lgamma.go

Documentation: math

     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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