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jeremybenn |
// Copyright 2010 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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/*
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Floating-point logarithm of the Gamma function.
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*/
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// The original C code and the long comment below are
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// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and
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// came with this notice. The go code is a simplified
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// version of the original C.
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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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// __ieee754_lgamma_r(x, signgamp)
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// Reentrant version of the logarithm of the Gamma function
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// with user provided 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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// minimum (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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// poly(z) is a 14 degree polynomial.
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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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// with accuracy
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// |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
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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 = 0.5772156649... is the Euler constant, which
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// is very 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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// where
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// |w - f(z)| < 2**-58.74
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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 computing 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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var _lgamA = [...]float64{
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7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8
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3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD
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6.73523010531292681824e-02, // 0x3FB13E001A5562A7
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2.05808084325167332806e-02, // 0x3F951322AC92547B
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7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
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2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B
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1.19270763183362067845e-03, // 0x3F538A94116F3F5D
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5.10069792153511336608e-04, // 0x3F40B6C689B99C00
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2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D
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1.08011567247583939954e-04, // 0x3F1C5088987DFB07
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2.52144565451257326939e-05, // 0x3EFA7074428CFA52
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4.48640949618915160150e-05, // 0x3F07858E90A45837
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}
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var _lgamR = [...]float64{
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1.0, // placeholder
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1.39200533467621045958e+00, // 0x3FF645A762C4AB74
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7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC
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1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27
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1.86459191715652901344e-02, // 0x3F9317EA742ED475
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7.77942496381893596434e-04, // 0x3F497DDACA41A95B
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7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140
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}
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var _lgamS = [...]float64{
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-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
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2.14982415960608852501e-01, // 0x3FCB848B36E20878
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3.25778796408930981787e-01, // 0x3FD4D98F4F139F59
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1.46350472652464452805e-01, // 0x3FC2BB9CBEE5F2F7
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2.66422703033638609560e-02, // 0x3F9B481C7E939961
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1.84028451407337715652e-03, // 0x3F5E26B67368F239
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3.19475326584100867617e-05, // 0x3F00BFECDD17E945
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}
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var _lgamT = [...]float64{
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4.83836122723810047042e-01, // 0x3FDEF72BC8EE38A2
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-1.47587722994593911752e-01, // 0xBFC2E4278DC6C509
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6.46249402391333854778e-02, // 0x3FB08B4294D5419B
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-3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713
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1.79706750811820387126e-02, // 0x3F9266E7970AF9EC
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-1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A
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6.10053870246291332635e-03, // 0x3F78FCE0E370E344
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-3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7
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2.25964780900612472250e-03, // 0x3F6282D32E15C915
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-1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1
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8.81081882437654011382e-04, // 0x3F4CDF0CEF61A8E9
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-5.38595305356740546715e-04, // 0xBF41A6109C73E0EC
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3.15632070903625950361e-04, // 0x3F34AF6D6C0EBBF7
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-3.12754168375120860518e-04, // 0xBF347F24ECC38C38
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3.35529192635519073543e-04, // 0x3F35FD3EE8C2D3F4
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}
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var _lgamU = [...]float64{
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-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
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6.32827064025093366517e-01, // 0x3FE4401E8B005DFF
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1.45492250137234768737e+00, // 0x3FF7475CD119BD6F
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9.77717527963372745603e-01, // 0x3FEF497644EA8450
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2.28963728064692451092e-01, // 0x3FCD4EAEF6010924
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1.33810918536787660377e-02, // 0x3F8B678BBF2BAB09
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}
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var _lgamV = [...]float64{
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1.0,
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2.45597793713041134822e+00, // 0x4003A5D7C2BD619C
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2.12848976379893395361e+00, // 0x40010725A42B18F5
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7.69285150456672783825e-01, // 0x3FE89DFBE45050AF
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1.04222645593369134254e-01, // 0x3FBAAE55D6537C88
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3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61
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}
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var _lgamW = [...]float64{
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4.18938533204672725052e-01, // 0x3FDACFE390C97D69
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8.33333333333329678849e-02, // 0x3FB555555555553B
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-2.77777777728775536470e-03, // 0xBF66C16C16B02E5C
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7.93650558643019558500e-04, // 0x3F4A019F98CF38B6
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-5.95187557450339963135e-04, // 0xBF4380CB8C0FE741
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8.36339918996282139126e-04, // 0x3F4B67BA4CDAD5D1
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-1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4
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}
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// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
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//
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// Special cases are:
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// Lgamma(+Inf) = +Inf
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// Lgamma(0) = +Inf
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// Lgamma(-integer) = +Inf
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// Lgamma(-Inf) = -Inf
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// Lgamma(NaN) = NaN
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func Lgamma(x float64) (lgamma float64, sign int) {
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const (
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Ymin = 1.461632144968362245
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Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
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Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
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Two58 = 1 << 58 // 0x4390000000000000 ~2.8823e+17
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Tiny = 1.0 / (1 << 70) // 0x3b90000000000000 ~8.47033e-22
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Tc = 1.46163214496836224576e+00 // 0x3FF762D86356BE3F
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Tf = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
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// Tt = -(tail of Tf)
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Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
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)
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// special cases
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sign = 1
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switch {
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case IsNaN(x):
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lgamma = x
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return
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case IsInf(x, 0):
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lgamma = x
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return
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case x == 0:
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lgamma = Inf(1)
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return
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}
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neg := false
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if x < 0 {
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x = -x
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neg = true
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}
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if x < Tiny { // if |x| < 2**-70, return -log(|x|)
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if neg {
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sign = -1
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}
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lgamma = -Log(x)
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return
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}
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var nadj float64
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if neg {
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if x >= Two52 { // |x| >= 2**52, must be -integer
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lgamma = Inf(1)
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return
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}
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t := sinPi(x)
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if t == 0 {
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lgamma = Inf(1) // -integer
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return
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}
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nadj = Log(Pi / Abs(t*x))
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if t < 0 {
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sign = -1
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}
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}
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switch {
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case x == 1 || x == 2: // purge off 1 and 2
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lgamma = 0
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return
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case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)
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var y float64
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var i int
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if x <= 0.9 {
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lgamma = -Log(x)
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switch {
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case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <= 0.9
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y = 1 - x
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i = 0
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case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316
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y = x - (Tc - 1)
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i = 1
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default: // 0 < x < 0.2316
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y = x
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i = 2
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}
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} else {
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lgamma = 0
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switch {
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case x >= (Ymin + 0.27): // 1.7316 <= x < 2
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y = 2 - x
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i = 0
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case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316
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y = x - Tc
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i = 1
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default: // 0.9 < x < 1.2316
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y = x - 1
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i = 2
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}
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}
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switch i {
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case 0:
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z := y * y
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p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))
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p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))
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p := y*p1 + p2
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lgamma += (p - 0.5*y)
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case 1:
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z := y * y
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w := z * y
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p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp
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p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))
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p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))
|
| 277 |
|
|
p := z*p1 - (Tt - w*(p2+y*p3))
|
| 278 |
|
|
lgamma += (Tf + p)
|
| 279 |
|
|
case 2:
|
| 280 |
|
|
p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))
|
| 281 |
|
|
p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))
|
| 282 |
|
|
lgamma += (-0.5*y + p1/p2)
|
| 283 |
|
|
}
|
| 284 |
|
|
case x < 8: // 2 <= x < 8
|
| 285 |
|
|
i := int(x)
|
| 286 |
|
|
y := x - float64(i)
|
| 287 |
|
|
p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))
|
| 288 |
|
|
q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))
|
| 289 |
|
|
lgamma = 0.5*y + p/q
|
| 290 |
|
|
z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
|
| 291 |
|
|
switch i {
|
| 292 |
|
|
case 7:
|
| 293 |
|
|
z *= (y + 6)
|
| 294 |
|
|
fallthrough
|
| 295 |
|
|
case 6:
|
| 296 |
|
|
z *= (y + 5)
|
| 297 |
|
|
fallthrough
|
| 298 |
|
|
case 5:
|
| 299 |
|
|
z *= (y + 4)
|
| 300 |
|
|
fallthrough
|
| 301 |
|
|
case 4:
|
| 302 |
|
|
z *= (y + 3)
|
| 303 |
|
|
fallthrough
|
| 304 |
|
|
case 3:
|
| 305 |
|
|
z *= (y + 2)
|
| 306 |
|
|
lgamma += Log(z)
|
| 307 |
|
|
}
|
| 308 |
|
|
case x < Two58: // 8 <= x < 2**58
|
| 309 |
|
|
t := Log(x)
|
| 310 |
|
|
z := 1 / x
|
| 311 |
|
|
y := z * z
|
| 312 |
|
|
w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))
|
| 313 |
|
|
lgamma = (x-0.5)*(t-1) + w
|
| 314 |
|
|
default: // 2**58 <= x <= Inf
|
| 315 |
|
|
lgamma = x * (Log(x) - 1)
|
| 316 |
|
|
}
|
| 317 |
|
|
if neg {
|
| 318 |
|
|
lgamma = nadj - lgamma
|
| 319 |
|
|
}
|
| 320 |
|
|
return
|
| 321 |
|
|
}
|
| 322 |
|
|
|
| 323 |
|
|
// sinPi(x) is a helper function for negative x
|
| 324 |
|
|
func sinPi(x float64) float64 {
|
| 325 |
|
|
const (
|
| 326 |
|
|
Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
|
| 327 |
|
|
Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
|
| 328 |
|
|
)
|
| 329 |
|
|
if x < 0.25 {
|
| 330 |
|
|
return -Sin(Pi * x)
|
| 331 |
|
|
}
|
| 332 |
|
|
|
| 333 |
|
|
// argument reduction
|
| 334 |
|
|
z := Floor(x)
|
| 335 |
|
|
var n int
|
| 336 |
|
|
if z != x { // inexact
|
| 337 |
|
|
x = Mod(x, 2)
|
| 338 |
|
|
n = int(x * 4)
|
| 339 |
|
|
} else {
|
| 340 |
|
|
if x >= Two53 { // x must be even
|
| 341 |
|
|
x = 0
|
| 342 |
|
|
n = 0
|
| 343 |
|
|
} else {
|
| 344 |
|
|
if x < Two52 {
|
| 345 |
|
|
z = x + Two52 // exact
|
| 346 |
|
|
}
|
| 347 |
|
|
n = int(1 & Float64bits(z))
|
| 348 |
|
|
x = float64(n)
|
| 349 |
|
|
n <<= 2
|
| 350 |
|
|
}
|
| 351 |
|
|
}
|
| 352 |
|
|
switch n {
|
| 353 |
|
|
case 0:
|
| 354 |
|
|
x = Sin(Pi * x)
|
| 355 |
|
|
case 1, 2:
|
| 356 |
|
|
x = Cos(Pi * (0.5 - x))
|
| 357 |
|
|
case 3, 4:
|
| 358 |
|
|
x = Sin(Pi * (1 - x))
|
| 359 |
|
|
case 5, 6:
|
| 360 |
|
|
x = -Cos(Pi * (x - 1.5))
|
| 361 |
|
|
default:
|
| 362 |
|
|
x = Sin(Pi * (x - 2))
|
| 363 |
|
|
}
|
| 364 |
|
|
return -x
|
| 365 |
|
|
}
|
