sysdeps/libm-ieee754/e_pow.c

   1 /* @(#)e_pow.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
  13    for performance improvement on pipelined processors.
  14 */
  15
  16 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
  18 #endif
  19
  20 /* __ieee754_pow(x,y) return x**y
  21  *
  22  *                    n
  23  * Method:  Let x =  2   * (1+f)
  24  *      1. Compute and return log2(x) in two pieces:
  25  *              log2(x) = w1 + w2,
  26  *         where w1 has 53-24 = 29 bit trailing zeros.
  27  *      2. Perform y*log2(x) = n+y' by simulating muti-precision
  28  *         arithmetic, where |y'|<=0.5.
  29  *      3. Return x**y = 2**n*exp(y'*log2)
  30  *
  31  * Special cases:
  32  *      1.  (anything) ** 0  is 1
  33  *      2.  (anything) ** 1  is itself
  34  *      3.  (anything) ** NAN is NAN
  35  *      4.  NAN ** (anything except 0) is NAN
  36  *      5.  +-(|x| > 1) **  +INF is +INF
  37  *      6.  +-(|x| > 1) **  -INF is +0
  38  *      7.  +-(|x| < 1) **  +INF is +0
  39  *      8.  +-(|x| < 1) **  -INF is +INF
  40  *      9.  +-1         ** +-INF is NAN
  41  *      10. +0 ** (+anything except 0, NAN)               is +0
  42  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  43  *      12. +0 ** (-anything except 0, NAN)               is +INF
  44  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  45  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  46  *      15. +INF ** (+anything except 0,NAN) is +INF
  47  *      16. +INF ** (-anything except 0,NAN) is +0
  48  *      17. -INF ** (anything)  = -0 ** (-anything)
  49  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  50  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
  51  *
  52  * Accuracy:
  53  *      pow(x,y) returns x**y nearly rounded. In particular
  54  *                      pow(integer,integer)
  55  *      always returns the correct integer provided it is
  56  *      representable.
  57  *
  58  * Constants :
  59  * The hexadecimal values are the intended ones for the following
  60  * constants. The decimal values may be used, provided that the
  61  * compiler will convert from decimal to binary accurately enough
  62  * to produce the hexadecimal values shown.
  63  */
  64
  65 #include "math.h"
  66 #include "math_private.h"
  67 #define zero      C[0]
  68 #define one       C[1]
  69 #define two       C[2]
  70 #define two53     C[3]
  71 #define huge      C[4]
  72 #define tiny      C[5]
  73 #define L1        C[6]
  74 #define L2        C[7]
  75 #define L3        C[8]
  76 #define L4        C[9]
  77 #define L5        C[10]
  78 #define L6        C[11]
  79 #define P1        C[12]
  80 #define P2        C[13]
  81 #define P3        C[14]
  82 #define P4        C[15]
  83 #define P5        C[16]
  84 #define lg2       C[17]
  85 #define lg2_h     C[18]
  86 #define lg2_l     C[19]
  87 #define ovt       C[20]
  88 #define cp        C[21]
  89 #define cp_h      C[22]
  90 #define cp_l      C[23]
  91 #define ivln2     C[24]
  92 #define ivln2_h   C[25]
  93 #define ivln2_l   C[26]
  94
  95 #ifdef __STDC__
  96 static const double
  97 #else
  98 static double
  99 #endif
 100 bp[] = {1.0, 1.5,},
 101 dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
 102 dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
 103 C[] = {
 104 0.0,
 105 1.0,
 106 2.0,
 107 9007199254740992.0      ,
 108 1.0e300,
 109 1.0e-300,
 110 5.99999999999994648725e-01 ,
 111 4.28571428578550184252e-01 ,
 112 3.33333329818377432918e-01 ,
 113 2.72728123808534006489e-01 ,
 114 2.30660745775561754067e-01 ,
 115 2.06975017800338417784e-01 ,
 116 1.66666666666666019037e-01 ,
 117 -2.77777777770155933842e-03 ,
 118 6.61375632143793436117e-05 ,
 119 -1.65339022054652515390e-06 ,
 120 4.13813679705723846039e-08 ,
 121 6.93147180559945286227e-01 ,
 122 6.93147182464599609375e-01 ,
 123 -1.90465429995776804525e-09 ,
 124 8.0085662595372944372e-0017 ,
 125 9.61796693925975554329e-01 ,
 126 9.61796700954437255859e-01 ,
 127 -7.02846165095275826516e-09 ,
 128 1.44269504088896338700e+00 ,
 129 1.44269502162933349609e+00 ,
 130 1.92596299112661746887e-08 };
 131
 132 #ifdef __STDC__
 133         double __ieee754_pow(double x, double y)
 134 #else
 135         double __ieee754_pow(x,y)
 136         double x, y;
 137 #endif
 138 {
 139         double z,ax,z_h,z_l,p_h,p_l;
 140         double y1,t1,t2,r,s,t,u,v,w, t12,t14,r_1,r_2,r_3;
 141         int32_t i,j,k,yisint,n;
 142         int32_t hx,hy,ix,iy;
 143         u_int32_t lx,ly;
 144
 145         EXTRACT_WORDS(hx,lx,x);
 146         EXTRACT_WORDS(hy,ly,y);
 147         ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
 148
 149     /* y==zero: x**0 = 1 */
 150         if((iy|ly)==0) return C[1];
 151
 152     /* +-NaN return x+y */
 153         if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
 154            iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
 155                 return x+y;
 156
 157     /* determine if y is an odd int when x < 0
 158      * yisint = 0       ... y is not an integer
 159      * yisint = 1       ... y is an odd int
 160      * yisint = 2       ... y is an even int
 161      */
 162         yisint  = 0;
 163         if(hx<0) {
 164             if(iy>=0x43400000) yisint = 2; /* even integer y */
 165             else if(iy>=0x3ff00000) {
 166                 k = (iy>>20)-0x3ff;        /* exponent */
 167                 if(k>20) {
 168                     j = ly>>(52-k);
 169                     if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
 170                 } else if(ly==0) {
 171                     j = iy>>(20-k);
 172                     if((int32_t)(j<<(20-k))==iy) yisint = 2-(j&1);
 173                 }
 174             }
 175         }
 176
 177     /* special value of y */
 178         if(ly==0) {
 179             if (iy==0x7ff00000) {       /* y is +-inf */
 180                 if(((ix-0x3ff00000)|lx)==0)
 181                     return  y - y;      /* inf**+-1 is NaN */
 182                 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
 183                     return (hy>=0)? y: C[0];
 184                 else                    /* (|x|<1)**-,+inf = inf,0 */
 185                     return (hy<0)?-y: C[0];
 186             }
 187             if(iy==0x3ff00000) {        /* y is  +-1 */
 188                 if(hy<0) return C[1]/x; else return x;
 189             }
 190             if(hy==0x40000000) return x*x; /* y is  2 */
 191             if(hy==0x3fe00000) {        /* y is  0.5 */
 192                 if(hx>=0)       /* x >= +0 */
 193                 return __ieee754_sqrt(x);
 194             }
 195         }
 196
 197         ax   = fabs(x);
 198     /* special value of x */
 199         if(lx==0) {
 200             if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
 201                 z = ax;                 /*x is +-0,+-inf,+-1*/
 202                 if(hy<0) z = C[1]/z;    /* z = (1/|x|) */
 203                 if(hx<0) {
 204                     if(((ix-0x3ff00000)|yisint)==0) {
 205                         z = (z-z)/(z-z); /* (-1)**non-int is NaN */
 206                     } else if(yisint==1)
 207                         z = -z;         /* (x<0)**odd = -(|x|**odd) */
 208                 }
 209                 return z;
 210             }
 211         }
 212
 213     /* (x<0)**(non-int) is NaN */
 214         if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
 215
 216     /* |y| is huge */
 217         if(iy>0x41e00000) { /* if |y| > 2**31 */
 218             if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
 219                 if(ix<=0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5];
 220                 if(ix>=0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5];
 221             }
 222         /* over/underflow if x is not close to one */
 223             if(ix<0x3fefffff) return (hy<0)? C[4]*C[4]:C[5]*C[5];
 224             if(ix>0x3ff00000) return (hy>0)? C[4]*C[4]:C[5]*C[5];
 225         /* now |1-x| is tiny <= 2**-20, suffice to compute
 226            log(x) by x-x^2/2+x^3/3-x^4/4 */
 227             t = x-1;            /* t has 20 trailing zeros */
 228             w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
 229             u = C[25]*t;        /* ivln2_h has 21 sig. bits */
 230             v = t*C[26]-w*C[24];
 231             t1 = u+v;
 232             SET_LOW_WORD(t1,0);
 233             t2 = v-(t1-u);
 234         } else {
 235             double s2,s_h,s_l,t_h,t_l,s22,s24,s26,r1,r2,r3;
 236             n = 0;
 237         /* take care subnormal number */
 238             if(ix<0x00100000)
 239                 {ax *= C[3]; n -= 53; GET_HIGH_WORD(ix,ax); }
 240             n  += ((ix)>>20)-0x3ff;
 241             j  = ix&0x000fffff;
 242         /* determine interval */
 243             ix = j|0x3ff00000;          /* normalize ix */
 244             if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
 245             else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
 246             else {k=0;n+=1;ix -= 0x00100000;}
 247             SET_HIGH_WORD(ax,ix);
 248
 249         /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
 250             u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
 251             v = C[1]/(ax+bp[k]);
 252             s = u*v;
 253             s_h = s;
 254             SET_LOW_WORD(s_h,0);
 255         /* t_h=ax+bp[k] High */
 256             t_h = C[0];
 257             SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
 258             t_l = ax - (t_h-bp[k]);
 259             s_l = v*((u-s_h*t_h)-s_h*t_l);
 260         /* compute log(ax) */
 261             s2 = s*s;
 262 #ifdef DO_NOT_USE_THIS
 263             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
 264 #else
 265             r1 = C[10]+s2*C[11]; s22=s2*s2;
 266             r2 = C[8]+s2*C[9]; s24=s22*s22;
 267             r3 = C[6]+s2*C[7]; s26=s24*s22;
 268             r = r3*s22 + r2*s24 + r1*s26;
 269 #endif
 270             r += s_l*(s_h+s);
 271             s2  = s_h*s_h;
 272             t_h = 3.0+s2+r;
 273             SET_LOW_WORD(t_h,0);
 274             t_l = r-((t_h-3.0)-s2);
 275         /* u+v = s*(1+...) */
 276             u = s_h*t_h;
 277             v = s_l*t_h+t_l*s;
 278         /* 2/(3log2)*(s+...) */
 279             p_h = u+v;
 280             SET_LOW_WORD(p_h,0);
 281             p_l = v-(p_h-u);
 282             z_h = C[22]*p_h;            /* cp_h+cp_l = 2/(3*log2) */
 283             z_l = C[23]*p_h+p_l*C[21]+dp_l[k];
 284         /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
 285             t = (double)n;
 286             t1 = (((z_h+z_l)+dp_h[k])+t);
 287             SET_LOW_WORD(t1,0);
 288             t2 = z_l-(((t1-t)-dp_h[k])-z_h);
 289         }
 290
 291         s = C[1]; /* s (sign of result -ve**odd) = -1 else = 1 */
 292         if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
 293             s = -C[1];/* (-ve)**(odd int) */
 294
 295     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
 296         y1  = y;
 297         SET_LOW_WORD(y1,0);
 298         p_l = (y-y1)*t1+y*t2;
 299         p_h = y1*t1;
 300         z = p_l+p_h;
 301         EXTRACT_WORDS(j,i,z);
 302         if (j>=0x40900000) {                            /* z >= 1024 */
 303             if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
 304                 return s*C[4]*C[4];                     /* overflow */
 305             else {
 306                 if(p_l+C[20]>z-p_h) return s*C[4]*C[4]; /* overflow */
 307             }
 308         } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
 309             if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
 310                 return s*C[5]*C[5];             /* underflow */
 311             else {
 312                 if(p_l<=z-p_h) return s*C[5]*C[5];      /* underflow */
 313             }
 314         }
 315     /*
 316      * compute 2**(p_h+p_l)
 317      */
 318         i = j&0x7fffffff;
 319         k = (i>>20)-0x3ff;
 320         n = 0;
 321         if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
 322             n = j+(0x00100000>>(k+1));
 323             k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
 324             t = C[0];
 325             SET_HIGH_WORD(t,n&~(0x000fffff>>k));
 326             n = ((n&0x000fffff)|0x00100000)>>(20-k);
 327             if(j<0) n = -n;
 328             p_h -= t;
 329         }
 330         t = p_l+p_h;
 331         SET_LOW_WORD(t,0);
 332         u = t*C[18];
 333         v = (p_l-(t-p_h))*C[17]+t*C[19];
 334         z = u+v;
 335         w = v-(z-u);
 336         t  = z*z;
 337 #ifdef DO_NOT_USE_THIS
 338         t1  = z - t*(C[12]+t*(C[13]+t*(C[14]+t*(C[15]+t*C[16]))));
 339 #else
 340         r_1 = C[15]+t*C[16]; t12 = t*t;
 341         r_2 = C[13]+t*C[14]; t14 = t12*t12;
 342         r_3 = t*C[12];
 343         t1 = z - r_3 - t12*r_2 - t14*r_1;
 344 #endif
 345         r  = (z*t1)/(t1-C[2])-(w+z*w);
 346         z  = C[1]-(r-z);
 347         GET_HIGH_WORD(j,z);
 348         j += (n<<20);
 349         if((j>>20)<=0) z = __scalbn(z,n);       /* subnormal output */
 350         else SET_HIGH_WORD(z,j);
 351         return s*z;
 352 }