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

 /* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
 /*
  * ====================================================
  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  *
  * Developed at SunSoft, a Sun Microsystems, Inc. business.
  * Permission to use, copy, modify, and distribute this
  * software is freely granted, provided that this notice
  * is preserved.
  * ====================================================
  *
  */
 /* lgamma_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)
  *      where
  *              poly(z) is a 14 degree polynomial.
  *   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)
  *      with accuracy
  *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
  *      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
  *      where
  *              |w - f(z)| < 2**-58.74
  *
  *   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(neg.integer) = inf and raise divide-by-zero
  *              lgamma(inf) = inf
  *              lgamma(-inf) = inf (bug for bug compatible with C99!?)
  *
  */

 #include "libm.h"
 #include "libc.h"

 static const double
 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
 a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
 a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
 a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
 a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
 a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
 a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
 a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
 a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
 a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
 a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
 tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
 tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
 /* tt = -(tail of tf) */
 tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
 t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
 t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
 t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
 t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
 t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
 t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
 t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
 t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
 t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
 t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
 t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
 t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
 t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
 u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
 u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
 u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
 u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
 u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
 v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
 v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
 v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
 v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
 v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
 s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
 s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
 s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
 s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
 s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
 s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
 r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
 r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
 r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
 r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
 r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
 r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
 w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
 w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
 w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
 w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
 w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */

 /* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
 static double sin_pi(double x)
 {
 	int n;

 	/* spurious inexact if odd int */
 	x = 2.0*(x*0.5 - floor(x*0.5));  /* 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 __sin(x, 0.0, 0);
 	case 1: return __cos(x, 0.0);
 	case 2: return __sin(-x, 0.0, 0);
 	case 3: return -__cos(x, 0.0);
 	}
 }

 double __lgamma_r(double x, int *signgamp)
 {
 	union {double f; uint64_t i;} u = {x};
 	double_t t,y,z,nadj,p,p1,p2,p3,q,r,w;
 	uint32_t ix;
 	int sign,i;

 	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
 	*signgamp = 1;
 	sign = u.i>>63;
 	ix = u.i>>32 & 0x7fffffff;
 	if (ix >= 0x7ff00000)
 		return x*x;
 	if (ix < (0x3ff-70)<<20) {  /* |x|<2**-70, return -log(|x|) */
 		if(sign) {
 			x = -x;
 			*signgamp = -1;
 		}
 		return -log(x);
 	}
 	if (sign) {
 		x = -x;
 		t = sin_pi(x);
 		if (t == 0.0) /* -integer */
 			return 1.0/(x-x);
 		if (t > 0.0)
 			*signgamp = -1;
 		else
 			t = -t;
 		nadj = log(pi/(t*x));
 	}

 	/* purge off 1 and 2 */
 	if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 0)
 		r = 0;
 	/* for x < 2.0 */
 	else if (ix < 0x40000000) {
 		if (ix <= 0x3feccccc) {   /* lgamma(x) = lgamma(x+1)-log(x) */
 			r = -log(x);
 			if (ix >= 0x3FE76944) {
 				y = 1.0 - x;
 				i = 0;
 			} else if (ix >= 0x3FCDA661) {
 				y = x - (tc-1.0);
 				i = 1;
 			} else {
 				y = x;
 				i = 2;
 			}
 		} else {
 			r = 0.0;
 			if (ix >= 0x3FFBB4C3) {  /* [1.7316,2] */
 				y = 2.0 - x;
 				i = 0;
 			} else if(ix >= 0x3FF3B4C4) {  /* [1.23,1.73] */
 				y = x - tc;
 				i = 1;
 			} else {
 				y = x - 1.0;
 				i = 2;
 			}
 		}
 		switch (i) {
 		case 0:
 			z = y*y;
 			p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
 			p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
 			p = y*p1+p2;
 			r += (p-0.5*y);
 			break;
 		case 1:
 			z = y*y;
 			w = z*y;
 			p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
 			p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
 			p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
 			p = z*p1-(tt-w*(p2+y*p3));
 			r += tf + p;
 			break;
 		case 2:
 			p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
 			p2 = 1.0+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
 			r += -0.5*y + p1/p2;
 		}
 	} else if (ix < 0x40200000) {  /* 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 = 1.0+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
 		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 += log(z);
 			break;
 		}
 	} else if (ix < 0x43900000) {  /* 8.0 <= x < 2**58 */
 		t = log(x);
 		z = 1.0/x;
 		y = z*z;
 		w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
 		r = (x-0.5)*(t-1.0)+w;
 	} else                         /* 2**58 <= x <= inf */
 		r =  x*(log(x)-1.0);
 	if (sign)
 		r = nadj - r;
 	return r;
 }

 weak_alias(__lgamma_r, lgamma_r);
	/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
	/*
	* ====================================================
	* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
	*
	* Developed at SunSoft, a Sun Microsystems, Inc. business.
	* Permission to use, copy, modify, and distribute this
	* software is freely granted, provided that this notice
	* is preserved.
	* ====================================================
	*
	*/
	/* lgamma_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(6.35.34.33.32.3) + 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)
	* where
	* poly(z) is a 14 degree polynomial.
	* 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)
	* with accuracy
	* \|P/Q - (lgamma(x)-0.5s)\| < 2**-61.71
	* 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
	* where
	* \|w - f(z)\| < 2**-58.74
	*
	* 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(neg.integer) = inf and raise divide-by-zero
	* lgamma(inf) = inf
	* lgamma(-inf) = inf (bug for bug compatible with C99!?)
	*
	*/

	#include "libm.h"
	#include "libc.h"

	static const double
	pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
	a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
	a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
	a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
	a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
	a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
	a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
	a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
	a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
	a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
	a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
	a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
	a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
	tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
	tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
	/* tt = -(tail of tf) */
	tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
	t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
	t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
	t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
	t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
	t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
	t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
	t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
	t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
	t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
	t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
	t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
	t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
	t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
	t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
	t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
	u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
	u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
	u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
	u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
	u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
	u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
	v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
	v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
	v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
	v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
	v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
	s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
	s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
	s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
	s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
	s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
	s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
	s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
	r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
	r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
	r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
	r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
	r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
	r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
	w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
	w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
	w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
	w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
	w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
	w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
	w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */

	/* sin(pix) assuming x > 2^-100, if sin(pix)==0 the sign is arbitrary */
	static double sin_pi(double x)
	{
	int n;

	/* spurious inexact if odd int */
	x = 2.0(x0.5 - floor(x0.5)); / 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 __sin(x, 0.0, 0);
	case 1: return __cos(x, 0.0);
	case 2: return __sin(-x, 0.0, 0);
	case 3: return -__cos(x, 0.0);
	}
	}

	double __lgamma_r(double x, int *signgamp)
	{
	union {double f; uint64_t i;} u = {x};
	double_t t,y,z,nadj,p,p1,p2,p3,q,r,w;
	uint32_t ix;
	int sign,i;

	/* purge off +-inf, NaN, +-0, tiny and negative arguments */
	*signgamp = 1;
	sign = u.i>>63;
	ix = u.i>>32 & 0x7fffffff;
	if (ix >= 0x7ff00000)
	return x*x;
	if (ix < (0x3ff-70)<<20) { /* \|x\|<2*-70, return -log(\|x\|) /
	if(sign) {
	x = -x;
	*signgamp = -1;
	}
	return -log(x);
	}
	if (sign) {
	x = -x;
	t = sin_pi(x);
	if (t == 0.0) /* -integer */
	return 1.0/(x-x);
	if (t > 0.0)
	*signgamp = -1;
	else
	t = -t;
	nadj = log(pi/(t*x));
	}

	/* purge off 1 and 2 */
	if ((ix == 0x3ff00000 \|\| ix == 0x40000000) && (uint32_t)u.i == 0)
	r = 0;
	/* for x < 2.0 */
	else if (ix < 0x40000000) {
	if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
	r = -log(x);
	if (ix >= 0x3FE76944) {
	y = 1.0 - x;
	i = 0;
	} else if (ix >= 0x3FCDA661) {
	y = x - (tc-1.0);
	i = 1;
	} else {
	y = x;
	i = 2;
	}
	} else {
	r = 0.0;
	if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */
	y = 2.0 - x;
	i = 0;
	} else if(ix >= 0x3FF3B4C4) { /* [1.23,1.73] */
	y = x - tc;
	i = 1;
	} else {
	y = x - 1.0;
	i = 2;
	}
	}
	switch (i) {
	case 0:
	z = y*y;
	p1 = a0+z(a2+z(a4+z(a6+z(a8+z*a10))));
	p2 = z(a1+z(a3+z(a5+z(a7+z(a9+za11)))));
	p = y*p1+p2;
	r += (p-0.5*y);
	break;
	case 1:
	z = y*y;
	w = z*y;
	p1 = t0+w(t3+w(t6+w(t9 +wt12))); /* parallel comp */
	p2 = t1+w(t4+w(t7+w(t10+wt13)));
	p3 = t2+w(t5+w(t8+w(t11+wt14)));
	p = zp1-(tt-w(p2+y*p3));
	r += tf + p;
	break;
	case 2:
	p1 = y(u0+y(u1+y(u2+y(u3+y(u4+yu5)))));
	p2 = 1.0+y(v1+y(v2+y(v3+y(v4+y*v5))));
	r += -0.5*y + p1/p2;
	}
	} else if (ix < 0x40200000) { /* 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 = 1.0+y(r1+y(r2+y(r3+y(r4+y(r5+yr6)))));
	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 += log(z);
	break;
	}
	} else if (ix < 0x43900000) { /* 8.0 <= x < 2*58 /
	t = log(x);
	z = 1.0/x;
	y = z*z;
	w = w0+z(w1+y(w2+y(w3+y(w4+y(w5+yw6)))));
	r = (x-0.5)*(t-1.0)+w;
	} else /* 2*58 <= x <= inf /
	r = x*(log(x)-1.0);
	if (sign)
	r = nadj - r;
	return r;
	}

	weak_alias(__lgamma_r, lgamma_r);