// 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
}
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