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/* 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(feature = "checked")]
_ => unreachable!(),
#[cfg(not(feature = "checked"))]
_ => {}
}
} 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);
}
|
