f9978195dc
(1) tgamma(-Inf) returned +Inf and failed to raise any exception, but
should always have raised an exception, and should behave like
tgamma(negative integer).
(2) tgamma(negative integer) returned +Inf and raised divide-by-zero,
but should return NaN and raise "invalid" on any IEEEish system.
(3) About half of the 2**52 negative intgers between -2**53 and -2**52
were misclassified as non-integers by using floor(x + 0.5) to round
to nearest, so tgamma(x) was wrong (+-0 instead of +Inf and now NaN)
on these args. The floor() expression is hard to use since rounding
of (x + 0.5) may give x or x + 1, depending on |x| and the current
rounding mode. The fixed version uses ceil(x) to classify x before
operating on x and ends up being more efficient since ceil(x) is
needed anyway.
(4) On at least the problematic args in (3), tgamma() raised a spurious
inexact.
(5) tgamma(large positive) raised divide-by-zero but should raise overflow.
(6) tgamma(+Inf) raised divide-by-zero but should not raise any exception.
(7) Raise inexact for tiny |x| in a way that has some chance of not being
optimized away.
The fix for (5) and (6), and probably for (2), also prevents -O optimizing
away the exception.
PR: 112180 (2)
Standards: Annex F in C99 (IEC 60559 binding) requires (1), (2) and (6).
320 lines
8.7 KiB
C
320 lines
8.7 KiB
C
/*-
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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. All advertising materials mentioning features or use of this software
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* must display the following acknowledgement:
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* This product includes software developed by the University of
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* California, Berkeley and its contributors.
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* 4. 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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#ifndef lint
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static char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 6/4/93";
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#endif /* not lint */
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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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 <math.h>
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#include "mathimpl.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(x)
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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 (!finite(x))
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return (x - x); /* x is NaN or -Inf */
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else
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return (neg_gam(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(x)
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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(x)
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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(x)
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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, 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), 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, 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, 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(z, c)
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double z, 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, 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, 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, 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(x)
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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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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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}
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