codekingpro/portable-devtools

1// Copyright 2010 The Go Authors. All rights reserved.2// Use of this source code is governed by a BSD-style3// license that can be found in the LICENSE file.4 5package math6 7/*8	Floating-point logarithm of the Gamma function.9*/10 11// The original C code and the long comment below are12// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and13// came with this notice. The go code is a simplified14// version of the original C.15//16// ====================================================17// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.18//19// Developed at SunPro, a Sun Microsystems, Inc. business.20// Permission to use, copy, modify, and distribute this21// software is freely granted, provided that this notice22// is preserved.23// ====================================================24//25// __ieee754_lgamma_r(x, signgamp)26// Reentrant version of the logarithm of the Gamma function27// with user provided pointer for the sign of Gamma(x).28//29// Method:30//   1. Argument Reduction for 0 < x <= 831//      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may32//      reduce x to a number in [1.5,2.5] by33//              lgamma(1+s) = log(s) + lgamma(s)34//      for example,35//              lgamma(7.3) = log(6.3) + lgamma(6.3)36//                          = log(6.3*5.3) + lgamma(5.3)37//                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)38//   2. Polynomial approximation of lgamma around its39//      minimum (ymin=1.461632144968362245) to maintain monotonicity.40//      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use41//              Let z = x-ymin;42//              lgamma(x) = -1.214862905358496078218 + z**2*poly(z)43//              poly(z) is a 14 degree polynomial.44//   2. Rational approximation in the primary interval [2,3]45//      We use the following approximation:46//              s = x-2.0;47//              lgamma(x) = 0.5*s + s*P(s)/Q(s)48//      with accuracy49//              |P/Q - (lgamma(x)-0.5s)| < 2**-61.7150//      Our algorithms are based on the following observation51//52//                             zeta(2)-1    2    zeta(3)-1    353// lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...54//                                 2                 355//56//      where Euler = 0.5772156649... is the Euler constant, which57//      is very close to 0.5.58//59//   3. For x>=8, we have60//      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....61//      (better formula:62//         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)63//      Let z = 1/x, then we approximation64//              f(z) = lgamma(x) - (x-0.5)(log(x)-1)65//      by66//                                  3       5             1167//              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z68//      where69//              |w - f(z)| < 2**-58.7470//71//   4. For negative x, since (G is gamma function)72//              -x*G(-x)*G(x) = pi/sin(pi*x),73//      we have74//              G(x) = pi/(sin(pi*x)*(-x)*G(-x))75//      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<076//      Hence, for x<0, signgam = sign(sin(pi*x)) and77//              lgamma(x) = log(|Gamma(x)|)78//                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);79//      Note: one should avoid computing pi*(-x) directly in the80//            computation of sin(pi*(-x)).81//82//   5. Special Cases83//              lgamma(2+s) ~ s*(1-Euler) for tiny s84//              lgamma(1)=lgamma(2)=085//              lgamma(x) ~ -log(x) for tiny x86//              lgamma(0) = lgamma(inf) = inf87//              lgamma(-integer) = +-inf88//89//90 91var _lgamA = [...]float64{92	7.72156649015328655494e-02, // 0x3FB3C467E37DB0C893	3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD94	6.73523010531292681824e-02, // 0x3FB13E001A5562A795	2.05808084325167332806e-02, // 0x3F951322AC92547B96	7.38555086081402883957e-03, // 0x3F7E404FB68FEFE897	2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B98	1.19270763183362067845e-03, // 0x3F538A94116F3F5D99	5.10069792153511336608e-04, // 0x3F40B6C689B99C00100	2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D101	1.08011567247583939954e-04, // 0x3F1C5088987DFB07102	2.52144565451257326939e-05, // 0x3EFA7074428CFA52103	4.48640949618915160150e-05, // 0x3F07858E90A45837104}105var _lgamR = [...]float64{106	1.0,                        // placeholder107	1.39200533467621045958e+00, // 0x3FF645A762C4AB74108	7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC109	1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27110	1.86459191715652901344e-02, // 0x3F9317EA742ED475111	7.77942496381893596434e-04, // 0x3F497DDACA41A95B112	7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140113}114var _lgamS = [...]float64{115	-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8116	2.14982415960608852501e-01,  // 0x3FCB848B36E20878117	3.25778796408930981787e-01,  // 0x3FD4D98F4F139F59118	1.46350472652464452805e-01,  // 0x3FC2BB9CBEE5F2F7119	2.66422703033638609560e-02,  // 0x3F9B481C7E939961120	1.84028451407337715652e-03,  // 0x3F5E26B67368F239121	3.19475326584100867617e-05,  // 0x3F00BFECDD17E945122}123var _lgamT = [...]float64{124	4.83836122723810047042e-01,  // 0x3FDEF72BC8EE38A2125	-1.47587722994593911752e-01, // 0xBFC2E4278DC6C509126	6.46249402391333854778e-02,  // 0x3FB08B4294D5419B127	-3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713128	1.79706750811820387126e-02,  // 0x3F9266E7970AF9EC129	-1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A130	6.10053870246291332635e-03,  // 0x3F78FCE0E370E344131	-3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7132	2.25964780900612472250e-03,  // 0x3F6282D32E15C915133	-1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1134	8.81081882437654011382e-04,  // 0x3F4CDF0CEF61A8E9135	-5.38595305356740546715e-04, // 0xBF41A6109C73E0EC136	3.15632070903625950361e-04,  // 0x3F34AF6D6C0EBBF7137	-3.12754168375120860518e-04, // 0xBF347F24ECC38C38138	3.35529192635519073543e-04,  // 0x3F35FD3EE8C2D3F4139}140var _lgamU = [...]float64{141	-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8142	6.32827064025093366517e-01,  // 0x3FE4401E8B005DFF143	1.45492250137234768737e+00,  // 0x3FF7475CD119BD6F144	9.77717527963372745603e-01,  // 0x3FEF497644EA8450145	2.28963728064692451092e-01,  // 0x3FCD4EAEF6010924146	1.33810918536787660377e-02,  // 0x3F8B678BBF2BAB09147}148var _lgamV = [...]float64{149	1.0,150	2.45597793713041134822e+00, // 0x4003A5D7C2BD619C151	2.12848976379893395361e+00, // 0x40010725A42B18F5152	7.69285150456672783825e-01, // 0x3FE89DFBE45050AF153	1.04222645593369134254e-01, // 0x3FBAAE55D6537C88154	3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61155}156var _lgamW = [...]float64{157	4.18938533204672725052e-01,  // 0x3FDACFE390C97D69158	8.33333333333329678849e-02,  // 0x3FB555555555553B159	-2.77777777728775536470e-03, // 0xBF66C16C16B02E5C160	7.93650558643019558500e-04,  // 0x3F4A019F98CF38B6161	-5.95187557450339963135e-04, // 0xBF4380CB8C0FE741162	8.36339918996282139126e-04,  // 0x3F4B67BA4CDAD5D1163	-1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4164}165 166// Lgamma returns the natural logarithm and sign (-1 or +1) of [Gamma](x).167//168// Special cases are:169//170//	Lgamma(+Inf) = +Inf171//	Lgamma(0) = +Inf172//	Lgamma(-integer) = +Inf173//	Lgamma(-Inf) = -Inf174//	Lgamma(NaN) = NaN175func Lgamma(x float64) (lgamma float64, sign int) {176	const (177		Ymin  = 1.461632144968362245178		Two52 = 1 << 52                     // 0x4330000000000000 ~4.5036e+15179		Two53 = 1 << 53                     // 0x4340000000000000 ~9.0072e+15180		Two58 = 1 << 58                     // 0x4390000000000000 ~2.8823e+17181		Tiny  = 1.0 / (1 << 70)             // 0x3b90000000000000 ~8.47033e-22182		Tc    = 1.46163214496836224576e+00  // 0x3FF762D86356BE3F183		Tf    = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42184		// Tt = -(tail of Tf)185		Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F186	)187	// special cases188	sign = 1189	switch {190	case IsNaN(x):191		lgamma = x192		return193	case IsInf(x, 0):194		lgamma = x195		return196	case x == 0:197		lgamma = Inf(1)198		return199	}200 201	neg := false202	if x < 0 {203		x = -x204		neg = true205	}206 207	if x < Tiny { // if |x| < 2**-70, return -log(|x|)208		if neg {209			sign = -1210		}211		lgamma = -Log(x)212		return213	}214	var nadj float64215	if neg {216		if x >= Two52 { // |x| >= 2**52, must be -integer217			lgamma = Inf(1)218			return219		}220		t := sinPi(x)221		if t == 0 {222			lgamma = Inf(1) // -integer223			return224		}225		nadj = Log(Pi / Abs(t*x))226		if t < 0 {227			sign = -1228		}229	}230 231	switch {232	case x == 1 || x == 2: // purge off 1 and 2233		lgamma = 0234		return235	case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)236		var y float64237		var i int238		if x <= 0.9 {239			lgamma = -Log(x)240			switch {241			case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <=  0.9242				y = 1 - x243				i = 0244			case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316245				y = x - (Tc - 1)246				i = 1247			default: // 0 < x < 0.2316248				y = x249				i = 2250			}251		} else {252			lgamma = 0253			switch {254			case x >= (Ymin + 0.27): // 1.7316 <= x < 2255				y = 2 - x256				i = 0257			case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316258				y = x - Tc259				i = 1260			default: // 0.9 < x < 1.2316261				y = x - 1262				i = 2263			}264		}265		switch i {266		case 0:267			z := y * y268			p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))269			p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))270			p := y*p1 + p2271			lgamma += (p - 0.5*y)272		case 1:273			z := y * y274			w := z * y275			p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp276			p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))277			p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))278			p := z*p1 - (Tt - w*(p2+y*p3))279			lgamma += (Tf + p)280		case 2:281			p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))282			p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))283			lgamma += (-0.5*y + p1/p2)284		}285	case x < 8: // 2 <= x < 8286		i := int(x)287		y := x - float64(i)288		p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))289		q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))290		lgamma = 0.5*y + p/q291		z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)292		switch i {293		case 7:294			z *= (y + 6)295			fallthrough296		case 6:297			z *= (y + 5)298			fallthrough299		case 5:300			z *= (y + 4)301			fallthrough302		case 4:303			z *= (y + 3)304			fallthrough305		case 3:306			z *= (y + 2)307			lgamma += Log(z)308		}309	case x < Two58: // 8 <= x < 2**58310		t := Log(x)311		z := 1 / x312		y := z * z313		w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))314		lgamma = (x-0.5)*(t-1) + w315	default: // 2**58 <= x <= Inf316		lgamma = x * (Log(x) - 1)317	}318	if neg {319		lgamma = nadj - lgamma320	}321	return322}323 324// sinPi(x) is a helper function for negative x325func sinPi(x float64) float64 {326	const (327		Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15328		Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15329	)330	if x < 0.25 {331		return -Sin(Pi * x)332	}333 334	// argument reduction335	z := Floor(x)336	var n int337	if z != x { // inexact338		x = Mod(x, 2)339		n = int(x * 4)340	} else {341		if x >= Two53 { // x must be even342			x = 0343			n = 0344		} else {345			if x < Two52 {346				z = x + Two52 // exact347			}348			n = int(1 & Float64bits(z))349			x = float64(n)350			n <<= 2351		}352	}353	switch n {354	case 0:355		x = Sin(Pi * x)356	case 1, 2:357		x = Cos(Pi * (0.5 - x))358	case 3, 4:359		x = Sin(Pi * (1 - x))360	case 5, 6:361		x = -Cos(Pi * (x - 1.5))362	default:363		x = Sin(Pi * (x - 2))364	}365	return -x366}367