prime/src/Runtime/e_pow.c

/* @(#)e_pow.c 1.2 95/01/04 */
/*
 * ====================================================
 * 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.
 * ====================================================
 */

/* __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 "fdlibm.h"

#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 */
    big = 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 __ieee754_pow(double x, double y)
#else
double __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 i, j, k, yisint, n;
  _INT32 hx, hy, ix, iy;
  _UINT32 lx, ly;

  hx = __HI(x);
  lx = __LO(x);
  hy = __HI(y);
  ly = __LO(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;
#ifdef __STDC__
    errno = EDOM; /* mf-- added to conform to old ANSI standard */
#endif
  }

  /* 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 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 ((((hx >> 31) + 1) | yisint) == 0) {
#ifdef __STDC__
    errno = EDOM; /* mf-- added to conform to old ANSI standard */
#endif
    return NAN;
  }

  /* |y| is big */
  if (iy > 0x41e00000) {   /* if |y| > 2**31 */
    if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
      if (ix <= 0x3fefffff)
        return (hy < 0) ? big * big : tiny * tiny;
      if (ix >= 0x3ff00000)
        return (hy > 0) ? big * big : tiny * tiny;
    }
    /* over/underflow if x is not close to one */
    if (ix < 0x3fefffff)
      return (hy < 0) ? big * big : tiny * tiny;
    if (ix > 0x3ff00000)
      return (hy > 0) ? big * big : 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;
    __LO(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;
      ix = __HI(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;
    }
    __HI(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;
    __LO(s_h) = 0;
    /* t_h=ax+bp[k] High */
    t_h = zero;
    __HI(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;
    __LO(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;
    __LO(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);
    __LO(t1) = 0;
    t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
  }

  s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
  if ((((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;
  __LO(y1) = 0;
  p_l = (y - y1) * t1 + y * t2;
  p_h = y1 * t1;
  z = p_l + p_h;
  j = __HI(z);
  i = __LO(z);
  if (j >= 0x40900000) {             /* z >= 1024 */
    if (((j - 0x40900000) | i) != 0) /* if z > 1024 */
      return s * big * big;          /* overflow */
    else {
      if (p_l + ovt > z - p_h)
        return s * big * big; /* 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;
    __HI(t) = (n & ~(0x000fffff >> k));
    n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
    if (j < 0)
      n = -n;
    p_h -= t;
  }
  t = p_l + p_h;
  __LO(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);
  j = __HI(z);
  j += (n << 20);
  if ((j >> 20) <= 0)
    z = scalbn(z, n); /* subnormal output */
  else
    __HI(z) += (n << 20);
  return s * z;
}