log.c

/* Copyright 2016 Samsung Electronics Co., Ltd.
 * Copyright 2016 University of Szeged
 *
 * 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://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) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 *     Developed at SunSoft, a Sun Microsystems, Inc. business.
 *     Permission to use, copy, modify, and distribute this
 *     software is freely granted, provided that this notice
 *     is preserved.
 *
 *     @(#)e_log.c 1.3 95/01/18
 */

#include "jerry-libm-internal.h"

/* log(x)
 * Return the logrithm of x
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *                      x = 2^k * (1+f),
 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *   2. Approximation of log(1+f).
 *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *               = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *      a polynomial of degree 14 to approximate R The maximum error
 *      of this polynomial approximation is bounded by 2**-58.45. In
 *      other words,
 *                      2      4      6      8      10      12      14
 *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 *      (the values of Lg1 to Lg7 are listed in the program)
 *      and
 *          |      2          14          |     -58.45
 *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
 *          |                             |
 *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *      In order to guarantee error in log below 1ulp, we compute log
 *      by
 *              log(1+f) = f - s*(f - R)                (if f is not too large)
 *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
 *
 *      3. Finally,  log(x) = k*ln2 + log(1+f).
 *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *         Here ln2 is split into two floating point number:
 *                      ln2_hi + ln2_lo,
 *         where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *      log(x) is NaN with signal if x < 0 (including -INF) ;
 *      log(+INF) is +INF; log(0) is -INF with signal;
 *      log(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * 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.
 */

#define zero   0.0
#define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
#define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
#define two54  1.80143985094819840000e+16 /* 43500000 00000000 */
#define Lg1    6.666666666666735130e-01 /* 3FE55555 55555593 */
#define Lg2    3.999999999940941908e-01 /* 3FD99999 9997FA04 */
#define Lg3    2.857142874366239149e-01 /* 3FD24924 94229359 */
#define Lg4    2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
#define Lg5    1.818357216161805012e-01 /* 3FC74664 96CB03DE */
#define Lg6    1.531383769920937332e-01 /* 3FC39A09 D078C69F */
#define Lg7    1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */

double
log (double x)
{
  double hfsq, f, s, z, R, w, t1, t2, dk;
  int k, hx, i, j;
  unsigned lx;

  hx = __HI (x); /* high word of x */
  lx = __LO (x); /* low  word of x */

  k = 0;
  if (hx < 0x00100000) /* x < 2**-1022  */
  {
    if (((hx & 0x7fffffff) | lx) == 0) /* log(+-0) = -inf */
    {
      return -two54 / zero;
    }
    if (hx < 0) /* log(-#) = NaN */
    {
      return (x - x) / zero;
    }
    k -= 54;
    x *= two54; /* subnormal number, scale up x */
    hx = __HI (x); /* high word of x */
  }
  if (hx >= 0x7ff00000)
  {
    return x + x;
  }
  k += (hx >> 20) - 1023;
  hx &= 0x000fffff;
  i = (hx + 0x95f64) & 0x100000;
  __HI (x) = hx | (i ^ 0x3ff00000); /* normalize x or x / 2 */
  k += (i >> 20);
  f = x - 1.0;
  if ((0x000fffff & (2 + hx)) < 3) /* |f| < 2**-20 */
  {
    if (f == zero)
    {
      if (k == 0)
      {
        return zero;
      }
      else
      {
        dk = (double) k;
        return dk * ln2_hi + dk * ln2_lo;
      }
    }
    R = f * f * (0.5 - 0.33333333333333333 * f);
    if (k == 0)
    {
      return f - R;
    }
    else
    {
      dk = (double) k;
      return dk * ln2_hi - ((R - dk * ln2_lo) - f);
    }
  }
  s = f / (2.0 + f);
  dk = (double) k;
  z = s * s;
  i = hx - 0x6147a;
  w = z * z;
  j = 0x6b851 - hx;
  t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
  t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
  i |= j;
  R = t2 + t1;
  if (i > 0)
  {
    hfsq = 0.5 * f * f;
    if (k == 0)
    {
      return f - (hfsq - s * (hfsq + R));
    }
    else
    {
      return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
    }
  }
  else
  {
    if (k == 0)
    {
      return f - s * (f - R);
    }
    else
    {
      return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
    }
  }
} /* log */

#undef zero
#undef ln2_hi
#undef ln2_lo
#undef two54
#undef Lg1
#undef Lg2
#undef Lg3
#undef Lg4
#undef Lg5
#undef Lg6
#undef Lg7