12116Sjkh/* e_powf.c -- float version of e_pow.c. 22116Sjkh * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 32116Sjkh */ 42116Sjkh 52116Sjkh/* 62116Sjkh * ==================================================== 72116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 82116Sjkh * 92116Sjkh * Developed at SunPro, a Sun Microsystems, Inc. business. 102116Sjkh * Permission to use, copy, modify, and distribute this 118870Srgrimes * software is freely granted, provided that this notice 122116Sjkh * is preserved. 132116Sjkh * ==================================================== 142116Sjkh */ 152116Sjkh 16176266Sbde#include <sys/cdefs.h> 17176266Sbde__FBSDID("$FreeBSD: stable/11/lib/msun/src/e_powf.c 369263 2021-02-15 18:17:29Z dim $"); 182116Sjkh 192116Sjkh#include "math.h" 202116Sjkh#include "math_private.h" 212116Sjkh 222116Sjkhstatic const float 232116Sjkhbp[] = {1.0, 1.5,}, 242116Sjkhdp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */ 252116Sjkhdp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */ 262116Sjkhzero = 0.0, 27336305Smarkjhalf = 0.5, 28336305Smarkjqrtr = 0.25, 29336305Smarkjthrd = 3.33333343e-01, /* 0x3eaaaaab */ 302116Sjkhone = 1.0, 312116Sjkhtwo = 2.0, 322116Sjkhtwo24 = 16777216.0, /* 0x4b800000 */ 332116Sjkhhuge = 1.0e30, 342116Sjkhtiny = 1.0e-30, 352116Sjkh /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 362116SjkhL1 = 6.0000002384e-01, /* 0x3f19999a */ 372116SjkhL2 = 4.2857143283e-01, /* 0x3edb6db7 */ 382116SjkhL3 = 3.3333334327e-01, /* 0x3eaaaaab */ 392116SjkhL4 = 2.7272811532e-01, /* 0x3e8ba305 */ 402116SjkhL5 = 2.3066075146e-01, /* 0x3e6c3255 */ 412116SjkhL6 = 2.0697501302e-01, /* 0x3e53f142 */ 422116SjkhP1 = 1.6666667163e-01, /* 0x3e2aaaab */ 432116SjkhP2 = -2.7777778450e-03, /* 0xbb360b61 */ 442116SjkhP3 = 6.6137559770e-05, /* 0x388ab355 */ 452116SjkhP4 = -1.6533901999e-06, /* 0xb5ddea0e */ 462116SjkhP5 = 4.1381369442e-08, /* 0x3331bb4c */ 472116Sjkhlg2 = 6.9314718246e-01, /* 0x3f317218 */ 482116Sjkhlg2_h = 6.93145752e-01, /* 0x3f317200 */ 492116Sjkhlg2_l = 1.42860654e-06, /* 0x35bfbe8c */ 502116Sjkhovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */ 512116Sjkhcp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */ 52176268Sbdecp_h = 9.6191406250e-01, /* 0x3f764000 =12b cp */ 53176268Sbdecp_l = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */ 542116Sjkhivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */ 552116Sjkhivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/ 562116Sjkhivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/ 572116Sjkh 5897413Salfredfloat 5997413Salfred__ieee754_powf(float x, float y) 602116Sjkh{ 612116Sjkh float z,ax,z_h,z_l,p_h,p_l; 62129959Sbde float y1,t1,t2,r,s,sn,t,u,v,w; 632116Sjkh int32_t i,j,k,yisint,n; 642116Sjkh int32_t hx,hy,ix,iy,is; 652116Sjkh 662116Sjkh GET_FLOAT_WORD(hx,x); 672116Sjkh GET_FLOAT_WORD(hy,y); 682116Sjkh ix = hx&0x7fffffff; iy = hy&0x7fffffff; 692116Sjkh 702116Sjkh /* y==zero: x**0 = 1 */ 718870Srgrimes if(iy==0) return one; 722116Sjkh 73226595Sdas /* x==1: 1**y = 1, even if y is NaN */ 74226595Sdas if (hx==0x3f800000) return one; 75226595Sdas 76176266Sbde /* y!=zero: result is NaN if either arg is NaN */ 772116Sjkh if(ix > 0x7f800000 || 782116Sjkh iy > 0x7f800000) 79176266Sbde return (x+0.0F)+(y+0.0F); 802116Sjkh 812116Sjkh /* determine if y is an odd int when x < 0 822116Sjkh * yisint = 0 ... y is not an integer 832116Sjkh * yisint = 1 ... y is an odd int 842116Sjkh * yisint = 2 ... y is an even int 852116Sjkh */ 862116Sjkh yisint = 0; 878870Srgrimes if(hx<0) { 882116Sjkh if(iy>=0x4b800000) yisint = 2; /* even integer y */ 892116Sjkh else if(iy>=0x3f800000) { 902116Sjkh k = (iy>>23)-0x7f; /* exponent */ 912116Sjkh j = iy>>(23-k); 922116Sjkh if((j<<(23-k))==iy) yisint = 2-(j&1); 938870Srgrimes } 948870Srgrimes } 952116Sjkh 962116Sjkh /* special value of y */ 972116Sjkh if (iy==0x7f800000) { /* y is +-inf */ 982116Sjkh if (ix==0x3f800000) 99226595Sdas return one; /* (-1)**+-inf is NaN */ 1002116Sjkh else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */ 1012116Sjkh return (hy>=0)? y: zero; 1022116Sjkh else /* (|x|<1)**-,+inf = inf,0 */ 1032116Sjkh return (hy<0)?-y: zero; 1048870Srgrimes } 1052116Sjkh if(iy==0x3f800000) { /* y is +-1 */ 1062116Sjkh if(hy<0) return one/x; else return x; 1072116Sjkh } 1082116Sjkh if(hy==0x40000000) return x*x; /* y is 2 */ 1092116Sjkh if(hy==0x3f000000) { /* y is 0.5 */ 1102116Sjkh if(hx>=0) /* x >= +0 */ 11123579Sbde return __ieee754_sqrtf(x); 1122116Sjkh } 1132116Sjkh 1142116Sjkh ax = fabsf(x); 1152116Sjkh /* special value of x */ 1162116Sjkh if(ix==0x7f800000||ix==0||ix==0x3f800000){ 1172116Sjkh z = ax; /*x is +-0,+-inf,+-1*/ 1182116Sjkh if(hy<0) z = one/z; /* z = (1/|x|) */ 1192116Sjkh if(hx<0) { 1202116Sjkh if(((ix-0x3f800000)|yisint)==0) { 1212116Sjkh z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 1228870Srgrimes } else if(yisint==1) 1232116Sjkh z = -z; /* (x<0)**odd = -(|x|**odd) */ 1242116Sjkh } 1252116Sjkh return z; 1262116Sjkh } 1278870Srgrimes 128129959Sbde n = ((u_int32_t)hx>>31)-1; 129129959Sbde 1302116Sjkh /* (x<0)**(non-int) is NaN */ 131129959Sbde if((n|yisint)==0) return (x-x)/(x-x); 1322116Sjkh 133129959Sbde sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 134129959Sbde if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */ 135129959Sbde 1362116Sjkh /* |y| is huge */ 1372116Sjkh if(iy>0x4d000000) { /* if |y| > 2**27 */ 1382116Sjkh /* over/underflow if x is not close to one */ 139369263Sdim if(ix<0x3f7ffff7) return (hy<0)? sn*huge*huge:sn*tiny*tiny; 140129959Sbde if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny; 1418870Srgrimes /* now |1-x| is tiny <= 2**-20, suffice to compute 1422116Sjkh log(x) by x-x^2/2+x^3/3-x^4/4 */ 14398349Sbde t = ax-1; /* t has 20 trailing zeros */ 144336305Smarkj w = (t*t)*(half-t*(thrd-t*qrtr)); 1452116Sjkh u = ivln2_h*t; /* ivln2_h has 16 sig. bits */ 1462116Sjkh v = t*ivln2_l-w*ivln2; 1472116Sjkh t1 = u+v; 1482116Sjkh GET_FLOAT_WORD(is,t1); 1492116Sjkh SET_FLOAT_WORD(t1,is&0xfffff000); 1502116Sjkh t2 = v-(t1-u); 1512116Sjkh } else { 1522116Sjkh float s2,s_h,s_l,t_h,t_l; 1532116Sjkh n = 0; 1542116Sjkh /* take care subnormal number */ 1552116Sjkh if(ix<0x00800000) 1562116Sjkh {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); } 1572116Sjkh n += ((ix)>>23)-0x7f; 1582116Sjkh j = ix&0x007fffff; 1592116Sjkh /* determine interval */ 1602116Sjkh ix = j|0x3f800000; /* normalize ix */ 1612116Sjkh if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */ 1622116Sjkh else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */ 1632116Sjkh else {k=0;n+=1;ix -= 0x00800000;} 1642116Sjkh SET_FLOAT_WORD(ax,ix); 1652116Sjkh 1662116Sjkh /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 1672116Sjkh u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 1682116Sjkh v = one/(ax+bp[k]); 1692116Sjkh s = u*v; 1702116Sjkh s_h = s; 1712116Sjkh GET_FLOAT_WORD(is,s_h); 1722116Sjkh SET_FLOAT_WORD(s_h,is&0xfffff000); 1732116Sjkh /* t_h=ax+bp[k] High */ 174129950Sbde is = ((ix>>1)&0xfffff000)|0x20000000; 175129950Sbde SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21)); 1762116Sjkh t_l = ax - (t_h-bp[k]); 1772116Sjkh s_l = v*((u-s_h*t_h)-s_h*t_l); 1782116Sjkh /* compute log(ax) */ 1792116Sjkh s2 = s*s; 1802116Sjkh r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 1812116Sjkh r += s_l*(s_h+s); 1822116Sjkh s2 = s_h*s_h; 183336305Smarkj t_h = 3+s2+r; 1842116Sjkh GET_FLOAT_WORD(is,t_h); 1852116Sjkh SET_FLOAT_WORD(t_h,is&0xfffff000); 186336305Smarkj t_l = r-((t_h-3)-s2); 1872116Sjkh /* u+v = s*(1+...) */ 1882116Sjkh u = s_h*t_h; 1892116Sjkh v = s_l*t_h+t_l*s; 1902116Sjkh /* 2/(3log2)*(s+...) */ 1912116Sjkh p_h = u+v; 1922116Sjkh GET_FLOAT_WORD(is,p_h); 1932116Sjkh SET_FLOAT_WORD(p_h,is&0xfffff000); 1942116Sjkh p_l = v-(p_h-u); 1952116Sjkh z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 1962116Sjkh z_l = cp_l*p_h+p_l*cp+dp_l[k]; 1972116Sjkh /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 198336305Smarkj t = n; 1992116Sjkh t1 = (((z_h+z_l)+dp_h[k])+t); 2002116Sjkh GET_FLOAT_WORD(is,t1); 2012116Sjkh SET_FLOAT_WORD(t1,is&0xfffff000); 2022116Sjkh t2 = z_l-(((t1-t)-dp_h[k])-z_h); 2032116Sjkh } 2042116Sjkh 2052116Sjkh /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 2062116Sjkh GET_FLOAT_WORD(is,y); 2072116Sjkh SET_FLOAT_WORD(y1,is&0xfffff000); 2082116Sjkh p_l = (y-y1)*t1+y*t2; 2092116Sjkh p_h = y1*t1; 2102116Sjkh z = p_l+p_h; 2112116Sjkh GET_FLOAT_WORD(j,z); 2122116Sjkh if (j>0x43000000) /* if z > 128 */ 213129959Sbde return sn*huge*huge; /* overflow */ 2142116Sjkh else if (j==0x43000000) { /* if z == 128 */ 215129959Sbde if(p_l+ovt>z-p_h) return sn*huge*huge; /* overflow */ 2162116Sjkh } 2172116Sjkh else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */ 218129959Sbde return sn*tiny*tiny; /* underflow */ 2192116Sjkh else if (j==0xc3160000){ /* z == -150 */ 220129959Sbde if(p_l<=z-p_h) return sn*tiny*tiny; /* underflow */ 2212116Sjkh } 2222116Sjkh /* 2232116Sjkh * compute 2**(p_h+p_l) 2242116Sjkh */ 2252116Sjkh i = j&0x7fffffff; 2262116Sjkh k = (i>>23)-0x7f; 2272116Sjkh n = 0; 2282116Sjkh if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */ 2292116Sjkh n = j+(0x00800000>>(k+1)); 2302116Sjkh k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */ 2312116Sjkh SET_FLOAT_WORD(t,n&~(0x007fffff>>k)); 2322116Sjkh n = ((n&0x007fffff)|0x00800000)>>(23-k); 2332116Sjkh if(j<0) n = -n; 2342116Sjkh p_h -= t; 2358870Srgrimes } 2362116Sjkh t = p_l+p_h; 2372116Sjkh GET_FLOAT_WORD(is,t); 238129955Sbde SET_FLOAT_WORD(t,is&0xffff8000); 2392116Sjkh u = t*lg2_h; 2402116Sjkh v = (p_l-(t-p_h))*lg2+t*lg2_l; 2412116Sjkh z = u+v; 2422116Sjkh w = v-(z-u); 2432116Sjkh t = z*z; 2442116Sjkh t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 2452116Sjkh r = (z*t1)/(t1-two)-(w+z*w); 2462116Sjkh z = one-(r-z); 2472116Sjkh GET_FLOAT_WORD(j,z); 2482116Sjkh j += (n<<23); 2492116Sjkh if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */ 2502116Sjkh else SET_FLOAT_WORD(z,j); 251129959Sbde return sn*z; 2522116Sjkh} 253