| /* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Developed at SunSoft, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| * |
| */ |
| /* lgamma_r(x, signgamp) |
| * Reentrant version of the logarithm of the Gamma function |
| * with user provide 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 |
| * minimun 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) |
| * where |
| * 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 = 0.5771... 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 compute 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(neg.integer) = inf and raise divide-by-zero |
| * lgamma(inf) = inf |
| * lgamma(-inf) = inf (bug for bug compatible with C99!?) |
| * |
| */ |
| |
| use super::{floor, k_cos, k_sin, log}; |
| |
| const PI: f64 = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */ |
| const A0: f64 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */ |
| const A1: f64 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */ |
| const A2: f64 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */ |
| const A3: f64 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */ |
| const A4: f64 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */ |
| const A5: f64 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */ |
| const A6: f64 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */ |
| const A7: f64 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */ |
| const A8: f64 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */ |
| const A9: f64 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */ |
| const A10: f64 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */ |
| const A11: f64 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */ |
| const TC: f64 = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */ |
| const TF: f64 = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */ |
| /* tt = -(tail of TF) */ |
| const TT: f64 = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */ |
| const T0: f64 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */ |
| const T1: f64 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */ |
| const T2: f64 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */ |
| const T3: f64 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */ |
| const T4: f64 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */ |
| const T5: f64 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */ |
| const T6: f64 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */ |
| const T7: f64 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */ |
| const T8: f64 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */ |
| const T9: f64 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */ |
| const T10: f64 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */ |
| const T11: f64 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */ |
| const T12: f64 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */ |
| const T13: f64 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */ |
| const T14: f64 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */ |
| const U0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ |
| const U1: f64 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */ |
| const U2: f64 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */ |
| const U3: f64 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */ |
| const U4: f64 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */ |
| const U5: f64 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */ |
| const V1: f64 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */ |
| const V2: f64 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */ |
| const V3: f64 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */ |
| const V4: f64 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */ |
| const V5: f64 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */ |
| const S0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */ |
| const S1: f64 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */ |
| const S2: f64 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */ |
| const S3: f64 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */ |
| const S4: f64 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */ |
| const S5: f64 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */ |
| const S6: f64 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */ |
| const R1: f64 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */ |
| const R2: f64 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */ |
| const R3: f64 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */ |
| const R4: f64 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */ |
| const R5: f64 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */ |
| const R6: f64 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */ |
| const W0: f64 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */ |
| const W1: f64 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */ |
| const W2: f64 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */ |
| const W3: f64 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */ |
| const W4: f64 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */ |
| const W5: f64 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */ |
| const W6: f64 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ |
| |
| /* sin(PI*x) assuming x > 2^-100, if sin(PI*x)==0 the sign is arbitrary */ |
| fn sin_pi(mut x: f64) -> f64 { |
| let mut n: i32; |
| |
| /* spurious inexact if odd int */ |
| x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */ |
| |
| n = (x * 4.0) as i32; |
| n = (n + 1) / 2; |
| x -= (n as f64) * 0.5; |
| x *= PI; |
| |
| match n { |
| 1 => k_cos(x, 0.0), |
| 2 => k_sin(-x, 0.0, 0), |
| 3 => -k_cos(x, 0.0), |
| 0 | _ => k_sin(x, 0.0, 0), |
| } |
| } |
| |
| pub fn lgamma_r(mut x: f64) -> (f64, i32) { |
| let u: u64 = x.to_bits(); |
| let mut t: f64; |
| let y: f64; |
| let mut z: f64; |
| let nadj: f64; |
| let p: f64; |
| let p1: f64; |
| let p2: f64; |
| let p3: f64; |
| let q: f64; |
| let mut r: f64; |
| let w: f64; |
| let ix: u32; |
| let sign: bool; |
| let i: i32; |
| let mut signgam: i32; |
| |
| /* purge off +-inf, NaN, +-0, tiny and negative arguments */ |
| signgam = 1; |
| sign = (u >> 63) != 0; |
| ix = ((u >> 32) as u32) & 0x7fffffff; |
| if ix >= 0x7ff00000 { |
| return (x * x, signgam); |
| } |
| if ix < (0x3ff - 70) << 20 { |
| /* |x|<2**-70, return -log(|x|) */ |
| if sign { |
| x = -x; |
| signgam = -1; |
| } |
| return (-log(x), signgam); |
| } |
| if sign { |
| x = -x; |
| t = sin_pi(x); |
| if t == 0.0 { |
| /* -integer */ |
| return (1.0 / (x - x), signgam); |
| } |
| if t > 0.0 { |
| signgam = -1; |
| } else { |
| t = -t; |
| } |
| nadj = log(PI / (t * x)); |
| } else { |
| nadj = 0.0; |
| } |
| |
| /* purge off 1 and 2 */ |
| if (ix == 0x3ff00000 || ix == 0x40000000) && (u & 0xffffffff) == 0 { |
| r = 0.0; |
| } |
| /* for x < 2.0 */ |
| else if ix < 0x40000000 { |
| if ix <= 0x3feccccc { |
| /* lgamma(x) = lgamma(x+1)-log(x) */ |
| r = -log(x); |
| if ix >= 0x3FE76944 { |
| y = 1.0 - x; |
| i = 0; |
| } else if ix >= 0x3FCDA661 { |
| y = x - (TC - 1.0); |
| i = 1; |
| } else { |
| y = x; |
| i = 2; |
| } |
| } else { |
| r = 0.0; |
| if ix >= 0x3FFBB4C3 { |
| /* [1.7316,2] */ |
| y = 2.0 - x; |
| i = 0; |
| } else if ix >= 0x3FF3B4C4 { |
| /* [1.23,1.73] */ |
| y = x - TC; |
| i = 1; |
| } else { |
| y = x - 1.0; |
| i = 2; |
| } |
| } |
| match i { |
| 0 => { |
| z = y * y; |
| p1 = A0 + z * (A2 + z * (A4 + z * (A6 + z * (A8 + z * A10)))); |
| p2 = z * (A1 + z * (A3 + z * (A5 + z * (A7 + z * (A9 + z * A11))))); |
| p = y * p1 + p2; |
| r += p - 0.5 * y; |
| } |
| 1 => { |
| z = y * y; |
| w = z * y; |
| p1 = T0 + w * (T3 + w * (T6 + w * (T9 + w * T12))); /* parallel comp */ |
| p2 = T1 + w * (T4 + w * (T7 + w * (T10 + w * T13))); |
| p3 = T2 + w * (T5 + w * (T8 + w * (T11 + w * T14))); |
| p = z * p1 - (TT - w * (p2 + y * p3)); |
| r += TF + p; |
| } |
| 2 => { |
| p1 = y * (U0 + y * (U1 + y * (U2 + y * (U3 + y * (U4 + y * U5))))); |
| p2 = 1.0 + y * (V1 + y * (V2 + y * (V3 + y * (V4 + y * V5)))); |
| r += -0.5 * y + p1 / p2; |
| } |
| #[cfg(debug_assertions)] |
| _ => unreachable!(), |
| #[cfg(not(debug_assertions))] |
| _ => {} |
| } |
| } else if ix < 0x40200000 { |
| /* x < 8.0 */ |
| i = x as i32; |
| y = x - (i as f64); |
| p = y * (S0 + y * (S1 + y * (S2 + y * (S3 + y * (S4 + y * (S5 + y * S6)))))); |
| q = 1.0 + y * (R1 + y * (R2 + y * (R3 + y * (R4 + y * (R5 + y * R6))))); |
| r = 0.5 * y + p / q; |
| z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ |
| // TODO: In C, this was implemented using switch jumps with fallthrough. |
| // Does this implementation have performance problems? |
| if i >= 7 { |
| z *= y + 6.0; |
| } |
| if i >= 6 { |
| z *= y + 5.0; |
| } |
| if i >= 5 { |
| z *= y + 4.0; |
| } |
| if i >= 4 { |
| z *= y + 3.0; |
| } |
| if i >= 3 { |
| z *= y + 2.0; |
| r += log(z); |
| } |
| } else if ix < 0x43900000 { |
| /* 8.0 <= x < 2**58 */ |
| t = log(x); |
| z = 1.0 / x; |
| y = z * z; |
| w = W0 + z * (W1 + y * (W2 + y * (W3 + y * (W4 + y * (W5 + y * W6))))); |
| r = (x - 0.5) * (t - 1.0) + w; |
| } else { |
| /* 2**58 <= x <= inf */ |
| r = x * (log(x) - 1.0); |
| } |
| if sign { |
| r = nadj - r; |
| } |
| return (r, signgam); |
| } |
