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+// Copyright 2010 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package math
+
+/*
+ Floating-point logarithm of the Gamma function.
+*/
+
+// The original C code and the long comment below are
+// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and
+// came with this notice. The go code is a simplified
+// version of the original C.
+//
+// ====================================================
+// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+//
+// Developed at SunPro, a Sun Microsystems, Inc. business.
+// Permission to use, copy, modify, and distribute this
+// software is freely granted, provided that this notice
+// is preserved.
+// ====================================================
+//
+// __ieee754_lgamma_r(x, signgamp)
+// Reentrant version of the logarithm of the Gamma function
+// with user provided pointer for the sign of Gamma(x).
+//
+// Method:
+// 1. Argument Reduction for 0 < x <= 8
+// Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
+// reduce x to a number in [1.5,2.5] by
+// lgamma(1+s) = log(s) + lgamma(s)
+// for example,
+// lgamma(7.3) = log(6.3) + lgamma(6.3)
+// = log(6.3*5.3) + lgamma(5.3)
+// = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
+// 2. Polynomial approximation of lgamma around its
+// minimum (ymin=1.461632144968362245) to maintain monotonicity.
+// On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
+// Let z = x-ymin;
+// lgamma(x) = -1.214862905358496078218 + z**2*poly(z)
+// poly(z) is a 14 degree polynomial.
+// 2. Rational approximation in the primary interval [2,3]
+// We use the following approximation:
+// s = x-2.0;
+// lgamma(x) = 0.5*s + s*P(s)/Q(s)
+// with accuracy
+// |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
+// Our algorithms are based on the following observation
+//
+// zeta(2)-1 2 zeta(3)-1 3
+// lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
+// 2 3
+//
+// where Euler = 0.5772156649... is the Euler constant, which
+// is very close to 0.5.
+//
+// 3. For x>=8, we have
+// lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
+// (better formula:
+// lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
+// Let z = 1/x, then we approximation
+// f(z) = lgamma(x) - (x-0.5)(log(x)-1)
+// by
+// 3 5 11
+// w = w0 + w1*z + w2*z + w3*z + ... + w6*z
+// where
+// |w - f(z)| < 2**-58.74
+//
+// 4. For negative x, since (G is gamma function)
+// -x*G(-x)*G(x) = pi/sin(pi*x),
+// we have
+// G(x) = pi/(sin(pi*x)*(-x)*G(-x))
+// since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
+// Hence, for x<0, signgam = sign(sin(pi*x)) and
+// lgamma(x) = log(|Gamma(x)|)
+// = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
+// Note: one should avoid computing pi*(-x) directly in the
+// computation of sin(pi*(-x)).
+//
+// 5. Special Cases
+// lgamma(2+s) ~ s*(1-Euler) for tiny s
+// lgamma(1)=lgamma(2)=0
+// lgamma(x) ~ -log(x) for tiny x
+// lgamma(0) = lgamma(inf) = inf
+// lgamma(-integer) = +-inf
+//
+//
+
+var _lgamA = [...]float64{
+ 7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8
+ 3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD
+ 6.73523010531292681824e-02, // 0x3FB13E001A5562A7
+ 2.05808084325167332806e-02, // 0x3F951322AC92547B
+ 7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
+ 2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B
+ 1.19270763183362067845e-03, // 0x3F538A94116F3F5D
+ 5.10069792153511336608e-04, // 0x3F40B6C689B99C00
+ 2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D
+ 1.08011567247583939954e-04, // 0x3F1C5088987DFB07
+ 2.52144565451257326939e-05, // 0x3EFA7074428CFA52
+ 4.48640949618915160150e-05, // 0x3F07858E90A45837
+}
+var _lgamR = [...]float64{
+ 1.0, // placeholder
+ 1.39200533467621045958e+00, // 0x3FF645A762C4AB74
+ 7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC
+ 1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27
+ 1.86459191715652901344e-02, // 0x3F9317EA742ED475
+ 7.77942496381893596434e-04, // 0x3F497DDACA41A95B
+ 7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140
+}
+var _lgamS = [...]float64{
+ -7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
+ 2.14982415960608852501e-01, // 0x3FCB848B36E20878
+ 3.25778796408930981787e-01, // 0x3FD4D98F4F139F59
+ 1.46350472652464452805e-01, // 0x3FC2BB9CBEE5F2F7
+ 2.66422703033638609560e-02, // 0x3F9B481C7E939961
+ 1.84028451407337715652e-03, // 0x3F5E26B67368F239
+ 3.19475326584100867617e-05, // 0x3F00BFECDD17E945
+}
+var _lgamT = [...]float64{
+ 4.83836122723810047042e-01, // 0x3FDEF72BC8EE38A2
+ -1.47587722994593911752e-01, // 0xBFC2E4278DC6C509
+ 6.46249402391333854778e-02, // 0x3FB08B4294D5419B
+ -3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713
+ 1.79706750811820387126e-02, // 0x3F9266E7970AF9EC
+ -1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A
+ 6.10053870246291332635e-03, // 0x3F78FCE0E370E344
+ -3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7
+ 2.25964780900612472250e-03, // 0x3F6282D32E15C915
+ -1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1
+ 8.81081882437654011382e-04, // 0x3F4CDF0CEF61A8E9
+ -5.38595305356740546715e-04, // 0xBF41A6109C73E0EC
+ 3.15632070903625950361e-04, // 0x3F34AF6D6C0EBBF7
+ -3.12754168375120860518e-04, // 0xBF347F24ECC38C38
+ 3.35529192635519073543e-04, // 0x3F35FD3EE8C2D3F4
+}
+var _lgamU = [...]float64{
+ -7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
+ 6.32827064025093366517e-01, // 0x3FE4401E8B005DFF
+ 1.45492250137234768737e+00, // 0x3FF7475CD119BD6F
+ 9.77717527963372745603e-01, // 0x3FEF497644EA8450
+ 2.28963728064692451092e-01, // 0x3FCD4EAEF6010924
+ 1.33810918536787660377e-02, // 0x3F8B678BBF2BAB09
+}
+var _lgamV = [...]float64{
+ 1.0,
+ 2.45597793713041134822e+00, // 0x4003A5D7C2BD619C
+ 2.12848976379893395361e+00, // 0x40010725A42B18F5
+ 7.69285150456672783825e-01, // 0x3FE89DFBE45050AF
+ 1.04222645593369134254e-01, // 0x3FBAAE55D6537C88
+ 3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61
+}
+var _lgamW = [...]float64{
+ 4.18938533204672725052e-01, // 0x3FDACFE390C97D69
+ 8.33333333333329678849e-02, // 0x3FB555555555553B
+ -2.77777777728775536470e-03, // 0xBF66C16C16B02E5C
+ 7.93650558643019558500e-04, // 0x3F4A019F98CF38B6
+ -5.95187557450339963135e-04, // 0xBF4380CB8C0FE741
+ 8.36339918996282139126e-04, // 0x3F4B67BA4CDAD5D1
+ -1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4
+}
+
+// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
+//
+// Special cases are:
+// Lgamma(+Inf) = +Inf
+// Lgamma(0) = +Inf
+// Lgamma(-integer) = +Inf
+// Lgamma(-Inf) = -Inf
+// Lgamma(NaN) = NaN
+func Lgamma(x float64) (lgamma float64, sign int) {
+ const (
+ Ymin = 1.461632144968362245
+ Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
+ Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
+ Two58 = 1 << 58 // 0x4390000000000000 ~2.8823e+17
+ Tiny = 1.0 / (1 << 70) // 0x3b90000000000000 ~8.47033e-22
+ Tc = 1.46163214496836224576e+00 // 0x3FF762D86356BE3F
+ Tf = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
+ // Tt = -(tail of Tf)
+ Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
+ )
+ // special cases
+ sign = 1
+ switch {
+ case IsNaN(x):
+ lgamma = x
+ return
+ case IsInf(x, 0):
+ lgamma = x
+ return
+ case x == 0:
+ lgamma = Inf(1)
+ return
+ }
+
+ neg := false
+ if x < 0 {
+ x = -x
+ neg = true
+ }
+
+ if x < Tiny { // if |x| < 2**-70, return -log(|x|)
+ if neg {
+ sign = -1
+ }
+ lgamma = -Log(x)
+ return
+ }
+ var nadj float64
+ if neg {
+ if x >= Two52 { // |x| >= 2**52, must be -integer
+ lgamma = Inf(1)
+ return
+ }
+ t := sinPi(x)
+ if t == 0 {
+ lgamma = Inf(1) // -integer
+ return
+ }
+ nadj = Log(Pi / Abs(t*x))
+ if t < 0 {
+ sign = -1
+ }
+ }
+
+ switch {
+ case x == 1 || x == 2: // purge off 1 and 2
+ lgamma = 0
+ return
+ case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)
+ var y float64
+ var i int
+ if x <= 0.9 {
+ lgamma = -Log(x)
+ switch {
+ case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <= 0.9
+ y = 1 - x
+ i = 0
+ case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316
+ y = x - (Tc - 1)
+ i = 1
+ default: // 0 < x < 0.2316
+ y = x
+ i = 2
+ }
+ } else {
+ lgamma = 0
+ switch {
+ case x >= (Ymin + 0.27): // 1.7316 <= x < 2
+ y = 2 - x
+ i = 0
+ case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316
+ y = x - Tc
+ i = 1
+ default: // 0.9 < x < 1.2316
+ y = x - 1
+ i = 2
+ }
+ }
+ switch i {
+ case 0:
+ z := y * y
+ p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))
+ p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))
+ p := y*p1 + p2
+ lgamma += (p - 0.5*y)
+ case 1:
+ z := y * y
+ w := z * y
+ p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp
+ p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))
+ p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))
+ p := z*p1 - (Tt - w*(p2+y*p3))
+ lgamma += (Tf + p)
+ case 2:
+ p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))
+ p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))
+ lgamma += (-0.5*y + p1/p2)
+ }
+ case x < 8: // 2 <= x < 8
+ i := int(x)
+ y := x - float64(i)
+ p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))
+ q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))
+ lgamma = 0.5*y + p/q
+ z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
+ switch i {
+ case 7:
+ z *= (y + 6)
+ fallthrough
+ case 6:
+ z *= (y + 5)
+ fallthrough
+ case 5:
+ z *= (y + 4)
+ fallthrough
+ case 4:
+ z *= (y + 3)
+ fallthrough
+ case 3:
+ z *= (y + 2)
+ lgamma += Log(z)
+ }
+ case x < Two58: // 8 <= x < 2**58
+ t := Log(x)
+ z := 1 / x
+ y := z * z
+ w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))
+ lgamma = (x-0.5)*(t-1) + w
+ default: // 2**58 <= x <= Inf
+ lgamma = x * (Log(x) - 1)
+ }
+ if neg {
+ lgamma = nadj - lgamma
+ }
+ return
+}
+
+// sinPi(x) is a helper function for negative x
+func sinPi(x float64) float64 {
+ const (
+ Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
+ Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
+ )
+ if x < 0.25 {
+ return -Sin(Pi * x)
+ }
+
+ // argument reduction
+ z := Floor(x)
+ var n int
+ if z != x { // inexact
+ x = Mod(x, 2)
+ n = int(x * 4)
+ } else {
+ if x >= Two53 { // x must be even
+ x = 0
+ n = 0
+ } else {
+ if x < Two52 {
+ z = x + Two52 // exact
+ }
+ n = int(1 & Float64bits(z))
+ x = float64(n)
+ n <<= 2
+ }
+ }
+ switch n {
+ case 0:
+ x = Sin(Pi * x)
+ case 1, 2:
+ x = Cos(Pi * (0.5 - x))
+ case 3, 4:
+ x = Sin(Pi * (1 - x))
+ case 5, 6:
+ x = -Cos(Pi * (x - 1.5))
+ default:
+ x = Sin(Pi * (x - 2))
+ }
+ return -x
+}