1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*- 2 * 3 * ***** BEGIN LICENSE BLOCK ***** 4 * Version: MPL 1.1/GPL 2.0/LGPL 2.1 5 * 6 * The contents of this file are subject to the Mozilla Public License Version 7 * 1.1 (the "License"); you may not use this file except in compliance with 8 * the License. You may obtain a copy of the License at 9 * http://www.mozilla.org/MPL/ 10 * 11 * Software distributed under the License is distributed on an "AS IS" basis, 12 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License 13 * for the specific language governing rights and limitations under the 14 * License. 15 * 16 * The Original Code is Mozilla Communicator client code, released 17 * March 31, 1998. 18 * 19 * The Initial Developer of the Original Code is 20 * Sun Microsystems, Inc. 21 * Portions created by the Initial Developer are Copyright (C) 1998 22 * the Initial Developer. All Rights Reserved. 23 * 24 * Contributor(s): 25 * 26 * Alternatively, the contents of this file may be used under the terms of 27 * either of the GNU General Public License Version 2 or later (the "GPL"), 28 * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"), 29 * in which case the provisions of the GPL or the LGPL are applicable instead 30 * of those above. If you wish to allow use of your version of this file only 31 * under the terms of either the GPL or the LGPL, and not to allow others to 32 * use your version of this file under the terms of the MPL, indicate your 33 * decision by deleting the provisions above and replace them with the notice 34 * and other provisions required by the GPL or the LGPL. If you do not delete 35 * the provisions above, a recipient may use your version of this file under 36 * the terms of any one of the MPL, the GPL or the LGPL. 37 * 38 * ***** END LICENSE BLOCK ***** */ 39 40 /* @(#)e_pow.c 1.3 95/01/18 */ 41 /* 42 * ==================================================== 43 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 44 * 45 * Developed at SunSoft, a Sun Microsystems, Inc. business. 46 * Permission to use, copy, modify, and distribute this 47 * software is freely granted, provided that this notice 48 * is preserved. 49 * ==================================================== 50 */ 51 52 /* __ieee754_pow(x,y) return x**y 53 * 54 * n 55 * Method: Let x = 2 * (1+f) 56 * 1. Compute and return log2(x) in two pieces: 57 * log2(x) = w1 + w2, 58 * where w1 has 53-24 = 29 bit trailing zeros. 59 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 60 * arithmetic, where |y'|<=0.5. 61 * 3. Return x**y = 2**n*exp(y'*log2) 62 * 63 * Special cases: 64 * 1. (anything) ** 0 is 1 65 * 2. (anything) ** 1 is itself 66 * 3. (anything) ** NAN is NAN 67 * 4. NAN ** (anything except 0) is NAN 68 * 5. +-(|x| > 1) ** +INF is +INF 69 * 6. +-(|x| > 1) ** -INF is +0 70 * 7. +-(|x| < 1) ** +INF is +0 71 * 8. +-(|x| < 1) ** -INF is +INF 72 * 9. +-1 ** +-INF is NAN 73 * 10. +0 ** (+anything except 0, NAN) is +0 74 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 75 * 12. +0 ** (-anything except 0, NAN) is +INF 76 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 77 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 78 * 15. +INF ** (+anything except 0,NAN) is +INF 79 * 16. +INF ** (-anything except 0,NAN) is +0 80 * 17. -INF ** (anything) = -0 ** (-anything) 81 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 82 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 83 * 84 * Accuracy: 85 * pow(x,y) returns x**y nearly rounded. In particular 86 * pow(integer,integer) 87 * always returns the correct integer provided it is 88 * representable. 89 * 90 * Constants : 91 * The hexadecimal values are the intended ones for the following 92 * constants. The decimal values may be used, provided that the 93 * compiler will convert from decimal to binary accurately enough 94 * to produce the hexadecimal values shown. 95 */ 96 97 #include "fdlibm.h" 98 99 #if defined(_MSC_VER) 100 /* Microsoft Compiler */ 101 #pragma warning( disable : 4723 ) /* disables potential divide by 0 warning */ 102 #endif 103 104 #ifdef __STDC__ 105 static const double 106 #else 107 static double 108 #endif 109 bp[] = {1.0, 1.5,}, 110 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */ 111 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */ 112 zero = 0.0, 113 one = 1.0, 114 two = 2.0, 115 two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */ 116 really_big = 1.0e300, 117 tiny = 1.0e-300, 118 /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */ 119 L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */ 120 L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */ 121 L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */ 122 L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */ 123 L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */ 124 L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */ 125 P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */ 126 P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */ 127 P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */ 128 P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */ 129 P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */ 130 lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */ 131 lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */ 132 lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */ 133 ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */ 134 cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */ 135 cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */ 136 cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/ 137 ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */ 138 ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/ 139 ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/ 140 141 #ifdef __STDC__ __ieee754_pow(double x,double y)142 double __ieee754_pow(double x, double y) 143 #else 144 double __ieee754_pow(x,y) 145 double x, y; 146 #endif 147 { 148 fd_twoints ux, uy, uz; 149 double y1,t1,p_h,t,z,ax; 150 double z_h,z_l,p_l; 151 double t2,r,s,u,v,w; 152 int i,j,k,yisint,n; 153 int hx,hy,ix,iy; 154 unsigned lx,ly; 155 156 ux.d = x; uy.d = y; 157 hx = __HI(ux); lx = __LO(ux); 158 hy = __HI(uy); ly = __LO(uy); 159 ix = hx&0x7fffffff; iy = hy&0x7fffffff; 160 161 /* y==zero: x**0 = 1 */ 162 if((iy|ly)==0) return one; 163 164 /* +-NaN return x+y */ 165 if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) || 166 iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0))) 167 return x+y; 168 169 /* determine if y is an odd int when x < 0 170 * yisint = 0 ... y is not an integer 171 * yisint = 1 ... y is an odd int 172 * yisint = 2 ... y is an even int 173 */ 174 yisint = 0; 175 if(hx<0) { 176 if(iy>=0x43400000) yisint = 2; /* even integer y */ 177 else if(iy>=0x3ff00000) { 178 k = (iy>>20)-0x3ff; /* exponent */ 179 if(k>20) { 180 j = ly>>(52-k); 181 if((j<<(52-k))==(int)ly) yisint = 2-(j&1); 182 } else if(ly==0) { 183 j = iy>>(20-k); 184 if((j<<(20-k))==iy) yisint = 2-(j&1); 185 } 186 } 187 } 188 189 /* special value of y */ 190 if(ly==0) { 191 if (iy==0x7ff00000) { /* y is +-inf */ 192 if(((ix-0x3ff00000)|lx)==0) 193 #ifdef _WIN32 194 /* VC++ optimizer reduces y - y to 0 */ 195 return y / y; 196 #else 197 return y - y; /* inf**+-1 is NaN */ 198 #endif 199 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */ 200 return (hy>=0)? y: zero; 201 else /* (|x|<1)**-,+inf = inf,0 */ 202 return (hy<0)?-y: zero; 203 } 204 if(iy==0x3ff00000) { /* y is +-1 */ 205 if(hy<0) return one/x; else return x; 206 } 207 if(hy==0x40000000) return x*x; /* y is 2 */ 208 if(hy==0x3fe00000) { /* y is 0.5 */ 209 if(hx>=0) /* x >= +0 */ 210 return fd_sqrt(x); 211 } 212 } 213 214 ax = fd_fabs(x); 215 /* special value of x */ 216 if(lx==0) { 217 if(ix==0x7ff00000||ix==0||ix==0x3ff00000){ 218 z = ax; /*x is +-0,+-inf,+-1*/ 219 if(hy<0) z = one/z; /* z = (1/|x|) */ 220 if(hx<0) { 221 if(((ix-0x3ff00000)|yisint)==0) { 222 z = (z-z)/(z-z); /* (-1)**non-int is NaN */ 223 } else if(yisint==1) { 224 #ifdef HPUX 225 uz.d = z; 226 __HI(uz) ^= 1<<31; /* some HPUXes cannot negate 0.. */ 227 z = uz.d; 228 #else 229 z = -z; /* (x<0)**odd = -(|x|**odd) */ 230 #endif 231 } 232 } 233 return z; 234 } 235 } 236 237 /* (x<0)**(non-int) is NaN */ 238 if((((hx>>31)+1)|yisint)==0) return (x-x)/(x-x); 239 240 /* |y| is really_big */ 241 if(iy>0x41e00000) { /* if |y| > 2**31 */ 242 if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */ 243 if(ix<=0x3fefffff) return (hy<0)? really_big*really_big:tiny*tiny; 244 if(ix>=0x3ff00000) return (hy>0)? really_big*really_big:tiny*tiny; 245 } 246 /* over/underflow if x is not close to one */ 247 if(ix<0x3fefffff) return (hy<0)? really_big*really_big:tiny*tiny; 248 if(ix>0x3ff00000) return (hy>0)? really_big*really_big:tiny*tiny; 249 /* now |1-x| is tiny <= 2**-20, suffice to compute 250 log(x) by x-x^2/2+x^3/3-x^4/4 */ 251 t = x-1; /* t has 20 trailing zeros */ 252 w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25)); 253 u = ivln2_h*t; /* ivln2_h has 21 sig. bits */ 254 v = t*ivln2_l-w*ivln2; 255 t1 = u+v; 256 uz.d = t1; 257 __LO(uz) = 0; 258 t1 = uz.d; 259 t2 = v-(t1-u); 260 } else { 261 double s_h,t_h; 262 double s2,s_l,t_l; 263 n = 0; 264 /* take care subnormal number */ 265 if(ix<0x00100000) 266 {ax *= two53; n -= 53; uz.d = ax; ix = __HI(uz); } 267 n += ((ix)>>20)-0x3ff; 268 j = ix&0x000fffff; 269 /* determine interval */ 270 ix = j|0x3ff00000; /* normalize ix */ 271 if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */ 272 else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */ 273 else {k=0;n+=1;ix -= 0x00100000;} 274 uz.d = ax; 275 __HI(uz) = ix; 276 ax = uz.d; 277 278 /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */ 279 u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */ 280 v = one/(ax+bp[k]); 281 s = u*v; 282 s_h = s; 283 uz.d = s_h; 284 __LO(uz) = 0; 285 s_h = uz.d; 286 /* t_h=ax+bp[k] High */ 287 t_h = zero; 288 uz.d = t_h; 289 __HI(uz)=((ix>>1)|0x20000000)+0x00080000+(k<<18); 290 t_h = uz.d; 291 t_l = ax - (t_h-bp[k]); 292 s_l = v*((u-s_h*t_h)-s_h*t_l); 293 /* compute log(ax) */ 294 s2 = s*s; 295 r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6))))); 296 r += s_l*(s_h+s); 297 s2 = s_h*s_h; 298 t_h = 3.0+s2+r; 299 uz.d = t_h; 300 __LO(uz) = 0; 301 t_h = uz.d; 302 t_l = r-((t_h-3.0)-s2); 303 /* u+v = s*(1+...) */ 304 u = s_h*t_h; 305 v = s_l*t_h+t_l*s; 306 /* 2/(3log2)*(s+...) */ 307 p_h = u+v; 308 uz.d = p_h; 309 __LO(uz) = 0; 310 p_h = uz.d; 311 p_l = v-(p_h-u); 312 z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */ 313 z_l = cp_l*p_h+p_l*cp+dp_l[k]; 314 /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ 315 t = (double)n; 316 t1 = (((z_h+z_l)+dp_h[k])+t); 317 uz.d = t1; 318 __LO(uz) = 0; 319 t1 = uz.d; 320 t2 = z_l-(((t1-t)-dp_h[k])-z_h); 321 } 322 323 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 324 if((((hx>>31)+1)|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */ 325 326 /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ 327 y1 = y; 328 uy.d = y1; 329 __LO(uy) = 0; 330 y1 = uy.d; 331 p_l = (y-y1)*t1+y*t2; 332 p_h = y1*t1; 333 z = p_l+p_h; 334 uz.d = z; 335 j = __HI(uz); 336 i = __LO(uz); 337 338 if (j>=0x40900000) { /* z >= 1024 */ 339 if(((j-0x40900000)|i)!=0) /* if z > 1024 */ 340 return s*really_big*really_big; /* overflow */ 341 else { 342 if(p_l+ovt>z-p_h) return s*really_big*really_big; /* overflow */ 343 } 344 } else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */ 345 if(((j-0xc090cc00)|i)!=0) /* z < -1075 */ 346 return s*tiny*tiny; /* underflow */ 347 else { 348 if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */ 349 } 350 } 351 /* 352 * compute 2**(p_h+p_l) 353 */ 354 i = j&0x7fffffff; 355 k = (i>>20)-0x3ff; 356 n = 0; 357 if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */ 358 n = j+(0x00100000>>(k+1)); 359 k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */ 360 t = zero; 361 uz.d = t; 362 __HI(uz) = (n&~(0x000fffff>>k)); 363 t = uz.d; 364 n = ((n&0x000fffff)|0x00100000)>>(20-k); 365 if(j<0) n = -n; 366 p_h -= t; 367 } 368 t = p_l+p_h; 369 uz.d = t; 370 __LO(uz) = 0; 371 t = uz.d; 372 u = t*lg2_h; 373 v = (p_l-(t-p_h))*lg2+t*lg2_l; 374 z = u+v; 375 w = v-(z-u); 376 t = z*z; 377 t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5)))); 378 r = (z*t1)/(t1-two)-(w+z*w); 379 z = one-(r-z); 380 uz.d = z; 381 j = __HI(uz); 382 j += (n<<20); 383 if((j>>20)<=0) z = fd_scalbn(z,n); /* subnormal output */ 384 else { uz.d = z; __HI(uz) += (n<<20); z = uz.d; } 385 return s*z; 386 } 387