/* @(#)e_pow.c 5.1 93/09/24 */ /* * ==================================================== * 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. * ==================================================== */ /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, for performance improvement on pipelined processors. */ #if defined(LIBM_SCCS) && !defined(lint) static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $"; #endif /* __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. (anything) ** 0 is 1 * 2. (anything) ** 1 is itself * 3. (anything) ** NAN is NAN * 4. NAN ** (anything except 0) is NAN * 5. +-(|x| > 1) ** +INF is +INF * 6. +-(|x| > 1) ** -INF is +0 * 7. +-(|x| < 1) ** +INF is +0 * 8. +-(|x| < 1) ** -INF is +INF * 9. +-1 ** +-INF is NAN * 10. +0 ** (+anything except 0, NAN) is +0 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 * 12. +0 ** (-anything except 0, NAN) is +INF * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) * 15. +INF ** (+anything except 0,NAN) is +INF * 16. +INF ** (-anything except 0,NAN) is +0 * 17. -INF ** (anything) = -0 ** (-anything) * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) * 19. (-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.h" #include "math_private.h" #define zero C[0] #define one C[1] #define two C[2] #define two53 C[3] #define huge C[4] #define tiny C[5] #define L1 C[6] #define L2 C[7] #define L3 C[8] #define L4 C[9] #define L5 C[10] #define L6 C[11] #define P1 C[12] #define P2 C[13] #define P3 C[14] #define P4 C[15] #define P5 C[16] #define lg2 C[17] #define lg2_h C[18] #define lg2_l C[19] #define ovt C[20] #define cp C[21] #define cp_h C[22] #define cp_l C[23] #define ivln2 C[24] #define ivln2_h C[25] #define ivln2_l C[26] #ifdef __STDC__ static const double #else static double #endif bp[] = {1.0, 1.5,}, dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ C[] = { 0.0, 1.0, 2.0, 9007199254740992.0 , 1.0e300, 1.0e-300, 5.99999999999994648725e-01 , 4.28571428578550184252e-01 , 3.33333329818377432918e-01 , 2.72728123808534006489e-01 , 2.30660745775561754067e-01 , 2.06975017800338417784e-01 , 1.66666666666666019037e-01 , -2.77777777770155933842e-03 , 6.61375632143793436117e-05 , -1.65339022054652515390e-06 , 4.13813679705723846039e-08 , 6.93147180559945286227e-01 , 6.93147182464599609375e-01 , -1.90465429995776804525e-09 , 8.0085662595372944372e-0017 , 9.61796693925975554329e-01 , 9.61796700954437255859e-01 , -7.02846165095275826516e-09 , 1.44269504088896338700e+00 , 1.44269502162933349609e+00 , 1.92596299112661746887e-08 }; #ifdef __STDC__ double __ieee754_pow(double x, double y) #else double __ieee754_pow(x,y) double x, y; #endif { double z,ax,z_h,z_l,p_h,p_l; double y1,t1,t2,r,s,t,u,v,w, t12,t14,r_1,r_2,r_3; int32_t i,j,k,yisint,n; int32_t hx,hy,ix,iy; u_int32_t lx,ly; EXTRACT_WORDS(hx,lx,x); EXTRACT_WORDS(hy,ly,y); ix = hx&0x7fffffff; iy = hy&0x7fffffff; /* y==zero: x**0 = 1 */ if((iy|ly)==0) return C[1]; /* +-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((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1); } else if(ly==0) { j = iy>>(20-k); if((int32_t)(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 y - y; /* inf**+-1 is NaN */ else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ return (hy>=0)? y: C[0]; else /* (|x|<1)**-,+inf = inf,0 */ return (hy<0)?-y: C[0]; } if(iy==0x3ff00000) { /* y is +-1 */ if(hy<0) return C[1]/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 = C[1]/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)? C[4]*C[4]:C[5]*C[5]; if(ix>=0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5]; } /* over/underflow if x is not close to one */ if(ix<0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5]; if(ix>0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5]; /* 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 = C[25]*t; /* ivln2_h has 21 sig. bits */ v = t*C[26]-w*C[24]; t1 = u+v; SET_LOW_WORD(t1,0); t2 = v-(t1-u); } else { double s2,s_h,s_l,t_h,t_l,s22,s24,s26,r1,r2,r3; n = 0; /* take care subnormal number */ if(ix<0x00100000) {ax *= C[3]; 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; #ifdef DO_NOT_USE_THIS r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); #else r1 = C[10]+s2*C[11]; s22=s2*s2; r2 = C[8]+s2*C[9]; s24=s22*s22; r3 = C[6]+s2*C[7]; s26=s24*s22; r = r3*s22 + r2*s24 + r1*s26; #endif 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 = C[22]*p_h; /* cp_h+cp_l = 2/(3*log2) */ z_l = C[23]*p_h+p_l*C[21]+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 = C[1]; /* s (sign of result -ve**odd) = -1 else = 1 */ if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) s = -C[1];/* (-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*C[4]*C[4]; /* overflow */ else { if(p_l+C[20]>z-p_h) return s*C[4]*C[4]; /* overflow */ } } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ return s*C[5]*C[5]; /* underflow */ else { if(p_l<=z-p_h) return s*C[5]*C[5]; /* 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 = C[0]; 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*C[18]; v = (p_l-(t-p_h))*C[17]+t*C[19]; z = u+v; w = v-(z-u); t = z*z; #ifdef DO_NOT_USE_THIS t1 = z - t*(C[12]+t*(C[13]+t*(C[14]+t*(C[15]+t*C[16])))); #else r_1 = C[15]+t*C[16]; t12 = t*t; r_2 = C[13]+t*C[14]; t14 = t12*t12; r_3 = t*C[12]; t1 = z - r_3 - t12*r_2 - t14*r_1; #endif r = (z*t1)/(t1-C[2])-(w+z*w); z = C[1]-(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; }