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
5package 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
91var _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}
105var _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}
114var _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}
123var _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}
140var _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}
148var _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}
156var _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
174func 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
324func 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