex/katex.min.js
new file mode 100644
index 0000000..66c0821
--- /dev/null
+++ b/specs/gl/katex/katex.min.js
@@ -0,0 +1,4 @@
+(function(e){if(typeof exports==="object"&&typeof module!=="undefined"){module.exports=e()}else if(typeof define==="function"&&define.amd){define([],e)}else{var t;if(typeof window!=="undefined"){t=window}else if(typeof global!=="undefined"){t=global}else if(typeof self!=="undefined"){t=self}else{t=this}t.katex=e()}})(function(){var e,t,r;return function a(e,t,r){function i(s,l){if(!t[s]){if(!e[s]){var o=typeof require=="function"&&require;if(!l&&o)return o(s,!0);if(n)return n(s,!0);var u=new Error("Cannot find module '"+s+"'");throw u.code="MODULE_NOT_FOUND",u}var p=t[s]={exports:{}};e[s][0].call(p.exports,function(t){var r=e[s][1][t];return i(r?r:t)},p,p.exports,a,e,t,r)}return t[s].exports}var n=typeof require=="function"&&require;for(var s=0;s<r.length;s++)i(r[s]);return i}({1:[function(e,t,r){var a=e("./src/ParseError");var i=e("./src/Settings");var n=e("./src/buildTree");var s=e("./src/parseTree");var l=e("./src/utils");var o=function(e,t,r){l.clearNode(t);var a=new i(r);var o=s(e,a);var u=n(o,e,a).toNode();t.appendChild(u)};if(typeof document!=="undefined"){if(document.compatMode!=="CSS1Compat"){typeof console!=="undefined"&&console.warn("Warning: KaTeX doesn't work in quirks mode. Make sure your "+"website has a suitable doctype.");o=function(){throw new a("KaTeX doesn't work in quirks mode.")}}}var u=function(e,t){var r=new i(t);var a=s(e,r);return n(a,e,r).toMarkup()};var p=function(e,t){var r=new i(t);return s(e,r)};t.exports={render:o,renderToString:u,__parse:p,ParseError:a}},{"./src/ParseError":6,"./src/Settings":8,"./src/buildTree":13,"./src/parseTree":22,"./src/utils":25}],2:[function(e,t,r){"use strict";function a(e){if(!e.__matchAtRelocatable){var t=e.source+"|()";var r="g"+(e.ignoreCase?"i":"")+(e.multiline?"m":"")+(e.unicode?"u":"");e.__matchAtRelocatable=new RegExp(t,r)}return e.__matchAtRelocatable}function i(e,t,r){if(e.global||e.sticky){throw new Error("matchAt(...): Only non-global regexes are supported")}var i=a(e);i.lastIndex=r;var n=i.exec(t);if(n[n.length-1]==null){n.length=n.length-1;return n}else{return null}}t.exports=i},{}],3:[function(e,t,r){var a=e("match-at");var i=e("./ParseError");function n(e){this.input=e;this.pos=0}function s(e,t,r,a){this.text=e;this.start=t;this.end=r;this.lexer=a}s.prototype.range=function(e,t){if(e.lexer!==this.lexer){return new s(t)}return new s(t,this.start,e.end,this.lexer)};var l=new RegExp("([ \r\n ]+)|"+"([!-\\[\\]-\u2027\u202a-\ud7ff\uf900-\uffff]"+"|[\ud800-\udbff][\udc00-\udfff]"+"|\\\\(?:[a-zA-Z]+|[^\ud800-\udfff])"+")");n.prototype.lex=function(){var e=this.input;var t=this.pos;if(t===e.length){return new s("EOF",t,t,this)}var r=a(l,e,t);if(r===null){throw new i("Unexpected character: '"+e[t]+"'",new s(e[t],t,t+1,this))}var n=r[2]||" ";var o=this.pos;this.pos+=r[0].length;var u=this.pos;return new s(n,o,u,this)};t.exports=n},{"./ParseError":6,"match-at":2}],4:[function(e,t,r){var a=e("./Lexer");function i(e,t){this.lexer=new a(e);this.macros=t;this.stack=[];this.discardedWhiteSpace=[]}i.prototype.nextToken=function(){for(;;){if(this.stack.length===0){this.stack.push(this.lexer.lex())}var e=this.stack.pop();var t=e.text;if(!(t.charAt(0)==="\\"&&this.macros.hasOwnProperty(t))){return e}var r=this.macros[t];if(typeof r==="string"){var i=new a(r);r=[];var n=i.lex();while(n.text!=="EOF"){r.push(n);n=i.lex()}r.reverse();this.macros[t]=r}this.stack=this.stack.concat(r)}};i.prototype.get=function(e){this.discardedWhiteSpace=[];var t=this.nextToken();if(e){while(t.text===" "){this.discardedWhiteSpace.push(t);t=this.nextToken()}}return t};i.prototype.unget=function(e){this.stack.push(e);while(this.discardedWhiteSpace.length!==0){this.stack.push(this.discardedWhiteSpace.pop())}};t.exports=i},{"./Lexer":3}],5:[function(e,t,r){function a(e){this.style=e.style;this.color=e.color;this.size=e.size;this.phantom=e.phantom;this.font=e.font;if(e.parentStyle===undefined){this.parentStyle=e.style}else{this.parentStyle=e.parentStyle}if(e.parentSize===undefined){this.parentSize=e.size}else{this.parentSize=e.parentSize}}a.prototype.extend=function(e){var t={style:this.style,size:this.size,color:this.color,parentStyle:this.style,parentSize:this.size,phantom:this.phantom,font:this.font};for(var r in e){if(e.hasOwnProperty(r)){t[r]=e[r]}}return new a(t)};a.prototype.withStyle=function(e){return this.extend({style:e})};a.prototype.withSize=function(e){return this.extend({size:e})};a.prototype.withColor=function(e){return this.extend({color:e})};a.prototype.withPhantom=function(){return this.extend({phantom:true})};a.prototype.withFont=function(e){return this.extend({font:e||this.font})};a.prototype.reset=function(){return this.extend({})};var i={"katex-blue":"#6495ed","katex-orange":"#ffa500","katex-pink":"#ff00af","katex-red":"#df0030","katex-green":"#28ae7b","katex-gray":"gray","katex-purple":"#9d38bd","katex-blueA":"#ccfaff","katex-blueB":"#80f6ff","katex-blueC":"#63d9ea","katex-blueD":"#11accd","katex-blueE":"#0c7f99","katex-tealA":"#94fff5","katex-tealB":"#26edd5","katex-tealC":"#01d1c1","katex-tealD":"#01a995","katex-tealE":"#208170","katex-greenA":"#b6ffb0","katex-greenB":"#8af281","katex-greenC":"#74cf70","katex-greenD":"#1fab54","katex-greenE":"#0d923f","katex-goldA":"#ffd0a9","katex-goldB":"#ffbb71","katex-goldC":"#ff9c39","katex-goldD":"#e07d10","katex-goldE":"#a75a05","katex-redA":"#fca9a9","katex-redB":"#ff8482","katex-redC":"#f9685d","katex-redD":"#e84d39","katex-redE":"#bc2612","katex-maroonA":"#ffbde0","katex-maroonB":"#ff92c6","katex-maroonC":"#ed5fa6","katex-maroonD":"#ca337c","katex-maroonE":"#9e034e","katex-purpleA":"#ddd7ff","katex-purpleB":"#c6b9fc","katex-purpleC":"#aa87ff","katex-purpleD":"#7854ab","katex-purpleE":"#543b78","katex-mintA":"#f5f9e8","katex-mintB":"#edf2df","katex-mintC":"#e0e5cc","katex-grayA":"#f6f7f7","katex-grayB":"#f0f1f2","katex-grayC":"#e3e5e6","katex-grayD":"#d6d8da","katex-grayE":"#babec2","katex-grayF":"#888d93","katex-grayG":"#626569","katex-grayH":"#3b3e40","katex-grayI":"#21242c","katex-kaBlue":"#314453","katex-kaGreen":"#71B307"};a.prototype.getColor=function(){if(this.phantom){return"transparent"}else{return i[this.color]||this.color}};t.exports=a},{}],6:[function(e,t,r){function a(e,t){var r="KaTeX parse error: "+e;var i;var n;if(t&&t.lexer&&t.start<=t.end){var s=t.lexer.input;i=t.start;n=t.end;if(i===s.length){r+=" at end of input: "}else{r+=" at position "+(i+1)+": "}var l=s.slice(i,n).replace(/[^]/g,"$&\u0332");var o;if(i>15){o="\u2026"+s.slice(i-15,i)}else{o=s.slice(0,i)}var u;if(n+15<s.length){u=s.slice(n,n+15)+"\u2026"}else{u=s.slice(n)}r+=o+l+u}var p=new Error(r);p.name="ParseError";p.__proto__=a.prototype;p.position=i;return p}a.prototype.__proto__=Error.prototype;t.exports=a},{}],7:[function(e,t,r){var a=e("./functions");var i=e("./environments");var n=e("./MacroExpander");var s=e("./symbols");var l=e("./utils");var o=e("./unicodeRegexes").cjkRegex;var u=e("./parseData");var p=e("./ParseError");function h(e,t){this.gullet=new n(e,t.macros);this.settings=t;this.leftrightDepth=0}var c=u.ParseNode;function 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this.handleInfixNodes(r)};h.prototype.handleInfixNodes=function(e){var t=-1;var r;for(var a=0;a<e.length;a++){var i=e[a];if(i.type==="infix"){if(t!==-1){throw new p("only one infix operator per group",i.value.token)}t=a;r=i.value.replaceWith}}if(t!==-1){var n;var s;var l=e.slice(0,t);var o=e.slice(t+1);if(l.length===1&&l[0].type==="ordgroup"){n=l[0]}else{n=new c("ordgroup",l,this.mode)}if(o.length===1&&o[0].type==="ordgroup"){s=o[0]}else{s=new c("ordgroup",o,this.mode)}var u=this.callFunction(r,[n,s],null);return[new c(u.type,u,this.mode)]}else{return e}};var v=1;h.prototype.handleSupSubscript=function(e){var t=this.nextToken;var r=t.text;this.consume();var i=this.parseGroup();if(!i){if(!this.settings.throwOnError&&this.nextToken.text[0]==="\\"){return this.handleUnsupportedCmd()}else{throw new p("Expected group after '"+r+"'",t)}}else if(i.isFunction){var n=a[i.result].greediness;if(n>v){return this.parseFunction(i)}else{throw new p("Got function '"+i.result+"' with no arguments "+"as "+e,t)}}else{return i.result}};h.prototype.handleUnsupportedCmd=function(){var e=this.nextToken.text;var t=[];for(var r=0;r<e.length;r++){t.push(new c("textord",e[r],"text"))}var a=new c("text",{body:t,type:"text"},this.mode);var i=new c("color",{color:this.settings.errorColor,value:[a],type:"color"},this.mode);this.consume();return i};h.prototype.parseAtom=function(){var e=this.parseImplicitGroup();if(this.mode==="text"){return e}var t;var r;while(true){var a=this.nextToken;if(a.text==="\\limits"||a.text==="\\nolimits"){if(!e||e.type!=="op"){throw new p("Limit controls must follow a math operator",a)}else{var i=a.text==="\\limits";e.value.limits=i;e.value.alwaysHandleSupSub=true}this.consume()}else if(a.text==="^"){if(t){throw new p("Double superscript",a)}t=this.handleSupSubscript("superscript")}else if(a.text==="_"){if(r){throw new p("Double subscript",a)}r=this.handleSupSubscript("subscript")}else if(a.text==="'"){var n=new c("textord","\\prime",this.mode);var s=[n];this.consume();while(this.nextToken.text==="'"){s.push(n);this.consume()}t=new c("ordgroup",s,this.mode)}else{break}}if(t||r){return new c("supsub",{base:e,sup:t,sub:r},this.mode)}else{return e}};var d=["\\tiny","\\scriptsize","\\footnotesize","\\small","\\normalsize","\\large","\\Large","\\LARGE","\\huge","\\Huge"];var g=["\\displaystyle","\\textstyle","\\scriptstyle","\\scriptscriptstyle"];h.prototype.parseImplicitGroup=function(){var e=this.parseSymbol();if(e==null){return this.parseFunction()}var t=e.result;var r;if(t==="\\left"){var a=this.parseFunction(e);++this.leftrightDepth;r=this.parseExpression(false);--this.leftrightDepth;this.expect("\\right",false);var n=this.parseFunction();return new c("leftright",{body:r,left:a.value.value,right:n.value.value},this.mode)}else if(t==="\\begin"){var s=this.parseFunction(e);var o=s.value.name;if(!i.hasOwnProperty(o)){throw new p("No such environment: "+o,s.value.nameGroup)}var u=i[o];var h=this.parseArguments("\\begin{"+o+"}",u);var m={mode:this.mode,envName:o,parser:this,positions:h.pop()};var f=u.handler(m,h);this.expect("\\end",false);var v=this.nextToken;var y=this.parseFunction();if(y.value.name!==o){throw new p("Mismatch: \\begin{"+o+"} matched "+"by \\end{"+y.value.name+"}",v)}f.position=y.position;return f}else if(l.contains(d,t)){r=this.parseExpression(false);return new c("sizing",{size:"size"+(l.indexOf(d,t)+1),value:r},this.mode)}else if(l.contains(g,t)){r=this.parseExpression(true);return new c("styling",{style:t.slice(1,t.length-5),value:r},this.mode)}else{return this.parseFunction(e)}};h.prototype.parseFunction=function(e){if(!e){e=this.parseGroup()}if(e){if(e.isFunction){var t=e.result;var r=a[t];if(this.mode==="text"&&!r.allowedInText){throw new p("Can't use function '"+t+"' in text mode",e.token)}var i=this.parseArguments(t,r);var n=e.token;var s=this.callFunction(t,i,i.pop(),n);return new c(s.type,s,this.mode)}else{return e.result}}else{return null}};h.prototype.callFunction=function(e,t,r,i){var n={funcName:e,parser:this,positions:r,token:i};return a[e].handler(n,t)};h.prototype.parseArguments=function(e,t){var r=t.numArgs+t.numOptionalArgs;if(r===0){return[[this.pos]]}var i=t.greediness;var n=[this.pos];var s=[];for(var l=0;l<r;l++){var o=this.nextToken;var u=t.argTypes&&t.argTypes[l];var h;if(l<t.numOptionalArgs){if(u){h=this.parseGroupOfType(u,true)}else{h=this.parseGroup(true)}if(!h){s.push(null);n.push(this.pos);continue}}else{if(u){h=this.parseGroupOfType(u)}else{h=this.parseGroup()}if(!h){if(!this.settings.throwOnError&&this.nextToken.text[0]==="\\"){h=new m(this.handleUnsupportedCmd(this.nextToken.text),false)}else{throw new p("Expected group after '"+e+"'",o)}}}var c;if(h.isFunction){var f=a[h.result].greediness;if(f>i){c=this.parseFunction(h)}else{throw new p("Got function '"+h.result+"' as "+"argument to '"+e+"'",o)}}else{c=h.result}s.push(c);n.push(this.pos)}s.push(n);return s};h.prototype.parseGroupOfType=function(e,t){var r=this.mode;if(e==="original"){e=r}if(e==="color"){return this.parseColorGroup(t)}if(e==="size"){return this.parseSizeGroup(t)}this.switchMode(e);if(e==="text"){while(this.nextToken.text===" "){this.consume()}}var a=this.parseGroup(t);this.switchMode(r);return a};h.prototype.parseStringGroup=function(e,t){if(t&&this.nextToken.text!=="["){return null}var r=this.mode;this.mode="text";this.expect(t?"[":"{");var a="";var i=this.nextToken;var n=i;while(this.nextToken.text!==(t?"]":"}")){if(this.nextToken.text==="EOF"){throw new p("Unexpected end of input in "+e,i.range(this.nextToken,a))}n=this.nextToken;a+=n.text;this.consume()}this.mode=r;this.expect(t?"]":"}");return i.range(n,a)};h.prototype.parseRegexGroup=function(e,t){var r=this.mode;this.mode="text";var a=this.nextToken;var i=a;var n="";while(this.nextToken.text!=="EOF"&&e.test(n+this.nextToken.text)){i=this.nextToken;n+=i.text;this.consume()}if(n===""){throw new p("Invalid "+t+": '"+a.text+"'",a)}this.mode=r;return a.range(i,n)};h.prototype.parseColorGroup=function(e){var t=this.parseStringGroup("color",e);if(!t){return null}var r=/^(#[a-z0-9]+|[a-z]+)$/i.exec(t.text);if(!r){throw new p("Invalid color: '"+t.text+"'",t)}return new m(new c("color",r[0],this.mode),false)};h.prototype.parseSizeGroup=function(e){var t;if(!e&&this.nextToken.text!=="{"){t=this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2}$/,"size")}else{t=this.parseStringGroup("size",e)}if(!t){return null}var r=/([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(t.text);if(!r){throw new p("Invalid size: '"+t.text+"'",t)}var a={number:+(r[1]+r[2]),unit:r[3]};if(a.unit!=="em"&&a.unit!=="ex"&&a.unit!=="mu"){throw new p("Invalid unit: '"+a.unit+"'",t)}return new m(new c("color",a,this.mode),false)};h.prototype.parseGroup=function(e){var t=this.nextToken;if(this.nextToken.text===(e?"[":"{")){this.consume();var r=this.parseExpression(false,e?"]":null);var a=this.nextToken;this.expect(e?"]":"}");if(this.mode==="text"){this.formLigatures(r)}return new m(new c("ordgroup",r,this.mode,t,a),false)}else{return e?null:this.parseSymbol()}};h.prototype.formLigatures=function(e){var t;var r=e.length-1;for(t=0;t<r;++t){var a=e[t];var i=a.value;if(i==="-"&&e[t+1].value==="-"){if(t+1<r&&e[t+2].value==="-"){e.splice(t,3,new c("textord","---","text",a,e[t+2]));r-=2}else{e.splice(t,2,new c("textord","--","text",a,e[t+1]));r-=1}}if((i==="'"||i==="`")&&e[t+1].value===i){e.splice(t,2,new c("textord",i+i,"text",a,e[t+1]));r-=1}}};h.prototype.parseSymbol=function(){var e=this.nextToken;if(a[e.text]){this.consume();return new m(e.text,true,e)}else if(s[this.mode][e.text]){this.consume();return new m(new c(s[this.mode][e.text].group,e.text,this.mode,e),false,e)}else if(this.mode==="text"&&o.test(e.text)){this.consume();return new m(new c("textord",e.text,this.mode,e),false,e)}else{return null}};h.prototype.ParseNode=c;t.exports=h},{"./MacroExpander":4,"./ParseError":6,"./environments":16,"./functions":19,"./parseData":21,"./symbols":23,"./unicodeRegexes":24,"./utils":25}],8:[function(e,t,r){function a(e,t){return e===undefined?t:e}function i(e){e=e||{};this.displayMode=a(e.displayMode,false);this.throwOnError=a(e.throwOnError,true);this.errorColor=a(e.errorColor,"#cc0000");this.macros=e.macros||{}}t.exports=i},{}],9:[function(e,t,r){var a=e("./fontMetrics.js").sigmas;var i=[{},{},{}];var n;for(var s in a){if(a.hasOwnProperty(s)){for(n=0;n<3;n++){i[n][s]=a[s][n]}}}for(n=0;n<3;n++){i[n].emPerEx=a.xHeight[n]/a.quad[n]}function l(e,t,r,a){this.id=e;this.size=t;this.cramped=a;this.sizeMultiplier=r;this.metrics=i[t>0?t-1:0]}l.prototype.sup=function(){return y[x[this.id]]};l.prototype.sub=function(){return y[b[this.id]]};l.prototype.fracNum=function(){return y[w[this.id]]};l.prototype.fracDen=function(){return y[k[this.id]]};l.prototype.cramp=function(){return 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l=["\\Gamma","\\Delta","\\Theta","\\Lambda","\\Xi","\\Pi","\\Sigma","\\Upsilon","\\Phi","\\Psi","\\Omega"];var o=["\u0131","\u0237","\xa3"];var u=function(e,t,r,s,l){if(n[r][e]&&n[r][e].replace){e=n[r][e].replace}var o=i.getCharacterMetrics(e,t);var u;if(o){var p=o.italic;if(r==="text"){p=0}u=new a.symbolNode(e,o.height,o.depth,p,o.skew,l)}else{typeof console!=="undefined"&&console.warn("No character metrics for '"+e+"' in style '"+t+"'");u=new a.symbolNode(e,0,0,0,0,l)}if(s){if(s.style.isTight()){u.classes.push("mtight")}if(s.getColor()){u.style.color=s.getColor()}}return u};var p=function(e,t,r,a){if(e==="\\"||n[t][e].font==="main"){return u(e,"Main-Regular",t,r,a)}else{return u(e,"AMS-Regular",t,r,a.concat(["amsrm"]))}};var h=function(e,t,r,a,i){if(i==="mathord"){return c(e,t,r,a)}else if(i==="textord"){return u(e,"Main-Regular",t,r,a.concat(["mathrm"]))}else{throw new Error("unexpected type: "+i+" in mathDefault")}};var 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r;if(e.value.index){r=new n.MathNode("mroot",[v(e.value.body,t),v(e.value.index,t)])}else{r=new n.MathNode("msqrt",[v(e.value.body,t)])}return r};m.leftright=function(e,t){var r=f(e.value.body,t);if(e.value.left!=="."){var a=new n.MathNode("mo",[h(e.value.left,e.mode)]);a.setAttribute("fence","true");r.unshift(a)}if(e.value.right!=="."){var i=new n.MathNode("mo",[h(e.value.right,e.mode)]);i.setAttribute("fence","true");r.push(i)}var s=new n.MathNode("mrow",r);return s};m.middle=function(e,t){var r=new n.MathNode("mo",[h(e.value.middle,e.mode)]);r.setAttribute("fence","true");return r};m.accent=function(e,t){var r=new n.MathNode("mo",[h(e.value.accent,e.mode)]);var a=new n.MathNode("mover",[v(e.value.base,t),r]);a.setAttribute("accent","true");return a};m.spacing=function(e){var t;if(e.value==="\\ "||e.value==="\\space"||e.value===" "||e.value==="~"){t=new n.MathNode("mtext",[new n.TextNode("\xa0")])}else{t=new 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t=[];if(e.value.value!=="."){t.push(h(e.value.value,e.mode))}var r=new n.MathNode("mo",t);if(e.value.mclass==="mopen"||e.value.mclass==="mclose"){r.setAttribute("fence","true")}else{r.setAttribute("fence","false")}return r};m.styling=function(e,t){var r=f(e.value.value,t);var a=new n.MathNode("mstyle",r);var i={display:["0","true"],text:["0","false"],script:["1","false"],scriptscript:["2","false"]};var s=i[e.value.style];a.setAttribute("scriptlevel",s[0]);a.setAttribute("displaystyle",s[1]);return a};m.sizing=function(e,t){var r=f(e.value.value,t);var i=new n.MathNode("mstyle",r);i.setAttribute("mathsize",a.sizingMultiplier[e.value.size]+"em");return i};m.overline=function(e,t){var r=new n.MathNode("mo",[new n.TextNode("\u203e")]);r.setAttribute("stretchy","true");var a=new n.MathNode("mover",[v(e.value.body,t),r]);a.setAttribute("accent","true");return a};m.underline=function(e,t){var r=new n.MathNode("mo",[new n.TextNode("\u203e")]);r.setAttribute("stretchy","true");var a=new n.MathNode("munder",[v(e.value.body,t),r]);a.setAttribute("accentunder","true");return a};m.rule=function(e){var t=new n.MathNode("mrow");return t};m.kern=function(e){var t=new n.MathNode("mrow");return t};m.llap=function(e,t){var r=new n.MathNode("mpadded",[v(e.value.body,t)]);r.setAttribute("lspace","-1width");r.setAttribute("width","0px");return r};m.rlap=function(e,t){var r=new n.MathNode("mpadded",[v(e.value.body,t)]);r.setAttribute("width","0px");return r};m.phantom=function(e,t){var r=f(e.value.value,t);return new n.MathNode("mphantom",r)};m.mclass=function(e,t){var r=f(e.value.value,t);return new n.MathNode("mstyle",r)};var f=function(e,t){var r=[];for(var a=0;a<e.length;a++){var i=e[a];r.push(v(i,t))}return r};var v=function(e,t){if(!e){return new n.MathNode("mrow")}if(m[e.type]){return m[e.type](e,t)}else{throw new s("Got group of unknown type: '"+e.type+"'")}};var d=function(e,t,r){var a=f(e,r);var i=new n.MathNode("mrow",a);var s=new n.MathNode("annotation",[new n.TextNode(t)]);s.setAttribute("encoding","application/x-tex");var l=new n.MathNode("semantics",[i,s]);var o=new n.MathNode("math",[l]);return u(["katex-mathml"],[o])};t.exports=d},{"./ParseError":6,"./buildCommon":10,"./fontMetrics":17,"./mathMLTree":20,"./symbols":23,"./utils":25}],13:[function(e,t,r){var a=e("./buildHTML");var i=e("./buildMathML");var n=e("./buildCommon");var s=e("./Options");var l=e("./Settings");var o=e("./Style");var u=n.makeSpan;var p=function(e,t,r){r=r||new l({});var n=o.TEXT;if(r.displayMode){n=o.DISPLAY}var p=new s({style:n,size:"size5"});var h=i(e,t,p);var c=a(e,p);var m=u(["katex"],[h,c]);if(r.displayMode){return u(["katex-display"],[m])}else{return m}};t.exports=p},{"./Options":5,"./Settings":8,"./Style":9,"./buildCommon":10,"./buildHTML":11,"./buildMathML":12}],14:[function(e,t,r){var a=e("./ParseError");var i=e("./Style");var n=e("./buildCommon");var s=e("./fontMetrics");var l=e("./symbols");var o=e("./utils");var u=n.makeSpan;var 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diff --git a/xml/glx.xml b/xml/glx.xml
index 810d55f..6fadd17 100644
--- a/xml/glx.xml
+++ b/xml/glx.xml
@@ -1086,7 +1086,7 @@
<param><ptype>Display</ptype> *<name>dpy</name></param>
<param><ptype>Window</ptype> <name>overlay</name></param>
<param><ptype>Window</ptype> <name>underlay</name></param>
- <param>long *<name>pTransparentIndex</name></param>
+ <param>unsigned long *<name>pTransparentIndex</name></param>
</command>
<command>
<proto>int <name>glXGetVideoDeviceNV</name></proto>
@@ -1252,7 +1252,7 @@
<param><ptype>GLuint</ptype> *<name>count</name></param>
</command>
<command>
- <proto>int <name>glXQueryGLXPbufferSGIX</name></proto>
+ <proto>void <name>glXQueryGLXPbufferSGIX</name></proto>
<param><ptype>Display</ptype> *<name>dpy</name></param>
<param><ptype>GLXPbufferSGIX</ptype> <name>pbuf</name></param>
<param>int <name>attribute</name></param>
@@ -1393,8 +1393,8 @@
<param><ptype>GLboolean</ptype> <name>bBlock</name></param>
</command>
<command>
- <proto><ptype>Bool</ptype> <name>glXSet3DfxModeMESA</name></proto>
- <param>int <name>mode</name></param>
+ <proto><ptype>GLboolean</ptype> <name>glXSet3DfxModeMESA</name></proto>
+ <param>GLint <name>mode</name></param>
</command>
<command>
<proto>void <name>glXSwapBuffers</name></proto>
diff --git a/xml/wgl.xml b/xml/wgl.xml
index 49277ed..c3b3630 100644
--- a/xml/wgl.xml
+++ b/xml/wgl.xml
@@ -538,7 +538,7 @@
</command>
<command>
<proto><ptype>BOOL</ptype> <name>wglBindVideoDeviceNV</name></proto>
- <param><ptype>HDC</ptype> <name>hDC</name></param>
+ <param><ptype>HDC</ptype> <name>hDc</name></param>
<param>unsigned int <name>uVideoSlot</name></param>
<param><ptype>HVIDEOOUTPUTDEVICENV</ptype> <name>hVideoDevice</name></param>
<param>const int *<name>piAttribList</name></param>
@@ -781,7 +781,7 @@
</command>
<command>
<proto>int <name>wglEnumerateVideoDevicesNV</name></proto>
- <param><ptype>HDC</ptype> <name>hDC</name></param>
+ <param><ptype>HDC</ptype> <name>hDc</name></param>
<param><ptype>HVIDEOOUTPUTDEVICENV</ptype> *<name>phDeviceList</name></param>
</command>
<command>
@@ -912,7 +912,7 @@
<command>
<proto><ptype>INT</ptype> <name>wglGetGPUInfoAMD</name></proto>
<param><ptype>UINT</ptype> <name>id</name></param>
- <param>int <name>property</name></param>
+ <param><ptype>INT</ptype> <name>property</name></param>
<param><ptype>GLenum</ptype> <name>dataType</name></param>
<param><ptype>UINT</ptype> <name>size</name></param>
<param>void *<name>data</name></param>
@@ -1235,7 +1235,7 @@
<command>
<proto><ptype>INT64</ptype> <name>wglSwapLayerBuffersMscOML</name></proto>
<param><ptype>HDC</ptype> <name>hdc</name></param>
- <param>int <name>fuPlanes</name></param>
+ <param><ptype>INT</ptype> <name>fuPlanes</name></param>
<param><ptype>INT64</ptype> <name>target_msc</name></param>
<param><ptype>INT64</ptype> <name>divisor</name></param>
<param><ptype>INT64</ptype> <name>remainder</name></param>
