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dc9dc135 XL |
1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */ |
2 | /* | |
3 | * ==================================================== | |
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. | |
5 | * | |
6 | * Developed at SunSoft, a Sun Microsystems, Inc. business. | |
7 | * Permission to use, copy, modify, and distribute this | |
8 | * software is freely granted, provided that this notice | |
9 | * is preserved. | |
10 | * ==================================================== | |
11 | * | |
12 | */ | |
13 | /* lgamma_r(x, signgamp) | |
14 | * Reentrant version of the logarithm of the Gamma function | |
15 | * with user provide pointer for the sign of Gamma(x). | |
16 | * | |
17 | * Method: | |
18 | * 1. Argument Reduction for 0 < x <= 8 | |
19 | * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may | |
20 | * reduce x to a number in [1.5,2.5] by | |
21 | * lgamma(1+s) = log(s) + lgamma(s) | |
22 | * for example, | |
23 | * lgamma(7.3) = log(6.3) + lgamma(6.3) | |
24 | * = log(6.3*5.3) + lgamma(5.3) | |
25 | * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) | |
26 | * 2. Polynomial approximation of lgamma around its | |
27 | * minimun ymin=1.461632144968362245 to maintain monotonicity. | |
28 | * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use | |
29 | * Let z = x-ymin; | |
30 | * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) | |
31 | * where | |
32 | * poly(z) is a 14 degree polynomial. | |
33 | * 2. Rational approximation in the primary interval [2,3] | |
34 | * We use the following approximation: | |
35 | * s = x-2.0; | |
36 | * lgamma(x) = 0.5*s + s*P(s)/Q(s) | |
37 | * with accuracy | |
38 | * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 | |
39 | * Our algorithms are based on the following observation | |
40 | * | |
41 | * zeta(2)-1 2 zeta(3)-1 3 | |
42 | * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... | |
43 | * 2 3 | |
44 | * | |
45 | * where Euler = 0.5771... is the Euler constant, which is very | |
46 | * close to 0.5. | |
47 | * | |
48 | * 3. For x>=8, we have | |
49 | * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... | |
50 | * (better formula: | |
51 | * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) | |
52 | * Let z = 1/x, then we approximation | |
53 | * f(z) = lgamma(x) - (x-0.5)(log(x)-1) | |
54 | * by | |
55 | * 3 5 11 | |
56 | * w = w0 + w1*z + w2*z + w3*z + ... + w6*z | |
57 | * where | |
58 | * |w - f(z)| < 2**-58.74 | |
59 | * | |
60 | * 4. For negative x, since (G is gamma function) | |
61 | * -x*G(-x)*G(x) = PI/sin(PI*x), | |
62 | * we have | |
63 | * G(x) = PI/(sin(PI*x)*(-x)*G(-x)) | |
64 | * since G(-x) is positive, sign(G(x)) = sign(sin(PI*x)) for x<0 | |
65 | * Hence, for x<0, signgam = sign(sin(PI*x)) and | |
66 | * lgamma(x) = log(|Gamma(x)|) | |
67 | * = log(PI/(|x*sin(PI*x)|)) - lgamma(-x); | |
68 | * Note: one should avoid compute PI*(-x) directly in the | |
69 | * computation of sin(PI*(-x)). | |
70 | * | |
71 | * 5. Special Cases | |
72 | * lgamma(2+s) ~ s*(1-Euler) for tiny s | |
73 | * lgamma(1) = lgamma(2) = 0 | |
74 | * lgamma(x) ~ -log(|x|) for tiny x | |
75 | * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero | |
76 | * lgamma(inf) = inf | |
77 | * lgamma(-inf) = inf (bug for bug compatible with C99!?) | |
78 | * | |
79 | */ | |
80 | ||
81 | use super::{floor, k_cos, k_sin, log}; | |
82 | ||
83 | const PI: f64 = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */ | |
84 | const A0: f64 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */ | |
85 | const A1: f64 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */ | |
86 | const A2: f64 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */ | |
87 | const A3: f64 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */ | |
88 | const A4: f64 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */ | |
89 | const A5: f64 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */ | |
90 | const A6: f64 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */ | |
91 | const A7: f64 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */ | |
92 | const A8: f64 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */ | |
93 | const A9: f64 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */ | |
94 | const A10: f64 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */ | |
95 | const A11: f64 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */ | |
96 | const TC: f64 = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */ | |
97 | const TF: f64 = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */ | |
98 | /* tt = -(tail of TF) */ | |
99 | const TT: f64 = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */ | |
100 | const T0: f64 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */ | |
101 | const T1: f64 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */ | |
102 | const T2: f64 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */ | |
103 | const T3: f64 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */ | |
104 | const T4: f64 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */ | |
105 | const T5: f64 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */ | |
106 | const T6: f64 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */ | |
107 | const T7: f64 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */ | |
108 | const T8: f64 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */ | |
109 | const T9: f64 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */ | |
110 | const T10: f64 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */ | |
111 | const T11: f64 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */ | |
112 | const T12: f64 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */ | |
113 | const T13: f64 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */ | |
114 | const T14: f64 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */ | |
115 | const U0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ | |
116 | const U1: f64 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */ | |
117 | const U2: f64 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */ | |
118 | const U3: f64 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */ | |
119 | const U4: f64 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */ | |
120 | const U5: f64 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */ | |
121 | const V1: f64 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */ | |
122 | const V2: f64 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */ | |
123 | const V3: f64 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */ | |
124 | const V4: f64 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */ | |
125 | const V5: f64 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */ | |
126 | const S0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ | |
127 | const S1: f64 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */ | |
128 | const S2: f64 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */ | |
129 | const S3: f64 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */ | |
130 | const S4: f64 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */ | |
131 | const S5: f64 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */ | |
132 | const S6: f64 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */ | |
133 | const R1: f64 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */ | |
134 | const R2: f64 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */ | |
135 | const R3: f64 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */ | |
136 | const R4: f64 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */ | |
137 | const R5: f64 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */ | |
138 | const R6: f64 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */ | |
139 | const W0: f64 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */ | |
140 | const W1: f64 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */ | |
141 | const W2: f64 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */ | |
142 | const W3: f64 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */ | |
143 | const W4: f64 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */ | |
144 | const W5: f64 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */ | |
145 | const W6: f64 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ | |
146 | ||
147 | /* sin(PI*x) assuming x > 2^-100, if sin(PI*x)==0 the sign is arbitrary */ | |
148 | fn sin_pi(mut x: f64) -> f64 { | |
149 | let mut n: i32; | |
150 | ||
151 | /* spurious inexact if odd int */ | |
152 | x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */ | |
153 | ||
154 | n = (x * 4.0) as i32; | |
f2b60f7d | 155 | n = div!(n + 1, 2); |
dc9dc135 XL |
156 | x -= (n as f64) * 0.5; |
157 | x *= PI; | |
158 | ||
159 | match n { | |
160 | 1 => k_cos(x, 0.0), | |
161 | 2 => k_sin(-x, 0.0, 0), | |
162 | 3 => -k_cos(x, 0.0), | |
163 | 0 | _ => k_sin(x, 0.0, 0), | |
164 | } | |
165 | } | |
166 | ||
f2b60f7d | 167 | #[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)] |
dc9dc135 XL |
168 | pub fn lgamma_r(mut x: f64) -> (f64, i32) { |
169 | let u: u64 = x.to_bits(); | |
170 | let mut t: f64; | |
171 | let y: f64; | |
172 | let mut z: f64; | |
173 | let nadj: f64; | |
174 | let p: f64; | |
175 | let p1: f64; | |
176 | let p2: f64; | |
177 | let p3: f64; | |
178 | let q: f64; | |
179 | let mut r: f64; | |
180 | let w: f64; | |
181 | let ix: u32; | |
182 | let sign: bool; | |
183 | let i: i32; | |
184 | let mut signgam: i32; | |
185 | ||
186 | /* purge off +-inf, NaN, +-0, tiny and negative arguments */ | |
187 | signgam = 1; | |
188 | sign = (u >> 63) != 0; | |
189 | ix = ((u >> 32) as u32) & 0x7fffffff; | |
190 | if ix >= 0x7ff00000 { | |
191 | return (x * x, signgam); | |
192 | } | |
193 | if ix < (0x3ff - 70) << 20 { | |
194 | /* |x|<2**-70, return -log(|x|) */ | |
195 | if sign { | |
196 | x = -x; | |
197 | signgam = -1; | |
198 | } | |
199 | return (-log(x), signgam); | |
200 | } | |
201 | if sign { | |
202 | x = -x; | |
203 | t = sin_pi(x); | |
204 | if t == 0.0 { | |
205 | /* -integer */ | |
206 | return (1.0 / (x - x), signgam); | |
207 | } | |
208 | if t > 0.0 { | |
209 | signgam = -1; | |
210 | } else { | |
211 | t = -t; | |
212 | } | |
213 | nadj = log(PI / (t * x)); | |
214 | } else { | |
215 | nadj = 0.0; | |
216 | } | |
217 | ||
218 | /* purge off 1 and 2 */ | |
219 | if (ix == 0x3ff00000 || ix == 0x40000000) && (u & 0xffffffff) == 0 { | |
220 | r = 0.0; | |
221 | } | |
222 | /* for x < 2.0 */ | |
223 | else if ix < 0x40000000 { | |
224 | if ix <= 0x3feccccc { | |
225 | /* lgamma(x) = lgamma(x+1)-log(x) */ | |
226 | r = -log(x); | |
227 | if ix >= 0x3FE76944 { | |
228 | y = 1.0 - x; | |
229 | i = 0; | |
230 | } else if ix >= 0x3FCDA661 { | |
231 | y = x - (TC - 1.0); | |
232 | i = 1; | |
233 | } else { | |
234 | y = x; | |
235 | i = 2; | |
236 | } | |
237 | } else { | |
238 | r = 0.0; | |
239 | if ix >= 0x3FFBB4C3 { | |
240 | /* [1.7316,2] */ | |
241 | y = 2.0 - x; | |
242 | i = 0; | |
243 | } else if ix >= 0x3FF3B4C4 { | |
244 | /* [1.23,1.73] */ | |
245 | y = x - TC; | |
246 | i = 1; | |
247 | } else { | |
248 | y = x - 1.0; | |
249 | i = 2; | |
250 | } | |
251 | } | |
252 | match i { | |
253 | 0 => { | |
254 | z = y * y; | |
255 | p1 = A0 + z * (A2 + z * (A4 + z * (A6 + z * (A8 + z * A10)))); | |
256 | p2 = z * (A1 + z * (A3 + z * (A5 + z * (A7 + z * (A9 + z * A11))))); | |
257 | p = y * p1 + p2; | |
258 | r += p - 0.5 * y; | |
259 | } | |
260 | 1 => { | |
261 | z = y * y; | |
262 | w = z * y; | |
263 | p1 = T0 + w * (T3 + w * (T6 + w * (T9 + w * T12))); /* parallel comp */ | |
264 | p2 = T1 + w * (T4 + w * (T7 + w * (T10 + w * T13))); | |
265 | p3 = T2 + w * (T5 + w * (T8 + w * (T11 + w * T14))); | |
266 | p = z * p1 - (TT - w * (p2 + y * p3)); | |
267 | r += TF + p; | |
268 | } | |
269 | 2 => { | |
270 | p1 = y * (U0 + y * (U1 + y * (U2 + y * (U3 + y * (U4 + y * U5))))); | |
271 | p2 = 1.0 + y * (V1 + y * (V2 + y * (V3 + y * (V4 + y * V5)))); | |
272 | r += -0.5 * y + p1 / p2; | |
273 | } | |
60c5eb7d | 274 | #[cfg(debug_assertions)] |
dc9dc135 | 275 | _ => unreachable!(), |
60c5eb7d | 276 | #[cfg(not(debug_assertions))] |
dc9dc135 XL |
277 | _ => {} |
278 | } | |
279 | } else if ix < 0x40200000 { | |
280 | /* x < 8.0 */ | |
281 | i = x as i32; | |
282 | y = x - (i as f64); | |
283 | p = y * (S0 + y * (S1 + y * (S2 + y * (S3 + y * (S4 + y * (S5 + y * S6)))))); | |
284 | q = 1.0 + y * (R1 + y * (R2 + y * (R3 + y * (R4 + y * (R5 + y * R6))))); | |
285 | r = 0.5 * y + p / q; | |
286 | z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ | |
287 | // TODO: In C, this was implemented using switch jumps with fallthrough. | |
288 | // Does this implementation have performance problems? | |
289 | if i >= 7 { | |
290 | z *= y + 6.0; | |
291 | } | |
292 | if i >= 6 { | |
293 | z *= y + 5.0; | |
294 | } | |
295 | if i >= 5 { | |
296 | z *= y + 4.0; | |
297 | } | |
298 | if i >= 4 { | |
299 | z *= y + 3.0; | |
300 | } | |
301 | if i >= 3 { | |
302 | z *= y + 2.0; | |
303 | r += log(z); | |
304 | } | |
305 | } else if ix < 0x43900000 { | |
306 | /* 8.0 <= x < 2**58 */ | |
307 | t = log(x); | |
308 | z = 1.0 / x; | |
309 | y = z * z; | |
310 | w = W0 + z * (W1 + y * (W2 + y * (W3 + y * (W4 + y * (W5 + y * W6))))); | |
311 | r = (x - 0.5) * (t - 1.0) + w; | |
312 | } else { | |
313 | /* 2**58 <= x <= inf */ | |
314 | r = x * (log(x) - 1.0); | |
315 | } | |
316 | if sign { | |
317 | r = nadj - r; | |
318 | } | |
319 | return (r, signgam); | |
320 | } |