13a8617a8SJordan K. Hubbard /* e_powf.c -- float version of e_pow.c.
23a8617a8SJordan K. Hubbard * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
33a8617a8SJordan K. Hubbard */
43a8617a8SJordan K. Hubbard
53a8617a8SJordan K. Hubbard /*
63a8617a8SJordan K. Hubbard * ====================================================
73a8617a8SJordan K. Hubbard * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
83a8617a8SJordan K. Hubbard *
93a8617a8SJordan K. Hubbard * Developed at SunPro, a Sun Microsystems, Inc. business.
103a8617a8SJordan K. Hubbard * Permission to use, copy, modify, and distribute this
113a8617a8SJordan K. Hubbard * software is freely granted, provided that this notice
123a8617a8SJordan K. Hubbard * is preserved.
133a8617a8SJordan K. Hubbard * ====================================================
143a8617a8SJordan K. Hubbard */
153a8617a8SJordan K. Hubbard
163a8617a8SJordan K. Hubbard #include "math.h"
173a8617a8SJordan K. Hubbard #include "math_private.h"
183a8617a8SJordan K. Hubbard
193a8617a8SJordan K. Hubbard static const float
203a8617a8SJordan K. Hubbard bp[] = {1.0, 1.5,},
213a8617a8SJordan K. Hubbard dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
223a8617a8SJordan K. Hubbard dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
233a8617a8SJordan K. Hubbard zero = 0.0,
247525d42fSMark Johnston half = 0.5,
257525d42fSMark Johnston qrtr = 0.25,
267525d42fSMark Johnston thrd = 3.33333343e-01, /* 0x3eaaaaab */
273a8617a8SJordan K. Hubbard one = 1.0,
283a8617a8SJordan K. Hubbard two = 2.0,
293a8617a8SJordan K. Hubbard two24 = 16777216.0, /* 0x4b800000 */
303a8617a8SJordan K. Hubbard huge = 1.0e30,
313a8617a8SJordan K. Hubbard tiny = 1.0e-30,
323a8617a8SJordan K. Hubbard /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
333a8617a8SJordan K. Hubbard L1 = 6.0000002384e-01, /* 0x3f19999a */
343a8617a8SJordan K. Hubbard L2 = 4.2857143283e-01, /* 0x3edb6db7 */
353a8617a8SJordan K. Hubbard L3 = 3.3333334327e-01, /* 0x3eaaaaab */
363a8617a8SJordan K. Hubbard L4 = 2.7272811532e-01, /* 0x3e8ba305 */
373a8617a8SJordan K. Hubbard L5 = 2.3066075146e-01, /* 0x3e6c3255 */
383a8617a8SJordan K. Hubbard L6 = 2.0697501302e-01, /* 0x3e53f142 */
393a8617a8SJordan K. Hubbard P1 = 1.6666667163e-01, /* 0x3e2aaaab */
403a8617a8SJordan K. Hubbard P2 = -2.7777778450e-03, /* 0xbb360b61 */
413a8617a8SJordan K. Hubbard P3 = 6.6137559770e-05, /* 0x388ab355 */
423a8617a8SJordan K. Hubbard P4 = -1.6533901999e-06, /* 0xb5ddea0e */
433a8617a8SJordan K. Hubbard P5 = 4.1381369442e-08, /* 0x3331bb4c */
443a8617a8SJordan K. Hubbard lg2 = 6.9314718246e-01, /* 0x3f317218 */
453a8617a8SJordan K. Hubbard lg2_h = 6.93145752e-01, /* 0x3f317200 */
463a8617a8SJordan K. Hubbard lg2_l = 1.42860654e-06, /* 0x35bfbe8c */
473a8617a8SJordan K. Hubbard ovt = 4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
483a8617a8SJordan K. Hubbard cp = 9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
49b4437c3dSBruce Evans cp_h = 9.6191406250e-01, /* 0x3f764000 =12b cp */
50b4437c3dSBruce Evans cp_l = -1.1736857402e-04, /* 0xb8f623c6 =tail of cp_h */
513a8617a8SJordan K. Hubbard ivln2 = 1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
523a8617a8SJordan K. Hubbard ivln2_h = 1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
533a8617a8SJordan K. Hubbard ivln2_l = 7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
543a8617a8SJordan K. Hubbard
5559b19ff1SAlfred Perlstein float
powf(float x,float y)5699843eb8SSteve Kargl powf(float x, float y)
573a8617a8SJordan K. Hubbard {
583a8617a8SJordan K. Hubbard float z,ax,z_h,z_l,p_h,p_l;
59f88a48ccSBruce Evans float y1,t1,t2,r,s,sn,t,u,v,w;
603a8617a8SJordan K. Hubbard int32_t i,j,k,yisint,n;
613a8617a8SJordan K. Hubbard int32_t hx,hy,ix,iy,is;
623a8617a8SJordan K. Hubbard
633a8617a8SJordan K. Hubbard GET_FLOAT_WORD(hx,x);
643a8617a8SJordan K. Hubbard GET_FLOAT_WORD(hy,y);
653a8617a8SJordan K. Hubbard ix = hx&0x7fffffff; iy = hy&0x7fffffff;
663a8617a8SJordan K. Hubbard
673a8617a8SJordan K. Hubbard /* y==zero: x**0 = 1 */
683a8617a8SJordan K. Hubbard if(iy==0) return one;
693a8617a8SJordan K. Hubbard
70cd24d798SDavid Schultz /* x==1: 1**y = 1, even if y is NaN */
71cd24d798SDavid Schultz if (hx==0x3f800000) return one;
72cd24d798SDavid Schultz
73011cbae1SBruce Evans /* y!=zero: result is NaN if either arg is NaN */
743a8617a8SJordan K. Hubbard if(ix > 0x7f800000 ||
753a8617a8SJordan K. Hubbard iy > 0x7f800000)
766f1b8a07SBruce Evans return nan_mix(x, y);
773a8617a8SJordan K. Hubbard
783a8617a8SJordan K. Hubbard /* determine if y is an odd int when x < 0
793a8617a8SJordan K. Hubbard * yisint = 0 ... y is not an integer
803a8617a8SJordan K. Hubbard * yisint = 1 ... y is an odd int
813a8617a8SJordan K. Hubbard * yisint = 2 ... y is an even int
823a8617a8SJordan K. Hubbard */
833a8617a8SJordan K. Hubbard yisint = 0;
843a8617a8SJordan K. Hubbard if(hx<0) {
853a8617a8SJordan K. Hubbard if(iy>=0x4b800000) yisint = 2; /* even integer y */
863a8617a8SJordan K. Hubbard else if(iy>=0x3f800000) {
873a8617a8SJordan K. Hubbard k = (iy>>23)-0x7f; /* exponent */
883a8617a8SJordan K. Hubbard j = iy>>(23-k);
893a8617a8SJordan K. Hubbard if((j<<(23-k))==iy) yisint = 2-(j&1);
903a8617a8SJordan K. Hubbard }
913a8617a8SJordan K. Hubbard }
923a8617a8SJordan K. Hubbard
933a8617a8SJordan K. Hubbard /* special value of y */
943a8617a8SJordan K. Hubbard if (iy==0x7f800000) { /* y is +-inf */
953a8617a8SJordan K. Hubbard if (ix==0x3f800000)
96cd24d798SDavid Schultz return one; /* (-1)**+-inf is NaN */
973a8617a8SJordan K. Hubbard else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
983a8617a8SJordan K. Hubbard return (hy>=0)? y: zero;
993a8617a8SJordan K. Hubbard else /* (|x|<1)**-,+inf = inf,0 */
1003a8617a8SJordan K. Hubbard return (hy<0)?-y: zero;
1013a8617a8SJordan K. Hubbard }
1023a8617a8SJordan K. Hubbard if(iy==0x3f800000) { /* y is +-1 */
1033a8617a8SJordan K. Hubbard if(hy<0) return one/x; else return x;
1043a8617a8SJordan K. Hubbard }
1053a8617a8SJordan K. Hubbard if(hy==0x40000000) return x*x; /* y is 2 */
1063a8617a8SJordan K. Hubbard if(hy==0x3f000000) { /* y is 0.5 */
1073a8617a8SJordan K. Hubbard if(hx>=0) /* x >= +0 */
10899843eb8SSteve Kargl return sqrtf(x);
1093a8617a8SJordan K. Hubbard }
1103a8617a8SJordan K. Hubbard
1113a8617a8SJordan K. Hubbard ax = fabsf(x);
1123a8617a8SJordan K. Hubbard /* special value of x */
1133a8617a8SJordan K. Hubbard if(ix==0x7f800000||ix==0||ix==0x3f800000){
1143a8617a8SJordan K. Hubbard z = ax; /*x is +-0,+-inf,+-1*/
1153a8617a8SJordan K. Hubbard if(hy<0) z = one/z; /* z = (1/|x|) */
1163a8617a8SJordan K. Hubbard if(hx<0) {
1173a8617a8SJordan K. Hubbard if(((ix-0x3f800000)|yisint)==0) {
1183a8617a8SJordan K. Hubbard z = (z-z)/(z-z); /* (-1)**non-int is NaN */
1193a8617a8SJordan K. Hubbard } else if(yisint==1)
1203a8617a8SJordan K. Hubbard z = -z; /* (x<0)**odd = -(|x|**odd) */
1213a8617a8SJordan K. Hubbard }
1223a8617a8SJordan K. Hubbard return z;
1233a8617a8SJordan K. Hubbard }
1243a8617a8SJordan K. Hubbard
125f88a48ccSBruce Evans n = ((u_int32_t)hx>>31)-1;
126f88a48ccSBruce Evans
1273a8617a8SJordan K. Hubbard /* (x<0)**(non-int) is NaN */
128f88a48ccSBruce Evans if((n|yisint)==0) return (x-x)/(x-x);
129f88a48ccSBruce Evans
130f88a48ccSBruce Evans sn = one; /* s (sign of result -ve**odd) = -1 else = 1 */
131f88a48ccSBruce Evans if((n|(yisint-1))==0) sn = -one;/* (-ve)**(odd int) */
1323a8617a8SJordan K. Hubbard
1333a8617a8SJordan K. Hubbard /* |y| is huge */
1343a8617a8SJordan K. Hubbard if(iy>0x4d000000) { /* if |y| > 2**27 */
1353a8617a8SJordan K. Hubbard /* over/underflow if x is not close to one */
136292815eaSMark Murray if(ix<0x3f7ffff6) return (hy<0)? sn*huge*huge:sn*tiny*tiny;
137f88a48ccSBruce Evans if(ix>0x3f800007) return (hy>0)? sn*huge*huge:sn*tiny*tiny;
1383a8617a8SJordan K. Hubbard /* now |1-x| is tiny <= 2**-20, suffice to compute
1393a8617a8SJordan K. Hubbard log(x) by x-x^2/2+x^3/3-x^4/4 */
1403e2ec6eaSBruce Evans t = ax-1; /* t has 20 trailing zeros */
1417525d42fSMark Johnston w = (t*t)*(half-t*(thrd-t*qrtr));
1423a8617a8SJordan K. Hubbard u = ivln2_h*t; /* ivln2_h has 16 sig. bits */
1433a8617a8SJordan K. Hubbard v = t*ivln2_l-w*ivln2;
1443a8617a8SJordan K. Hubbard t1 = u+v;
1453a8617a8SJordan K. Hubbard GET_FLOAT_WORD(is,t1);
1463a8617a8SJordan K. Hubbard SET_FLOAT_WORD(t1,is&0xfffff000);
1473a8617a8SJordan K. Hubbard t2 = v-(t1-u);
1483a8617a8SJordan K. Hubbard } else {
1493a8617a8SJordan K. Hubbard float s2,s_h,s_l,t_h,t_l;
1503a8617a8SJordan K. Hubbard n = 0;
1513a8617a8SJordan K. Hubbard /* take care subnormal number */
1523a8617a8SJordan K. Hubbard if(ix<0x00800000)
1533a8617a8SJordan K. Hubbard {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
1543a8617a8SJordan K. Hubbard n += ((ix)>>23)-0x7f;
1553a8617a8SJordan K. Hubbard j = ix&0x007fffff;
1563a8617a8SJordan K. Hubbard /* determine interval */
1573a8617a8SJordan K. Hubbard ix = j|0x3f800000; /* normalize ix */
1583a8617a8SJordan K. Hubbard if(j<=0x1cc471) k=0; /* |x|<sqrt(3/2) */
1593a8617a8SJordan K. Hubbard else if(j<0x5db3d7) k=1; /* |x|<sqrt(3) */
1603a8617a8SJordan K. Hubbard else {k=0;n+=1;ix -= 0x00800000;}
1613a8617a8SJordan K. Hubbard SET_FLOAT_WORD(ax,ix);
1623a8617a8SJordan K. Hubbard
1633a8617a8SJordan K. Hubbard /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
1643a8617a8SJordan K. Hubbard u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
1653a8617a8SJordan K. Hubbard v = one/(ax+bp[k]);
1663a8617a8SJordan K. Hubbard s = u*v;
1673a8617a8SJordan K. Hubbard s_h = s;
1683a8617a8SJordan K. Hubbard GET_FLOAT_WORD(is,s_h);
1693a8617a8SJordan K. Hubbard SET_FLOAT_WORD(s_h,is&0xfffff000);
1703a8617a8SJordan K. Hubbard /* t_h=ax+bp[k] High */
17112be4e0dSBruce Evans is = ((ix>>1)&0xfffff000)|0x20000000;
17212be4e0dSBruce Evans SET_FLOAT_WORD(t_h,is+0x00400000+(k<<21));
1733a8617a8SJordan K. Hubbard t_l = ax - (t_h-bp[k]);
1743a8617a8SJordan K. Hubbard s_l = v*((u-s_h*t_h)-s_h*t_l);
1753a8617a8SJordan K. Hubbard /* compute log(ax) */
1763a8617a8SJordan K. Hubbard s2 = s*s;
1773a8617a8SJordan K. Hubbard r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
1783a8617a8SJordan K. Hubbard r += s_l*(s_h+s);
1793a8617a8SJordan K. Hubbard s2 = s_h*s_h;
1807525d42fSMark Johnston t_h = 3+s2+r;
1813a8617a8SJordan K. Hubbard GET_FLOAT_WORD(is,t_h);
1823a8617a8SJordan K. Hubbard SET_FLOAT_WORD(t_h,is&0xfffff000);
1837525d42fSMark Johnston t_l = r-((t_h-3)-s2);
1843a8617a8SJordan K. Hubbard /* u+v = s*(1+...) */
1853a8617a8SJordan K. Hubbard u = s_h*t_h;
1863a8617a8SJordan K. Hubbard v = s_l*t_h+t_l*s;
1873a8617a8SJordan K. Hubbard /* 2/(3log2)*(s+...) */
1883a8617a8SJordan K. Hubbard p_h = u+v;
1893a8617a8SJordan K. Hubbard GET_FLOAT_WORD(is,p_h);
1903a8617a8SJordan K. Hubbard SET_FLOAT_WORD(p_h,is&0xfffff000);
1913a8617a8SJordan K. Hubbard p_l = v-(p_h-u);
1923a8617a8SJordan K. Hubbard z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
1933a8617a8SJordan K. Hubbard z_l = cp_l*p_h+p_l*cp+dp_l[k];
1943a8617a8SJordan K. Hubbard /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
1957525d42fSMark Johnston t = n;
1963a8617a8SJordan K. Hubbard t1 = (((z_h+z_l)+dp_h[k])+t);
1973a8617a8SJordan K. Hubbard GET_FLOAT_WORD(is,t1);
1983a8617a8SJordan K. Hubbard SET_FLOAT_WORD(t1,is&0xfffff000);
1993a8617a8SJordan K. Hubbard t2 = z_l-(((t1-t)-dp_h[k])-z_h);
2003a8617a8SJordan K. Hubbard }
2013a8617a8SJordan K. Hubbard
2023a8617a8SJordan K. Hubbard /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
2033a8617a8SJordan K. Hubbard GET_FLOAT_WORD(is,y);
2043a8617a8SJordan K. Hubbard SET_FLOAT_WORD(y1,is&0xfffff000);
2053a8617a8SJordan K. Hubbard p_l = (y-y1)*t1+y*t2;
2063a8617a8SJordan K. Hubbard p_h = y1*t1;
2073a8617a8SJordan K. Hubbard z = p_l+p_h;
2083a8617a8SJordan K. Hubbard GET_FLOAT_WORD(j,z);
2093a8617a8SJordan K. Hubbard if (j>0x43000000) /* if z > 128 */
210f88a48ccSBruce Evans return sn*huge*huge; /* overflow */
2113a8617a8SJordan K. Hubbard else if (j==0x43000000) { /* if z == 128 */
212f88a48ccSBruce Evans if(p_l+ovt>z-p_h) return sn*huge*huge; /* overflow */
2133a8617a8SJordan K. Hubbard }
2143a8617a8SJordan K. Hubbard else if ((j&0x7fffffff)>0x43160000) /* z <= -150 */
215f88a48ccSBruce Evans return sn*tiny*tiny; /* underflow */
2163a8617a8SJordan K. Hubbard else if (j==0xc3160000){ /* z == -150 */
217f88a48ccSBruce Evans if(p_l<=z-p_h) return sn*tiny*tiny; /* underflow */
2183a8617a8SJordan K. Hubbard }
2193a8617a8SJordan K. Hubbard /*
2203a8617a8SJordan K. Hubbard * compute 2**(p_h+p_l)
2213a8617a8SJordan K. Hubbard */
2223a8617a8SJordan K. Hubbard i = j&0x7fffffff;
2233a8617a8SJordan K. Hubbard k = (i>>23)-0x7f;
2243a8617a8SJordan K. Hubbard n = 0;
2253a8617a8SJordan K. Hubbard if(i>0x3f000000) { /* if |z| > 0.5, set n = [z+0.5] */
2263a8617a8SJordan K. Hubbard n = j+(0x00800000>>(k+1));
2273a8617a8SJordan K. Hubbard k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */
2283a8617a8SJordan K. Hubbard SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
2293a8617a8SJordan K. Hubbard n = ((n&0x007fffff)|0x00800000)>>(23-k);
2303a8617a8SJordan K. Hubbard if(j<0) n = -n;
2313a8617a8SJordan K. Hubbard p_h -= t;
2323a8617a8SJordan K. Hubbard }
2333a8617a8SJordan K. Hubbard t = p_l+p_h;
2343a8617a8SJordan K. Hubbard GET_FLOAT_WORD(is,t);
2355f20e5ceSBruce Evans SET_FLOAT_WORD(t,is&0xffff8000);
2363a8617a8SJordan K. Hubbard u = t*lg2_h;
2373a8617a8SJordan K. Hubbard v = (p_l-(t-p_h))*lg2+t*lg2_l;
2383a8617a8SJordan K. Hubbard z = u+v;
2393a8617a8SJordan K. Hubbard w = v-(z-u);
2403a8617a8SJordan K. Hubbard t = z*z;
2413a8617a8SJordan K. Hubbard t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
2423a8617a8SJordan K. Hubbard r = (z*t1)/(t1-two)-(w+z*w);
2433a8617a8SJordan K. Hubbard z = one-(r-z);
2443a8617a8SJordan K. Hubbard GET_FLOAT_WORD(j,z);
245e75a1bbcSKarl Tomlinson /*
246e75a1bbcSKarl Tomlinson * sign bit of z is 0.
247e75a1bbcSKarl Tomlinson * sign bit of j will indicate sign of 0x7f-biased exponent.
248e75a1bbcSKarl Tomlinson */
249e75a1bbcSKarl Tomlinson j += (int32_t)((u_int32_t)n<<23);
2503a8617a8SJordan K. Hubbard if((j>>23)<=0) z = scalbnf(z,n); /* subnormal output */
2513a8617a8SJordan K. Hubbard else SET_FLOAT_WORD(z,j);
252f88a48ccSBruce Evans return sn*z;
2533a8617a8SJordan K. Hubbard }
254