1/* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */ 2/* 3 * ==================================================== 4 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11/* pow(x,y) return x**y 12 * 13 * n 14 * Method: Let x = 2 * (1+f) 15 * 1. Compute and return log2(x) in two pieces: 16 * log2(x) = w1 + w2, 17 * where w1 has 53-24 = 29 bit trailing zeros. 18 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 19 * arithmetic, where |y'|<=0.5. 20 * 3. Return x**y = 2**n*exp(y'*log2) 21 * 22 * Special cases: 23 * 1. (anything) ** 0 is 1 24 * 2. 1 ** (anything) is 1 25 * 3. (anything except 1) ** NAN is NAN 26 * 4. NAN ** (anything except 0) is NAN 27 * 5. +-(|x| > 1) ** +INF is +INF 28 * 6. +-(|x| > 1) ** -INF is +0 29 * 7. +-(|x| < 1) ** +INF is +0 30 * 8. +-(|x| < 1) ** -INF is +INF 31 * 9. -1 ** +-INF is 1 32 * 10. +0 ** (+anything except 0, NAN) is +0 33 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 34 * 12. +0 ** (-anything except 0, NAN) is +INF, raise divbyzero 35 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise divbyzero 36 * 14. -0 ** (+odd integer) is -0 37 * 15. -0 ** (-odd integer) is -INF, raise divbyzero 38 * 16. +INF ** (+anything except 0,NAN) is +INF 39 * 17. +INF ** (-anything except 0,NAN) is +0 40 * 18. -INF ** (+odd integer) is -INF 41 * 19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer) 42 * 20. (anything) ** 1 is (anything) 43 * 21. (anything) ** -1 is 1/(anything) 44 * 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 45 * 23. (-anything except 0 and inf) ** (non-integer) is NAN 46 * 47 * Accuracy: 48 * pow(x,y) returns x**y nearly rounded. In particular 49 * pow(integer,integer) 50 * always returns the correct integer provided it is 51 * representable. 52 * 53 * Constants : 54 * The hexadecimal values are the intended ones for the following 55 * constants. The decimal values may be used, provided that the 56 * compiler will convert from decimal to binary accurately enough 57 * to produce the hexadecimal values shown. 58 */ 59 60#include "libm.h" 61 62static const double 63bp[] = {1.0, 1.5,}, 64dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 65dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 66two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 67huge = 1.0e300, 68tiny = 1.0e-300, 69/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 70L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 71L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 72L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 73L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 74L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 75L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 76P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 77P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 78P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 79P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 80P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 81lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 82lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 83lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 84ovt = 8.0085662595372944372e-017, /* -(1024-log2(ovfl+.5ulp)) */ 85cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 86cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 87cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 88ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 89ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 90ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 91 92double pow(double x, double y) 93{ 94 double z,ax,z_h,z_l,p_h,p_l; 95 double y1,t1,t2,r,s,t,u,v,w; 96 int32_t i,j,k,yisint,n; 97 int32_t hx,hy,ix,iy; 98 uint32_t lx,ly; 99 100 EXTRACT_WORDS(hx, lx, x); 101 EXTRACT_WORDS(hy, ly, y); 102 ix = hx & 0x7fffffff; 103 iy = hy & 0x7fffffff; 104 105 /* x**0 = 1, even if x is NaN */ 106 if ((iy|ly) == 0) 107 return 1.0; 108 /* 1**y = 1, even if y is NaN */ 109 if (hx == 0x3ff00000 && lx == 0) 110 return 1.0; 111 /* NaN if either arg is NaN */ 112 if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) || 113 iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0)) 114 return x + y; 115 116 /* determine if y is an odd int when x < 0 117 * yisint = 0 ... y is not an integer 118 * yisint = 1 ... y is an odd int 119 * yisint = 2 ... y is an even int 120 */ 121 yisint = 0; 122 if (hx < 0) { 123 if (iy >= 0x43400000) 124 yisint = 2; /* even integer y */ 125 else if (iy >= 0x3ff00000) { 126 k = (iy>>20) - 0x3ff; /* exponent */ 127 if (k > 20) { 128 uint32_t j = ly>>(52-k); 129 if ((j<<(52-k)) == ly) 130 yisint = 2 - (j&1); 131 } else if (ly == 0) { 132 uint32_t j = iy>>(20-k); 133 if ((j<<(20-k)) == iy) 134 yisint = 2 - (j&1); 135 } 136 } 137 } 138 139 /* special value of y */ 140 if (ly == 0) { 141 if (iy == 0x7ff00000) { /* y is +-inf */ 142 if (((ix-0x3ff00000)|lx) == 0) /* (-1)**+-inf is 1 */ 143 return 1.0; 144 else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */ 145 return hy >= 0 ? y : 0.0; 146 else /* (|x|<1)**+-inf = 0,inf */ 147 return hy >= 0 ? 0.0 : -y; 148 } 149 if (iy == 0x3ff00000) { /* y is +-1 */ 150 if (hy >= 0) 151 return x; 152 y = 1/x; 153#if FLT_EVAL_METHOD!=0 154 { 155 union {double f; uint64_t i;} u = {y}; 156 uint64_t i = u.i & -1ULL/2; 157 if (i>>52 == 0 && (i&(i-1))) 158 FORCE_EVAL((float)y); 159 } 160#endif 161 return y; 162 } 163 if (hy == 0x40000000) /* y is 2 */ 164 return x*x; 165 if (hy == 0x3fe00000) { /* y is 0.5 */ 166 if (hx >= 0) /* x >= +0 */ 167 return sqrt(x); 168 } 169 } 170 171 ax = fabs(x); 172 /* special value of x */ 173 if (lx == 0) { 174 if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { /* x is +-0,+-inf,+-1 */ 175 z = ax; 176 if (hy < 0) /* z = (1/|x|) */ 177 z = 1.0/z; 178 if (hx < 0) { 179 if (((ix-0x3ff00000)|yisint) == 0) { 180 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 181 } else if (yisint == 1) 182 z = -z; /* (x<0)**odd = -(|x|**odd) */ 183 } 184 return z; 185 } 186 } 187 188 s = 1.0; /* sign of result */ 189 if (hx < 0) { 190 if (yisint == 0) /* (x<0)**(non-int) is NaN */ 191 return (x-x)/(x-x); 192 if (yisint == 1) /* (x<0)**(odd int) */ 193 s = -1.0; 194 } 195 196 /* |y| is huge */ 197 if (iy > 0x41e00000) { /* if |y| > 2**31 */ 198 if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */ 199 if (ix <= 0x3fefffff) 200 return hy < 0 ? huge*huge : tiny*tiny; 201 if (ix >= 0x3ff00000) 202 return hy > 0 ? huge*huge : tiny*tiny; 203 } 204 /* over/underflow if x is not close to one */ 205 if (ix < 0x3fefffff) 206 return hy < 0 ? s*huge*huge : s*tiny*tiny; 207 if (ix > 0x3ff00000) 208 return hy > 0 ? s*huge*huge : s*tiny*tiny; 209 /* now |1-x| is tiny <= 2**-20, suffice to compute 210 log(x) by x-x^2/2+x^3/3-x^4/4 */ 211 t = ax - 1.0; /* t has 20 trailing zeros */ 212 w = (t*t)*(0.5 - t*(0.3333333333333333333333-t*0.25)); 213 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 214 v = t*ivln2_l - w*ivln2; 215 t1 = u + v; 216 SET_LOW_WORD(t1, 0); 217 t2 = v - (t1-u); 218 } else { 219 double ss,s2,s_h,s_l,t_h,t_l; 220 n = 0; 221 /* take care subnormal number */ 222 if (ix < 0x00100000) { 223 ax *= two53; 224 n -= 53; 225 GET_HIGH_WORD(ix,ax); 226 } 227 n += ((ix)>>20) - 0x3ff; 228 j = ix & 0x000fffff; 229 /* determine interval */ 230 ix = j | 0x3ff00000; /* normalize ix */ 231 if (j <= 0x3988E) /* |x|<sqrt(3/2) */ 232 k = 0; 233 else if (j < 0xBB67A) /* |x|<sqrt(3) */ 234 k = 1; 235 else { 236 k = 0; 237 n += 1; 238 ix -= 0x00100000; 239 } 240 SET_HIGH_WORD(ax, ix); 241 242 /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 243 u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 244 v = 1.0/(ax+bp[k]); 245 ss = u*v; 246 s_h = ss; 247 SET_LOW_WORD(s_h, 0); 248 /* t_h=ax+bp[k] High */ 249 t_h = 0.0; 250 SET_HIGH_WORD(t_h, ((ix>>1)|0x20000000) + 0x00080000 + (k<<18)); 251 t_l = ax - (t_h-bp[k]); 252 s_l = v*((u-s_h*t_h)-s_h*t_l); 253 /* compute log(ax) */ 254 s2 = ss*ss; 255 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 256 r += s_l*(s_h+ss); 257 s2 = s_h*s_h; 258 t_h = 3.0 + s2 + r; 259 SET_LOW_WORD(t_h, 0); 260 t_l = r - ((t_h-3.0)-s2); 261 /* u+v = ss*(1+...) */ 262 u = s_h*t_h; 263 v = s_l*t_h + t_l*ss; 264 /* 2/(3log2)*(ss+...) */ 265 p_h = u + v; 266 SET_LOW_WORD(p_h, 0); 267 p_l = v - (p_h-u); 268 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 269 z_l = cp_l*p_h+p_l*cp + dp_l[k]; 270 /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 271 t = (double)n; 272 t1 = ((z_h + z_l) + dp_h[k]) + t; 273 SET_LOW_WORD(t1, 0); 274 t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); 275 } 276 277 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 278 y1 = y; 279 SET_LOW_WORD(y1, 0); 280 p_l = (y-y1)*t1 + y*t2; 281 p_h = y1*t1; 282 z = p_l + p_h; 283 EXTRACT_WORDS(j, i, z); 284 if (j >= 0x40900000) { /* z >= 1024 */ 285 if (((j-0x40900000)|i) != 0) /* if z > 1024 */ 286 return s*huge*huge; /* overflow */ 287 if (p_l + ovt > z - p_h) 288 return s*huge*huge; /* overflow */ 289 } else if ((j&0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j 290 if (((j-0xc090cc00)|i) != 0) /* z < -1075 */ 291 return s*tiny*tiny; /* underflow */ 292 if (p_l <= z - p_h) 293 return s*tiny*tiny; /* underflow */ 294 } 295 /* 296 * compute 2**(p_h+p_l) 297 */ 298 i = j & 0x7fffffff; 299 k = (i>>20) - 0x3ff; 300 n = 0; 301 if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 302 n = j + (0x00100000>>(k+1)); 303 k = ((n&0x7fffffff)>>20) - 0x3ff; /* new k for n */ 304 t = 0.0; 305 SET_HIGH_WORD(t, n & ~(0x000fffff>>k)); 306 n = ((n&0x000fffff)|0x00100000)>>(20-k); 307 if (j < 0) 308 n = -n; 309 p_h -= t; 310 } 311 t = p_l + p_h; 312 SET_LOW_WORD(t, 0); 313 u = t*lg2_h; 314 v = (p_l-(t-p_h))*lg2 + t*lg2_l; 315 z = u + v; 316 w = v - (z-u); 317 t = z*z; 318 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 319 r = (z*t1)/(t1-2.0) - (w + z*w); 320 z = 1.0 - (r-z); 321 GET_HIGH_WORD(j, z); 322 j += n<<20; 323 if ((j>>20) <= 0) /* subnormal output */ 324 z = scalbn(z,n); 325 else 326 SET_HIGH_WORD(z, j); 327 return s*z; 328} 329