Black Lives Matter. Support the Equal Justice Initiative.

# Source file src/math/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
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  // ====================================================
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
94  	6.73523010531292681824e-02, // 0x3FB13E001A5562A7
95  	2.05808084325167332806e-02, // 0x3F951322AC92547B
96  	7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
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
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
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  	}
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
```

View as plain text