426 lines
12 KiB
C
426 lines
12 KiB
C
/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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*
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* Permission to use, copy, modify, and distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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* copyright notice and this permission notice appear in all copies.
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*
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* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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*/
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/* lgammal_r(x, signgamp)
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* Reentrant version of the logarithm of the Gamma function
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* with user provide pointer for the sign of Gamma(x).
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*
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* Method:
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* 1. Argument Reduction for 0 < x <= 8
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* Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
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* reduce x to a number in [1.5,2.5] by
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* lgamma(1+s) = log(s) + lgamma(s)
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* for example,
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* lgamma(7.3) = log(6.3) + lgamma(6.3)
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* = log(6.3*5.3) + lgamma(5.3)
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* = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
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* 2. Polynomial approximation of lgamma around its
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* minimun ymin=1.461632144968362245 to maintain monotonicity.
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* On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
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* Let z = x-ymin;
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* lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
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* 2. Rational approximation in the primary interval [2,3]
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* We use the following approximation:
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* s = x-2.0;
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* lgamma(x) = 0.5*s + s*P(s)/Q(s)
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* Our algorithms are based on the following observation
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*
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* zeta(2)-1 2 zeta(3)-1 3
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* lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
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* 2 3
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*
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* where Euler = 0.5771... is the Euler constant, which is very
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* close to 0.5.
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*
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* 3. For x>=8, we have
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* lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
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* (better formula:
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* lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
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* Let z = 1/x, then we approximation
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* f(z) = lgamma(x) - (x-0.5)(log(x)-1)
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* by
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* 3 5 11
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* w = w0 + w1*z + w2*z + w3*z + ... + w6*z
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*
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* 4. For negative x, since (G is gamma function)
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* -x*G(-x)*G(x) = pi/sin(pi*x),
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* we have
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* G(x) = pi/(sin(pi*x)*(-x)*G(-x))
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* since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
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* Hence, for x<0, signgam = sign(sin(pi*x)) and
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* lgamma(x) = log(|Gamma(x)|)
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* = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
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* Note: one should avoid compute pi*(-x) directly in the
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* computation of sin(pi*(-x)).
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*
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* 5. Special Cases
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* lgamma(2+s) ~ s*(1-Euler) for tiny s
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* lgamma(1)=lgamma(2)=0
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* lgamma(x) ~ -log(x) for tiny x
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* lgamma(0) = lgamma(inf) = inf
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* lgamma(-integer) = +-inf
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*
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*/
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#include <openlibm_math.h>
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#include "math_private.h"
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static const long double
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half = 0.5L,
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one = 1.0L,
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pi = 3.14159265358979323846264L,
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two63 = 9.223372036854775808e18L,
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/* lgam(1+x) = 0.5 x + x a(x)/b(x)
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-0.268402099609375 <= x <= 0
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peak relative error 6.6e-22 */
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a0 = -6.343246574721079391729402781192128239938E2L,
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a1 = 1.856560238672465796768677717168371401378E3L,
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a2 = 2.404733102163746263689288466865843408429E3L,
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a3 = 8.804188795790383497379532868917517596322E2L,
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a4 = 1.135361354097447729740103745999661157426E2L,
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a5 = 3.766956539107615557608581581190400021285E0L,
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b0 = 8.214973713960928795704317259806842490498E3L,
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b1 = 1.026343508841367384879065363925870888012E4L,
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b2 = 4.553337477045763320522762343132210919277E3L,
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b3 = 8.506975785032585797446253359230031874803E2L,
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b4 = 6.042447899703295436820744186992189445813E1L,
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/* b5 = 1.000000000000000000000000000000000000000E0 */
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tc = 1.4616321449683623412626595423257213284682E0L,
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tf = -1.2148629053584961146050602565082954242826E-1,/* double precision */
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/* tt = (tail of tf), i.e. tf + tt has extended precision. */
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tt = 3.3649914684731379602768989080467587736363E-18L,
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/* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
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-1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
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/* lgam (x + tc) = tf + tt + x g(x)/h(x)
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- 0.230003726999612341262659542325721328468 <= x
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<= 0.2699962730003876587373404576742786715318
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peak relative error 2.1e-21 */
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g0 = 3.645529916721223331888305293534095553827E-18L,
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g1 = 5.126654642791082497002594216163574795690E3L,
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g2 = 8.828603575854624811911631336122070070327E3L,
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g3 = 5.464186426932117031234820886525701595203E3L,
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g4 = 1.455427403530884193180776558102868592293E3L,
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g5 = 1.541735456969245924860307497029155838446E2L,
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g6 = 4.335498275274822298341872707453445815118E0L,
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h0 = 1.059584930106085509696730443974495979641E4L,
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h1 = 2.147921653490043010629481226937850618860E4L,
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h2 = 1.643014770044524804175197151958100656728E4L,
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h3 = 5.869021995186925517228323497501767586078E3L,
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h4 = 9.764244777714344488787381271643502742293E2L,
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h5 = 6.442485441570592541741092969581997002349E1L,
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/* h6 = 1.000000000000000000000000000000000000000E0 */
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/* lgam (x+1) = -0.5 x + x u(x)/v(x)
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-0.100006103515625 <= x <= 0.231639862060546875
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peak relative error 1.3e-21 */
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u0 = -8.886217500092090678492242071879342025627E1L,
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u1 = 6.840109978129177639438792958320783599310E2L,
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u2 = 2.042626104514127267855588786511809932433E3L,
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u3 = 1.911723903442667422201651063009856064275E3L,
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u4 = 7.447065275665887457628865263491667767695E2L,
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u5 = 1.132256494121790736268471016493103952637E2L,
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u6 = 4.484398885516614191003094714505960972894E0L,
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v0 = 1.150830924194461522996462401210374632929E3L,
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v1 = 3.399692260848747447377972081399737098610E3L,
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v2 = 3.786631705644460255229513563657226008015E3L,
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v3 = 1.966450123004478374557778781564114347876E3L,
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v4 = 4.741359068914069299837355438370682773122E2L,
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v5 = 4.508989649747184050907206782117647852364E1L,
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/* v6 = 1.000000000000000000000000000000000000000E0 */
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/* lgam (x+2) = .5 x + x s(x)/r(x)
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0 <= x <= 1
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peak relative error 7.2e-22 */
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s0 = 1.454726263410661942989109455292824853344E6L,
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s1 = -3.901428390086348447890408306153378922752E6L,
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s2 = -6.573568698209374121847873064292963089438E6L,
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s3 = -3.319055881485044417245964508099095984643E6L,
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s4 = -7.094891568758439227560184618114707107977E5L,
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s5 = -6.263426646464505837422314539808112478303E4L,
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s6 = -1.684926520999477529949915657519454051529E3L,
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r0 = -1.883978160734303518163008696712983134698E7L,
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r1 = -2.815206082812062064902202753264922306830E7L,
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r2 = -1.600245495251915899081846093343626358398E7L,
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r3 = -4.310526301881305003489257052083370058799E6L,
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r4 = -5.563807682263923279438235987186184968542E5L,
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r5 = -3.027734654434169996032905158145259713083E4L,
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r6 = -4.501995652861105629217250715790764371267E2L,
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/* r6 = 1.000000000000000000000000000000000000000E0 */
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/* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
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x >= 8
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Peak relative error 1.51e-21
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w0 = LS2PI - 0.5 */
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w0 = 4.189385332046727417803e-1L,
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w1 = 8.333333333333331447505E-2L,
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w2 = -2.777777777750349603440E-3L,
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w3 = 7.936507795855070755671E-4L,
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w4 = -5.952345851765688514613E-4L,
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w5 = 8.412723297322498080632E-4L,
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w6 = -1.880801938119376907179E-3L,
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w7 = 4.885026142432270781165E-3L;
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static const long double zero = 0.0L;
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static long double
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sin_pi(long double x)
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{
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long double y, z;
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int n, ix;
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u_int32_t se, i0, i1;
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GET_LDOUBLE_WORDS (se, i0, i1, x);
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ix = se & 0x7fff;
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ix = (ix << 16) | (i0 >> 16);
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if (ix < 0x3ffd8000) /* 0.25 */
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return sinl (pi * x);
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y = -x; /* x is assume negative */
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/*
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* argument reduction, make sure inexact flag not raised if input
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* is an integer
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*/
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z = floorl (y);
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if (z != y)
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{ /* inexact anyway */
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y *= 0.5;
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y = 2.0*(y - floorl(y)); /* y = |x| mod 2.0 */
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n = (int) (y*4.0);
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}
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else
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{
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if (ix >= 0x403f8000) /* 2^64 */
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{
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y = zero; n = 0; /* y must be even */
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}
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else
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{
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if (ix < 0x403e8000) /* 2^63 */
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z = y + two63; /* exact */
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GET_LDOUBLE_WORDS (se, i0, i1, z);
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n = i1 & 1;
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y = n;
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n <<= 2;
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}
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}
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switch (n)
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{
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case 0:
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y = sinl (pi * y);
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break;
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case 1:
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case 2:
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y = cosl (pi * (half - y));
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break;
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case 3:
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case 4:
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y = sinl (pi * (one - y));
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break;
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case 5:
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case 6:
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y = -cosl (pi * (y - 1.5));
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break;
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default:
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y = sinl (pi * (y - 2.0));
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break;
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}
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return -y;
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}
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long double
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lgammal_r(long double x, int *signgamp)
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{
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long double t, y, z, nadj, p, p1, p2, q, r, w;
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int i, ix;
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u_int32_t se, i0, i1;
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*signgamp = 1;
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GET_LDOUBLE_WORDS (se, i0, i1, x);
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ix = se & 0x7fff;
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if ((ix | i0 | i1) == 0)
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{
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if (se & 0x8000)
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*signgamp = -1;
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return one / fabsl (x);
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}
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ix = (ix << 16) | (i0 >> 16);
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/* purge off +-inf, NaN, +-0, and negative arguments */
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if (ix >= 0x7fff0000)
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return x * x;
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if (ix < 0x3fc08000) /* 2^-63 */
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{ /* |x|<2**-63, return -log(|x|) */
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if (se & 0x8000)
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{
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*signgamp = -1;
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return -logl (-x);
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}
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else
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return -logl (x);
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}
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if (se & 0x8000)
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{
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t = sin_pi (x);
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if (t == zero)
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return one / fabsl (t); /* -integer */
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nadj = logl (pi / fabsl (t * x));
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if (t < zero)
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*signgamp = -1;
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x = -x;
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}
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/* purge off 1 and 2 */
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if ((((ix - 0x3fff8000) | i0 | i1) == 0)
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|| (((ix - 0x40008000) | i0 | i1) == 0))
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r = 0;
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else if (ix < 0x40008000) /* 2.0 */
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{
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/* x < 2.0 */
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if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */
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{
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/* lgamma(x) = lgamma(x+1) - log(x) */
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r = -logl (x);
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if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */
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{
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y = x - one;
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i = 0;
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}
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else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */
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{
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y = x - (tc - one);
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i = 1;
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}
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else
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{
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/* x < 0.23 */
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y = x;
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i = 2;
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}
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}
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else
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{
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r = zero;
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if (ix >= 0x3fffdda6) /* 1.73162841796875 */
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{
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/* [1.7316,2] */
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y = x - 2.0;
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i = 0;
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}
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else if (ix >= 0x3fff9da6)/* 1.23162841796875 */
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{
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/* [1.23,1.73] */
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y = x - tc;
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i = 1;
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}
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else
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{
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/* [0.9, 1.23] */
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y = x - one;
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i = 2;
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}
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}
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switch (i)
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{
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case 0:
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p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
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p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
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r += half * y + y * p1/p2;
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break;
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case 1:
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p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
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p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
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p = tt + y * p1/p2;
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r += (tf + p);
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break;
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case 2:
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p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
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p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
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r += (-half * y + p1 / p2);
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}
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}
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else if (ix < 0x40028000) /* 8.0 */
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{
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/* x < 8.0 */
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i = (int) x;
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t = zero;
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y = x - (double) i;
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p = y *
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(s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
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q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
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r = half * y + p / q;
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z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
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switch (i)
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{
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case 7:
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z *= (y + 6.0); /* FALLTHRU */
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case 6:
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z *= (y + 5.0); /* FALLTHRU */
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case 5:
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z *= (y + 4.0); /* FALLTHRU */
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case 4:
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z *= (y + 3.0); /* FALLTHRU */
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case 3:
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z *= (y + 2.0); /* FALLTHRU */
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r += logl (z);
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break;
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}
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}
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else if (ix < 0x40418000) /* 2^66 */
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{
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/* 8.0 <= x < 2**66 */
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t = logl (x);
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z = one / x;
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y = z * z;
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w = w0 + z * (w1
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+ y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
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r = (x - half) * (t - one) + w;
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}
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else
|
|
/* 2**66 <= x <= inf */
|
|
r = x * (logl (x) - one);
|
|
if (se & 0x8000)
|
|
r = nadj - r;
|
|
return r;
|
|
}
|