summaryrefslogtreecommitdiff
path: root/libgo/go/math/lgamma.go
blob: 19ac3ffafcaf81d027e3145b4d5c66934cccae45 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
// 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
//
//

var _lgamA = [...]float64{
	7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8
	3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD
	6.73523010531292681824e-02, // 0x3FB13E001A5562A7
	2.05808084325167332806e-02, // 0x3F951322AC92547B
	7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
	2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B
	1.19270763183362067845e-03, // 0x3F538A94116F3F5D
	5.10069792153511336608e-04, // 0x3F40B6C689B99C00
	2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D
	1.08011567247583939954e-04, // 0x3F1C5088987DFB07
	2.52144565451257326939e-05, // 0x3EFA7074428CFA52
	4.48640949618915160150e-05, // 0x3F07858E90A45837
}
var _lgamR = [...]float64{
	1.0, // placeholder
	1.39200533467621045958e+00, // 0x3FF645A762C4AB74
	7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC
	1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27
	1.86459191715652901344e-02, // 0x3F9317EA742ED475
	7.77942496381893596434e-04, // 0x3F497DDACA41A95B
	7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140
}
var _lgamS = [...]float64{
	-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
	2.14982415960608852501e-01,  // 0x3FCB848B36E20878
	3.25778796408930981787e-01,  // 0x3FD4D98F4F139F59
	1.46350472652464452805e-01,  // 0x3FC2BB9CBEE5F2F7
	2.66422703033638609560e-02,  // 0x3F9B481C7E939961
	1.84028451407337715652e-03,  // 0x3F5E26B67368F239
	3.19475326584100867617e-05,  // 0x3F00BFECDD17E945
}
var _lgamT = [...]float64{
	4.83836122723810047042e-01,  // 0x3FDEF72BC8EE38A2
	-1.47587722994593911752e-01, // 0xBFC2E4278DC6C509
	6.46249402391333854778e-02,  // 0x3FB08B4294D5419B
	-3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713
	1.79706750811820387126e-02,  // 0x3F9266E7970AF9EC
	-1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A
	6.10053870246291332635e-03,  // 0x3F78FCE0E370E344
	-3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7
	2.25964780900612472250e-03,  // 0x3F6282D32E15C915
	-1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1
	8.81081882437654011382e-04,  // 0x3F4CDF0CEF61A8E9
	-5.38595305356740546715e-04, // 0xBF41A6109C73E0EC
	3.15632070903625950361e-04,  // 0x3F34AF6D6C0EBBF7
	-3.12754168375120860518e-04, // 0xBF347F24ECC38C38
	3.35529192635519073543e-04,  // 0x3F35FD3EE8C2D3F4
}
var _lgamU = [...]float64{
	-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
	6.32827064025093366517e-01,  // 0x3FE4401E8B005DFF
	1.45492250137234768737e+00,  // 0x3FF7475CD119BD6F
	9.77717527963372745603e-01,  // 0x3FEF497644EA8450
	2.28963728064692451092e-01,  // 0x3FCD4EAEF6010924
	1.33810918536787660377e-02,  // 0x3F8B678BBF2BAB09
}
var _lgamV = [...]float64{
	1.0,
	2.45597793713041134822e+00, // 0x4003A5D7C2BD619C
	2.12848976379893395361e+00, // 0x40010725A42B18F5
	7.69285150456672783825e-01, // 0x3FE89DFBE45050AF
	1.04222645593369134254e-01, // 0x3FBAAE55D6537C88
	3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61
}
var _lgamW = [...]float64{
	4.18938533204672725052e-01,  // 0x3FDACFE390C97D69
	8.33333333333329678849e-02,  // 0x3FB555555555553B
	-2.77777777728775536470e-03, // 0xBF66C16C16B02E5C
	7.93650558643019558500e-04,  // 0x3F4A019F98CF38B6
	-5.95187557450339963135e-04, // 0xBF4380CB8C0FE741
	8.36339918996282139126e-04,  // 0x3F4B67BA4CDAD5D1
	-1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4
}

// 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
		Tc    = 1.46163214496836224576e+00  // 0x3FF762D86356BE3F
		Tf    = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
		// Tt = -(tail of Tf)
		Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
	)
	// special cases
	sign = 1
	switch {
	case IsNaN(x):
		lgamma = x
		return
	case 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 / Abs(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 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))
			p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))
			p := y*p1 + p2
			lgamma += (p - 0.5*y)
		case 1:
			z := y * y
			w := z * y
			p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp
			p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))
			p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))
			p := z*p1 - (Tt - w*(p2+y*p3))
			lgamma += (Tf + p)
		case 2:
			p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))
			p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))
			lgamma += (-0.5*y + p1/p2)
		}
	case x < 8: // 2 <= x < 8
		i := int(x)
		y := x - float64(i)
		p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))
		q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))
		lgamma = 0.5*y + p/q
		z := 1.0 // 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 := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))
		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 = Mod(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
}