exp.c

/* Copyright JS Foundation and other contributors, http:https://js.foundation
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http:https://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *
 * This file is based on work under the following copyright and permission
 * notice:
 *
 *     Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
 *
 *     Permission to use, copy, modify, and distribute this
 *     software is freely granted, provided that this notice
 *     is preserved.
 *
 *     @(#)e_exp.c 1.6 04/04/22
 */

#include "jerry-math-internal.h"

/* 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 Remes 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.
 */

static const double halF[2] = {
  0.5,
  -0.5,
};
static const double ln2HI[2] = {
  6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
  -6.93147180369123816490e-01, /* 0xbfe62e42, 0xfee00000 */
};
static const double ln2LO[2] = {
  1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
  -1.90821492927058770002e-10, /* 0xbdea39ef, 0x35793c76 */
};

#define one         1.0
#define huge        1.0e+300
#define twom1000    9.33263618503218878990e-302 /* 2**-1000=0x01700000,0 */
#define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */
#define u_threshold -7.45133219101941108420e+02 /* 0xc0874910, 0xD52D3051 */
#define invln2      1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */
#define P1          1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */
#define P2          -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */
#define P3          6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */
#define P4          -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */
#define P5          4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */

double
exp (double x) /* default IEEE double exp */
{
  double hi, lo, c, t;
  int k = 0, xsb;
  unsigned hx;

  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) /* NaN */
      {
        return x + x;
      }
      else /* exp(+-inf) = {inf,0} */
      {
        return (xsb == 0) ? x : 0.0;
      }
    }
    if (x > o_threshold) /* overflow */
    {
      return huge * huge;
    }
    if (x < u_threshold) /* underflow */
    {
      return twom1000 * twom1000;
    }
  }

  /* 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 = (int) (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 (huge + x > one) /* trigger inexact */
    {
      return one + x;
    }
  }
  else
  {
    k = 0;
  }

  double_accessor ret;

  /* 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
  {
    ret.dbl = one - ((lo - (x * c) / (2.0 - c)) - hi);
  }
  if (k >= -1021)
  {
    ret.as_int.hi += (((unsigned int) k) << 20); /* add k to y's exponent */
    return ret.dbl;
  }
  else
  {
    ret.as_int.hi += ((k + 1000) << 20); /* add k to y's exponent */
    return ret.dbl * twom1000;
  }
} /* exp */

#undef one
#undef huge
#undef twom1000
#undef o_threshold
#undef u_threshold
#undef invln2
#undef P1
#undef P2
#undef P3
#undef P4
#undef P5