e_pow.c

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/* @(#)e_pow.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
#endif

/* __ieee754_pow(x,y) return x**y
 *
 *          n
 * Method:  Let x =  2   * (1+f)
 *  1. Compute and return log2(x) in two pieces:
 *    log2(x) = w1 + w2,
 *     where w1 has 53-24 = 29 bit trailing zeros.
 *  2. Perform y*log2(x) = n+y' by simulating muti-precision
 *     arithmetic, where |y'|<=0.5.
 *  3. Return x**y = 2**n*exp(y'*log2)
 *
 * Special cases:
 *  1.  (anything) ** 0  is 1
 *  2.  (anything) ** 1  is itself
 *  3.  (anything) ** NAN is NAN
 *  4.  NAN ** (anything except 0) is NAN
 *  5.  +-(|x| > 1) **  +INF is +INF
 *  6.  +-(|x| > 1) **  -INF is +0
 *  7.  +-(|x| < 1) **  +INF is +0
 *  8.  +-(|x| < 1) **  -INF is +INF
 *  9.  +-1         ** +-INF is NAN
 *  10. +0 ** (+anything except 0, NAN)               is +0
 *  11. -0 ** (+anything except 0, NAN, odd integer)  is +0
 *  12. +0 ** (-anything except 0, NAN)               is +INF
 *  13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
 *  14. -0 ** (odd integer) = -( +0 ** (odd integer) )
 *  15. +INF ** (+anything except 0,NAN) is +INF
 *  16. +INF ** (-anything except 0,NAN) is +0
 *  17. -INF ** (anything)  = -0 ** (-anything)
 *  18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
 *  19. (-anything except 0 and inf) ** (non-integer) is NAN
 *
 * Accuracy:
 *  pow(x,y) returns x**y nearly rounded. In particular
 *      pow(integer,integer)
 *  always returns the correct integer provided it is
 *  representable.
 *
 * Constants :
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

#include "math_libm.h"
#include "math_private.h"

libm_hidden_proto(scalbn)
    libm_hidden_proto(fabs)
#ifdef __STDC__
     static const double
#else
     static double
#endif
       bp[] = { 1.0, 1.5, }, dp_h[] = {
     0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */

         dp_l[] = {
     0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */

         zero = 0.0, one = 1.0, two = 2.0, two53 = 9007199254740992.0,  /* 0x43400000, 0x00000000 */
         huge_val = 1.0e300, tiny = 1.0e-300,
         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
         L1 = 5.99999999999994648725e-01,       /* 0x3FE33333, 0x33333303 */
         L2 = 4.28571428578550184252e-01,       /* 0x3FDB6DB6, 0xDB6FABFF */
         L3 = 3.33333329818377432918e-01,       /* 0x3FD55555, 0x518F264D */
         L4 = 2.72728123808534006489e-01,       /* 0x3FD17460, 0xA91D4101 */
         L5 = 2.30660745775561754067e-01,       /* 0x3FCD864A, 0x93C9DB65 */
         L6 = 2.06975017800338417784e-01,       /* 0x3FCA7E28, 0x4A454EEF */
         P1 = 1.66666666666666019037e-01,       /* 0x3FC55555, 0x5555553E */
         P2 = -2.77777777770155933842e-03,      /* 0xBF66C16C, 0x16BEBD93 */
         P3 = 6.61375632143793436117e-05,       /* 0x3F11566A, 0xAF25DE2C */
         P4 = -1.65339022054652515390e-06,      /* 0xBEBBBD41, 0xC5D26BF1 */
         P5 = 4.13813679705723846039e-08,       /* 0x3E663769, 0x72BEA4D0 */
         lg2 = 6.93147180559945286227e-01,      /* 0x3FE62E42, 0xFEFA39EF */
         lg2_h = 6.93147182464599609375e-01,    /* 0x3FE62E43, 0x00000000 */
         lg2_l = -1.90465429995776804525e-09,   /* 0xBE205C61, 0x0CA86C39 */
         ovt = 8.0085662595372944372e-0017,     /* -(1024-log2(ovfl+.5ulp)) */
         cp = 9.61796693925975554329e-01,       /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
         cp_h = 9.61796700954437255859e-01,     /* 0x3FEEC709, 0xE0000000 =(float)cp */
         cp_l = -7.02846165095275826516e-09,    /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h */
         ivln2 = 1.44269504088896338700e+00,    /* 0x3FF71547, 0x652B82FE =1/ln2 */
         ivln2_h = 1.44269502162933349609e+00,  /* 0x3FF71547, 0x60000000 =24b 1/ln2 */
         ivln2_l = 1.92596299112661746887e-08;  /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail */

#ifdef __STDC__
     double attribute_hidden __ieee754_pow(double x, double y)
#else
     double attribute_hidden __ieee754_pow(x, y)
     double x, y;
#endif
     {
         double z, ax, z_h, z_l, p_h, p_l;
         double y1, t1, t2, r, s, t, u, v, w;
         int32_t i, j, k, yisint, n;
         int32_t hx, hy, ix, iy;
         u_int32_t lx, ly;

         EXTRACT_WORDS(hx, lx, x);
         EXTRACT_WORDS(hy, ly, y);
         ix = hx & 0x7fffffff;
         iy = hy & 0x7fffffff;

         /* y==zero: x**0 = 1 */
         if ((iy | ly) == 0)
             return one;

         /* +-NaN return x+y */
         if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) ||
             iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
             return x + y;

         /* determine if y is an odd int when x < 0
          * yisint = 0       ... y is not an integer
          * yisint = 1       ... y is an odd int
          * yisint = 2       ... y is an even int
          */
         yisint = 0;
         if (hx < 0) {
             if (iy >= 0x43400000)
                 yisint = 2;    /* even integer y */
             else if (iy >= 0x3ff00000) {
                 k = (iy >> 20) - 0x3ff;        /* exponent */
                 if (k > 20) {
                     j = ly >> (52 - k);
                     if ((j << (52 - k)) == ly)
                         yisint = 2 - (j & 1);
                 } else if (ly == 0) {
                     j = iy >> (20 - k);
                     if ((j << (20 - k)) == iy)
                         yisint = 2 - (j & 1);
                 }
             }
         }

         /* special value of y */
         if (ly == 0) {
             if (iy == 0x7ff00000) {    /* y is +-inf */
                 if (((ix - 0x3ff00000) | lx) == 0)
                     return y - y;      /* inf**+-1 is NaN */
                 else if (ix >= 0x3ff00000)     /* (|x|>1)**+-inf = inf,0 */
                     return (hy >= 0) ? y : zero;
                 else           /* (|x|<1)**-,+inf = inf,0 */
                     return (hy < 0) ? -y : zero;
             }
             if (iy == 0x3ff00000) {    /* y is  +-1 */
                 if (hy < 0)
                     return one / x;
                 else
                     return x;
             }
             if (hy == 0x40000000)
                 return x * x;  /* y is  2 */
             if (hy == 0x3fe00000) {    /* y is  0.5 */
                 if (hx >= 0)   /* x >= +0 */
                     return __ieee754_sqrt(x);
             }
         }

         ax = fabs(x);
         /* special value of x */
         if (lx == 0) {
             if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) {
                 z = ax;        /* x is +-0,+-inf,+-1 */
                 if (hy < 0)
                     z = one / z;       /* z = (1/|x|) */
                 if (hx < 0) {
                     if (((ix - 0x3ff00000) | yisint) == 0) {
                         z = (z - z) / (z - z); /* (-1)**non-int is NaN */
                     } else if (yisint == 1)
                         z = -z;        /* (x<0)**odd = -(|x|**odd) */
                 }
                 return z;
             }
         }

         /* (x<0)**(non-int) is NaN */
         if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0)
             return (x - x) / (x - x);

         /* |y| is huge */
         if (iy > 0x41e00000) { /* if |y| > 2**31 */
             if (iy > 0x43f00000) {     /* if |y| > 2**64, must o/uflow */
                 if (ix <= 0x3fefffff)
                     return (hy < 0) ? huge_val * huge_val : tiny * tiny;
                 if (ix >= 0x3ff00000)
                     return (hy > 0) ? huge_val * huge_val : tiny * tiny;
             }
             /* over/underflow if x is not close to one */
             if (ix < 0x3fefffff)
                 return (hy < 0) ? huge_val * huge_val : tiny * tiny;
             if (ix > 0x3ff00000)
                 return (hy > 0) ? huge_val * huge_val : tiny * tiny;
             /* now |1-x| is tiny <= 2**-20, suffice to compute
                log(x) by x-x^2/2+x^3/3-x^4/4 */
             t = x - 1;         /* t has 20 trailing zeros */
             w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
             u = ivln2_h * t;   /* ivln2_h has 21 sig. bits */
             v = t * ivln2_l - w * ivln2;
             t1 = u + v;
             SET_LOW_WORD(t1, 0);
             t2 = v - (t1 - u);
         } else {
             double s2, s_h, s_l, t_h, t_l;
             n = 0;
             /* take care subnormal number */
             if (ix < 0x00100000) {
                 ax *= two53;
                 n -= 53;
                 GET_HIGH_WORD(ix, ax);
             }
             n += ((ix) >> 20) - 0x3ff;
             j = ix & 0x000fffff;
             /* determine interval */
             ix = j | 0x3ff00000;       /* normalize ix */
             if (j <= 0x3988E)
                 k = 0;         /* |x|<sqrt(3/2) */
             else if (j < 0xBB67A)
                 k = 1;         /* |x|<sqrt(3)   */
             else {
                 k = 0;
                 n += 1;
                 ix -= 0x00100000;
             }
             SET_HIGH_WORD(ax, ix);

             /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
             u = ax - bp[k];    /* bp[0]=1.0, bp[1]=1.5 */
             v = one / (ax + bp[k]);
             s = u * v;
             s_h = s;
             SET_LOW_WORD(s_h, 0);
             /* t_h=ax+bp[k] High */
             t_h = zero;
             SET_HIGH_WORD(t_h,
                           ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18));
             t_l = ax - (t_h - bp[k]);
             s_l = v * ((u - s_h * t_h) - s_h * t_l);
             /* compute log(ax) */
             s2 = s * s;
             r = s2 * s2 * (L1 +
                            s2 * (L2 +
                                  s2 * (L3 +
                                        s2 * (L4 + s2 * (L5 + s2 * L6)))));
             r += s_l * (s_h + s);
             s2 = s_h * s_h;
             t_h = 3.0 + s2 + r;
             SET_LOW_WORD(t_h, 0);
             t_l = r - ((t_h - 3.0) - s2);
             /* u+v = s*(1+...) */
             u = s_h * t_h;
             v = s_l * t_h + t_l * s;
             /* 2/(3log2)*(s+...) */
             p_h = u + v;
             SET_LOW_WORD(p_h, 0);
             p_l = v - (p_h - u);
             z_h = cp_h * p_h;  /* cp_h+cp_l = 2/(3*log2) */
             z_l = cp_l * p_h + p_l * cp + dp_l[k];
             /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
             t = (double) n;
             t1 = (((z_h + z_l) + dp_h[k]) + t);
             SET_LOW_WORD(t1, 0);
             t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
         }

         s = one;               /* s (sign of result -ve**odd) = -1 else = 1 */
         if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
             s = -one;          /* (-ve)**(odd int) */

         /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
         y1 = y;
         SET_LOW_WORD(y1, 0);
         p_l = (y - y1) * t1 + y * t2;
         p_h = y1 * t1;
         z = p_l + p_h;
         EXTRACT_WORDS(j, i, z);
         if (j >= 0x40900000) { /* z >= 1024 */
             if (((j - 0x40900000) | i) != 0)   /* if z > 1024 */
                 return s * huge_val * huge_val;        /* overflow */
             else {
                 if (p_l + ovt > z - p_h)
                     return s * huge_val * huge_val;    /* overflow */
             }
         } else if ((j & 0x7fffffff) >= 0x4090cc00) {   /* z <= -1075 */
             if (((j - 0xc090cc00) | i) != 0)   /* z < -1075 */
                 return s * tiny * tiny;        /* underflow */
             else {
                 if (p_l <= z - p_h)
                     return s * tiny * tiny;    /* underflow */
             }
         }
         /*
          * compute 2**(p_h+p_l)
          */
         i = j & 0x7fffffff;
         k = (i >> 20) - 0x3ff;
         n = 0;
         if (i > 0x3fe00000) {  /* if |z| > 0.5, set n = [z+0.5] */
             n = j + (0x00100000 >> (k + 1));
             k = ((n & 0x7fffffff) >> 20) - 0x3ff;      /* new k for n */
             t = zero;
             SET_HIGH_WORD(t, n & ~(0x000fffff >> k));
             n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
             if (j < 0)
                 n = -n;
             p_h -= t;
         }
         t = p_l + p_h;
         SET_LOW_WORD(t, 0);
         u = t * lg2_h;
         v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
         z = u + v;
         w = v - (z - u);
         t = z * z;
         t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
         r = (z * t1) / (t1 - two) - (w + z * w);
         z = one - (r - z);
         GET_HIGH_WORD(j, z);
         j += (n << 20);
         if ((j >> 20) <= 0)
             z = scalbn(z, n);  /* subnormal output */
         else
             SET_HIGH_WORD(z, j);
         return s * z;
     }