s_log1p.c

/* @(#)s_log1p.c 5.1 93/09/24 */
/*
 * ====================================================
 * 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.
 * ====================================================
 */
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
   for performance improvement on pipelined processors.
 */

/* double log1p(double 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.
 */

#include <math.h>
#include <math_private.h>

static const double
  ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  ln2_lo = 1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  two54 = 1.80143985094819840000e+16,   /* 43500000 00000000 */
  Lp[] = { 0.0, 6.666666666666735130e-01, /* 3FE55555 55555593 */
	   3.999999999940941908e-01, /* 3FD99999 9997FA04 */
	   2.857142874366239149e-01, /* 3FD24924 94229359 */
	   2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
	   1.818357216161805012e-01, /* 3FC74664 96CB03DE */
	   1.531383769920937332e-01, /* 3FC39A09 D078C69F */
	   1.479819860511658591e-01 }; /* 3FC2F112 DF3E5244 */

static const double zero = 0.0;

double
__log1p (double x)
{
  double hfsq, f, c, s, z, R, u, z2, z4, z6, R1, R2, R3, R4;
  int32_t k, hx, hu, ax;

  GET_HIGH_WORD (hx, x);
  ax = hx & 0x7fffffff;

  k = 1;
  if (hx < 0x3FDA827A)                          /* x < 0.41422  */
    {
      if (__glibc_unlikely (ax >= 0x3ff00000))           /* x <= -1.0 */
	{
	  if (x == -1.0)
	    return -two54 / zero;               /* log1p(-1)=-inf */
	  else
	    return (x - x) / (x - x);           /* log1p(x<-1)=NaN */
	}
      if (__glibc_unlikely (ax < 0x3e200000))           /* |x| < 2**-29 */
	{
	  math_force_eval (two54 + x);          /* raise inexact */
	  if (ax < 0x3c900000)                  /* |x| < 2**-54 */
	    return x;
	  else
	    return x - x * x * 0.5;
	}
      if (hx > 0 || hx <= ((int32_t) 0xbfd2bec3))
	{
	  k = 0; f = x; hu = 1;
	}                       /* -0.2929<x<0.41422 */
    }
  else if (__glibc_unlikely (hx >= 0x7ff00000))
    return x + x;
  if (k != 0)
    {
      if (hx < 0x43400000)
	{
	  u = 1.0 + x;
	  GET_HIGH_WORD (hu, u);
	  k = (hu >> 20) - 1023;
	  c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0); /* correction term */
	  c /= u;
	}
      else
	{
	  u = x;
	  GET_HIGH_WORD (hu, u);
	  k = (hu >> 20) - 1023;
	  c = 0;
	}
      hu &= 0x000fffff;
      if (hu < 0x6a09e)
	{
	  SET_HIGH_WORD (u, hu | 0x3ff00000);   /* normalize u */
	}
      else
	{
	  k += 1;
	  SET_HIGH_WORD (u, hu | 0x3fe00000);   /* normalize u/2 */
	  hu = (0x00100000 - hu) >> 2;
	}
      f = u - 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;
  R1 = z * Lp[1]; z2 = z * z;
  R2 = Lp[2] + z * Lp[3]; z4 = z2 * z2;
  R3 = Lp[4] + z * Lp[5]; z6 = z4 * z2;
  R4 = Lp[6] + z * Lp[7];
  R = R1 + z2 * R2 + z4 * R3 + z6 * R4;
  if (k == 0)
    return f - (hfsq - s * (hfsq + R));
  else
    return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
}