third_party/musl/src/math/lgammal.c - cobalt - Git at Google

 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_lgammal.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 <steve@moshier.net>
  *
  * 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(x)
  * 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
  *
  */

 #define _GNU_SOURCE
 #include "libm.h"

 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
 long double __lgammal_r(long double x, int *sg)
 {
 	return __lgamma_r(x, sg);
 }
 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
 static const long double
 pi = 3.14159265358979323846264L,

 /* 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;

 /* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */
 static long double sin_pi(long double x)
 {
 	int n;

 	/* spurious inexact if odd int */
 	x *= 0.5;
 	x = 2.0*(x - floorl(x));  /* x mod 2.0 */

 	n = (int)(x*4.0);
 	n = (n+1)/2;
 	x -= n*0.5f;
 	x *= pi;

 	switch (n) {
 	default: /* case 4: */
 	case 0: return __sinl(x, 0.0, 0);
 	case 1: return __cosl(x, 0.0);
 	case 2: return __sinl(-x, 0.0, 0);
 	case 3: return -__cosl(x, 0.0);
 	}
 }

 long double __lgammal_r(long double x, int *sg) {
 	long double t, y, z, nadj, p, p1, p2, q, r, w;
 	union ldshape u = {x};
 	uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
 	int sign = u.i.se >> 15;
 	int i;

 	*sg = 1;

 	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
 	if (ix >= 0x7fff0000)
 		return x * x;
 	if (ix < 0x3fc08000) {  /* |x|<2**-63, return -log(|x|) */
 		if (sign) {
 			*sg = -1;
 			x = -x;
 		}
 		return -logl(x);
 	}
 	if (sign) {
 		x = -x;
 		t = sin_pi(x);
 		if (t == 0.0)
 			return 1.0 / (x-x); /* -integer */
 		if (t > 0.0)
 			*sg = -1;
 		else
 			t = -t;
 		nadj = logl(pi / (t * x));
 	}

 	/* purge off 1 and 2 (so the sign is ok with downward rounding) */
 	if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) {
 		r = 0;
 	} else if (ix < 0x40008000) {  /* 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 - 1.0;
 				i = 0;
 			} else if (ix >= 0x3ffced33) {  /* 2.31639862060546875e-1 */
 				y = x - (tc - 1.0);
 				i = 1;
 			} else { /* x < 0.23 */
 				y = x;
 				i = 2;
 			}
 		} else {
 			r = 0.0;
 			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 - 1.0;
 				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 += 0.5 * 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 += (-0.5 * y + p1 / p2);
 		}
 	} else if (ix < 0x40028000) {  /* 8.0 */
 		/* x < 8.0 */
 		i = (int)x;
 		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 = 0.5 * y + p / q;
 		z = 1.0;
 		/* 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 = 1.0 / x;
 		y = z * z;
 		w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
 		r = (x - 0.5) * (t - 1.0) + w;
 	} else /* 2**66 <= x <= inf */
 		r = x * (logl(x) - 1.0);
 	if (sign)
 		r = nadj - r;
 	return r;
 }
 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
 // TODO: broken implementation to make things compile
 long double __lgammal_r(long double x, int *sg)
 {
 	return __lgamma_r(x, sg);
 }
 #endif

 long double lgammal(long double x)
 {
 	return __lgammal_r(x, &__signgam);
 }

 weak_alias(__lgammal_r, lgammal_r);
	/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_lgammal.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 <steve@moshier.net>
	*
	* 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(x)
	* 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(6.35.34.33.32.3) + 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) = 0.5s + sP(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+0.5log(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 + w1z + w2z + w3z + ... + w6z
	*
	* 4. For negative x, since (G is gamma function)
	* -xG(-x)G(x) = pi/sin(pi*x),
	* we have
	* G(x) = pi/(sin(pix)(-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/(\|xsin(pix)\|)) - 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
	*
	*/

	#define _GNU_SOURCE
	#include "libm.h"

	#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
	long double __lgammal_r(long double x, int *sg)
	{
	return __lgamma_r(x, sg);
	}
	#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
	static const long double
	pi = 3.14159265358979323846264L,

	/* 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;

	/* sin(pix) assuming x > 2^-1000, if sin(pix)==0 the sign is arbitrary */
	static long double sin_pi(long double x)
	{
	int n;

	/* spurious inexact if odd int */
	x *= 0.5;
	x = 2.0(x - floorl(x)); / x mod 2.0 */

	n = (int)(x*4.0);
	n = (n+1)/2;
	x -= n*0.5f;
	x *= pi;

	switch (n) {
	default: /* case 4: */
	case 0: return __sinl(x, 0.0, 0);
	case 1: return __cosl(x, 0.0);
	case 2: return __sinl(-x, 0.0, 0);
	case 3: return -__cosl(x, 0.0);
	}
	}

	long double __lgammal_r(long double x, int *sg) {
	long double t, y, z, nadj, p, p1, p2, q, r, w;
	union ldshape u = {x};
	uint32_t ix = (u.i.se & 0x7fffU)<<16 \| u.i.m>>48;
	int sign = u.i.se >> 15;
	int i;

	*sg = 1;

	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
	if (ix >= 0x7fff0000)
	return x * x;
	if (ix < 0x3fc08000) { /* \|x\|<2*-63, return -log(\|x\|) /
	if (sign) {
	*sg = -1;
	x = -x;
	}
	return -logl(x);
	}
	if (sign) {
	x = -x;
	t = sin_pi(x);
	if (t == 0.0)
	return 1.0 / (x-x); /* -integer */
	if (t > 0.0)
	*sg = -1;
	else
	t = -t;
	nadj = logl(pi / (t * x));
	}

	/* purge off 1 and 2 (so the sign is ok with downward rounding) */
	if ((ix == 0x3fff8000 \|\| ix == 0x40008000) && u.i.m == 0) {
	r = 0;
	} else if (ix < 0x40008000) { /* 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 - 1.0;
	i = 0;
	} else if (ix >= 0x3ffced33) { /* 2.31639862060546875e-1 */
	y = x - (tc - 1.0);
	i = 1;
	} else { /* x < 0.23 */
	y = x;
	i = 2;
	}
	} else {
	r = 0.0;
	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 - 1.0;
	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 += 0.5 * 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 += (-0.5 * y + p1 / p2);
	}
	} else if (ix < 0x40028000) { /* 8.0 */
	/* x < 8.0 */
	i = (int)x;
	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 = 0.5 * y + p / q;
	z = 1.0;
	/* 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 = 1.0 / x;
	y = z * z;
	w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
	r = (x - 0.5) * (t - 1.0) + w;
	} else /* 2*66 <= x <= inf /
	r = x * (logl(x) - 1.0);
	if (sign)
	r = nadj - r;
	return r;
	}
	#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
	// TODO: broken implementation to make things compile
	long double __lgammal_r(long double x, int *sg)
	{
	return __lgamma_r(x, sg);
	}
	#endif

	long double lgammal(long double x)
	{
	return __lgammal_r(x, &__signgam);
	}

	weak_alias(__lgammal_r, lgammal_r);