| // 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 |
| // |
| // |
| |
| // 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 |
| A0 = 7.72156649015328655494e-02 // 0x3FB3C467E37DB0C8 |
| A1 = 3.22467033424113591611e-01 // 0x3FD4A34CC4A60FAD |
| A2 = 6.73523010531292681824e-02 // 0x3FB13E001A5562A7 |
| A3 = 2.05808084325167332806e-02 // 0x3F951322AC92547B |
| A4 = 7.38555086081402883957e-03 // 0x3F7E404FB68FEFE8 |
| A5 = 2.89051383673415629091e-03 // 0x3F67ADD8CCB7926B |
| A6 = 1.19270763183362067845e-03 // 0x3F538A94116F3F5D |
| A7 = 5.10069792153511336608e-04 // 0x3F40B6C689B99C00 |
| A8 = 2.20862790713908385557e-04 // 0x3F2CF2ECED10E54D |
| A9 = 1.08011567247583939954e-04 // 0x3F1C5088987DFB07 |
| A10 = 2.52144565451257326939e-05 // 0x3EFA7074428CFA52 |
| A11 = 4.48640949618915160150e-05 // 0x3F07858E90A45837 |
| Tc = 1.46163214496836224576e+00 // 0x3FF762D86356BE3F |
| Tf = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42 |
| // Tt = -(tail of Tf) |
| Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F |
| T0 = 4.83836122723810047042e-01 // 0x3FDEF72BC8EE38A2 |
| T1 = -1.47587722994593911752e-01 // 0xBFC2E4278DC6C509 |
| T2 = 6.46249402391333854778e-02 // 0x3FB08B4294D5419B |
| T3 = -3.27885410759859649565e-02 // 0xBFA0C9A8DF35B713 |
| T4 = 1.79706750811820387126e-02 // 0x3F9266E7970AF9EC |
| T5 = -1.03142241298341437450e-02 // 0xBF851F9FBA91EC6A |
| T6 = 6.10053870246291332635e-03 // 0x3F78FCE0E370E344 |
| T7 = -3.68452016781138256760e-03 // 0xBF6E2EFFB3E914D7 |
| T8 = 2.25964780900612472250e-03 // 0x3F6282D32E15C915 |
| T9 = -1.40346469989232843813e-03 // 0xBF56FE8EBF2D1AF1 |
| T10 = 8.81081882437654011382e-04 // 0x3F4CDF0CEF61A8E9 |
| T11 = -5.38595305356740546715e-04 // 0xBF41A6109C73E0EC |
| T12 = 3.15632070903625950361e-04 // 0x3F34AF6D6C0EBBF7 |
| T13 = -3.12754168375120860518e-04 // 0xBF347F24ECC38C38 |
| T14 = 3.35529192635519073543e-04 // 0x3F35FD3EE8C2D3F4 |
| U0 = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8 |
| U1 = 6.32827064025093366517e-01 // 0x3FE4401E8B005DFF |
| U2 = 1.45492250137234768737e+00 // 0x3FF7475CD119BD6F |
| U3 = 9.77717527963372745603e-01 // 0x3FEF497644EA8450 |
| U4 = 2.28963728064692451092e-01 // 0x3FCD4EAEF6010924 |
| U5 = 1.33810918536787660377e-02 // 0x3F8B678BBF2BAB09 |
| V1 = 2.45597793713041134822e+00 // 0x4003A5D7C2BD619C |
| V2 = 2.12848976379893395361e+00 // 0x40010725A42B18F5 |
| V3 = 7.69285150456672783825e-01 // 0x3FE89DFBE45050AF |
| V4 = 1.04222645593369134254e-01 // 0x3FBAAE55D6537C88 |
| V5 = 3.21709242282423911810e-03 // 0x3F6A5ABB57D0CF61 |
| S0 = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8 |
| S1 = 2.14982415960608852501e-01 // 0x3FCB848B36E20878 |
| S2 = 3.25778796408930981787e-01 // 0x3FD4D98F4F139F59 |
| S3 = 1.46350472652464452805e-01 // 0x3FC2BB9CBEE5F2F7 |
| S4 = 2.66422703033638609560e-02 // 0x3F9B481C7E939961 |
| S5 = 1.84028451407337715652e-03 // 0x3F5E26B67368F239 |
| S6 = 3.19475326584100867617e-05 // 0x3F00BFECDD17E945 |
| R1 = 1.39200533467621045958e+00 // 0x3FF645A762C4AB74 |
| R2 = 7.21935547567138069525e-01 // 0x3FE71A1893D3DCDC |
| R3 = 1.71933865632803078993e-01 // 0x3FC601EDCCFBDF27 |
| R4 = 1.86459191715652901344e-02 // 0x3F9317EA742ED475 |
| R5 = 7.77942496381893596434e-04 // 0x3F497DDACA41A95B |
| R6 = 7.32668430744625636189e-06 // 0x3EDEBAF7A5B38140 |
| W0 = 4.18938533204672725052e-01 // 0x3FDACFE390C97D69 |
| W1 = 8.33333333333329678849e-02 // 0x3FB555555555553B |
| W2 = -2.77777777728775536470e-03 // 0xBF66C16C16B02E5C |
| W3 = 7.93650558643019558500e-04 // 0x3F4A019F98CF38B6 |
| W4 = -5.95187557450339963135e-04 // 0xBF4380CB8C0FE741 |
| W5 = 8.36339918996282139126e-04 // 0x3F4B67BA4CDAD5D1 |
| W6 = -1.63092934096575273989e-03 // 0xBF5AB89D0B9E43E4 |
| ) |
| // TODO(rsc): Remove manual inlining of IsNaN, IsInf |
| // when compiler does it for us |
| // special cases |
| sign = 1 |
| switch { |
| case x != x: // IsNaN(x): |
| lgamma = x |
| return |
| case x < -MaxFloat64 || x > MaxFloat64: // 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 / Fabs(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 := A0 + z*(A2+z*(A4+z*(A6+z*(A8+z*A10)))) |
| p2 := z * (A1 + z*(A3+z*(A5+z*(A7+z*(A9+z*A11))))) |
| p := y*p1 + p2 |
| lgamma += (p - 0.5*y) |
| case 1: |
| z := y * y |
| w := z * y |
| p1 := T0 + w*(T3+w*(T6+w*(T9+w*T12))) // parallel comp |
| p2 := T1 + w*(T4+w*(T7+w*(T10+w*T13))) |
| p3 := T2 + w*(T5+w*(T8+w*(T11+w*T14))) |
| p := z*p1 - (Tt - w*(p2+y*p3)) |
| lgamma += (Tf + p) |
| case 2: |
| p1 := y * (U0 + y*(U1+y*(U2+y*(U3+y*(U4+y*U5))))) |
| p2 := 1 + y*(V1+y*(V2+y*(V3+y*(V4+y*V5)))) |
| lgamma += (-0.5*y + p1/p2) |
| } |
| case x < 8: // 2 <= x < 8 |
| i := int(x) |
| y := x - float64(i) |
| p := y * (S0 + y*(S1+y*(S2+y*(S3+y*(S4+y*(S5+y*S6)))))) |
| q := 1 + y*(R1+y*(R2+y*(R3+y*(R4+y*(R5+y*R6))))) |
| lgamma = 0.5*y + p/q |
| z := float64(1) // 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 := W0 + z*(W1+y*(W2+y*(W3+y*(W4+y*(W5+y*W6))))) |
| 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 = Fmod(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 |
| } |