relibc_openlibm/ld80/e_lgammal_r.c

/*
 * ====================================================
 * 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.
 * ====================================================
 */

/*
 * Copyright (c) 2008 Stephen L. Moshier <[email protected]>
 *
 * Permission to use, copy, modify, and distribute this software for any
 * purpose with or without fee is hereby granted, provided that the above
 * copyright notice and this permission notice appear in all copies.
 *
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
 */

/* 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 [1.5,2.5] by
 *		lgamma(1+s) = log(s) + lgamma(s)
 *	for example,
 *		lgamma(7.3) = log(6.3) + lgamma(6.3)
 *			    = log(6.3*5.3) + lgamma(5.3)
 *			    = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
 *   2. Polynomial approximation of lgamma around its
 *	minimun ymin=1.461632144968362245 to maintain monotonicity.
 *	On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
 *		Let z = x-ymin;
 *		lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
 *   2. Rational approximation in the primary interval [2,3]
 *	We use the following approximation:
 *		s = x-2.0;
 *		lgamma(x) = 0.5*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 = 0.5771... is the Euler constant, which is very
 *	close to 0.5.
 *
 *   3. For x>=8, we have
 *	lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
 *	(better formula:
 *	   lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
 *	Let z = 1/x, then we approximation
 *		f(z) = lgamma(x) - (x-0.5)(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 <openlibm_math.h>

#include "math_private.h"

static const long double
  half = 0.5L,
  one = 1.0L,
  pi = 3.14159265358979323846264L,
  two63 = 9.223372036854775808e18L,

  /* lgam(1+x) = 0.5 x + x a(x)/b(x)
     -0.268402099609375 <= x <= 0
     peak relative error 6.6e-22 */
  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 =  1.000000000000000000000000000000000000000E0 */


  tc =  1.4616321449683623412626595423257213284682E0L,
  tf = -1.2148629053584961146050602565082954242826E-1,/* double precision */
/* tt = (tail of tf), i.e. tf + tt has extended precision. */
  tt = 3.3649914684731379602768989080467587736363E-18L,
  /* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
-1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */

  /* lgam (x + tc) = tf + tt + x g(x)/h(x)
     - 0.230003726999612341262659542325721328468 <= x
     <= 0.2699962730003876587373404576742786715318
     peak relative error 2.1e-21 */
  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 = 1.000000000000000000000000000000000000000E0 */


  /* lgam (x+1) = -0.5 x + x u(x)/v(x)
     -0.100006103515625 <= x <= 0.231639862060546875
     peak relative error 1.3e-21 */
  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 =  1.000000000000000000000000000000000000000E0 */


  /* lgam (x+2) = .5 x + x s(x)/r(x)
     0 <= x <= 1
     peak relative error 7.2e-22 */
  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 =  1.000000000000000000000000000000000000000E0 */


/* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
   x >= 8
   Peak relative error 1.51e-21
   w0 = LS2PI - 0.5 */
  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;

static const long double zero = 0.0L;

static long double
sin_pi(long double x)
{
  long double 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) /* 0.25 */
    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  *= 0.5;
      y = 2.0*(y - floorl(y));		/* y = |x| mod 2.0 */
      n = (int) (y*4.0);
    }
  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 - 1.5));
      break;
    default:
      y = sinl (pi * (y - 2.0));
      break;
    }
  return -y;
}


long double
lgammal_r(long double x, int *signgamp)
{
  long double 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)
    {
      if (se & 0x8000)
	*signgamp = -1;
      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 -logl (-x);
	}
      else
	return -logl (x);
    }
  if (se & 0x8000)
    {
      t = sin_pi (x);
      if (t == zero)
	return one / fabsl (t);	/* -integer */
      nadj = 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) /* 2.0 */
    {
      /* x < 2.0 */
      if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */
	{
	  /* lgamma(x) = lgamma(x+1) - log(x) */
	  r = -logl (x);
	  if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */
	    {
	      y = x - one;
	      i = 0;
	    }
	  else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */
	    {
	      y = x - (tc - one);
	      i = 1;
	    }
	  else
	    {
	      /* x < 0.23 */
	      y = x;
	      i = 2;
	    }
	}
      else
	{
	  r = zero;
	  if (ix >= 0x3fffdda6) /* 1.73162841796875 */
	    {
	      /* [1.7316,2] */
	      y = x - 2.0;
	      i = 0;
	    }
	  else if (ix >= 0x3fff9da6)/* 1.23162841796875 */
	    {
	      /* [1.23,1.73] */
	      y = x - tc;
	      i = 1;
	    }
	  else
	    {
	      /* [0.9, 1.23] */
	      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) /* 8.0 */
    {
      /* x < 8.0 */
      i = (int) x;
      t = zero;
      y = x - (double) 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 + 6.0);	/* FALLTHRU */
	case 6:
	  z *= (y + 5.0);	/* FALLTHRU */
	case 5:
	  z *= (y + 4.0);	/* FALLTHRU */
	case 4:
	  z *= (y + 3.0);	/* FALLTHRU */
	case 3:
	  z *= (y + 2.0);	/* FALLTHRU */
	  r += logl (z);
	  break;
	}
    }
  else if (ix < 0x40418000) /* 2^66 */
    {
      /* 8.0 <= x < 2**66 */
      t = 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 * (logl (x) - one);
  if (se & 0x8000)
    r = nadj - r;
  return r;
}