GCC Code Coverage Report
Directory: ./ Exec Total Coverage
File: lib/libm/src/b_tgamma.c Lines: 103 132 78.0 %
Date: 2017-11-13 Branches: 30 38 78.9 %

Line Branch Exec Source
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/*	$OpenBSD: b_tgamma.c,v 1.10 2016/09/12 19:47:02 guenther Exp $	*/
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/*-
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 * Copyright (c) 1992, 1993
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 *	The Regents of the University of California.  All rights reserved.
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 *
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 * Redistribution and use in source and binary forms, with or without
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 * modification, are permitted provided that the following conditions
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 * are met:
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 * 1. Redistributions of source code must retain the above copyright
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 *    notice, this list of conditions and the following disclaimer.
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 * 2. Redistributions in binary form must reproduce the above copyright
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 *    notice, this list of conditions and the following disclaimer in the
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 *    documentation and/or other materials provided with the distribution.
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 * 3. Neither the name of the University nor the names of its contributors
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 *    may be used to endorse or promote products derived from this software
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 *    without specific prior written permission.
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 *
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 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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 * SUCH DAMAGE.
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 */
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/*
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 * This code by P. McIlroy, Oct 1992;
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 *
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 * The financial support of UUNET Communications Services is greatfully
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 * acknowledged.
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 */
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#include <float.h>
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#include <math.h>
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#include "math_private.h"
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/* METHOD:
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 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
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 *	At negative integers, return NaN and raise invalid.
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 *
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 * x < 6.5:
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 *	Use argument reduction G(x+1) = xG(x) to reach the
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 *	range [1.066124,2.066124].  Use a rational
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 *	approximation centered at the minimum (x0+1) to
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 *	ensure monotonicity.
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 *
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 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
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 *	adjusted for equal-ripples:
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 *
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 *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
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 *
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 *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
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 *	avoid premature round-off.
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 *
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 * Special values:
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 *	-Inf:			return NaN and raise invalid;
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 *	negative integer:	return NaN and raise invalid;
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 *	other x ~< -177.79:	return +-0 and raise underflow;
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 *	+-0:			return +-Inf and raise divide-by-zero;
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 *	finite x ~> 171.63:	return +Inf and raise overflow;
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 *	+Inf:			return +Inf;
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 *	NaN: 			return NaN.
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 *
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 * Accuracy: tgamma(x) is accurate to within
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 *	x > 0:  error provably < 0.9ulp.
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 *	Maximum observed in 1,000,000 trials was .87ulp.
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 *	x < 0:
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 *	Maximum observed error < 4ulp in 1,000,000 trials.
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 */
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static double neg_gam(double);
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static double small_gam(double);
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static double smaller_gam(double);
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static struct Double large_gam(double);
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static struct Double ratfun_gam(double, double);
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/*
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 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
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 * [1.066.., 2.066..] accurate to 4.25e-19.
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 */
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#define LEFT -.3955078125	/* left boundary for rat. approx */
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#define x0 .461632144968362356785	/* xmin - 1 */
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#define a0_hi 0.88560319441088874992
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#define a0_lo -.00000000000000004996427036469019695
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#define P0	 6.21389571821820863029017800727e-01
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#define P1	 2.65757198651533466104979197553e-01
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#define P2	 5.53859446429917461063308081748e-03
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#define P3	 1.38456698304096573887145282811e-03
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#define P4	 2.40659950032711365819348969808e-03
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#define Q0	 1.45019531250000000000000000000e+00
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#define Q1	 1.06258521948016171343454061571e+00
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#define Q2	-2.07474561943859936441469926649e-01
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#define Q3	-1.46734131782005422506287573015e-01
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#define Q4	 3.07878176156175520361557573779e-02
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#define Q5	 5.12449347980666221336054633184e-03
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#define Q6	-1.76012741431666995019222898833e-03
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#define Q7	 9.35021023573788935372153030556e-05
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#define Q8	 6.13275507472443958924745652239e-06
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/*
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 * Constants for large x approximation (x in [6, Inf])
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 * (Accurate to 2.8*10^-19 absolute)
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 */
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#define lns2pi_hi 0.418945312500000
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#define lns2pi_lo -.000006779295327258219670263595
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#define Pa0	 8.33333333333333148296162562474e-02
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#define Pa1	-2.77777777774548123579378966497e-03
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#define Pa2	 7.93650778754435631476282786423e-04
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#define Pa3	-5.95235082566672847950717262222e-04
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#define Pa4	 8.41428560346653702135821806252e-04
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#define Pa5	-1.89773526463879200348872089421e-03
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#define Pa6	 5.69394463439411649408050664078e-03
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#define Pa7	-1.44705562421428915453880392761e-02
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static const double zero = 0., one = 1.0, tiny = 1e-300;
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double
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tgamma(double x)
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{
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	struct Double u;
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	if (x >= 6) {
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		if(x > 171.63)
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			return(x/zero);
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		u = large_gam(x);
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		return(__exp__D(u.a, u.b));
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	} else if (x >= 1.0 + LEFT + x0)
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		return (small_gam(x));
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	else if (x > 1.e-17)
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		return (smaller_gam(x));
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	else if (x > -1.e-17) {
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		if (x != 0.0)
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			u.a = one - tiny;	/* raise inexact */
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		return (one/x);
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	} else if (!isfinite(x)) {
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		return (x - x);			/* x = NaN, -Inf */
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	 } else
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		return (neg_gam(x));
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}
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DEF_STD(tgamma);
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LDBL_MAYBE_UNUSED_CLONE(tgamma);
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/*
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 * We simply call tgamma() rather than bloating the math library
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 * with a float-optimized version of it.  The reason is that tgammaf()
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 * is essentially useless, since the function is superexponential
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 * and floats have very limited range.  -- das@freebsd.org
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 */
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float
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tgammaf(float x)
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{
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	return tgamma(x);
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}
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/*
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 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
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 */
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static struct Double
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large_gam(double x)
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{
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	double z, p;
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	struct Double t, u, v;
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	z = one/(x*x);
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	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
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	p = p/x;
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	u = __log__D(x);
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	u.a -= one;
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	v.a = (x -= .5);
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	TRUNC(v.a);
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	v.b = x - v.a;
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	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
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	t.b = v.b*u.a + x*u.b;
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	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
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	t.b += lns2pi_lo; t.b += p;
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	u.a = lns2pi_hi + t.b; u.a += t.a;
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	u.b = t.a - u.a;
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	u.b += lns2pi_hi; u.b += t.b;
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	return (u);
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}
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/*
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 * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
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 * It also has correct monotonicity.
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 */
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static double
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small_gam(double x)
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{
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	double y, ym1, t;
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	struct Double yy, r;
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	y = x - one;
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	ym1 = y - one;
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	if (y <= 1.0 + (LEFT + x0)) {
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		yy = ratfun_gam(y - x0, 0);
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		return (yy.a + yy.b);
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	}
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	r.a = y;
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	TRUNC(r.a);
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	yy.a = r.a - one;
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	y = ym1;
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	yy.b = r.b = y - yy.a;
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	/* Argument reduction: G(x+1) = x*G(x) */
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	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
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		t = r.a*yy.a;
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		r.b = r.a*yy.b + y*r.b;
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		r.a = t;
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		TRUNC(r.a);
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		r.b += (t - r.a);
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	}
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	/* Return r*tgamma(y). */
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	yy = ratfun_gam(y - x0, 0);
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	y = r.b*(yy.a + yy.b) + r.a*yy.b;
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	y += yy.a*r.a;
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	return (y);
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}
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/*
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 * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
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 */
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static double
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smaller_gam(double x)
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{
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	double t, d;
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	struct Double r, xx;
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	if (x < x0 + LEFT) {
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		t = x;
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		TRUNC(t);
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		d = (t+x)*(x-t);
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		t *= t;
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		xx.a = (t + x);
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		TRUNC(xx.a);
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		xx.b = x - xx.a; xx.b += t; xx.b += d;
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		t = (one-x0); t += x;
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		d = (one-x0); d -= t; d += x;
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		x = xx.a + xx.b;
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	} else {
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		xx.a =  x;
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		TRUNC(xx.a);
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		xx.b = x - xx.a;
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		t = x - x0;
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		d = (-x0 -t); d += x;
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	}
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	r = ratfun_gam(t, d);
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	d = r.a/x;
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	TRUNC(d);
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	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
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	return (d + r.a/x);
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}
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/*
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 * returns (z+c)^2 * P(z)/Q(z) + a0
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 */
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static struct Double
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ratfun_gam(double z, double c)
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{
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	double p, q;
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	struct Double r, t;
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	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
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	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
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	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
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	p = p/q;
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	t.a = z;
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	TRUNC(t.a);			/* t ~= z + c */
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	t.b = (z - t.a) + c;
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	t.b *= (t.a + z);
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	q = (t.a *= t.a);		/* t = (z+c)^2 */
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	TRUNC(t.a);
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	t.b += (q - t.a);
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	r.a = p;
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	TRUNC(r.a);			/* r = P/Q */
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	r.b = p - r.a;
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	t.b = t.b*p + t.a*r.b + a0_lo;
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	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
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	r.a = t.a + a0_hi;
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	TRUNC(r.a);
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	r.b = ((a0_hi-r.a) + t.a) + t.b;
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	return (r);			/* r = a0 + t */
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}
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static double
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neg_gam(double x)
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{
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	int sgn = 1;
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	struct Double lg, lsine;
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	double y, z;
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	y = ceil(x);
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	if (y == x)		/* Negative integer. */
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		return ((x - x) / zero);
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	z = y - x;
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	if (z > 0.5)
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		z = one - z;
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	y = 0.5 * y;
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	if (y == ceil(y))
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		sgn = -1;
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	if (z < .25)
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		z = sin(M_PI*z);
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	else
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		z = cos(M_PI*(0.5-z));
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	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
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	if (x < -170) {
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		if (x < -190)
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			return ((double)sgn*tiny*tiny);
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		y = one - x;		/* exact: 128 < |x| < 255 */
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		lg = large_gam(y);
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		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
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		lg.a -= lsine.a;		/* exact (opposite signs) */
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		lg.b -= lsine.b;
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		y = -(lg.a + lg.b);
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		z = (y + lg.a) + lg.b;
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		y = __exp__D(y, z);
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		if (sgn < 0) y = -y;
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		return (y);
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	}
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	y = one-x;
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3
	if (one-y == x)
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		y = tgamma(y);
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	else		/* 1-x is inexact */
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		y = -x*tgamma(-x);
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	if (sgn < 0) y = -y;
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	return (M_PI / (y*z));
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}