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.  +-1 ** anything  is 1.0
  25  *      2.  +-1 ** +-INF     is 1.0
  26  *      3.  (anything) ** 0  is 1
  27  *      4.  (anything) ** 1  is itself
  28  *      5.  (anything) ** NAN is NAN
  29  *      6.  NAN ** (anything except 0) is NAN
  30  *      7.  +-(|x| > 1) **  +INF is +INF
  31  *      8.  +-(|x| > 1) **  -INF is +0
  32  *      9.  +-(|x| < 1) **  +INF is +0
  33  *      10  +-(|x| < 1) **  -INF is +INF
  34  *      11. +0 ** (+anything except 0, NAN)               is +0
  35  *      12. -0 ** (+anything except 0, NAN, odd integer)  is +0
  36  *      13. +0 ** (-anything except 0, NAN)               is +INF
  37  *      14. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  38  *      15. -0 ** (odd integer) = -( +0 ** (odd integer) )
  39  *      16. +INF ** (+anything except 0,NAN) is +INF
  40  *      17. +INF ** (-anything except 0,NAN) is +0
  41  *      18. -INF ** (anything)  = -0 ** (-anything)
  42  *      19. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  43  *      20. (-anything except 0 and inf) ** (non-integer) is NAN
  44  *
  45  * Accuracy:
  46  *      pow(x,y) returns x**y nearly rounded. In particular
  47  *                      pow(integer,integer)
  48  *      always returns the correct integer provided it is
  49  *      representable.
  50  *
  51  * Constants :
  52  * The hexadecimal values are the intended ones for the following
  53  * constants. The decimal values may be used, provided that the
  54  * compiler will convert from decimal to binary accurately enough
  55  * to produce the hexadecimal values shown.
  56  */
  57
  58 #include "math.h"
  59 #include "math_private.h"
  60
  61 static const double
  62 bp[] = {1.0, 1.5,},
  63 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  64 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  65 zero    =  0.0,
  66 one     =  1.0,
  67 two     =  2.0,
  68 two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
  69 huge    =  1.0e300,
  70 tiny    =  1.0e-300,
  71         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  72 L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  73 L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  74 L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  75 L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  76 L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
  77 L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
  78 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  79 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  80 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  81 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  82 P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
  83 lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
  84 lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
  85 lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
  86 ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
  87 cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
  88 cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
  89 cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
  90 ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
  91 ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
  92 ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
  93
  94 double __ieee754_pow(double x, double y)
  95 {
  96         double z,ax,z_h,z_l,p_h,p_l;
  97         double _y1,t1,t2,r,s,t,u,v,w;
  98         int32_t i,j,k,yisint,n;
  99         int32_t hx,hy,ix,iy;
 100         u_int32_t lx,ly;
 101
 102         EXTRACT_WORDS(hx,lx,x);
 103     /* x==1: 1**y = 1 (even if y is NaN) */
 104         if (hx==0x3ff00000 && lx==0) {
 105                 return x;
 106         }
 107         ix = hx&0x7fffffff;
 108
 109         EXTRACT_WORDS(hy,ly,y);
 110         iy = hy&0x7fffffff;
 111
 112     /* y==zero: x**0 = 1 */
 113         if((iy|ly)==0) return one;
 114
 115     /* +-NaN return x+y */
 116         if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
 117            iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
 118                 return x+y;
 119
 120     /* determine if y is an odd int when x < 0
 121      * yisint = 0       ... y is not an integer
 122      * yisint = 1       ... y is an odd int
 123      * yisint = 2       ... y is an even int
 124      */
 125         yisint  = 0;
 126         if(hx<0) {
 127             if(iy>=0x43400000) yisint = 2; /* even integer y */
 128             else if(iy>=0x3ff00000) {
 129                 k = (iy>>20)-0x3ff;        /* exponent */
 130                 if(k>20) {
 131                     j = ly>>(52-k);
 132                     if((j<<(52-k))==ly) yisint = 2-(j&1);
 133                 } else if(ly==0) {
 134                     j = iy>>(20-k);
 135                     if((j<<(20-k))==iy) yisint = 2-(j&1);
 136                 }
 137             }
 138         }
 139
 140     /* special value of y */
 141         if(ly==0) {
 142             if (iy==0x7ff00000) {       /* y is +-inf */
 143                 if (((ix-0x3ff00000)|lx)==0)
 144                     return one;         /* +-1**+-inf is 1 (yes, weird rule) */
 145                 if (ix >= 0x3ff00000)   /* (|x|>1)**+-inf = inf,0 */
 146                     return (hy>=0) ? y : zero;
 147                 /* (|x|<1)**-,+inf = inf,0 */
 148                 return (hy<0) ? -y : zero;
 149             }
 150             if(iy==0x3ff00000) {        /* y is  +-1 */
 151                 if(hy<0) return one/x; else return x;
 152             }
 153             if(hy==0x40000000) return x*x; /* y is  2 */
 154             if(hy==0x3fe00000) {        /* y is  0.5 */
 155                 if(hx>=0)       /* x >= +0 */
 156                     return __ieee754_sqrt(x);
 157             }
 158         }
 159
 160         ax   = fabs(x);
 161     /* special value of x */
 162         if(lx==0) {
 163             if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
 164                 z = ax;                 /*x is +-0,+-inf,+-1*/
 165                 if(hy<0) z = one/z;     /* z = (1/|x|) */
 166                 if(hx<0) {
 167                     if(((ix-0x3ff00000)|yisint)==0) {
 168                         z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 169                     } else if(yisint==1)
 170                         z = -z;         /* (x<0)**odd = -(|x|**odd) */
 171                 }
 172                 return z;
 173             }
 174         }
 175
 176     /* (x<0)**(non-int) is NaN */
 177         if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
 178
 179     /* |y| is huge */
 180         if(iy>0x41e00000) { /* if |y| > 2**31 */
 181             if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
 182                 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
 183                 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
 184             }
 185         /* over/underflow if x is not close to one */
 186             if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
 187             if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
 188         /* now |1-x| is tiny <= 2**-20, suffice to compute
 189            log(x) by x-x^2/2+x^3/3-x^4/4 */
 190             t = x-1;            /* t has 20 trailing zeros */
 191             w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
 192             u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
 193             v = t*ivln2_l-w*ivln2;
 194             t1 = u+v;
 195             SET_LOW_WORD(t1,0);
 196             t2 = v-(t1-u);
 197         } else {
 198             double s2,s_h,s_l,t_h,t_l;
 199             n = 0;
 200         /* take care subnormal number */
 201             if(ix<0x00100000)
 202                 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
 203             n  += ((ix)>>20)-0x3ff;
 204             j  = ix&0x000fffff;
 205         /* determine interval */
 206             ix = j|0x3ff00000;          /* normalize ix */
 207             if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
 208             else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
 209             else {k=0;n+=1;ix -= 0x00100000;}
 210             SET_HIGH_WORD(ax,ix);
 211
 212         /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 213             u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 214             v = one/(ax+bp[k]);
 215             s = u*v;
 216             s_h = s;
 217             SET_LOW_WORD(s_h,0);
 218         /* t_h=ax+bp[k] High */
 219             t_h = zero;
 220             SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
 221             t_l = ax - (t_h-bp[k]);
 222             s_l = v*((u-s_h*t_h)-s_h*t_l);
 223         /* compute log(ax) */
 224             s2 = s*s;
 225             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
 226             r += s_l*(s_h+s);
 227             s2  = s_h*s_h;
 228             t_h = 3.0+s2+r;
 229             SET_LOW_WORD(t_h,0);
 230             t_l = r-((t_h-3.0)-s2);
 231         /* u+v = s*(1+...) */
 232             u = s_h*t_h;
 233             v = s_l*t_h+t_l*s;
 234         /* 2/(3log2)*(s+...) */
 235             p_h = u+v;
 236             SET_LOW_WORD(p_h,0);
 237             p_l = v-(p_h-u);
 238             z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
 239             z_l = cp_l*p_h+p_l*cp+dp_l[k];
 240         /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 241             t = (double)n;
 242             t1 = (((z_h+z_l)+dp_h[k])+t);
 243             SET_LOW_WORD(t1,0);
 244             t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 245         }
 246
 247         s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
 248         if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
 249             s = -one;/* (-ve)**(odd int) */
 250
 251     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 252         _y1  = y;
 253         SET_LOW_WORD(_y1,0);
 254         p_l = (y-_y1)*t1+y*t2;
 255         p_h = _y1*t1;
 256         z = p_l+p_h;
 257         EXTRACT_WORDS(j,i,z);
 258         if (j>=0x40900000) {                            /* z >= 1024 */
 259             if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
 260                 return s*huge*huge;                     /* overflow */
 261             else {
 262                 if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
 263             }
 264         } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
 265             if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
 266                 return s*tiny*tiny;             /* underflow */
 267             else {
 268                 if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
 269             }
 270         }
 271     /*
 272      * compute 2**(p_h+p_l)
 273      */
 274         i = j&0x7fffffff;
 275         k = (i>>20)-0x3ff;
 276         n = 0;
 277         if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
 278             n = j+(0x00100000>>(k+1));
 279             k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
 280             t = zero;
 281             SET_HIGH_WORD(t,n&~(0x000fffff>>k));
 282             n = ((n&0x000fffff)|0x00100000)>>(20-k);
 283             if(j<0) n = -n;
 284             p_h -= t;
 285         }
 286         t = p_l+p_h;
 287         SET_LOW_WORD(t,0);
 288         u = t*lg2_h;
 289         v = (p_l-(t-p_h))*lg2+t*lg2_l;
 290         z = u+v;
 291         w = v-(z-u);
 292         t  = z*z;
 293         t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 294         r  = (z*t1)/(t1-two)-(w+z*w);
 295         z  = one-(r-z);
 296         GET_HIGH_WORD(j,z);
 297         j += (n<<20);
 298         if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
 299         else SET_HIGH_WORD(z,j);
 300         return s*z;
 301 }