js/src/fdlibm/e_pow.c

   1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
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  16  * The Original Code is Mozilla Communicator client code, released
  17  * March 31, 1998.
  18  *
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  21  * Portions created by the Initial Developer are Copyright (C) 1998
  22  * the Initial Developer. All Rights Reserved.
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  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__
 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 }