prime/src/Runtime/e_exp.c

/* @(#)e_exp.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_exp(x)
 * Returns the exponential of x.
 *
 * Method
 *   1. Argument reduction:
 *      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
 *	Given x, find r and integer k such that
 *
 *               x = k*ln2 + r,  |r| <= 0.5*ln2.
 *
 *      Here r will be represented as r = hi-lo for better
 *	accuracy.
 *
 *   2. Approximation of exp(r) by a special rational function on
 *	the interval [0,0.34658]:
 *	Write
 *	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
 *      We use a special Reme algorithm on [0,0.34658] to generate
 * 	a polynomial of degree 5 to approximate R. The maximum error
 *	of this polynomial approximation is bounded by 2**-59. In
 *	other words,
 *	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
 *  	(where z=r*r, and the values of P1 to P5 are listed below)
 *	and
 *	    |                  5          |     -59
 *	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
 *	    |                             |
 *	The computation of exp(r) thus becomes
 *                             2*r
 *		exp(r) = 1 + -------
 *		              R - r
 *                                 r*R1(r)
 *		       = 1 + r + ----------- (for better accuracy)
 *		                  2 - R1(r)
 *	where
 *			         2       4             10
 *		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
 *
 *   3. Scale back to obtain exp(x):
 *	From step 1, we have
 *	   exp(x) = 2^k * exp(r)
 *
 * Special cases:
 *	exp(INF) is INF, exp(NaN) is NaN;
 *	exp(-INF) is 0, and
 *	for finite argument, only exp(0)=1 is exact.
 *
 * Accuracy:
 *	according to an error analysis, the error is always less than
 *	1 ulp (unit in the last place).
 *
 * Misc. info.
 *	For IEEE double
 *	    if x >  7.09782712893383973096e+02 then exp(x) overflow
 *	    if x < -7.45133219101941108420e+02 then exp(x) underflow
 *
 * 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
    one = 1.0,
    halF[2] =
        {
            0.5,
            -0.5,
},
    big = 1.0e+300, twom1000 = 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
    o_threshold = 7.09782712893383973096e+02,               /* 0x40862E42, 0xFEFA39EF */
    u_threshold = -7.45133219101941108420e+02,              /* 0xc0874910, 0xD52D3051 */
    ln2HI[2] =
        {
            6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
            -6.93147180369123816490e-01,
}, /* 0xbfe62e42, 0xfee00000 */
    ln2LO[2] =
        {
            1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
            -1.90821492927058770002e-10,
},                                       /* 0xbdea39ef, 0x35793c76 */
    invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
    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 */

#ifdef __STDC__
double __ieee754_exp(double x) /* default IEEE double exp */
#else
double __ieee754_exp(x) /* default IEEE double exp */
double x;
#endif
{
  double y, hi, lo, c, t;
  _INT32 k, xsb; /*- cc 020130 -*/
  _UINT32 hx;    /*- cc 020130 -*/

  hx = __HI(x);         /* high word of x */
  xsb = (hx >> 31) & 1; /* sign bit of x */
  hx &= 0x7fffffff;     /* high word of |x| */

  /* filter out non-finite argument */
  if (hx >= 0x40862E42) { /* if |x|>=709.78... */
    if (hx >= 0x7ff00000) {
      if (((hx & 0xfffff) | __LO(x)) != 0)
        return x + x; /* NaN */
      else
        return (xsb == 0) ? x : 0.0; /* exp(+-inf)={inf,0} */
    }
    if (x > o_threshold)
      return big * big; /* overflow */
    if (x < u_threshold)
      return twom1000 * twom1000; /* underflow */
  }

  /* argument reduction */
  if (hx > 0x3fd62e42) {   /* if  |x| > 0.5 ln2 */
    if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
      hi = x - ln2HI[xsb];
      lo = ln2LO[xsb];
      k = 1 - xsb - xsb;
    } else {
      k = invln2 * x + halF[xsb];
      t = k;
      hi = x - t * ln2HI[0]; /* t*ln2HI is exact here */
      lo = t * ln2LO[0];
    }
    x = hi - lo;
  } else if (hx < 0x3e300000) { /* when |x|<2**-28 */
    if (big + x > one)
      return one + x; /* trigger inexact */
  } else
    k = 0;

  /* x is now in primary range */
  t = x * x;
  c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
  if (k == 0)
    return one - ((x * c) / (c - 2.0) - x);
  else
    y = one - ((lo - (x * c) / (2.0 - c)) - hi);
  if (k >= -1021) {
    __HI(y) += (k << 20); /* add k to y's exponent */
    return y;
  } else {
    __HI(y) += ((k + 1000) << 20); /* add k to y's exponent */
    return y * twom1000;
  }
}