| 0,0 → 1,309 |
|---|
| /* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */ |
| /* @(#)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. |
| * ==================================================== |
| */ |
| |
| #if defined(LIBM_SCCS) && !defined(lint) |
| static char rcsid[] = "$Id: e_pow.c,v 1.6 1994/09/13 00:40:33 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" |
| |
| #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 */ |
| 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*/ |
| |
| #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; |
| 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 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 y - y; /* inf**+-1 is NaN */ |
| else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ |
| return (hy>=0)? y: zero; |
| else /* (|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 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|<sqrt(3/2) */ |
| else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ |
| else {k=0;n+=1;ix -= 0x00100000;} |
| SET_HIGH_WORD(ax,ix); |
| |
| /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ |
| u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ |
| v = one/(ax+bp[k]); |
| s = u*v; |
| s_h = s; |
| SET_LOW_WORD(s_h,0); |
| /* t_h=ax+bp[k] High */ |
| t_h = zero; |
| SET_HIGH_WORD(t_h,((ix>>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; |
| } |