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/* |
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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(x) |
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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 <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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136 |
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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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145 |
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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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156 |
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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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164 |
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165 |
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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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185 |
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186 |
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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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199 |
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static const long double zero = 0.0L; |
200 |
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201 |
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static long double |
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sin_pi(long double x) |
203 |
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{ |
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long double y, z; |
205 |
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int n, ix; |
206 |
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u_int32_t se, i0, i1; |
207 |
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208 |
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GET_LDOUBLE_WORDS (se, i0, i1, x); |
209 |
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ix = se & 0x7fff; |
210 |
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ix = (ix << 16) | (i0 >> 16); |
211 |
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if (ix < 0x3ffd8000) /* 0.25 */ |
212 |
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return sinl (pi * x); |
213 |
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y = -x; /* x is assume negative */ |
214 |
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215 |
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/* |
216 |
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* argument reduction, make sure inexact flag not raised if input |
217 |
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* is an integer |
218 |
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*/ |
219 |
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z = floorl (y); |
220 |
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if (z != y) |
221 |
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{ /* inexact anyway */ |
222 |
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y *= 0.5; |
223 |
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y = 2.0*(y - floorl(y)); /* y = |x| mod 2.0 */ |
224 |
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n = (int) (y*4.0); |
225 |
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} |
226 |
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else |
227 |
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{ |
228 |
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if (ix >= 0x403f8000) /* 2^64 */ |
229 |
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{ |
230 |
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y = zero; n = 0; /* y must be even */ |
231 |
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} |
232 |
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else |
233 |
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{ |
234 |
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if (ix < 0x403e8000) /* 2^63 */ |
235 |
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z = y + two63; /* exact */ |
236 |
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GET_LDOUBLE_WORDS (se, i0, i1, z); |
237 |
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n = i1 & 1; |
238 |
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y = n; |
239 |
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n <<= 2; |
240 |
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} |
241 |
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} |
242 |
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243 |
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switch (n) |
244 |
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{ |
245 |
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case 0: |
246 |
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y = sinl (pi * y); |
247 |
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break; |
248 |
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case 1: |
249 |
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case 2: |
250 |
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y = cosl (pi * (half - y)); |
251 |
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break; |
252 |
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case 3: |
253 |
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case 4: |
254 |
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y = sinl (pi * (one - y)); |
255 |
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break; |
256 |
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case 5: |
257 |
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case 6: |
258 |
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y = -cosl (pi * (y - 1.5)); |
259 |
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break; |
260 |
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default: |
261 |
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y = sinl (pi * (y - 2.0)); |
262 |
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break; |
263 |
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} |
264 |
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return -y; |
265 |
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} |
266 |
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267 |
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268 |
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long double |
269 |
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lgammal(long double x) |
270 |
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{ |
271 |
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long double t, y, z, nadj, p, p1, p2, q, r, w; |
272 |
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int i, ix; |
273 |
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u_int32_t se, i0, i1; |
274 |
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275 |
|
60 |
signgam = 1; |
276 |
|
30 |
GET_LDOUBLE_WORDS (se, i0, i1, x); |
277 |
|
30 |
ix = se & 0x7fff; |
278 |
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|
279 |
✓✓ |
30 |
if ((ix | i0 | i1) == 0) |
280 |
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{ |
281 |
✓✓ |
10 |
if (se & 0x8000) |
282 |
|
5 |
signgam = -1; |
283 |
|
10 |
return one / fabsl (x); |
284 |
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} |
285 |
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|
286 |
|
20 |
ix = (ix << 16) | (i0 >> 16); |
287 |
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|
288 |
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/* purge off +-inf, NaN, +-0, and negative arguments */ |
289 |
✓✓ |
20 |
if (ix >= 0x7fff0000) |
290 |
|
10 |
return x * x; |
291 |
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|
292 |
✗✓ |
10 |
if (ix < 0x3fc08000) /* 2^-63 */ |
293 |
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{ /* |x|<2**-63, return -log(|x|) */ |
294 |
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if (se & 0x8000) |
295 |
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{ |
296 |
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signgam = -1; |
297 |
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return -logl (-x); |
298 |
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} |
299 |
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else |
300 |
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return -logl (x); |
301 |
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} |
302 |
✗✓ |
10 |
if (se & 0x8000) |
303 |
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{ |
304 |
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t = sin_pi (x); |
305 |
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if (t == zero) |
306 |
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return one / fabsl (t); /* -integer */ |
307 |
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nadj = logl (pi / fabsl (t * x)); |
308 |
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if (t < zero) |
309 |
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signgam = -1; |
310 |
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x = -x; |
311 |
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} |
312 |
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|
313 |
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/* purge off 1 and 2 */ |
314 |
✗✓ |
20 |
if ((((ix - 0x3fff8000) | i0 | i1) == 0) |
315 |
✓✗ |
20 |
|| (((ix - 0x40008000) | i0 | i1) == 0)) |
316 |
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r = 0; |
317 |
✓✓ |
10 |
else if (ix < 0x40008000) /* 2.0 */ |
318 |
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{ |
319 |
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/* x < 2.0 */ |
320 |
✗✓ |
5 |
if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */ |
321 |
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{ |
322 |
|
|
/* lgamma(x) = lgamma(x+1) - log(x) */ |
323 |
|
|
r = -logl (x); |
324 |
|
|
if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */ |
325 |
|
|
{ |
326 |
|
|
y = x - one; |
327 |
|
|
i = 0; |
328 |
|
|
} |
329 |
|
|
else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */ |
330 |
|
|
{ |
331 |
|
|
y = x - (tc - one); |
332 |
|
|
i = 1; |
333 |
|
|
} |
334 |
|
|
else |
335 |
|
|
{ |
336 |
|
|
/* x < 0.23 */ |
337 |
|
|
y = x; |
338 |
|
|
i = 2; |
339 |
|
|
} |
340 |
|
|
} |
341 |
|
|
else |
342 |
|
|
{ |
343 |
|
|
r = zero; |
344 |
✗✓ |
5 |
if (ix >= 0x3fffdda6) /* 1.73162841796875 */ |
345 |
|
|
{ |
346 |
|
|
/* [1.7316,2] */ |
347 |
|
|
y = x - 2.0; |
348 |
|
|
i = 0; |
349 |
|
|
} |
350 |
✗✓ |
5 |
else if (ix >= 0x3fff9da6)/* 1.23162841796875 */ |
351 |
|
|
{ |
352 |
|
|
/* [1.23,1.73] */ |
353 |
|
|
y = x - tc; |
354 |
|
|
i = 1; |
355 |
|
|
} |
356 |
|
|
else |
357 |
|
|
{ |
358 |
|
|
/* [0.9, 1.23] */ |
359 |
|
5 |
y = x - one; |
360 |
|
|
i = 2; |
361 |
|
|
} |
362 |
|
|
} |
363 |
✗✗✓✓
|
10 |
switch (i) |
364 |
|
|
{ |
365 |
|
|
case 0: |
366 |
|
|
p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5)))); |
367 |
|
|
p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y)))); |
368 |
|
|
r += half * y + y * p1/p2; |
369 |
|
|
break; |
370 |
|
|
case 1: |
371 |
|
|
p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6))))); |
372 |
|
|
p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y))))); |
373 |
|
|
p = tt + y * p1/p2; |
374 |
|
|
r += (tf + p); |
375 |
|
|
break; |
376 |
|
|
case 2: |
377 |
|
5 |
p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6)))))); |
378 |
|
5 |
p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y))))); |
379 |
|
5 |
r += (-half * y + p1 / p2); |
380 |
|
5 |
} |
381 |
|
|
} |
382 |
✓✗ |
5 |
else if (ix < 0x40028000) /* 8.0 */ |
383 |
|
|
{ |
384 |
|
|
/* x < 8.0 */ |
385 |
|
10 |
i = (int) x; |
386 |
|
|
t = zero; |
387 |
|
10 |
y = x - (double) i; |
388 |
|
10 |
p = y * |
389 |
|
10 |
(s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6)))))); |
390 |
|
10 |
q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y)))))); |
391 |
|
10 |
r = half * y + p / q; |
392 |
|
|
z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ |
393 |
✗✗✗✗
✓✓ |
10 |
switch (i) |
394 |
|
|
{ |
395 |
|
|
case 7: |
396 |
|
|
z *= (y + 6.0); /* FALLTHRU */ |
397 |
|
|
case 6: |
398 |
|
|
z *= (y + 5.0); /* FALLTHRU */ |
399 |
|
|
case 5: |
400 |
|
|
z *= (y + 4.0); /* FALLTHRU */ |
401 |
|
|
case 4: |
402 |
|
|
z *= (y + 3.0); /* FALLTHRU */ |
403 |
|
|
case 3: |
404 |
|
5 |
z *= (y + 2.0); /* FALLTHRU */ |
405 |
|
5 |
r += logl (z); |
406 |
|
5 |
break; |
407 |
|
|
} |
408 |
|
|
} |
409 |
|
|
else if (ix < 0x40418000) /* 2^66 */ |
410 |
|
|
{ |
411 |
|
|
/* 8.0 <= x < 2**66 */ |
412 |
|
|
t = logl (x); |
413 |
|
|
z = one / x; |
414 |
|
|
y = z * z; |
415 |
|
|
w = w0 + z * (w1 |
416 |
|
|
+ y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7)))))); |
417 |
|
|
r = (x - half) * (t - one) + w; |
418 |
|
|
} |
419 |
|
|
else |
420 |
|
|
/* 2**66 <= x <= inf */ |
421 |
|
|
r = x * (logl (x) - one); |
422 |
✗✓ |
10 |
if (se & 0x8000) |
423 |
|
|
r = nadj - r; |
424 |
|
10 |
return r; |
425 |
|
30 |
} |
426 |
|
|
DEF_STD(lgammal); |