blob: a3d24ced9c65e3e2d44027b3c016870e5822707e [file] [log] [blame]
 /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* __ieee754_pow(x,y) return x**y * * n * Method: Let x = 2 * (1+f) * 1. Compute and return log2(x) in two pieces: * log2(x) = w1 + w2, * where w1 has 53-24 = 29 bit trailing zeros. * 2. Perform y*log2(x) = n+y' by simulating muti-precision * arithmetic, where |y'|<=0.5. * 3. Return x**y = 2**n*exp(y'*log2) * * Special cases: * 1. +-1 ** anything is 1.0 * 2. +-1 ** +-INF is 1.0 * 3. (anything) ** 0 is 1 * 4. (anything) ** 1 is itself * 5. (anything) ** NAN is NAN * 6. NAN ** (anything except 0) is NAN * 7. +-(|x| > 1) ** +INF is +INF * 8. +-(|x| > 1) ** -INF is +0 * 9. +-(|x| < 1) ** +INF is +0 * 10 +-(|x| < 1) ** -INF is +INF * 11. +0 ** (+anything except 0, NAN) is +0 * 12. -0 ** (+anything except 0, NAN, odd integer) is +0 * 13. +0 ** (-anything except 0, NAN) is +INF * 14. -0 ** (-anything except 0, NAN, odd integer) is +INF * 15. -0 ** (odd integer) = -( +0 ** (odd integer) ) * 16. +INF ** (+anything except 0,NAN) is +INF * 17. +INF ** (-anything except 0,NAN) is +0 * 18. -INF ** (anything) = -0 ** (-anything) * 19. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) * 20. (-anything except 0 and inf) ** (non-integer) is NAN * * Accuracy: * pow(x,y) returns x**y nearly rounded. In particular * pow(integer,integer) * always returns the correct integer provided it is * representable. * * Constants : * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include "math_libm.h" #include "math_private.h" #if defined(_MSC_VER) /* Handle Microsoft VC++ compiler specifics. */ /* C4756: overflow in constant arithmetic */ #pragma warning ( disable : 4756 ) #endif #ifdef __WATCOMC__ /* Watcom defines huge=__huge */ #undef huge #endif static const double bp[] = {1.0, 1.5,}, dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ huge = 1.0e300, tiny = 1.0e-300, /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ double attribute_hidden __ieee754_pow(double x, double y) { double z,ax,z_h,z_l,p_h,p_l; double y1,t1,t2,r,s,t,u,v,w; int32_t i,j,k,yisint,n; int32_t hx,hy,ix,iy; u_int32_t lx,ly; EXTRACT_WORDS(hx,lx,x); /* x==1: 1**y = 1 (even if y is NaN) */ if (hx==0x3ff00000 && lx==0) { return x; } ix = hx&0x7fffffff; EXTRACT_WORDS(hy,ly,y); iy = hy&0x7fffffff; /* y==zero: x**0 = 1 */ if((iy|ly)==0) return one; /* +-NaN return x+y */ if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) return x+y; /* determine if y is an odd int when x < 0 * yisint = 0 ... y is not an integer * yisint = 1 ... y is an odd int * yisint = 2 ... y is an even int */ yisint = 0; if(hx<0) { if(iy>=0x43400000) yisint = 2; /* even integer y */ else if(iy>=0x3ff00000) { k = (iy>>20)-0x3ff; /* exponent */ if(k>20) { j = ly>>(52-k); if((j<<(52-k))==ly) yisint = 2-(j&1); } else if(ly==0) { j = iy>>(20-k); if((j<<(20-k))==iy) yisint = 2-(j&1); } } } /* special value of y */ if(ly==0) { if (iy==0x7ff00000) { /* y is +-inf */ if (((ix-0x3ff00000)|lx)==0) return one; /* +-1**+-inf is 1 (yes, weird rule) */ if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */ return (hy>=0) ? y : zero; /* (|x|<1)**-,+inf = inf,0 */ return (hy<0) ? -y : zero; } if(iy==0x3ff00000) { /* y is +-1 */ if(hy<0) return one/x; else return x; } if(hy==0x40000000) return x*x; /* y is 2 */ if(hy==0x3fe00000) { /* y is 0.5 */ if(hx>=0) /* x >= +0 */ return __ieee754_sqrt(x); } } ax = fabs(x); /* special value of x */ if(lx==0) { if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ z = ax; /*x is +-0,+-inf,+-1*/ if(hy<0) z = one/z; /* z = (1/|x|) */ if(hx<0) { if(((ix-0x3ff00000)|yisint)==0) { z = (z-z)/(z-z); /* (-1)**non-int is NaN */ } else if(yisint==1) z = -z; /* (x<0)**odd = -(|x|**odd) */ } return z; } } /* (x<0)**(non-int) is NaN */ if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); /* |y| is huge */ if(iy>0x41e00000) { /* if |y| > 2**31 */ if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; } /* over/underflow if x is not close to one */ if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; /* now |1-x| is tiny <= 2**-20, suffice to compute log(x) by x-x^2/2+x^3/3-x^4/4 */ t = x-1; /* t has 20 trailing zeros */ w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ v = t*ivln2_l-w*ivln2; t1 = u+v; SET_LOW_WORD(t1,0); t2 = v-(t1-u); } else { double s2,s_h,s_l,t_h,t_l; n = 0; /* take care subnormal number */ if(ix<0x00100000) {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } n += ((ix)>>20)-0x3ff; j = ix&0x000fffff; /* determine interval */ ix = j|0x3ff00000; /* normalize ix */ if(j<=0x3988E) k=0; /* |x|>1)|0x20000000)+0x00080000+(k<<18)); t_l = ax - (t_h-bp[k]); s_l = v*((u-s_h*t_h)-s_h*t_l); /* compute log(ax) */ s2 = s*s; r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); r += s_l*(s_h+s); s2 = s_h*s_h; t_h = 3.0+s2+r; SET_LOW_WORD(t_h,0); t_l = r-((t_h-3.0)-s2); /* u+v = s*(1+...) */ u = s_h*t_h; v = s_l*t_h+t_l*s; /* 2/(3log2)*(s+...) */ p_h = u+v; SET_LOW_WORD(p_h,0); p_l = v-(p_h-u); z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ z_l = cp_l*p_h+p_l*cp+dp_l[k]; /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ t = (double)n; t1 = (((z_h+z_l)+dp_h[k])+t); SET_LOW_WORD(t1,0); t2 = z_l-(((t1-t)-dp_h[k])-z_h); } s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ y1 = y; SET_LOW_WORD(y1,0); p_l = (y-y1)*t1+y*t2; p_h = y1*t1; z = p_l+p_h; EXTRACT_WORDS(j,i,z); if (j>=0x40900000) { /* z >= 1024 */ if(((j-0x40900000)|i)!=0) /* if z > 1024 */ return s*huge*huge; /* overflow */ else { if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ } } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ return s*tiny*tiny; /* underflow */ else { if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ } } /* * compute 2**(p_h+p_l) */ i = j&0x7fffffff; k = (i>>20)-0x3ff; n = 0; if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ n = j+(0x00100000>>(k+1)); k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ t = zero; SET_HIGH_WORD(t,n&~(0x000fffff>>k)); n = ((n&0x000fffff)|0x00100000)>>(20-k); if(j<0) n = -n; p_h -= t; } t = p_l+p_h; SET_LOW_WORD(t,0); u = t*lg2_h; v = (p_l-(t-p_h))*lg2+t*lg2_l; z = u+v; w = v-(z-u); t = z*z; t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); r = (z*t1)/(t1-two)-(w+z*w); z = one-(r-z); GET_HIGH_WORD(j,z); j += (n<<20); if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ else SET_HIGH_WORD(z,j); return s*z; } /* * wrapper pow(x,y) return x**y */ #ifndef _IEEE_LIBM double pow(double x, double y) { double z = __ieee754_pow(x, y); if (_LIB_VERSION == _IEEE_|| isnan(y)) return z; if (isnan(x)) { if (y == 0.0) return __kernel_standard(x, y, 42); /* pow(NaN,0.0) */ return z; } if (x == 0.0) { if (y == 0.0) return __kernel_standard(x, y, 20); /* pow(0.0,0.0) */ if (isfinite(y) && y < 0.0) return __kernel_standard(x,y,23); /* pow(0.0,negative) */ return z; } if (!isfinite(z)) { if (isfinite(x) && isfinite(y)) { if (isnan(z)) return __kernel_standard(x, y, 24); /* pow neg**non-int */ return __kernel_standard(x, y, 21); /* pow overflow */ } } if (z == 0.0 && isfinite(x) && isfinite(y)) return __kernel_standard(x, y, 22); /* pow underflow */ return z; } #else strong_alias(__ieee754_pow, pow) #endif libm_hidden_def(pow)