1 /* @(#)e_pow.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #if defined(LIBM_SCCS) && !defined(lint) 14 static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $"; 15 #endif 16 17 /* __ieee754_pow(x,y) return x**y 18 * 19 * n 20 * Method: Let x = 2 * (1+f) 21 * 1. Compute and return log2(x) in two pieces: 22 * log2(x) = w1 + w2, 23 * where w1 has 53-24 = 29 bit trailing zeros. 24 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 25 * arithmetic, where |y'|<=0.5. 26 * 3. Return x**y = 2**n*exp(y'*log2) 27 * 28 * Special cases: 29 * 1. (anything) ** 0 is 1 30 * 2. (anything) ** 1 is itself 31 * 3. (anything) ** NAN is NAN 32 * 4. NAN ** (anything except 0) is NAN 33 * 5. +-(|x| > 1) ** +INF is +INF 34 * 6. +-(|x| > 1) ** -INF is +0 35 * 7. +-(|x| < 1) ** +INF is +0 36 * 8. +-(|x| < 1) ** -INF is +INF 37 * 9. +-1 ** +-INF is NAN 38 * 10. +0 ** (+anything except 0, NAN) is +0 39 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 40 * 12. +0 ** (-anything except 0, NAN) is +INF 41 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 42 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 43 * 15. +INF ** (+anything except 0,NAN) is +INF 44 * 16. +INF ** (-anything except 0,NAN) is +0 45 * 17. -INF ** (anything) = -0 ** (-anything) 46 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 47 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 48 * 49 * Accuracy: 50 * pow(x,y) returns x**y nearly rounded. In particular 51 * pow(integer,integer) 52 * always returns the correct integer provided it is 53 * representable. 54 * 55 * Constants : 56 * The hexadecimal values are the intended ones for the following 57 * constants. The decimal values may be used, provided that the 58 * compiler will convert from decimal to binary accurately enough 59 * to produce the hexadecimal values shown. 60 */ 61 62 /*#include "math.h"*/ 63 #include "math_private.h" 64 65 #ifdef __STDC__ 66 static const double 67 #else 68 static double 69 #endif 70 bp[] = {1.0, 1.5,}, 71 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 72 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 73 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 74 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 75 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 76 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 77 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 78 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 79 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 80 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 81 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 82 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 83 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 84 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 85 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 86 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 87 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 88 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 89 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 90 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 91 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 92 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 93 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 94 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 95 96 #ifdef __STDC__ 97 double __ieee754_pow(double x, double y) 98 #else 99 double __ieee754_pow(x,y) 100 double x, y; 101 #endif 102 { 103 double z,ax,z_h,z_l,p_h,p_l; 104 double y1,t1,t2,r,s,t,u,v,w; 105 int32_t i,j,k,yisint,n; 106 int32_t hx,hy,ix,iy; 107 u_int32_t lx,ly; 108 109 EXTRACT_WORDS(hx,lx,x); 110 EXTRACT_WORDS(hy,ly,y); 111 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 112 113 /* y==zero: x**0 = 1 */ 114 if((iy|ly)==0) return one; 115 116 /* +-NaN return x+y */ 117 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 118 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 119 return x+y; 120 121 /* determine if y is an odd int when x < 0 122 * yisint = 0 ... y is not an integer 123 * yisint = 1 ... y is an odd int 124 * yisint = 2 ... y is an even int 125 */ 126 yisint = 0; 127 if(hx<0) { 128 if(iy>=0x43400000) yisint = 2; /* even integer y */ 129 else if(iy>=0x3ff00000) { 130 k = (iy>>20)-0x3ff; /* exponent */ 131 if(k>20) { 132 j = ly>>(52-k); 133 if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1); 134 } else if(ly==0) { 135 j = iy>>(20-k); 136 if((j<<(20-k))==iy) yisint = 2-(j&1); 137 } 138 } 139 } 140 141 /* special value of y */ 142 if(ly==0) { 143 if (iy==0x7ff00000) { /* y is +-inf */ 144 if(((ix-0x3ff00000)|lx)==0) 145 return y - y; /* inf**+-1 is NaN */ 146 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 147 return (hy>=0)? y: zero; 148 else /* (|x|<1)**-,+inf = inf,0 */ 149 return (hy<0)?-y: zero; 150 } 151 if(iy==0x3ff00000) { /* y is +-1 */ 152 if(hy<0) return one/x; else return x; 153 } 154 if(hy==0x40000000) return x*x; /* y is 2 */ 155 if(hy==0x3fe00000) { /* y is 0.5 */ 156 if(hx>=0) /* x >= +0 */ 157 return __ieee754_sqrt(x); 158 } 159 } 160 161 ax = x < 0 ? -x : x; /*fabs(x);*/ 162 /* special value of x */ 163 if(lx==0) { 164 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 165 z = ax; /*x is +-0,+-inf,+-1*/ 166 if(hy<0) z = one/z; /* z = (1/|x|) */ 167 if(hx<0) { 168 if(((ix-0x3ff00000)|yisint)==0) { 169 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 170 } else if(yisint==1) 171 z = -z; /* (x<0)**odd = -(|x|**odd) */ 172 } 173 return z; 174 } 175 } 176 177 /* (x<0)**(non-int) is NaN */ 178 if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x); 179 180 /* |y| is huge */ 181 if(iy>0x41e00000) { /* if |y| > 2**31 */ 182 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 183 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 184 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 185 } 186 /* over/underflow if x is not close to one */ 187 if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny; 188 if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny; 189 /* now |1-x| is tiny <= 2**-20, suffice to compute 190 log(x) by x-x^2/2+x^3/3-x^4/4 */ 191 t = x-1; /* t has 20 trailing zeros */ 192 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 193 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 194 v = t*ivln2_l-w*ivln2; 195 t1 = u+v; 196 SET_LOW_WORD(t1,0); 197 t2 = v-(t1-u); 198 } else { 199 double s2,s_h,s_l,t_h,t_l; 200 n = 0; 201 /* take care subnormal number */ 202 if(ix<0x00100000) 203 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); } 204 n += ((ix)>>20)-0x3ff; 205 j = ix&0x000fffff; 206 /* determine interval */ 207 ix = j|0x3ff00000; /* normalize ix */ 208 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 209 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 210 else {k=0;n+=1;ix -= 0x00100000;} 211 SET_HIGH_WORD(ax,ix); 212 213 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 214 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 215 v = one/(ax+bp[k]); 216 s = u*v; 217 s_h = s; 218 SET_LOW_WORD(s_h,0); 219 /* t_h=ax+bp[k] High */ 220 t_h = zero; 221 SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18)); 222 t_l = ax - (t_h-bp[k]); 223 s_l = v*((u-s_h*t_h)-s_h*t_l); 224 /* compute log(ax) */ 225 s2 = s*s; 226 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 227 r += s_l*(s_h+s); 228 s2 = s_h*s_h; 229 t_h = 3.0+s2+r; 230 SET_LOW_WORD(t_h,0); 231 t_l = r-((t_h-3.0)-s2); 232 /* u+v = s*(1+...) */ 233 u = s_h*t_h; 234 v = s_l*t_h+t_l*s; 235 /* 2/(3log2)*(s+...) */ 236 p_h = u+v; 237 SET_LOW_WORD(p_h,0); 238 p_l = v-(p_h-u); 239 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 240 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 241 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 242 t = (double)n; 243 t1 = (((z_h+z_l)+dp_h[k])+t); 244 SET_LOW_WORD(t1,0); 245 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 246 } 247 248 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 249 if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0) 250 s = -one;/* (-ve)**(odd int) */ 251 252 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 253 y1 = y; 254 SET_LOW_WORD(y1,0); 255 p_l = (y-y1)*t1+y*t2; 256 p_h = y1*t1; 257 z = p_l+p_h; 258 EXTRACT_WORDS(j,i,z); 259 if (j>=0x40900000) { /* z >= 1024 */ 260 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 261 return s*huge*huge; /* overflow */ 262 else { 263 if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */ 264 } 265 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 266 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 267 return s*tiny*tiny; /* underflow */ 268 else { 269 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 270 } 271 } 272 /* 273 * compute 2**(p_h+p_l) 274 */ 275 i = j&0x7fffffff; 276 k = (i>>20)-0x3ff; 277 n = 0; 278 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 279 n = j+(0x00100000>>(k+1)); 280 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 281 t = zero; 282 SET_HIGH_WORD(t,n&~(0x000fffff>>k)); 283 n = ((n&0x000fffff)|0x00100000)>>(20-k); 284 if(j<0) n = -n; 285 p_h -= t; 286 } 287 t = p_l+p_h; 288 SET_LOW_WORD(t,0); 289 u = t*lg2_h; 290 v = (p_l-(t-p_h))*lg2+t*lg2_l; 291 z = u+v; 292 w = v-(z-u); 293 t = z*z; 294 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 295 r = (z*t1)/(t1-two)-(w+z*w); 296 z = one-(r-z); 297 GET_HIGH_WORD(j,z); 298 j += (n<<20); 299 if((j>>20)<=0) z = SDL_NAME(scalbn)(z,n); /* subnormal output */ 300 else SET_HIGH_WORD(z,j); 301 return s*z; 302 } 303
