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Source file src/math/lgamma.go

     1	// Copyright 2010 The Go Authors. All rights reserved.
     2	// Use of this source code is governed by a BSD-style
     3	// license that can be found in the LICENSE file.
     4	
     5	package math
     6	
     7	/*
     8		Floating-point logarithm of the Gamma function.
     9	*/
    10	
    11	// The original C code and the long comment below are
    12	// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and
    13	// came with this notice. The go code is a simplified
    14	// 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 this
    21	// software is freely granted, provided that this notice
    22	// is preserved.
    23	// ====================================================
    24	//
    25	// __ieee754_lgamma_r(x, signgamp)
    26	// Reentrant version of the logarithm of the Gamma function
    27	// with user provided pointer for the sign of Gamma(x).
    28	//
    29	// Method:
    30	//   1. Argument Reduction for 0 < x <= 8
    31	//      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
    32	//      reduce x to a number in [1.5,2.5] by
    33	//              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 its
    39	//      minimum (ymin=1.461632144968362245) to maintain monotonicity.
    40	//      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
    41	//              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 accuracy
    49	//              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
    50	//      Our algorithms are based on the following observation
    51	//
    52	//                             zeta(2)-1    2    zeta(3)-1    3
    53	// lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
    54	//                                 2                 3
    55	//
    56	//      where Euler = 0.5772156649... is the Euler constant, which
    57	//      is very close to 0.5.
    58	//
    59	//   3. For x>=8, we have
    60	//      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 approximation
    64	//              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
    65	//      by
    66	//                                  3       5             11
    67	//              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
    68	//      where
    69	//              |w - f(z)| < 2**-58.74
    70	//
    71	//   4. For negative x, since (G is gamma function)
    72	//              -x*G(-x)*G(x) = pi/sin(pi*x),
    73	//      we have
    74	//              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
    75	//      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
    76	//      Hence, for x<0, signgam = sign(sin(pi*x)) and
    77	//              lgamma(x) = log(|Gamma(x)|)
    78	//                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
    79	//      Note: one should avoid computing pi*(-x) directly in the
    80	//            computation of sin(pi*(-x)).
    81	//
    82	//   5. Special Cases
    83	//              lgamma(2+s) ~ s*(1-Euler) for tiny s
    84	//              lgamma(1)=lgamma(2)=0
    85	//              lgamma(x) ~ -log(x) for tiny x
    86	//              lgamma(0) = lgamma(inf) = inf
    87	//              lgamma(-integer) = +-inf
    88	//
    89	//
    90	
    91	var _lgamA = [...]float64{
    92		7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8
    93		3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD
    94		6.73523010531292681824e-02, // 0x3FB13E001A5562A7
    95		2.05808084325167332806e-02, // 0x3F951322AC92547B
    96		7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
    97		2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B
    98		1.19270763183362067845e-03, // 0x3F538A94116F3F5D
    99		5.10069792153511336608e-04, // 0x3F40B6C689B99C00
   100		2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D
   101		1.08011567247583939954e-04, // 0x3F1C5088987DFB07
   102		2.52144565451257326939e-05, // 0x3EFA7074428CFA52
   103		4.48640949618915160150e-05, // 0x3F07858E90A45837
   104	}
   105	var _lgamR = [...]float64{
   106		1.0, // placeholder
   107		1.39200533467621045958e+00, // 0x3FF645A762C4AB74
   108		7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC
   109		1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27
   110		1.86459191715652901344e-02, // 0x3F9317EA742ED475
   111		7.77942496381893596434e-04, // 0x3F497DDACA41A95B
   112		7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140
   113	}
   114	var _lgamS = [...]float64{
   115		-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
   116		2.14982415960608852501e-01,  // 0x3FCB848B36E20878
   117		3.25778796408930981787e-01,  // 0x3FD4D98F4F139F59
   118		1.46350472652464452805e-01,  // 0x3FC2BB9CBEE5F2F7
   119		2.66422703033638609560e-02,  // 0x3F9B481C7E939961
   120		1.84028451407337715652e-03,  // 0x3F5E26B67368F239
   121		3.19475326584100867617e-05,  // 0x3F00BFECDD17E945
   122	}
   123	var _lgamT = [...]float64{
   124		4.83836122723810047042e-01,  // 0x3FDEF72BC8EE38A2
   125		-1.47587722994593911752e-01, // 0xBFC2E4278DC6C509
   126		6.46249402391333854778e-02,  // 0x3FB08B4294D5419B
   127		-3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713
   128		1.79706750811820387126e-02,  // 0x3F9266E7970AF9EC
   129		-1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A
   130		6.10053870246291332635e-03,  // 0x3F78FCE0E370E344
   131		-3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7
   132		2.25964780900612472250e-03,  // 0x3F6282D32E15C915
   133		-1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1
   134		8.81081882437654011382e-04,  // 0x3F4CDF0CEF61A8E9
   135		-5.38595305356740546715e-04, // 0xBF41A6109C73E0EC
   136		3.15632070903625950361e-04,  // 0x3F34AF6D6C0EBBF7
   137		-3.12754168375120860518e-04, // 0xBF347F24ECC38C38
   138		3.35529192635519073543e-04,  // 0x3F35FD3EE8C2D3F4
   139	}
   140	var _lgamU = [...]float64{
   141		-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
   142		6.32827064025093366517e-01,  // 0x3FE4401E8B005DFF
   143		1.45492250137234768737e+00,  // 0x3FF7475CD119BD6F
   144		9.77717527963372745603e-01,  // 0x3FEF497644EA8450
   145		2.28963728064692451092e-01,  // 0x3FCD4EAEF6010924
   146		1.33810918536787660377e-02,  // 0x3F8B678BBF2BAB09
   147	}
   148	var _lgamV = [...]float64{
   149		1.0,
   150		2.45597793713041134822e+00, // 0x4003A5D7C2BD619C
   151		2.12848976379893395361e+00, // 0x40010725A42B18F5
   152		7.69285150456672783825e-01, // 0x3FE89DFBE45050AF
   153		1.04222645593369134254e-01, // 0x3FBAAE55D6537C88
   154		3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61
   155	}
   156	var _lgamW = [...]float64{
   157		4.18938533204672725052e-01,  // 0x3FDACFE390C97D69
   158		8.33333333333329678849e-02,  // 0x3FB555555555553B
   159		-2.77777777728775536470e-03, // 0xBF66C16C16B02E5C
   160		7.93650558643019558500e-04,  // 0x3F4A019F98CF38B6
   161		-5.95187557450339963135e-04, // 0xBF4380CB8C0FE741
   162		8.36339918996282139126e-04,  // 0x3F4B67BA4CDAD5D1
   163		-1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4
   164	}
   165	
   166	// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
   167	//
   168	// Special cases are:
   169	//	Lgamma(+Inf) = +Inf
   170	//	Lgamma(0) = +Inf
   171	//	Lgamma(-integer) = +Inf
   172	//	Lgamma(-Inf) = -Inf
   173	//	Lgamma(NaN) = NaN
   174	func Lgamma(x float64) (lgamma float64, sign int) {
   175		const (
   176			Ymin  = 1.461632144968362245
   177			Two52 = 1 << 52                     // 0x4330000000000000 ~4.5036e+15
   178			Two53 = 1 << 53                     // 0x4340000000000000 ~9.0072e+15
   179			Two58 = 1 << 58                     // 0x4390000000000000 ~2.8823e+17
   180			Tiny  = 1.0 / (1 << 70)             // 0x3b90000000000000 ~8.47033e-22
   181			Tc    = 1.46163214496836224576e+00  // 0x3FF762D86356BE3F
   182			Tf    = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
   183			// Tt = -(tail of Tf)
   184			Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
   185		)
   186		// special cases
   187		sign = 1
   188		switch {
   189		case IsNaN(x):
   190			lgamma = x
   191			return
   192		case IsInf(x, 0):
   193			lgamma = x
   194			return
   195		case x == 0:
   196			lgamma = Inf(1)
   197			return
   198		}
   199	
   200		neg := false
   201		if x < 0 {
   202			x = -x
   203			neg = true
   204		}
   205	
   206		if x < Tiny { // if |x| < 2**-70, return -log(|x|)
   207			if neg {
   208				sign = -1
   209			}
   210			lgamma = -Log(x)
   211			return
   212		}
   213		var nadj float64
   214		if neg {
   215			if x >= Two52 { // |x| >= 2**52, must be -integer
   216				lgamma = Inf(1)
   217				return
   218			}
   219			t := sinPi(x)
   220			if t == 0 {
   221				lgamma = Inf(1) // -integer
   222				return
   223			}
   224			nadj = Log(Pi / Abs(t*x))
   225			if t < 0 {
   226				sign = -1
   227			}
   228		}
   229	
   230		switch {
   231		case x == 1 || x == 2: // purge off 1 and 2
   232			lgamma = 0
   233			return
   234		case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)
   235			var y float64
   236			var i int
   237			if x <= 0.9 {
   238				lgamma = -Log(x)
   239				switch {
   240				case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <=  0.9
   241					y = 1 - x
   242					i = 0
   243				case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316
   244					y = x - (Tc - 1)
   245					i = 1
   246				default: // 0 < x < 0.2316
   247					y = x
   248					i = 2
   249				}
   250			} else {
   251				lgamma = 0
   252				switch {
   253				case x >= (Ymin + 0.27): // 1.7316 <= x < 2
   254					y = 2 - x
   255					i = 0
   256				case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316
   257					y = x - Tc
   258					i = 1
   259				default: // 0.9 < x < 1.2316
   260					y = x - 1
   261					i = 2
   262				}
   263			}
   264			switch i {
   265			case 0:
   266				z := y * y
   267				p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))
   268				p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))
   269				p := y*p1 + p2
   270				lgamma += (p - 0.5*y)
   271			case 1:
   272				z := y * y
   273				w := z * y
   274				p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp
   275				p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))
   276				p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))
   277				p := z*p1 - (Tt - w*(p2+y*p3))
   278				lgamma += (Tf + p)
   279			case 2:
   280				p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))
   281				p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))
   282				lgamma += (-0.5*y + p1/p2)
   283			}
   284		case x < 8: // 2 <= x < 8
   285			i := int(x)
   286			y := x - float64(i)
   287			p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))
   288			q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))
   289			lgamma = 0.5*y + p/q
   290			z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
   291			switch i {
   292			case 7:
   293				z *= (y + 6)
   294				fallthrough
   295			case 6:
   296				z *= (y + 5)
   297				fallthrough
   298			case 5:
   299				z *= (y + 4)
   300				fallthrough
   301			case 4:
   302				z *= (y + 3)
   303				fallthrough
   304			case 3:
   305				z *= (y + 2)
   306				lgamma += Log(z)
   307			}
   308		case x < Two58: // 8 <= x < 2**58
   309			t := Log(x)
   310			z := 1 / x
   311			y := z * z
   312			w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))
   313			lgamma = (x-0.5)*(t-1) + w
   314		default: // 2**58 <= x <= Inf
   315			lgamma = x * (Log(x) - 1)
   316		}
   317		if neg {
   318			lgamma = nadj - lgamma
   319		}
   320		return
   321	}
   322	
   323	// sinPi(x) is a helper function for negative x
   324	func sinPi(x float64) float64 {
   325		const (
   326			Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
   327			Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
   328		)
   329		if x < 0.25 {
   330			return -Sin(Pi * x)
   331		}
   332	
   333		// argument reduction
   334		z := Floor(x)
   335		var n int
   336		if z != x { // inexact
   337			x = Mod(x, 2)
   338			n = int(x * 4)
   339		} else {
   340			if x >= Two53 { // x must be even
   341				x = 0
   342				n = 0
   343			} else {
   344				if x < Two52 {
   345					z = x + Two52 // exact
   346				}
   347				n = int(1 & Float64bits(z))
   348				x = float64(n)
   349				n <<= 2
   350			}
   351		}
   352		switch n {
   353		case 0:
   354			x = Sin(Pi * x)
   355		case 1, 2:
   356			x = Cos(Pi * (0.5 - x))
   357		case 3, 4:
   358			x = Sin(Pi * (1 - x))
   359		case 5, 6:
   360			x = -Cos(Pi * (x - 1.5))
   361		default:
   362			x = Sin(Pi * (x - 2))
   363		}
   364		return -x
   365	}
   366	

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