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; |
155 | n = div!(n + 1, 2); |
156 | x -= (n as f64) * 0.5; |
157 | x *= PI; |
158 | |
159 | match n { |
160 | 1 => k_cos(x, y:0.0), |
161 | 2 => k_sin(-x, y:0.0, iy:0), |
162 | 3 => -k_cos(x, y:0.0), |
163 | 0 | _ => k_sin(x, y:0.0, iy:0), |
164 | } |
165 | } |
166 | |
167 | #[cfg_attr (all(test, assert_no_panic), no_panic::no_panic)] |
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 | } |
274 | #[cfg (debug_assertions)] |
275 | _ => unreachable!(), |
276 | #[cfg (not(debug_assertions))] |
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 | } |
321 | |