1141296Sdas/* @(#)e_pow.c 1.5 04/04/22 SMI */ 22116Sjkh/* 32116Sjkh * ==================================================== 4141296Sdas * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 52116Sjkh * 62116Sjkh * Permission to use, copy, modify, and distribute this 7141296Sdas * software is freely granted, provided that this notice 82116Sjkh * is preserved. 92116Sjkh * ==================================================== 102116Sjkh */ 112116Sjkh 12176266Sbde#include <sys/cdefs.h> 13176266Sbde__FBSDID("$FreeBSD: releng/10.3/lib/msun/src/e_pow.c 271779 2014-09-18 15:10:22Z tijl $"); 142116Sjkh 152116Sjkh/* __ieee754_pow(x,y) return x**y 162116Sjkh * 172116Sjkh * n 182116Sjkh * Method: Let x = 2 * (1+f) 192116Sjkh * 1. Compute and return log2(x) in two pieces: 202116Sjkh * log2(x) = w1 + w2, 212116Sjkh * where w1 has 53-24 = 29 bit trailing zeros. 22271779Stijl * 2. Perform y*log2(x) = n+y' by simulating multi-precision 232116Sjkh * arithmetic, where |y'|<=0.5. 242116Sjkh * 3. Return x**y = 2**n*exp(y'*log2) 252116Sjkh * 262116Sjkh * Special cases: 272116Sjkh * 1. (anything) ** 0 is 1 282116Sjkh * 2. (anything) ** 1 is itself 29271779Stijl * 3. (anything) ** NAN is NAN except 1 ** NAN = 1 302116Sjkh * 4. NAN ** (anything except 0) is NAN 312116Sjkh * 5. +-(|x| > 1) ** +INF is +INF 322116Sjkh * 6. +-(|x| > 1) ** -INF is +0 332116Sjkh * 7. +-(|x| < 1) ** +INF is +0 342116Sjkh * 8. +-(|x| < 1) ** -INF is +INF 35271779Stijl * 9. +-1 ** +-INF is 1 362116Sjkh * 10. +0 ** (+anything except 0, NAN) is +0 372116Sjkh * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 382116Sjkh * 12. +0 ** (-anything except 0, NAN) is +INF 392116Sjkh * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 402116Sjkh * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 412116Sjkh * 15. +INF ** (+anything except 0,NAN) is +INF 422116Sjkh * 16. +INF ** (-anything except 0,NAN) is +0 432116Sjkh * 17. -INF ** (anything) = -0 ** (-anything) 442116Sjkh * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 452116Sjkh * 19. (-anything except 0 and inf) ** (non-integer) is NAN 462116Sjkh * 472116Sjkh * Accuracy: 482116Sjkh * pow(x,y) returns x**y nearly rounded. In particular 492116Sjkh * pow(integer,integer) 50141296Sdas * always returns the correct integer provided it is 512116Sjkh * representable. 522116Sjkh * 532116Sjkh * Constants : 54141296Sdas * The hexadecimal values are the intended ones for the following 55141296Sdas * constants. The decimal values may be used, provided that the 56141296Sdas * compiler will convert from decimal to binary accurately enough 572116Sjkh * to produce the hexadecimal values shown. 582116Sjkh */ 592116Sjkh 602116Sjkh#include "math.h" 612116Sjkh#include "math_private.h" 622116Sjkh 638870Srgrimesstatic const double 642116Sjkhbp[] = {1.0, 1.5,}, 652116Sjkhdp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 662116Sjkhdp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 672116Sjkhzero = 0.0, 682116Sjkhone = 1.0, 692116Sjkhtwo = 2.0, 702116Sjkhtwo53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 712116Sjkhhuge = 1.0e300, 722116Sjkhtiny = 1.0e-300, 732116Sjkh /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 742116SjkhL1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 752116SjkhL2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 762116SjkhL3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 772116SjkhL4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 782116SjkhL5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 792116SjkhL6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 802116SjkhP1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 812116SjkhP2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 822116SjkhP3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 832116SjkhP4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 842116SjkhP5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 852116Sjkhlg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 862116Sjkhlg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 872116Sjkhlg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 882116Sjkhovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 892116Sjkhcp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 902116Sjkhcp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 912116Sjkhcp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 922116Sjkhivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 932116Sjkhivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 942116Sjkhivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 952116Sjkh 9697413Salfreddouble 9797413Salfred__ieee754_pow(double x, double y) 982116Sjkh{ 992116Sjkh double z,ax,z_h,z_l,p_h,p_l; 100141296Sdas double y1,t1,t2,r,s,t,u,v,w; 1012116Sjkh int32_t i,j,k,yisint,n; 1022116Sjkh int32_t hx,hy,ix,iy; 1032116Sjkh u_int32_t lx,ly; 1042116Sjkh 1052116Sjkh EXTRACT_WORDS(hx,lx,x); 1062116Sjkh EXTRACT_WORDS(hy,ly,y); 1072116Sjkh ix = hx&0x7fffffff; iy = hy&0x7fffffff; 1082116Sjkh 1092116Sjkh /* y==zero: x**0 = 1 */ 110141296Sdas if((iy|ly)==0) return one; 1112116Sjkh 112226595Sdas /* x==1: 1**y = 1, even if y is NaN */ 113226595Sdas if (hx==0x3ff00000 && lx == 0) return one; 114226595Sdas 115176266Sbde /* y!=zero: result is NaN if either arg is NaN */ 1162116Sjkh if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 117141296Sdas iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 118176266Sbde return (x+0.0)+(y+0.0); 1192116Sjkh 1202116Sjkh /* determine if y is an odd int when x < 0 1212116Sjkh * yisint = 0 ... y is not an integer 1222116Sjkh * yisint = 1 ... y is an odd int 1232116Sjkh * yisint = 2 ... y is an even int 1242116Sjkh */ 1252116Sjkh yisint = 0; 126141296Sdas if(hx<0) { 1272116Sjkh if(iy>=0x43400000) yisint = 2; /* even integer y */ 1282116Sjkh else if(iy>=0x3ff00000) { 1292116Sjkh k = (iy>>20)-0x3ff; /* exponent */ 1302116Sjkh if(k>20) { 1312116Sjkh j = ly>>(52-k); 1322116Sjkh if((j<<(52-k))==ly) yisint = 2-(j&1); 1332116Sjkh } else if(ly==0) { 1342116Sjkh j = iy>>(20-k); 1352116Sjkh if((j<<(20-k))==iy) yisint = 2-(j&1); 1362116Sjkh } 137141296Sdas } 138141296Sdas } 1392116Sjkh 1402116Sjkh /* special value of y */ 141141296Sdas if(ly==0) { 1422116Sjkh if (iy==0x7ff00000) { /* y is +-inf */ 1432116Sjkh if(((ix-0x3ff00000)|lx)==0) 144271779Stijl return one; /* (-1)**+-inf is 1 */ 1452116Sjkh else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 1462116Sjkh return (hy>=0)? y: zero; 1472116Sjkh else /* (|x|<1)**-,+inf = inf,0 */ 1482116Sjkh return (hy<0)?-y: zero; 149141296Sdas } 1502116Sjkh if(iy==0x3ff00000) { /* y is +-1 */ 1512116Sjkh if(hy<0) return one/x; else return x; 1522116Sjkh } 1532116Sjkh if(hy==0x40000000) return x*x; /* y is 2 */ 1542116Sjkh if(hy==0x3fe00000) { /* y is 0.5 */ 1552116Sjkh if(hx>=0) /* x >= +0 */ 156141296Sdas return sqrt(x); 1572116Sjkh } 1582116Sjkh } 1592116Sjkh 1602116Sjkh ax = fabs(x); 1612116Sjkh /* special value of x */ 1622116Sjkh if(lx==0) { 1632116Sjkh if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 1642116Sjkh z = ax; /*x is +-0,+-inf,+-1*/ 1652116Sjkh if(hy<0) z = one/z; /* z = (1/|x|) */ 1662116Sjkh if(hx<0) { 1672116Sjkh if(((ix-0x3ff00000)|yisint)==0) { 1682116Sjkh z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 169141296Sdas } else if(yisint==1) 1702116Sjkh z = -z; /* (x<0)**odd = -(|x|**odd) */ 1712116Sjkh } 1722116Sjkh return z; 1732116Sjkh } 1742116Sjkh } 175141296Sdas 176129956Sbde /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be 177129956Sbde n = (hx>>31)+1; 1782116Sjkh but ANSI C says a right shift of a signed negative quantity is 1792116Sjkh implementation defined. */ 180129956Sbde n = ((u_int32_t)hx>>31)-1; 1812116Sjkh 182129956Sbde /* (x<0)**(non-int) is NaN */ 183129956Sbde if((n|yisint)==0) return (x-x)/(x-x); 184129956Sbde 185141296Sdas s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 186141296Sdas if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 187129956Sbde 1882116Sjkh /* |y| is huge */ 1892116Sjkh if(iy>0x41e00000) { /* if |y| > 2**31 */ 1902116Sjkh if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 1912116Sjkh if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 1922116Sjkh if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 1932116Sjkh } 1942116Sjkh /* over/underflow if x is not close to one */ 195141296Sdas if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny; 196141296Sdas if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny; 197141296Sdas /* now |1-x| is tiny <= 2**-20, suffice to compute 1982116Sjkh log(x) by x-x^2/2+x^3/3-x^4/4 */ 199141296Sdas t = ax-one; /* t has 20 trailing zeros */ 2002116Sjkh w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 2012116Sjkh u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 2022116Sjkh v = t*ivln2_l-w*ivln2; 2032116Sjkh t1 = u+v; 2042116Sjkh SET_LOW_WORD(t1,0); 2052116Sjkh t2 = v-(t1-u); 2062116Sjkh } else { 207141296Sdas double ss,s2,s_h,s_l,t_h,t_l; 2082116Sjkh n = 0; 2092116Sjkh /* take care subnormal number */ 2102116Sjkh if(ix<0x00100000) 2112116Sjkh {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 2122116Sjkh n += ((ix)>>20)-0x3ff; 2132116Sjkh j = ix&0x000fffff; 2142116Sjkh /* determine interval */ 2152116Sjkh ix = j|0x3ff00000; /* normalize ix */ 2162116Sjkh if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 2172116Sjkh else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 2182116Sjkh else {k=0;n+=1;ix -= 0x00100000;} 2192116Sjkh SET_HIGH_WORD(ax,ix); 2202116Sjkh 221141296Sdas /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 2222116Sjkh u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 2232116Sjkh v = one/(ax+bp[k]); 224141296Sdas ss = u*v; 225141296Sdas s_h = ss; 2262116Sjkh SET_LOW_WORD(s_h,0); 2272116Sjkh /* t_h=ax+bp[k] High */ 2282116Sjkh t_h = zero; 2292116Sjkh SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 2302116Sjkh t_l = ax - (t_h-bp[k]); 2312116Sjkh s_l = v*((u-s_h*t_h)-s_h*t_l); 2322116Sjkh /* compute log(ax) */ 233141296Sdas s2 = ss*ss; 2342116Sjkh r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 235141296Sdas r += s_l*(s_h+ss); 2362116Sjkh s2 = s_h*s_h; 2372116Sjkh t_h = 3.0+s2+r; 2382116Sjkh SET_LOW_WORD(t_h,0); 2392116Sjkh t_l = r-((t_h-3.0)-s2); 240141296Sdas /* u+v = ss*(1+...) */ 2412116Sjkh u = s_h*t_h; 242141296Sdas v = s_l*t_h+t_l*ss; 243141296Sdas /* 2/(3log2)*(ss+...) */ 2442116Sjkh p_h = u+v; 2452116Sjkh SET_LOW_WORD(p_h,0); 2462116Sjkh p_l = v-(p_h-u); 2472116Sjkh z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 2482116Sjkh z_l = cp_l*p_h+p_l*cp+dp_l[k]; 249141296Sdas /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 2502116Sjkh t = (double)n; 2512116Sjkh t1 = (((z_h+z_l)+dp_h[k])+t); 2522116Sjkh SET_LOW_WORD(t1,0); 2532116Sjkh t2 = z_l-(((t1-t)-dp_h[k])-z_h); 2542116Sjkh } 2552116Sjkh 2562116Sjkh /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 2572116Sjkh y1 = y; 2582116Sjkh SET_LOW_WORD(y1,0); 2592116Sjkh p_l = (y-y1)*t1+y*t2; 2602116Sjkh p_h = y1*t1; 2612116Sjkh z = p_l+p_h; 2622116Sjkh EXTRACT_WORDS(j,i,z); 2632116Sjkh if (j>=0x40900000) { /* z >= 1024 */ 2642116Sjkh if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 265141296Sdas return s*huge*huge; /* overflow */ 2662116Sjkh else { 267141296Sdas if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 2682116Sjkh } 2692116Sjkh } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 270141296Sdas if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 271141296Sdas return s*tiny*tiny; /* underflow */ 2722116Sjkh else { 273141296Sdas if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 2742116Sjkh } 2752116Sjkh } 2762116Sjkh /* 2772116Sjkh * compute 2**(p_h+p_l) 2782116Sjkh */ 2792116Sjkh i = j&0x7fffffff; 2802116Sjkh k = (i>>20)-0x3ff; 2812116Sjkh n = 0; 2822116Sjkh if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 2832116Sjkh n = j+(0x00100000>>(k+1)); 2842116Sjkh k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 2852116Sjkh t = zero; 2862116Sjkh SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 2872116Sjkh n = ((n&0x000fffff)|0x00100000)>>(20-k); 2882116Sjkh if(j<0) n = -n; 2892116Sjkh p_h -= t; 290141296Sdas } 2912116Sjkh t = p_l+p_h; 2922116Sjkh SET_LOW_WORD(t,0); 2932116Sjkh u = t*lg2_h; 2942116Sjkh v = (p_l-(t-p_h))*lg2+t*lg2_l; 2952116Sjkh z = u+v; 2962116Sjkh w = v-(z-u); 2972116Sjkh t = z*z; 2982116Sjkh t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 2992116Sjkh r = (z*t1)/(t1-two)-(w+z*w); 3002116Sjkh z = one-(r-z); 3012116Sjkh GET_HIGH_WORD(j,z); 3022116Sjkh j += (n<<20); 3032116Sjkh if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */ 3042116Sjkh else SET_HIGH_WORD(z,j); 305141296Sdas return s*z; 3062116Sjkh} 307