03a88e3de9
. Disconnect imprecise.c from the build. This file can be deleted.
. Add b_tgammal.c to the build for ld80 and ld128 targets. The ld128
is a 'git mv' of imprecise.c to ld128/b_tgammal.c.
* lib/msun/ld80/b_expl.c:
. New file. Implement __exp__D for ld80 targets. This is based on
bsdsrc/b_exp.c.
* lib/msun/ld80/b_logl.c:
. New file. Implement __log__D for ld80 targets. This is based on
bsdsrc/b_log.c.
* lib/msun/ld80/b_tgammal.c b/lib/msun/ld80/b_tgammal.c
. New file. Implement tgammal(x) for ld80 targets.
Submitted by: Steve Kargl
Differential Revision: https://reviews.freebsd.org/D33444
Reviewed by: pfg
420 lines
11 KiB
C
420 lines
11 KiB
C
/*-
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* SPDX-License-Identifier: BSD-3-Clause
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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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* The original code, FreeBSD's old svn r93211, contain the following
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* attribution:
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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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* bsdrc/b_tgamma.c converted to long double by Steven G. Kargl.
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*/
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/*
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* See bsdsrc/t_tgamma.c for implementation details.
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*/
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#include <float.h>
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#if LDBL_MAX_EXP != 0x4000
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#error "Unsupported long double format"
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#endif
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#ifdef __i386__
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#include <ieeefp.h>
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#endif
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#include "fpmath.h"
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#include "math.h"
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#include "math_private.h"
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/* Used in b_log.c and below. */
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struct Double {
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long double a;
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long double b;
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};
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#include "b_logl.c"
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#include "b_expl.c"
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static const double zero = 0.;
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static const volatile double tiny = 1e-300;
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/*
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* x >= 6
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*
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* Use the asymptotic approximation (Stirling's formula) adjusted for
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* equal-ripples:
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*
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* log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(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 avoid
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* premature round-off.
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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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/*
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* The following is a decomposition of 0.5 * (log(2*pi) - 1) into the
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* first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The
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* variables are clearly misnamed.
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*/
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static const union IEEEl2bits
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ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L),
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ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
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#define ln2pi_hi (ln2pi_hiu.e)
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#define ln2pi_lo (ln2pi_lou.e)
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static const union IEEEl2bits
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Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L),
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Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L),
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Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L),
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Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L),
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Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L),
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Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L),
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Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L),
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Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L),
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Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L),
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Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L);
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#define Pa0 (Pa0u.e)
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#define Pa1 (Pa1u.e)
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#define Pa2 (Pa2u.e)
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#define Pa3 (Pa3u.e)
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#define Pa4 (Pa4u.e)
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#define Pa5 (Pa5u.e)
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#define Pa6 (Pa6u.e)
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#define Pa7 (Pa7u.e)
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#define Pa8 (Pa8u.e)
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#define Pa9 (Pa9u.e)
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static struct Double
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large_gam(long double x)
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{
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long double p, z, thi, tlo, xhi, xlo;
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long double logx;
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struct Double u;
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z = 1 / (x * x);
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p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
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z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9))))))));
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p = p / x;
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u = __log__D(x);
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u.a -= 1;
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/* Split (x - 0.5) in high and low parts. */
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x -= 0.5L;
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xhi = (float)x;
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xlo = x - xhi;
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/* Compute t = (x-.5)*(log(x)-1) in extra precision. */
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thi = xhi * u.a;
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tlo = xlo * u.a + x * u.b;
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/* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
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tlo += ln2pi_lo;
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tlo += p;
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u.a = ln2pi_hi + tlo;
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u.a += thi;
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u.b = thi - u.a;
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u.b += ln2pi_hi;
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u.b += tlo;
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return (u);
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}
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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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* Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
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*/
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static const union IEEEl2bits
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a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L),
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a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L);
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#define a0_hi (a0_hiu.e)
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#define a0_lo (a0_lou.e)
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static const union IEEEl2bits
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P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L),
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P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L),
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P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L),
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P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L),
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P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L),
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P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L),
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P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L),
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P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L),
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P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L),
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Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L),
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Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L),
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Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L),
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Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L),
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Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L),
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Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L),
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Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L),
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Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L);
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#define P0 (P0u.e)
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#define P1 (P1u.e)
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#define P2 (P2u.e)
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#define P3 (P3u.e)
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#define P4 (P4u.e)
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#define P5 (P5u.e)
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#define P6 (P6u.e)
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#define P7 (P7u.e)
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#define P8 (P8u.e)
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#define Q1 (Q1u.e)
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#define Q2 (Q2u.e)
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#define Q3 (Q3u.e)
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#define Q4 (Q4u.e)
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#define Q5 (Q5u.e)
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#define Q6 (Q6u.e)
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#define Q7 (Q7u.e)
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#define Q8 (Q8u.e)
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static struct Double
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ratfun_gam(long double z, long double c)
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{
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long double p, q, thi, tlo;
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struct Double r;
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q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
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z * (Q6 + z * (Q7 + z * Q8)))))));
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p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 +
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z * (P6 + z * (P7 + z * P8)))))));
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p = p / q;
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/* Split z into high and low parts. */
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thi = (float)z;
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tlo = (z - thi) + c;
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tlo *= (thi + z);
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/* Split (z+c)^2 into high and low parts. */
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thi *= thi;
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q = thi;
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thi = (float)thi;
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tlo += (q - thi);
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/* Split p/q into high and low parts. */
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r.a = (float)p;
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r.b = p - r.a;
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tlo = tlo * p + thi * r.b + a0_lo;
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thi *= r.a; /* t = (z+c)^2*(P/Q) */
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r.a = (float)(thi + a0_hi);
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r.b = ((a0_hi - r.a) + thi) + tlo;
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return (r); /* r = a0 + t */
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}
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/*
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* x < 6
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*
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* Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
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* 2.066124]. Use a rational approximation centered at the minimum
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* (x0+1) to ensure monotonicity.
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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 const union IEEEl2bits
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xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
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#define x0 (xm1u.e)
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static const double
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left = -0.3955078125; /* left boundary for rat. approx */
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static long double
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small_gam(long double x)
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{
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long double t, y, ym1;
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struct Double yy, r;
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y = x - 1;
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if (y <= 1 + (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 = (float)y;
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yy.a = r.a - 1;
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y = y - 1 ;
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r.b = yy.b = y - yy.a;
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/* Argument reduction: G(x+1) = x*G(x) */
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for (ym1 = y - 1; 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 = (float)t;
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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 1 ulp.
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*/
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static long double
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smaller_gam(long double x)
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{
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long double d, rhi, rlo, t, xhi, xlo;
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struct Double r;
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if (x < x0 + left) {
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t = (float)x;
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d = (t + x) * (x - t);
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t *= t;
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xhi = (float)(t + x);
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xlo = x - xhi;
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xlo += t;
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xlo += d;
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t = 1 - x0;
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t += x;
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d = 1 - x0;
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d -= t;
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d += x;
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x = xhi + xlo;
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} else {
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xhi = (float)x;
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xlo = x - xhi;
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t = x - x0;
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d = - x0 - t;
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d += x;
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}
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r = ratfun_gam(t, d);
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d = (float)(r.a / x);
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r.a -= d * xhi;
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r.a -= d * xlo;
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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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* x < 0
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*
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* 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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static const union IEEEl2bits
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piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
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#define pi (piu.e)
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static long double
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neg_gam(long 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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long double y, z;
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y = ceill(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 = 1 - z;
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y = y / 2;
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if (y == ceill(y))
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sgn = -1;
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if (z < 0.25)
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z = sinpil(z);
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else
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z = cospil(0.5 - z);
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/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
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if (x < -1753) {
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if (x < -1760)
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return (sgn * tiny * tiny);
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y = expl(lgammal(x) / 2);
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y *= y;
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return (sgn < 0 ? -y : y);
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}
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y = 1 - x;
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if (1 - y == x)
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y = tgammal(y);
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else /* 1-x is inexact */
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y = - x * tgammal(-x);
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if (sgn < 0) y = -y;
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return (pi / (y * z));
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}
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/*
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* xmax comes from lgamma(xmax) - emax * log(2) = 0.
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* static const float xmax = 35.040095f
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* static const double xmax = 171.624376956302725;
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* ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
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* ld128: 1.75554834290446291700388921607020320e+03L,
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*
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* iota is a sloppy threshold to isolate x = 0.
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*/
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static const double xmax = 1755.54834290446291689;
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static const double iota = 0x1p-116;
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long double
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tgammal(long double x)
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{
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struct Double u;
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ENTERI();
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if (x >= 6) {
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if (x > xmax)
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RETURNI(x / zero);
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u = large_gam(x);
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RETURNI(__exp__D(u.a, u.b));
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}
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if (x >= 1 + left + x0)
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RETURNI(small_gam(x));
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if (x > iota)
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RETURNI(smaller_gam(x));
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if (x > -iota) {
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if (x != 0)
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u.a = 1 - tiny; /* raise inexact */
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RETURNI(1 / x);
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}
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if (!isfinite(x))
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RETURNI(x - x); /* x is NaN or -Inf */
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RETURNI(neg_gam(x));
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}
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