modules/fdlibm/src/e_pow.cpp

   1 /* @(#)e_pow.c 1.5 04/04/22 SMI */
   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
  12 //#include <sys/cdefs.h>
  13 //__FBSDID("$FreeBSD$");
  14
  15 /* __ieee754_pow(x,y) return x**y
  16  *
  17  *                    n
  18  * Method:  Let x =  2   * (1+f)
  19  *      1. Compute and return log2(x) in two pieces:
  20  *              log2(x) = w1 + w2,
  21  *         where w1 has 53-24 = 29 bit trailing zeros.
  22  *      2. Perform y*log2(x) = n+y' by simulating multi-precision
  23  *         arithmetic, where |y'|<=0.5.
  24  *      3. Return x**y = 2**n*exp(y'*log2)
  25  *
  26  * Special cases:
  27  *      1.  (anything) ** 0  is 1
  28  *      2.  (anything) ** 1  is itself
  29  *      3.  (anything) ** NAN is NAN except 1 ** NAN = 1
  30  *      4.  NAN ** (anything except 0) is NAN
  31  *      5.  +-(|x| > 1) **  +INF is +INF
  32  *      6.  +-(|x| > 1) **  -INF is +0
  33  *      7.  +-(|x| < 1) **  +INF is +0
  34  *      8.  +-(|x| < 1) **  -INF is +INF
  35  *      9.  +-1         ** +-INF is 1
  36  *      10. +0 ** (+anything except 0, NAN)               is +0
  37  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  38  *      12. +0 ** (-anything except 0, NAN)               is +INF
  39  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  40  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  41  *      15. +INF ** (+anything except 0,NAN) is +INF
  42  *      16. +INF ** (-anything except 0,NAN) is +0
  43  *      17. -INF ** (anything)  = -0 ** (-anything)
  44  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  45  *      19. (-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 <cmath>
  61 #include <float.h>
  62 #include "math_private.h"
  63
  64 static const double
  65 bp[] = {1.0, 1.5,},
  66 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  67 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  68 zero    =  0.0,
  69 half    =  0.5,
  70 qrtr    =  0.25,
  71 thrd    =  3.3333333333333331e-01, /* 0x3fd55555, 0x55555555 */
  72 one     =  1.0,
  73 two     =  2.0,
  74 two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
  75 huge    =  1.0e300,
  76 tiny    =  1.0e-300,
  77         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  78 L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  79 L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  80 L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  81 L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  82 L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
  83 L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
  84 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  85 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  86 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  87 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  88 P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
  89 lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
  90 lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
  91 lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
  92 ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
  93 cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
  94 cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
  95 cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
  96 ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
  97 ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
  98 ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
  99
 100 double
 101 __ieee754_pow(double x, double y)
 102 {
 103         double z,ax,z_h,z_l,p_h,p_l;
 104         double y1,t1,t2,r,s,t,u,v,w;
 105         int32_t i,j,k,yisint,n;
 106         int32_t hx,hy,ix,iy;
 107         u_int32_t lx,ly;
 108
 109         EXTRACT_WORDS(hx,lx,x);
 110         EXTRACT_WORDS(hy,ly,y);
 111         ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
 112
 113     /* y==zero: x**0 = 1 */
 114         if((iy|ly)==0) return one;
 115
 116     /* x==1: 1**y = 1, even if y is NaN */
 117         if (hx==0x3ff00000 && lx == 0) return one;
 118
 119     /* y!=zero: result is NaN if either arg is NaN */
 120         if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
 121            iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
 122             return nan_mix(x, y);
 123
 124     /* determine if y is an odd int when x < 0
 125      * yisint = 0       ... y is not an integer
 126      * yisint = 1       ... y is an odd int
 127      * yisint = 2       ... y is an even int
 128      */
 129         yisint  = 0;
 130         if(hx<0) {
 131             if(iy>=0x43400000) yisint = 2; /* even integer y */
 132             else if(iy>=0x3ff00000) {
 133                 k = (iy>>20)-0x3ff;        /* exponent */
 134                 if(k>20) {
 135                     j = ly>>(52-k);
 136                     if(((u_int32_t)j<<(52-k))==ly) yisint = 2-(j&1);
 137                 } else if(ly==0) {
 138                     j = iy>>(20-k);
 139                     if((j<<(20-k))==iy) yisint = 2-(j&1);
 140                 }
 141             }
 142         }
 143
 144     /* special value of y */
 145         if(ly==0) {
 146             if (iy==0x7ff00000) {       /* y is +-inf */
 147                 if(((ix-0x3ff00000)|lx)==0)
 148                     return  one;        /* (-1)**+-inf is 1 */
 149                 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
 150                     return (hy>=0)? y: zero;
 151                 else                    /* (|x|<1)**-,+inf = inf,0 */
 152                     return (hy<0)?-y: zero;
 153             }
 154             if(iy==0x3ff00000) {        /* y is  +-1 */
 155                 if(hy<0) return one/x; else return x;
 156             }
 157             if(hy==0x40000000) return x*x; /* y is  2 */
 158             if(hy==0x3fe00000) {        /* y is  0.5 */
 159                 if(hx>=0)       /* x >= +0 */
 160                 return std::sqrt(x);
 161             }
 162         }
 163
 164         ax   = fabs(x);
 165     /* special value of x */
 166         if(lx==0) {
 167             if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
 168                 z = ax;                 /*x is +-0,+-inf,+-1*/
 169                 if(hy<0) z = one/z;     /* z = (1/|x|) */
 170                 if(hx<0) {
 171                     if(((ix-0x3ff00000)|yisint)==0) {
 172                         z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 173                     } else if(yisint==1)
 174                         z = -z;         /* (x<0)**odd = -(|x|**odd) */
 175                 }
 176                 return z;
 177             }
 178         }
 179
 180     /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
 181         n = (hx>>31)+1;
 182        but ANSI C says a right shift of a signed negative quantity is
 183        implementation defined.  */
 184         n = ((u_int32_t)hx>>31)-1;
 185
 186     /* (x<0)**(non-int) is NaN */
 187         if((n|yisint)==0) return (x-x)/(x-x);
 188
 189         s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
 190         if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
 191
 192     /* |y| is huge */
 193         if(iy>0x41e00000) { /* if |y| > 2**31 */
 194             if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
 195                 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
 196                 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
 197             }
 198         /* over/underflow if x is not close to one */
 199             if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
 200             if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
 201         /* now |1-x| is tiny <= 2**-20, suffice to compute
 202            log(x) by x-x^2/2+x^3/3-x^4/4 */
 203             t = ax-one;         /* t has 20 trailing zeros */
 204             w = (t*t)*(half-t*(thrd-t*qrtr));
 205             u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
 206             v = t*ivln2_l-w*ivln2;
 207             t1 = u+v;
 208             SET_LOW_WORD(t1,0);
 209             t2 = v-(t1-u);
 210         } else {
 211             double ss,s2,s_h,s_l,t_h,t_l;
 212             n = 0;
 213         /* take care subnormal number */
 214             if(ix<0x00100000)
 215                 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
 216             n  += ((ix)>>20)-0x3ff;
 217             j  = ix&0x000fffff;
 218         /* determine interval */
 219             ix = j|0x3ff00000;          /* normalize ix */
 220             if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
 221             else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
 222             else {k=0;n+=1;ix -= 0x00100000;}
 223             SET_HIGH_WORD(ax,ix);
 224
 225         /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 226             u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 227             v = one/(ax+bp[k]);
 228             ss = u*v;
 229             s_h = ss;
 230             SET_LOW_WORD(s_h,0);
 231         /* t_h=ax+bp[k] High */
 232             t_h = zero;
 233             SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
 234             t_l = ax - (t_h-bp[k]);
 235             s_l = v*((u-s_h*t_h)-s_h*t_l);
 236         /* compute log(ax) */
 237             s2 = ss*ss;
 238             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
 239             r += s_l*(s_h+ss);
 240             s2  = s_h*s_h;
 241             t_h = 3+s2+r;
 242             SET_LOW_WORD(t_h,0);
 243             t_l = r-((t_h-3)-s2);
 244         /* u+v = ss*(1+...) */
 245             u = s_h*t_h;
 246             v = s_l*t_h+t_l*ss;
 247         /* 2/(3log2)*(ss+...) */
 248             p_h = u+v;
 249             SET_LOW_WORD(p_h,0);
 250             p_l = v-(p_h-u);
 251             z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
 252             z_l = cp_l*p_h+p_l*cp+dp_l[k];
 253         /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 254             t = n;
 255             t1 = (((z_h+z_l)+dp_h[k])+t);
 256             SET_LOW_WORD(t1,0);
 257             t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 258         }
 259
 260     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 261         y1  = y;
 262         SET_LOW_WORD(y1,0);
 263         p_l = (y-y1)*t1+y*t2;
 264         p_h = y1*t1;
 265         z = p_l+p_h;
 266         EXTRACT_WORDS(j,i,z);
 267         if (j>=0x40900000) {                            /* z >= 1024 */
 268             if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
 269                 return s*huge*huge;                     /* overflow */
 270             else {
 271                 if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
 272             }
 273         } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
 274             if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
 275                 return s*tiny*tiny;             /* underflow */
 276             else {
 277                 if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
 278             }
 279         }
 280     /*
 281      * compute 2**(p_h+p_l)
 282      */
 283         i = j&0x7fffffff;
 284         k = (i>>20)-0x3ff;
 285         n = 0;
 286         if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
 287             n = j+(0x00100000>>(k+1));
 288             k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
 289             t = zero;
 290             SET_HIGH_WORD(t,n&~(0x000fffff>>k));
 291             n = ((n&0x000fffff)|0x00100000)>>(20-k);
 292             if(j<0) n = -n;
 293             p_h -= t;
 294         }
 295         t = p_l+p_h;
 296         SET_LOW_WORD(t,0);
 297         u = t*lg2_h;
 298         v = (p_l-(t-p_h))*lg2+t*lg2_l;
 299         z = u+v;
 300         w = v-(z-u);
 301         t  = z*z;
 302         t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
 303         r  = (z*t1)/(t1-two)-(w+z*w);
 304         z  = one-(r-z);
 305         GET_HIGH_WORD(j,z);
 306         j += (n<<20);
 307         if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
 308         else SET_HIGH_WORD(z,j);
 309         return s*z;
 310 }