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Diffstat (limited to 'src/pkg/math/lgamma.go')
| -rw-r--r-- | src/pkg/math/lgamma.go | 365 |
1 files changed, 0 insertions, 365 deletions
diff --git a/src/pkg/math/lgamma.go b/src/pkg/math/lgamma.go deleted file mode 100644 index 6a02c412d..000000000 --- a/src/pkg/math/lgamma.go +++ /dev/null @@ -1,365 +0,0 @@ -// 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 -} |