[388] | 1 | /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_lgammal.c */
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| 2 | /*
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| 3 | * ====================================================
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 5 | *
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| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business.
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| 7 | * Permission to use, copy, modify, and distribute this
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| 8 | * software is freely granted, provided that this notice
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| 9 | * is preserved.
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| 10 | * ====================================================
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| 11 | */
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| 12 | /*
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| 13 | * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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| 14 | *
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| 15 | * Permission to use, copy, modify, and distribute this software for any
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| 16 | * purpose with or without fee is hereby granted, provided that the above
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| 17 | * copyright notice and this permission notice appear in all copies.
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| 18 | *
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| 19 | * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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| 20 | * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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| 21 | * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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| 22 | * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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| 23 | * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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| 24 | * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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| 25 | * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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| 26 | */
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| 27 | /* lgammal(x)
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| 28 | * Reentrant version of the logarithm of the Gamma function
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| 29 | * with user provide pointer for the sign of Gamma(x).
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| 30 | *
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| 31 | * Method:
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| 32 | * 1. Argument Reduction for 0 < x <= 8
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| 33 | * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
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| 34 | * reduce x to a number in [1.5,2.5] by
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| 35 | * lgamma(1+s) = log(s) + lgamma(s)
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| 36 | * for example,
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| 37 | * lgamma(7.3) = log(6.3) + lgamma(6.3)
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| 38 | * = log(6.3*5.3) + lgamma(5.3)
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| 39 | * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
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| 40 | * 2. Polynomial approximation of lgamma around its
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| 41 | * minimun ymin=1.461632144968362245 to maintain monotonicity.
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| 42 | * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
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| 43 | * Let z = x-ymin;
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| 44 | * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
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| 45 | * 2. Rational approximation in the primary interval [2,3]
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| 46 | * We use the following approximation:
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| 47 | * s = x-2.0;
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| 48 | * lgamma(x) = 0.5*s + s*P(s)/Q(s)
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| 49 | * Our algorithms are based on the following observation
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| 50 | *
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| 51 | * zeta(2)-1 2 zeta(3)-1 3
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| 52 | * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
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| 53 | * 2 3
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| 54 | *
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| 55 | * where Euler = 0.5771... is the Euler constant, which is very
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| 56 | * close to 0.5.
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| 57 | *
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| 58 | * 3. For x>=8, we have
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| 59 | * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
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| 60 | * (better formula:
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| 61 | * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
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| 62 | * Let z = 1/x, then we approximation
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| 63 | * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
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| 64 | * by
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| 65 | * 3 5 11
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| 66 | * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
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| 67 | *
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| 68 | * 4. For negative x, since (G is gamma function)
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| 69 | * -x*G(-x)*G(x) = pi/sin(pi*x),
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| 70 | * we have
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| 71 | * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
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| 72 | * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
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| 73 | * Hence, for x<0, signgam = sign(sin(pi*x)) and
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| 74 | * lgamma(x) = log(|Gamma(x)|)
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| 75 | * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
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| 76 | * Note: one should avoid compute pi*(-x) directly in the
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| 77 | * computation of sin(pi*(-x)).
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| 78 | *
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| 79 | * 5. Special Cases
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| 80 | * lgamma(2+s) ~ s*(1-Euler) for tiny s
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| 81 | * lgamma(1)=lgamma(2)=0
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| 82 | * lgamma(x) ~ -log(x) for tiny x
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| 83 | * lgamma(0) = lgamma(inf) = inf
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| 84 | * lgamma(-integer) = +-inf
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| 85 | *
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| 86 | */
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| 87 |
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| 88 | #define _GNU_SOURCE
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| 89 | #include "libm.h"
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| 90 | #include "libc.h"
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| 91 |
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| 92 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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| 93 | double __lgamma_r(double x, int *sg);
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| 94 |
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| 95 | long double __lgammal_r(long double x, int *sg)
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| 96 | {
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| 97 | return __lgamma_r(x, sg);
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| 98 | }
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| 99 | #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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| 100 | static const long double
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| 101 | pi = 3.14159265358979323846264L,
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| 102 |
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| 103 | /* lgam(1+x) = 0.5 x + x a(x)/b(x)
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| 104 | -0.268402099609375 <= x <= 0
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| 105 | peak relative error 6.6e-22 */
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| 106 | a0 = -6.343246574721079391729402781192128239938E2L,
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| 107 | a1 = 1.856560238672465796768677717168371401378E3L,
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| 108 | a2 = 2.404733102163746263689288466865843408429E3L,
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| 109 | a3 = 8.804188795790383497379532868917517596322E2L,
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| 110 | a4 = 1.135361354097447729740103745999661157426E2L,
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| 111 | a5 = 3.766956539107615557608581581190400021285E0L,
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| 112 |
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| 113 | b0 = 8.214973713960928795704317259806842490498E3L,
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| 114 | b1 = 1.026343508841367384879065363925870888012E4L,
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| 115 | b2 = 4.553337477045763320522762343132210919277E3L,
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| 116 | b3 = 8.506975785032585797446253359230031874803E2L,
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| 117 | b4 = 6.042447899703295436820744186992189445813E1L,
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| 118 | /* b5 = 1.000000000000000000000000000000000000000E0 */
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| 119 |
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| 120 |
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| 121 | tc = 1.4616321449683623412626595423257213284682E0L,
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| 122 | tf = -1.2148629053584961146050602565082954242826E-1, /* double precision */
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| 123 | /* tt = (tail of tf), i.e. tf + tt has extended precision. */
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| 124 | tt = 3.3649914684731379602768989080467587736363E-18L,
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| 125 | /* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
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| 126 | -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
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| 127 |
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| 128 | /* lgam (x + tc) = tf + tt + x g(x)/h(x)
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| 129 | -0.230003726999612341262659542325721328468 <= x
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| 130 | <= 0.2699962730003876587373404576742786715318
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| 131 | peak relative error 2.1e-21 */
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| 132 | g0 = 3.645529916721223331888305293534095553827E-18L,
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| 133 | g1 = 5.126654642791082497002594216163574795690E3L,
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| 134 | g2 = 8.828603575854624811911631336122070070327E3L,
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| 135 | g3 = 5.464186426932117031234820886525701595203E3L,
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| 136 | g4 = 1.455427403530884193180776558102868592293E3L,
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| 137 | g5 = 1.541735456969245924860307497029155838446E2L,
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| 138 | g6 = 4.335498275274822298341872707453445815118E0L,
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| 139 |
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| 140 | h0 = 1.059584930106085509696730443974495979641E4L,
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| 141 | h1 = 2.147921653490043010629481226937850618860E4L,
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| 142 | h2 = 1.643014770044524804175197151958100656728E4L,
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| 143 | h3 = 5.869021995186925517228323497501767586078E3L,
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| 144 | h4 = 9.764244777714344488787381271643502742293E2L,
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| 145 | h5 = 6.442485441570592541741092969581997002349E1L,
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| 146 | /* h6 = 1.000000000000000000000000000000000000000E0 */
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| 147 |
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| 148 |
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| 149 | /* lgam (x+1) = -0.5 x + x u(x)/v(x)
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| 150 | -0.100006103515625 <= x <= 0.231639862060546875
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| 151 | peak relative error 1.3e-21 */
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| 152 | u0 = -8.886217500092090678492242071879342025627E1L,
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| 153 | u1 = 6.840109978129177639438792958320783599310E2L,
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| 154 | u2 = 2.042626104514127267855588786511809932433E3L,
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| 155 | u3 = 1.911723903442667422201651063009856064275E3L,
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| 156 | u4 = 7.447065275665887457628865263491667767695E2L,
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| 157 | u5 = 1.132256494121790736268471016493103952637E2L,
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| 158 | u6 = 4.484398885516614191003094714505960972894E0L,
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| 159 |
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| 160 | v0 = 1.150830924194461522996462401210374632929E3L,
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| 161 | v1 = 3.399692260848747447377972081399737098610E3L,
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| 162 | v2 = 3.786631705644460255229513563657226008015E3L,
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| 163 | v3 = 1.966450123004478374557778781564114347876E3L,
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| 164 | v4 = 4.741359068914069299837355438370682773122E2L,
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| 165 | v5 = 4.508989649747184050907206782117647852364E1L,
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| 166 | /* v6 = 1.000000000000000000000000000000000000000E0 */
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| 167 |
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| 168 |
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| 169 | /* lgam (x+2) = .5 x + x s(x)/r(x)
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| 170 | 0 <= x <= 1
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| 171 | peak relative error 7.2e-22 */
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| 172 | s0 = 1.454726263410661942989109455292824853344E6L,
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| 173 | s1 = -3.901428390086348447890408306153378922752E6L,
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| 174 | s2 = -6.573568698209374121847873064292963089438E6L,
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| 175 | s3 = -3.319055881485044417245964508099095984643E6L,
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| 176 | s4 = -7.094891568758439227560184618114707107977E5L,
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| 177 | s5 = -6.263426646464505837422314539808112478303E4L,
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| 178 | s6 = -1.684926520999477529949915657519454051529E3L,
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| 179 |
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| 180 | r0 = -1.883978160734303518163008696712983134698E7L,
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| 181 | r1 = -2.815206082812062064902202753264922306830E7L,
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| 182 | r2 = -1.600245495251915899081846093343626358398E7L,
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| 183 | r3 = -4.310526301881305003489257052083370058799E6L,
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| 184 | r4 = -5.563807682263923279438235987186184968542E5L,
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| 185 | r5 = -3.027734654434169996032905158145259713083E4L,
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| 186 | r6 = -4.501995652861105629217250715790764371267E2L,
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| 187 | /* r6 = 1.000000000000000000000000000000000000000E0 */
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| 188 |
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| 189 |
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| 190 | /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
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| 191 | x >= 8
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| 192 | Peak relative error 1.51e-21
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| 193 | w0 = LS2PI - 0.5 */
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| 194 | w0 = 4.189385332046727417803e-1L,
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| 195 | w1 = 8.333333333333331447505E-2L,
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| 196 | w2 = -2.777777777750349603440E-3L,
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| 197 | w3 = 7.936507795855070755671E-4L,
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| 198 | w4 = -5.952345851765688514613E-4L,
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| 199 | w5 = 8.412723297322498080632E-4L,
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| 200 | w6 = -1.880801938119376907179E-3L,
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| 201 | w7 = 4.885026142432270781165E-3L;
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| 202 |
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| 203 | /* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */
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| 204 | static long double sin_pi(long double x)
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| 205 | {
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| 206 | int n;
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| 207 |
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| 208 | /* spurious inexact if odd int */
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| 209 | x *= 0.5;
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| 210 | x = 2.0*(x - floorl(x)); /* x mod 2.0 */
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| 211 |
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| 212 | n = (int)(x*4.0);
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| 213 | n = (n+1)/2;
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| 214 | x -= n*0.5f;
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| 215 | x *= pi;
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| 216 |
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| 217 | switch (n) {
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| 218 | default: /* case 4: */
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| 219 | case 0: return __sinl(x, 0.0, 0);
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| 220 | case 1: return __cosl(x, 0.0);
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| 221 | case 2: return __sinl(-x, 0.0, 0);
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| 222 | case 3: return -__cosl(x, 0.0);
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| 223 | }
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| 224 | }
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| 225 |
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| 226 | long double __lgammal_r(long double x, int *sg) {
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| 227 | long double t, y, z, nadj, p, p1, p2, q, r, w;
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| 228 | union ldshape u = {x};
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| 229 | uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
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| 230 | int sign = u.i.se >> 15;
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| 231 | int i;
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| 232 |
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| 233 | *sg = 1;
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| 234 |
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| 235 | /* purge off +-inf, NaN, +-0, tiny and negative arguments */
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| 236 | if (ix >= 0x7fff0000)
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| 237 | return x * x;
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| 238 | if (ix < 0x3fc08000) { /* |x|<2**-63, return -log(|x|) */
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| 239 | if (sign) {
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| 240 | *sg = -1;
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| 241 | x = -x;
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| 242 | }
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| 243 | return -logl(x);
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| 244 | }
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| 245 | if (sign) {
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| 246 | x = -x;
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| 247 | t = sin_pi(x);
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| 248 | if (t == 0.0)
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| 249 | return 1.0 / (x-x); /* -integer */
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| 250 | if (t > 0.0)
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| 251 | *sg = -1;
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| 252 | else
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| 253 | t = -t;
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| 254 | nadj = logl(pi / (t * x));
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| 255 | }
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| 256 |
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| 257 | /* purge off 1 and 2 (so the sign is ok with downward rounding) */
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| 258 | if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) {
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| 259 | r = 0;
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| 260 | } else if (ix < 0x40008000) { /* x < 2.0 */
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| 261 | if (ix <= 0x3ffee666) { /* 8.99993896484375e-1 */
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| 262 | /* lgamma(x) = lgamma(x+1) - log(x) */
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| 263 | r = -logl(x);
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| 264 | if (ix >= 0x3ffebb4a) { /* 7.31597900390625e-1 */
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| 265 | y = x - 1.0;
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| 266 | i = 0;
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| 267 | } else if (ix >= 0x3ffced33) { /* 2.31639862060546875e-1 */
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| 268 | y = x - (tc - 1.0);
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| 269 | i = 1;
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| 270 | } else { /* x < 0.23 */
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| 271 | y = x;
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| 272 | i = 2;
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| 273 | }
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| 274 | } else {
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| 275 | r = 0.0;
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| 276 | if (ix >= 0x3fffdda6) { /* 1.73162841796875 */
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| 277 | /* [1.7316,2] */
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| 278 | y = x - 2.0;
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| 279 | i = 0;
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| 280 | } else if (ix >= 0x3fff9da6) { /* 1.23162841796875 */
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| 281 | /* [1.23,1.73] */
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| 282 | y = x - tc;
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| 283 | i = 1;
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| 284 | } else {
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| 285 | /* [0.9, 1.23] */
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| 286 | y = x - 1.0;
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| 287 | i = 2;
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| 288 | }
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| 289 | }
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| 290 | switch (i) {
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| 291 | case 0:
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| 292 | p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
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| 293 | p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
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| 294 | r += 0.5 * y + y * p1/p2;
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| 295 | break;
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| 296 | case 1:
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| 297 | p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
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| 298 | p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
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| 299 | p = tt + y * p1/p2;
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| 300 | r += (tf + p);
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| 301 | break;
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| 302 | case 2:
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| 303 | p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
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| 304 | p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
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| 305 | r += (-0.5 * y + p1 / p2);
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| 306 | }
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| 307 | } else if (ix < 0x40028000) { /* 8.0 */
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| 308 | /* x < 8.0 */
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| 309 | i = (int)x;
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| 310 | y = x - (double)i;
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| 311 | p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
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| 312 | q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
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| 313 | r = 0.5 * y + p / q;
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| 314 | z = 1.0;
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| 315 | /* lgamma(1+s) = log(s) + lgamma(s) */
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| 316 | switch (i) {
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| 317 | case 7:
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| 318 | z *= (y + 6.0); /* FALLTHRU */
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| 319 | case 6:
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| 320 | z *= (y + 5.0); /* FALLTHRU */
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| 321 | case 5:
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| 322 | z *= (y + 4.0); /* FALLTHRU */
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| 323 | case 4:
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| 324 | z *= (y + 3.0); /* FALLTHRU */
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| 325 | case 3:
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| 326 | z *= (y + 2.0); /* FALLTHRU */
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| 327 | r += logl(z);
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| 328 | break;
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| 329 | }
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| 330 | } else if (ix < 0x40418000) { /* 2^66 */
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| 331 | /* 8.0 <= x < 2**66 */
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| 332 | t = logl(x);
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| 333 | z = 1.0 / x;
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| 334 | y = z * z;
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| 335 | w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
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| 336 | r = (x - 0.5) * (t - 1.0) + w;
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| 337 | } else /* 2**66 <= x <= inf */
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| 338 | r = x * (logl(x) - 1.0);
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| 339 | if (sign)
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| 340 | r = nadj - r;
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| 341 | return r;
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| 342 | }
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| 343 | #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
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| 344 | // TODO: broken implementation to make things compile
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| 345 | double __lgamma_r(double x, int *sg);
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| 346 |
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| 347 | long double __lgammal_r(long double x, int *sg)
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| 348 | {
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| 349 | return __lgamma_r(x, sg);
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| 350 | }
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| 351 | #endif
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| 352 |
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| 353 | extern int __signgam;
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| 354 |
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| 355 | long double lgammal(long double x)
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| 356 | {
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| 357 | return __lgammal_r(x, &__signgam);
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| 358 | }
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| 359 |
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| 360 | weak_alias(__lgammal_r, lgammal_r);
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