blob: 5f785a0358a96942dcd84a3c3845d53e7f6d3afb [file] [log] [blame]
 /* * Copyright 2012 Google Inc. * * Use of this source code is governed by a BSD-style license that can be * found in the LICENSE file. */ // http://metamerist.com/cbrt/CubeRoot.cpp // #include #include "CubicUtilities.h" #define TEST_ALTERNATIVES 0 #if TEST_ALTERNATIVES typedef float (*cuberootfnf) (float); typedef double (*cuberootfnd) (double); // estimate bits of precision (32-bit float case) inline int bits_of_precision(float a, float b) { const double kd = 1.0 / log(2.0); if (a==b) return 23; const double kdmin = pow(2.0, -23.0); double d = fabs(a-b); if (d < kdmin) return 23; return int(-log(d)*kd); } // estiamte bits of precision (64-bit double case) inline int bits_of_precision(double a, double b) { const double kd = 1.0 / log(2.0); if (a==b) return 52; const double kdmin = pow(2.0, -52.0); double d = fabs(a-b); if (d < kdmin) return 52; return int(-log(d)*kd); } // cube root via x^(1/3) static float pow_cbrtf(float x) { return (float) pow(x, 1.0f/3.0f); } // cube root via x^(1/3) static double pow_cbrtd(double x) { return pow(x, 1.0/3.0); } // cube root approximation using bit hack for 32-bit float static float cbrt_5f(float f) { unsigned int* p = (unsigned int *) &f; *p = *p/3 + 709921077; return f; } #endif // cube root approximation using bit hack for 64-bit float // adapted from Kahan's cbrt static double cbrt_5d(double d) { const unsigned int B1 = 715094163; double t = 0.0; unsigned int* pt = (unsigned int*) &t; unsigned int* px = (unsigned int*) &d; pt[1]=px[1]/3+B1; return t; } #if TEST_ALTERNATIVES // cube root approximation using bit hack for 64-bit float // adapted from Kahan's cbrt #if 0 static double quint_5d(double d) { return sqrt(sqrt(d)); const unsigned int B1 = 71509416*5/3; double t = 0.0; unsigned int* pt = (unsigned int*) &t; unsigned int* px = (unsigned int*) &d; pt[1]=px[1]/5+B1; return t; } #endif // iterative cube root approximation using Halley's method (float) static float cbrta_halleyf(const float a, const float R) { const float a3 = a*a*a; const float b= a * (a3 + R + R) / (a3 + a3 + R); return b; } #endif // iterative cube root approximation using Halley's method (double) static double cbrta_halleyd(const double a, const double R) { const double a3 = a*a*a; const double b= a * (a3 + R + R) / (a3 + a3 + R); return b; } #if TEST_ALTERNATIVES // iterative cube root approximation using Newton's method (float) static float cbrta_newtonf(const float a, const float x) { // return (1.0 / 3.0) * ((a + a) + x / (a * a)); return a - (1.0f / 3.0f) * (a - x / (a*a)); } // iterative cube root approximation using Newton's method (double) static double cbrta_newtond(const double a, const double x) { return (1.0/3.0) * (x / (a*a) + 2*a); } // cube root approximation using 1 iteration of Halley's method (double) static double halley_cbrt1d(double d) { double a = cbrt_5d(d); return cbrta_halleyd(a, d); } // cube root approximation using 1 iteration of Halley's method (float) static float halley_cbrt1f(float d) { float a = cbrt_5f(d); return cbrta_halleyf(a, d); } // cube root approximation using 2 iterations of Halley's method (double) static double halley_cbrt2d(double d) { double a = cbrt_5d(d); a = cbrta_halleyd(a, d); return cbrta_halleyd(a, d); } #endif // cube root approximation using 3 iterations of Halley's method (double) static double halley_cbrt3d(double d) { double a = cbrt_5d(d); a = cbrta_halleyd(a, d); a = cbrta_halleyd(a, d); return cbrta_halleyd(a, d); } #if TEST_ALTERNATIVES // cube root approximation using 2 iterations of Halley's method (float) static float halley_cbrt2f(float d) { float a = cbrt_5f(d); a = cbrta_halleyf(a, d); return cbrta_halleyf(a, d); } // cube root approximation using 1 iteration of Newton's method (double) static double newton_cbrt1d(double d) { double a = cbrt_5d(d); return cbrta_newtond(a, d); } // cube root approximation using 2 iterations of Newton's method (double) static double newton_cbrt2d(double d) { double a = cbrt_5d(d); a = cbrta_newtond(a, d); return cbrta_newtond(a, d); } // cube root approximation using 3 iterations of Newton's method (double) static double newton_cbrt3d(double d) { double a = cbrt_5d(d); a = cbrta_newtond(a, d); a = cbrta_newtond(a, d); return cbrta_newtond(a, d); } // cube root approximation using 4 iterations of Newton's method (double) static double newton_cbrt4d(double d) { double a = cbrt_5d(d); a = cbrta_newtond(a, d); a = cbrta_newtond(a, d); a = cbrta_newtond(a, d); return cbrta_newtond(a, d); } // cube root approximation using 2 iterations of Newton's method (float) static float newton_cbrt1f(float d) { float a = cbrt_5f(d); return cbrta_newtonf(a, d); } // cube root approximation using 2 iterations of Newton's method (float) static float newton_cbrt2f(float d) { float a = cbrt_5f(d); a = cbrta_newtonf(a, d); return cbrta_newtonf(a, d); } // cube root approximation using 3 iterations of Newton's method (float) static float newton_cbrt3f(float d) { float a = cbrt_5f(d); a = cbrta_newtonf(a, d); a = cbrta_newtonf(a, d); return cbrta_newtonf(a, d); } // cube root approximation using 4 iterations of Newton's method (float) static float newton_cbrt4f(float d) { float a = cbrt_5f(d); a = cbrta_newtonf(a, d); a = cbrta_newtonf(a, d); a = cbrta_newtonf(a, d); return cbrta_newtonf(a, d); } static double TestCubeRootf(const char* szName, cuberootfnf cbrt, double rA, double rB, int rN) { const int N = rN; float dd = float((rB-rA) / N); // calculate 1M numbers int i=0; float d = (float) rA; double s = 0.0; for(d=(float) rA, i=0; i 1.0e-6) { if (bc < minbits) { minbits = bc; worstx = d; worsty = a; } } } bits /= N; printf(" %3d mbp %6.3f abp\n", minbits, bits); return s; } static double TestCubeRootd(const char* szName, cuberootfnd cbrt, double rA, double rB, int rN) { const int N = rN; double dd = (rB-rA) / N; int i=0; double s = 0.0; double d = 0.0; for(d=rA, i=0; i 1.0e-6) { if (bc < minbits) { bits_of_precision(a, b); minbits = bc; worstx = d; worsty = a; } } } bits /= N; printf(" %3d mbp %6.3f abp\n", minbits, bits); return s; } static int _tmain() { // a million uniform steps through the range from 0.0 to 1.0 // (doing uniform steps in the log scale would be better) double a = 0.0; double b = 1.0; int n = 1000000; printf("32-bit float tests\n"); printf("----------------------------------------\n"); TestCubeRootf("cbrt_5f", cbrt_5f, a, b, n); TestCubeRootf("pow", pow_cbrtf, a, b, n); TestCubeRootf("halley x 1", halley_cbrt1f, a, b, n); TestCubeRootf("halley x 2", halley_cbrt2f, a, b, n); TestCubeRootf("newton x 1", newton_cbrt1f, a, b, n); TestCubeRootf("newton x 2", newton_cbrt2f, a, b, n); TestCubeRootf("newton x 3", newton_cbrt3f, a, b, n); TestCubeRootf("newton x 4", newton_cbrt4f, a, b, n); printf("\n\n"); printf("64-bit double tests\n"); printf("----------------------------------------\n"); TestCubeRootd("cbrt_5d", cbrt_5d, a, b, n); TestCubeRootd("pow", pow_cbrtd, a, b, n); TestCubeRootd("halley x 1", halley_cbrt1d, a, b, n); TestCubeRootd("halley x 2", halley_cbrt2d, a, b, n); TestCubeRootd("halley x 3", halley_cbrt3d, a, b, n); TestCubeRootd("newton x 1", newton_cbrt1d, a, b, n); TestCubeRootd("newton x 2", newton_cbrt2d, a, b, n); TestCubeRootd("newton x 3", newton_cbrt3d, a, b, n); TestCubeRootd("newton x 4", newton_cbrt4d, a, b, n); printf("\n\n"); return 0; } #endif double cube_root(double x) { if (approximately_zero_cubed(x)) { return 0; } double result = halley_cbrt3d(fabs(x)); if (x < 0) { result = -result; } return result; } #if TEST_ALTERNATIVES // http://bytes.com/topic/c/answers/754588-tips-find-cube-root-program-using-c /* cube root */ int icbrt(int n) { int t=0, x=(n+2)/3; /* works for n=0 and n>=1 */ for(; t!=x;) { int x3=x*x*x; t=x; x*=(2*n + x3); x/=(2*x3 + n); } return x ; /* always(?) equal to floor(n^(1/3)) */ } #endif