e_lgammal_r.cpp

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/* See the import.pl script for potential modifications */
/*
 * ====================================================
 * 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.
 * ====================================================
 */

/* Long Double expansions are
  Copyright (C) 2001 Stephen L. Moshier <[e-mail]>
  and are incorporated herein by permission of the author.  The author 
  reserves the right to distribute this material elsewhere under different
  copying permissions.  These modifications are distributed here under 
  the following terms:

    This library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
    License as published by the Free Software Foundation; either
    version 2.1l of the License, or (at your option) any later version.

    This library is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Lesser General Public License for more details.

    You should have received a copy of the GNU Lesser General Public
    License along with this library; if not, write to the Free Software
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */

/* __ieee754_lgammal_r(x, signgamp)
 * Reentrant version of the logarithm of the Gamma function
 * with user provide pointer for the sign of Gamma(x).
 *
 * Method:
 *   1. Argument Reduction for 0 < x <= 8
 *	Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
 *	reduce x to a number in [1.5l,2.5l] by
 *		lgamma(1+s) = log(s) + lgamma(s)
 *	for example,
 *		lgamma(7.3l) = log(6.3l) + lgamma(6.3l)
 *			    = log(6.3l*5.3l) + lgamma(5.3l)
 *			    = log(6.3l*5.3l*4.3l*3.3l*2.3l) + lgamma(2.3l)
 *   2. Polynomial approximation of lgamma around its
 *	minimun ymin=1.461632144968362245l to maintain monotonicity.
 *	On [ymin-0.23l, ymin+0.27l] (i.e., [1.23164l,1.73163l]), use
 *		Let z = x-ymin;
 *		lgamma(x) = -1.214862905358496078218l + z^2*poly(z)
 *   2. Rational approximation in the primary interval [2,3]
 *	We use the following approximation:
 *		s = x-2.0l;
 *		lgamma(x) = 0.5l*s + s*P(s)/Q(s)
 *	Our algorithms are based on the following observation
 *
 *                             zeta(2)-1    2    zeta(3)-1    3
 * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
 *                                 2                 3
 *
 *	where Euler = 0.5771l... is the Euler constant, which is very
 *	close to 0.5l.
 *
 *   3. For x>=8, we have
 *	lgamma(x)~(x-0.5l)log(x)-x+0.5l*log(2pi)+1/(12x)-1/(360x**3)+....
 *	(better formula:
 *	   lgamma(x)~(x-0.5l)*(log(x)-1)-.5*(log(2pi)-1) + ...)
 *	Let z = 1/x, then we approximation
 *		f(z) = lgamma(x) - (x-0.5l)(log(x)-1)
 *	by
 *				    3       5             11
 *		w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
 *
 *   4. For negative x, since (G is gamma function)
 *		-x*G(-x)*G(x) = pi/sin(pi*x),
 *	we have
 *		G(x) = pi/(sin(pi*x)*(-x)*G(-x))
 *	since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
 *	Hence, for x<0, signgam = sign(sin(pi*x)) and
 *		lgamma(x) = log(|Gamma(x)|)
 *			  = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
 *	Note: one should avoid compute pi*(-x) directly in the
 *	      computation of sin(pi*(-x)).
 *
 *   5. Special Cases
 *		lgamma(2+s) ~ s*(1-Euler) for tiny s
 *		lgamma(1)=lgamma(2)=0
 *		lgamma(x) ~ -log(x) for tiny x
 *		lgamma(0) = lgamma(inf) = inf
 *		lgamma(-integer) = +-inf
 *
 */

#include "math.h"
#include "math_private.h"

namespace streflop_libm {
#ifdef __STDC__
static const Extended
#else
static Extended
#endif
  half = 0.5l,
  one = 1.0l,
  pi = 3.14159265358979323846264l,
  two63 = 9.223372036854775808e18l,

  /* lgam(1+x) = 0.5l x + x a(x)/b(x)
     -0.268402099609375l <= x <= 0
     peak relative error 6.6e-22l */
  a0 = -6.343246574721079391729402781192128239938E2l,
  a1 =  1.856560238672465796768677717168371401378E3l,
  a2 =  2.404733102163746263689288466865843408429E3l,
  a3 =  8.804188795790383497379532868917517596322E2l,
  a4 =  1.135361354097447729740103745999661157426E2l,
  a5 =  3.766956539107615557608581581190400021285E0l,

  b0 =  8.214973713960928795704317259806842490498E3l,
  b1 =  1.026343508841367384879065363925870888012E4l,
  b2 =  4.553337477045763320522762343132210919277E3l,
  b3 =  8.506975785032585797446253359230031874803E2l,
  b4 =  6.042447899703295436820744186992189445813E1l,
  /* b5 =  1.000000000000000000000000000000000000000E0l */


  tc =  1.4616321449683623412626595423257213284682E0l,
  tf = -1.2148629053584961146050602565082954242826E-1l,/* Double precision */
/* tt = (tail of tf), i.e. tf + tt has extended precision. */
  tt = 3.3649914684731379602768989080467587736363E-18l,
  /* lgam ( 1.4616321449683623412626595423257213284682E0l ) =
-1.2148629053584960809551455717769158215135617312999903886372437313313530E-1l */

  /* lgam (x + tc) = tf + tt + x g(x)/h(x)
     - 0.230003726999612341262659542325721328468l <= x
     <= 0.2699962730003876587373404576742786715318l
     peak relative error 2.1e-21l */
  g0 = 3.645529916721223331888305293534095553827E-18l,
  g1 = 5.126654642791082497002594216163574795690E3l,
  g2 = 8.828603575854624811911631336122070070327E3l,
  g3 = 5.464186426932117031234820886525701595203E3l,
  g4 = 1.455427403530884193180776558102868592293E3l,
  g5 = 1.541735456969245924860307497029155838446E2l,
  g6 = 4.335498275274822298341872707453445815118E0l,

  h0 = 1.059584930106085509696730443974495979641E4l,
  h1 =  2.147921653490043010629481226937850618860E4l,
  h2 = 1.643014770044524804175197151958100656728E4l,
  h3 =  5.869021995186925517228323497501767586078E3l,
  h4 =  9.764244777714344488787381271643502742293E2l,
  h5 =  6.442485441570592541741092969581997002349E1l,
  /* h6 = 1.000000000000000000000000000000000000000E0l */


  /* lgam (x+1) = -0.5l x + x u(x)/v(x)
     -0.100006103515625l <= x <= 0.231639862060546875l
     peak relative error 1.3e-21l */
  u0 = -8.886217500092090678492242071879342025627E1l,
  u1 =  6.840109978129177639438792958320783599310E2l,
  u2 =  2.042626104514127267855588786511809932433E3l,
  u3 =  1.911723903442667422201651063009856064275E3l,
  u4 =  7.447065275665887457628865263491667767695E2l,
  u5 =  1.132256494121790736268471016493103952637E2l,
  u6 =  4.484398885516614191003094714505960972894E0l,

  v0 =  1.150830924194461522996462401210374632929E3l,
  v1 =  3.399692260848747447377972081399737098610E3l,
  v2 =  3.786631705644460255229513563657226008015E3l,
  v3 =  1.966450123004478374557778781564114347876E3l,
  v4 =  4.741359068914069299837355438370682773122E2l,
  v5 =  4.508989649747184050907206782117647852364E1l,
  /* v6 =  1.000000000000000000000000000000000000000E0l */


  /* lgam (x+2) = .5 x + x s(x)/r(x)
     0 <= x <= 1
     peak relative error 7.2e-22l */
  s0 =  1.454726263410661942989109455292824853344E6l,
  s1 = -3.901428390086348447890408306153378922752E6l,
  s2 = -6.573568698209374121847873064292963089438E6l,
  s3 = -3.319055881485044417245964508099095984643E6l,
  s4 = -7.094891568758439227560184618114707107977E5l,
  s5 = -6.263426646464505837422314539808112478303E4l,
  s6 = -1.684926520999477529949915657519454051529E3l,

  r0 = -1.883978160734303518163008696712983134698E7l,
  r1 = -2.815206082812062064902202753264922306830E7l,
  r2 = -1.600245495251915899081846093343626358398E7l,
  r3 = -4.310526301881305003489257052083370058799E6l,
  r4 = -5.563807682263923279438235987186184968542E5l,
  r5 = -3.027734654434169996032905158145259713083E4l,
  r6 = -4.501995652861105629217250715790764371267E2l,
  /* r6 =  1.000000000000000000000000000000000000000E0l */


/* lgam(x) = ( x - 0.5l ) * log(x) - x + LS2PI + 1/x w(1/x^2)
   x >= 8
   Peak relative error 1.51e-21l
   w0 = LS2PI - 0.5l */
  w0 =  4.189385332046727417803e-1l,
  w1 =  8.333333333333331447505E-2l,
  w2 = -2.777777777750349603440E-3l,
  w3 =  7.936507795855070755671E-4l,
  w4 = -5.952345851765688514613E-4l,
  w5 =  8.412723297322498080632E-4l,
  w6 = -1.880801938119376907179E-3l,
  w7 =  4.885026142432270781165E-3l;

#ifdef __STDC__
static const Extended zero = 0.0l;
#else
static Extended zero = 0.0l;
#endif

#ifdef __STDC__
static Extended
sin_pi (Extended x)
#else
static Extended
sin_pi (x)
     Extended x;
#endif
{
  Extended y, z;
  int n, ix;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  ix = (ix << 16) | (i0 >> 16);
  if (ix < 0x3ffd8000) /* 0.25l */
    return __sinl (pi * x);
  y = -x;			/* x is assume negative */

  /*
   * argument reduction, make sure inexact flag not raised if input
   * is an integer
   */
  z = __floorl (y);
  if (z != y)
    {				/* inexact anyway */
      y  *= 0.5l;
      y = 2.0l*(y - __floorl(y));		/* y = |x| mod 2.0l */
      n = (int) (y*4.0l);
    }
  else
    {
      if (ix >= 0x403f8000)  /* 2^64 */
	{
	  y = zero; n = 0;                 /* y must be even */
	}
      else
	{
	if (ix < 0x403e8000)  /* 2^63 */
	  z = y + two63;	/* exact */
	GET_LDOUBLE_WORDS (se, i0, i1, z);
	n = i1 & 1;
	y  = n;
	n <<= 2;
      }
    }

  switch (n)
    {
    case 0:
      y = __sinl (pi * y);
      break;
    case 1:
    case 2:
      y = __cosl (pi * (half - y));
      break;
    case 3:
    case 4:
      y = __sinl (pi * (one - y));
      break;
    case 5:
    case 6:
      y = -__cosl (pi * (y - 1.5l));
      break;
    default:
      y = __sinl (pi * (y - 2.0l));
      break;
    }
  return -y;
}


#ifdef __STDC__
Extended
__ieee754_lgammal_r (Extended x, int *signgamp)
#else
Extended
__ieee754_lgammal_r (x, signgamp)
     Extended x;
     int *signgamp;
#endif
{
  Extended t, y, z, nadj, p, p1, p2, q, r, w;
  int i, ix;
  u_int32_t se, i0, i1;

  *signgamp = 1;
  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;

  if ((ix | i0 | i1) == 0)
    return one / fabsl (x);

  ix = (ix << 16) | (i0 >> 16);

  /* purge off +-inf, NaN, +-0, and negative arguments */
  if (ix >= 0x7fff0000)
    return x * x;

  if (ix < 0x3fc08000) /* 2^-63 */
    {				/* |x|<2**-63, return -log(|x|) */
      if (se & 0x8000)
	{
	  *signgamp = -1;
	  return -__ieee754_logl (-x);
	}
      else
	return -__ieee754_logl (x);
    }
  if (se & 0x8000)
    {
      t = sin_pi (x);
      if (t == zero)
	return one / fabsl (t);	/* -integer */
      nadj = __ieee754_logl (pi / fabsl (t * x));
      if (t < zero)
	*signgamp = -1;
      x = -x;
    }

  /* purge off 1 and 2 */
  if ((((ix - 0x3fff8000) | i0 | i1) == 0)
      || (((ix - 0x40008000) | i0 | i1) == 0))
    r = 0;
  else if (ix < 0x40008000) /* 2.0l */
    {
      /* x < 2.0l */
      if (ix <= 0x3ffee666) /* 8.99993896484375e-1l */
	{
	  /* lgamma(x) = lgamma(x+1) - log(x) */
	  r = -__ieee754_logl (x);
	  if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1l */
	    {
	      y = x - one;
	      i = 0;
	    }
	  else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1l */
	    {
	      y = x - (tc - one);
	      i = 1;
	    }
	  else
	    {
	      /* x < 0.23l */
	      y = x;
	      i = 2;
	    }
	}
      else
	{
	  r = zero;
	  if (ix >= 0x3fffdda6) /* 1.73162841796875l */
	    {
	      /* [1.7316l,2] */
	      y = x - 2.0l;
	      i = 0;
	    }
	  else if (ix >= 0x3fff9da6)/* 1.23162841796875l */
	    {
	      /* [1.23l,1.73l] */
	      y = x - tc;
	      i = 1;
	    }
	  else
	    {
	      /* [0.9l, 1.23l] */
	      y = x - one;
	      i = 2;
	    }
	}
      switch (i)
	{
	case 0:
	  p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
	  p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
	  r += half * y + y * p1/p2;
	  break;
	case 1:
    p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
    p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
    p = tt + y * p1/p2;
	  r += (tf + p);
	  break;
	case 2:
 p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
      p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
	  r += (-half * y + p1 / p2);
	}
    }
  else if (ix < 0x40028000) /* 8.0l */
    {
      /* x < 8.0l */
      i = (int) x;
      t = zero;
      y = x - (Extended) i;
  p = y *
     (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
  q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
      r = half * y + p / q;
      z = one;			/* lgamma(1+s) = log(s) + lgamma(s) */
      switch (i)
	{
	case 7:
	  z *= (y + 6.0l);	/* FALLTHRU */
	case 6:
	  z *= (y + 5.0l);	/* FALLTHRU */
	case 5:
	  z *= (y + 4.0l);	/* FALLTHRU */
	case 4:
	  z *= (y + 3.0l);	/* FALLTHRU */
	case 3:
	  z *= (y + 2.0l);	/* FALLTHRU */
	  r += __ieee754_logl (z);
	  break;
	}
    }
  else if (ix < 0x40418000) /* 2^66 */
    {
      /* 8.0l <= x < 2**66 */
      t = __ieee754_logl (x);
      z = one / x;
      y = z * z;
      w = w0 + z * (w1
          + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
      r = (x - half) * (t - one) + w;
    }
  else
    /* 2**66 <= x <= inf */
    r = x * (__ieee754_logl (x) - one);
  if (se & 0x8000)
    r = nadj - r;
  return r;
}
}