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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 }