libm/e_pow.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /* __ieee754_pow(x,y) return x**y
  13  *
  14  *                    n
  15  * Method:  Let x =  2   * (1+f)
  16  *      1. Compute and return log2(x) in two pieces:
  17  *              log2(x) = w1 + w2,
  18  *         where w1 has 53-24 = 29 bit trailing zeros.
  19  *      2. Perform y*log2(x) = n+y' by simulating muti-precision
  20  *         arithmetic, where |y'|<=0.5.
  21  *      3. Return x**y = 2**n*exp(y'*log2)
  22  *
  23  * Special cases:
  24  *      1.  (anything) ** 0  is 1
  25  *      2.  (anything) ** 1  is itself
  26  *      3.  (anything) ** NAN is NAN
  27  *      4.  NAN ** (anything except 0) is NAN
  28  *      5.  +-(|x| > 1) **  +INF is +INF
  29  *      6.  +-(|x| > 1) **  -INF is +0
  30  *      7.  +-(|x| < 1) **  +INF is +0
  31  *      8.  +-(|x| < 1) **  -INF is +INF
  32  *      9.  +-1         ** +-INF is NAN
  33  *      10. +0 ** (+anything except 0, NAN)               is +0
  34  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  35  *      12. +0 ** (-anything except 0, NAN)               is +INF
  36  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  37  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  38  *      15. +INF ** (+anything except 0,NAN) is +INF
  39  *      16. +INF ** (-anything except 0,NAN) is +0
  40  *      17. -INF ** (anything)  = -0 ** (-anything)
  41  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  42  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
  43  *
  44  * Accuracy:
  45  *      pow(x,y) returns x**y nearly rounded. In particular
  46  *                      pow(integer,integer)
  47  *      always returns the correct integer provided it is
  48  *      representable.
  49  *
  50  * Constants :
  51  * The hexadecimal values are the intended ones for the following
  52  * constants. The decimal values may be used, provided that the
  53  * compiler will convert from decimal to binary accurately enough
  54  * to produce the hexadecimal values shown.
  55  */
  56
  57 #include "math.h"
  58 #include "math_private.h"
  59
  60 static const double
  61 bp[] = {1.0, 1.5,},
  62 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  63 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  64 zero    =  0.0,
  65 one     =  1.0,
  66 two     =  2.0,
  67 two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
  68 huge    =  1.0e300,
  69 tiny    =  1.0e-300,
  70         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  71 L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  72 L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  73 L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  74 L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  75 L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
  76 L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
  77 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  78 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  79 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  80 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  81 P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
  82 lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
  83 lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
  84 lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
  85 ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
  86 cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
  87 cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
  88 cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
  89 ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
  90 ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
  91 ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
  92
  93 double attribute_hidden __ieee754_pow(double x, double y)
  94 {
  95         double z,ax,z_h,z_l,p_h,p_l;
  96         double y1,t1,t2,r,s,t,u,v,w;
  97         int32_t i,j,k,yisint,n;
  98         int32_t hx,hy,ix,iy;
  99         u_int32_t lx,ly;
 100
 101         EXTRACT_WORDS(hx,lx,x);
 102         EXTRACT_WORDS(hy,ly,y);
 103         ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
 104
 105     /* y==zero: x**0 = 1 */
 106         if((iy|ly)==0) return one;
 107
 108     /* +-NaN return x+y */
 109         if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
 110            iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
 111                 return x+y;
 112
 113     /* determine if y is an odd int when x < 0
 114      * yisint = 0       ... y is not an integer
 115      * yisint = 1       ... y is an odd int
 116      * yisint = 2       ... y is an even int
 117      */
 118         yisint  = 0;
 119         if(hx<0) {
 120             if(iy>=0x43400000) yisint = 2; /* even integer y */
 121             else if(iy>=0x3ff00000) {
 122                 k = (iy>>20)-0x3ff;        /* exponent */
 123                 if(k>20) {
 124                     j = ly>>(52-k);
 125                     if((j<<(52-k))==ly) yisint = 2-(j&1);
 126                 } else if(ly==0) {
 127                     j = iy>>(20-k);
 128                     if((j<<(20-k))==iy) yisint = 2-(j&1);
 129                 }
 130             }
 131         }
 132
 133     /* special value of y */
 134         if(ly==0) {
 135             if (iy==0x7ff00000) {       /* y is +-inf */
 136                 if(((ix-0x3ff00000)|lx)==0)
 137                     return  y - y;      /* inf**+-1 is NaN */
 138                 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
 139                     return (hy>=0)? y: zero;
 140                 else                    /* (|x|<1)**-,+inf = inf,0 */
 141                     return (hy<0)?-y: zero;
 142             }
 143             if(iy==0x3ff00000) {        /* y is  +-1 */
 144                 if(hy<0) return one/x; else return x;
 145             }
 146             if(hy==0x40000000) return x*x; /* y is  2 */
 147             if(hy==0x3fe00000) {        /* y is  0.5 */
 148                 if(hx>=0)       /* x >= +0 */
 149                 return __ieee754_sqrt(x);
 150             }
 151         }
 152
 153         ax   = fabs(x);
 154     /* special value of x */
 155         if(lx==0) {
 156             if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
 157                 z = ax;                 /*x is +-0,+-inf,+-1*/
 158                 if(hy<0) z = one/z;     /* z = (1/|x|) */
 159                 if(hx<0) {
 160                     if(((ix-0x3ff00000)|yisint)==0) {
 161                         z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 162                     } else if(yisint==1)
 163                         z = -z;         /* (x<0)**odd = -(|x|**odd) */
 164                 }
 165                 return z;
 166             }
 167         }
 168
 169     /* (x<0)**(non-int) is NaN */
 170         if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
 171
 172     /* |y| is huge */
 173         if(iy>0x41e00000) { /* if |y| > 2**31 */
 174             if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
 175                 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
 176                 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
 177             }
 178         /* over/underflow if x is not close to one */
 179             if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
 180             if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
 181         /* now |1-x| is tiny <= 2**-20, suffice to compute
 182            log(x) by x-x^2/2+x^3/3-x^4/4 */
 183             t = x-1;            /* t has 20 trailing zeros */
 184             w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
 185             u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
 186             v = t*ivln2_l-w*ivln2;
 187             t1 = u+v;
 188             SET_LOW_WORD(t1,0);
 189             t2 = v-(t1-u);
 190         } else {
 191             double s2,s_h,s_l,t_h,t_l;
 192             n = 0;
 193         /* take care subnormal number */
 194             if(ix<0x00100000)
 195                 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
 196             n  += ((ix)>>20)-0x3ff;
 197             j  = ix&0x000fffff;
 198         /* determine interval */
 199             ix = j|0x3ff00000;          /* normalize ix */
 200             if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
 201             else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
 202             else {k=0;n+=1;ix -= 0x00100000;}
 203             SET_HIGH_WORD(ax,ix);
 204
 205         /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 206             u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 207             v = one/(ax+bp[k]);
 208             s = u*v;
 209             s_h = s;
 210             SET_LOW_WORD(s_h,0);
 211         /* t_h=ax+bp[k] High */
 212             t_h = zero;
 213             SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
 214             t_l = ax - (t_h-bp[k]);
 215             s_l = v*((u-s_h*t_h)-s_h*t_l);
 216         /* compute log(ax) */
 217             s2 = s*s;
 218             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
 219             r += s_l*(s_h+s);
 220             s2  = s_h*s_h;
 221             t_h = 3.0+s2+r;
 222             SET_LOW_WORD(t_h,0);
 223             t_l = r-((t_h-3.0)-s2);
 224         /* u+v = s*(1+...) */
 225             u = s_h*t_h;
 226             v = s_l*t_h+t_l*s;
 227         /* 2/(3log2)*(s+...) */
 228             p_h = u+v;
 229             SET_LOW_WORD(p_h,0);
 230             p_l = v-(p_h-u);
 231             z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
 232             z_l = cp_l*p_h+p_l*cp+dp_l[k];
 233         /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 234             t = (double)n;
 235             t1 = (((z_h+z_l)+dp_h[k])+t);
 236             SET_LOW_WORD(t1,0);
 237             t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 238         }
 239
 240         s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
 241         if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
 242             s = -one;/* (-ve)**(odd int) */
 243
 244     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 245         y1  = y;
 246         SET_LOW_WORD(y1,0);
 247         p_l = (y-y1)*t1+y*t2;
 248         p_h = y1*t1;
 249         z = p_l+p_h;
 250         EXTRACT_WORDS(j,i,z);
 251         if (j>=0x40900000) {                            /* z >= 1024 */
 252             if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
 253                 return s*huge*huge;                     /* overflow */
 254             else {
 255                 if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
 256             }
 257         } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
 258             if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
 259                 return s*tiny*tiny;             /* underflow */
 260             else {
 261                 if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
 262             }
 263         }
 264     /*
 265      * compute 2**(p_h+p_l)
 266      */
 267         i = j&0x7fffffff;
 268         k = (i>>20)-0x3ff;
 269         n = 0;
 270         if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
 271             n = j+(0x00100000>>(k+1));
 272             k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
 273             t = zero;
 274             SET_HIGH_WORD(t,n&~(0x000fffff>>k));
 275             n = ((n&0x000fffff)|0x00100000)>>(20-k);
 276             if(j<0) n = -n;
 277             p_h -= t;
 278         }
 279         t = p_l+p_h;
 280         SET_LOW_WORD(t,0);
 281         u = t*lg2_h;
 282         v = (p_l-(t-p_h))*lg2+t*lg2_l;
 283         z = u+v;
 284         w = v-(z-u);
 285         t  = z*z;
 286         t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 287         r  = (z*t1)/(t1-two)-(w+z*w);
 288         z  = one-(r-z);
 289         GET_HIGH_WORD(j,z);
 290         j += (n<<20);
 291         if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
 292         else SET_HIGH_WORD(z,j);
 293         return s*z;
 294 }
 295
 296 /*
 297  * wrapper pow(x,y) return x**y
 298  */
 299 #ifndef _IEEE_LIBM
 300 double pow(double x, double y)
 301 {
 302         double z = __ieee754_pow(x, y);
 303         if (_LIB_VERSION == _IEEE_|| isnan(y))
 304                 return z;
 305         if (isnan(x)) {
 306                 if (y == 0.0)
 307                         return __kernel_standard(x, y, 42); /* pow(NaN,0.0) */
 308                 return z;
 309         }
 310         if (x == 0.0) {
 311                 if (y == 0.0)
 312                         return __kernel_standard(x, y, 20); /* pow(0.0,0.0) */
 313                 if (isfinite(y) && y < 0.0)
 314                         return __kernel_standard(x,y,23); /* pow(0.0,negative) */
 315                 return z;
 316         }
 317         if (!isfinite(z)) {
 318                 if (isfinite(x) && isfinite(y)) {
 319                         if (isnan(z))
 320                                 return __kernel_standard(x, y, 24); /* pow neg**non-int */
 321                         return __kernel_standard(x, y, 21); /* pow overflow */
 322                 }
 323         }
 324         if (z == 0.0 && isfinite(x) && isfinite(y))
 325                 return __kernel_standard(x, y, 22); /* pow underflow */
 326         return z;
 327 }
 328 #else
 329 strong_alias(__ieee754_pow, pow)
 330 #endif
 331 libm_hidden_def(pow)