log1p.c

/* Copyright JS Foundation and other contributors, http://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://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.
 *
 *     Permission to use, copy, modify, and distribute this
 *     software is freely granted, provided that this notice
 *     is preserved.
 *
 *     @(#)s_log1p.c 5.1 93/09/24
 */

#include "jerry-math-internal.h"

/* log1p(x)
 * Method :
 *   1. Argument Reduction: find k and f such that
 *      1+x = 2^k * (1+f),
 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
 *  may not be representable exactly. In that case, a correction
 *  term is need. Let u=1+x rounded. Let c = (1+x)-u, then
 *  log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
 *  and add back the correction term c/u.
 *  (Note: when x > 2**53, one can simply return log(x))
 *
 *   2. Approximation of log1p(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) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
 *    (the values of Lp1 to Lp7 are listed in the program)
 *  and
 *      |      2          14          |     -58.45
 *      | Lp1*s +...+Lp7*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
 *    log1p(f) = f - (hfsq - s*(hfsq+R)).
 *
 *  3. Finally, log1p(x) = k*ln2 + log1p(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:
 *  log1p(x) is NaN with signal if x < -1 (including -INF) ;
 *  log1p(+INF) is +INF; log1p(-1) is -INF with signal;
 *  log1p(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.
 *
 * Note: Assuming log() return accurate answer, the following
 *    algorithm can be used to compute log1p(x) to within a few ULP:
 *
 *    u = 1+x;
 *    if(u==1.0) return x ; else
 *         return log(u)*(x/(u-1.0));
 *
 *   See HP-15C Advanced Functions Handbook, p.193.
 */

#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 Lp1    6.666666666666735130e-01 /* 3FE55555 55555593 */
#define Lp2    3.999999999940941908e-01 /* 3FD99999 9997FA04 */
#define Lp3    2.857142874366239149e-01 /* 3FD24924 94229359 */
#define Lp4    2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
#define Lp5    1.818357216161805012e-01 /* 3FC74664 96CB03DE */
#define Lp6    1.531383769920937332e-01 /* 3FC39A09 D078C69F */
#define Lp7    1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */

double
log1p (double x)
{
  double hfsq, f, c, s, z, R;
  double_accessor u;
  int k, hx, hu, ax;

  hx = __HI (x);
  ax = hx & 0x7fffffff;
  c = 0;
  k = 1;
  if (hx < 0x3FDA827A)
  {
    /* 1+x < sqrt(2)+ */
    if (ax >= 0x3ff00000)
    {
      /* x <= -1.0 */
      if (x == -1.0)
      {
        /* log1p(-1) = -inf */
        return -INFINITY;
      }
      else
      {
        /* log1p(x<-1) = NaN */
        return NAN;
      }
    }
    if (ax < 0x3e200000)
    { /* |x| < 2**-29 */
      if ((two54 + x > zero) /* raise inexact */
          && (ax < 0x3c900000)) /* |x| < 2**-54 */
      {
        return x;
      }
      else
      {
        return x - x * x * 0.5;
      }
    }
    if ((hx > 0) || hx <= ((int) 0xbfd2bec4))
    {
      /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
      k = 0;
      f = x;
      hu = 1;
    }
  }
  if (hx >= 0x7ff00000)
  {
    return x + x;
  }
  if (k != 0)
  {
    if (hx < 0x43400000)
    {
      u.dbl = 1.0 + x;
      hu = u.as_int.hi;
      k = (hu >> 20) - 1023;
      c = (k > 0) ? 1.0 - (u.dbl - x) : x - (u.dbl - 1.0); /* correction term */
      c /= u.dbl;
    }
    else
    {
      u.dbl = x;
      hu = u.as_int.hi;
      k = (hu >> 20) - 1023;
      c = 0;
    }
    hu &= 0x000fffff;
    /*
     * The approximation to sqrt(2) used in thresholds is not
     * critical.  However, the ones used above must give less
     * strict bounds than the one here so that the k==0 case is
     * never reached from here, since here we have committed to
     * using the correction term but don't use it if k==0.
     */
    if (hu < 0x6a09e)
    {
      /* u ~< sqrt(2) */
      u.as_int.hi = hu | 0x3ff00000; /* normalize u */
    }
    else
    {
      k += 1;
      u.as_int.hi = hu | 0x3fe00000; /* normalize u/2 */
      hu = (0x00100000 - hu) >> 2;
    }
    f = u.dbl - 1.0;
  }
  hfsq = 0.5 * f * f;
  if (hu == 0)
  {
    /* |f| < 2**-20 */
    if (f == zero)
    {
      if (k == 0)
      {
        return zero;
      }
      else
      {
        c += k * ln2_lo;
        return k * ln2_hi + c;
      }
    }
    R = hfsq * (1.0 - 0.66666666666666666 * f);
    if (k == 0)
    {
      return f - R;
    }
    else
    {
      return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
    }
  }
  s = f / (2.0 + f);
  z = s * s;
  R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
  if (k == 0)
  {
    return f - (hfsq - s * (hfsq + R));
  }
  else
  {
    return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
  }
} /* log1p */

#undef zero
#undef ln2_hi
#undef ln2_lo
#undef two54
#undef Lp1
#undef Lp2
#undef Lp3
#undef Lp4
#undef Lp5
#undef Lp6
#undef Lp7