From 1b28226a6788d095c66925276c7a28041f8bb2de Mon Sep 17 00:00:00 2001 From: gingerBill Date: Tue, 16 Nov 2021 15:32:32 +0000 Subject: [PATCH] Add `math.lgamma` based off FreeBSD's `/usr/src/lib/msun/src/e_lgamma_r.c` --- core/math/math_lgamma.odin | 361 +++++++++++++++++++++++++++++++++++++ 1 file changed, 361 insertions(+) create mode 100644 core/math/math_lgamma.odin diff --git a/core/math/math_lgamma.odin b/core/math/math_lgamma.odin new file mode 100644 index 000000000..e6cbdf6cd --- /dev/null +++ b/core/math/math_lgamma.odin @@ -0,0 +1,361 @@ +package math + +// The original C code and the long comment below are +// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and +// came with this notice. +// +// ==================================================== +// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. +// +// Developed at SunPro, a Sun Microsystems, Inc. business. +// Permission to use, copy, modify, and distribute this +// software is freely granted, provided that this notice +// is preserved. +// ==================================================== +// +// __ieee754_lgamma_r(x, signgamp) +// Reentrant version of the logarithm of the Gamma function +// with user provided 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 +// minimum (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) +// 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 = 0.5772156649... 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 computing 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(inf) = inf +// lgamma(-integer) = +-inf +// +// + + +lgamma_f64 :: proc "contextless" (x: f64) -> (lgamma: f64, sign: int) { + sin_pi :: proc "contextless" (x: f64) -> f64 { + if x < 0.25 { + return -sin(PI * x) + } + x := x + + // argument reduction + z := floor(x) + n: int + if z != x { // inexact + x = mod(x, 2) + n = int(x * 4) + } else { + if x >= TWO_53 { // x must be even + x = 0 + n = 0 + } else { + if x < TWO_52 { + z = x + TWO_52 // exact + } + n = int(1 & transmute(u64)z) + x = f64(n) + n <<= 2 + } + } + switch n { + case 0: + x = sin(PI * x) + case 1, 2: + x = cos(PI * (0.5 - x)) + case 3, 4: + x = sin(PI * (1 - x)) + case 5, 6: + x = -cos(PI * (x - 1.5)) + case: + x = sin(PI * (x - 2)) + } + return -x + } + + @static lgamA := [?]f64{ + 0h3FB3C467E37DB0C8, + 0h3FD4A34CC4A60FAD, + 0h3FB13E001A5562A7, + 0h3F951322AC92547B, + 0h3F7E404FB68FEFE8, + 0h3F67ADD8CCB7926B, + 0h3F538A94116F3F5D, + 0h3F40B6C689B99C00, + 0h3F2CF2ECED10E54D, + 0h3F1C5088987DFB07, + 0h3EFA7074428CFA52, + 0h3F07858E90A45837, + } + @static lgamR := [?]f64{ + 1.0, + 0h3FF645A762C4AB74, + 0h3FE71A1893D3DCDC, + 0h3FC601EDCCFBDF27, + 0h3F9317EA742ED475, + 0h3F497DDACA41A95B, + 0h3EDEBAF7A5B38140, + } + @static lgamS := [?]f64{ + 0hBFB3C467E37DB0C8, + 0h3FCB848B36E20878, + 0h3FD4D98F4F139F59, + 0h3FC2BB9CBEE5F2F7, + 0h3F9B481C7E939961, + 0h3F5E26B67368F239, + 0h3F00BFECDD17E945, + } + @static lgamT := [?]f64{ + 0h3FDEF72BC8EE38A2, + 0hBFC2E4278DC6C509, + 0h3FB08B4294D5419B, + 0hBFA0C9A8DF35B713, + 0h3F9266E7970AF9EC, + 0hBF851F9FBA91EC6A, + 0h3F78FCE0E370E344, + 0hBF6E2EFFB3E914D7, + 0h3F6282D32E15C915, + 0hBF56FE8EBF2D1AF1, + 0h3F4CDF0CEF61A8E9, + 0hBF41A6109C73E0EC, + 0h3F34AF6D6C0EBBF7, + 0hBF347F24ECC38C38, + 0h3F35FD3EE8C2D3F4, + } + @static lgamU := [?]f64{ + 0hBFB3C467E37DB0C8, + 0h3FE4401E8B005DFF, + 0h3FF7475CD119BD6F, + 0h3FEF497644EA8450, + 0h3FCD4EAEF6010924, + 0h3F8B678BBF2BAB09, + } + @static lgamV := [?]f64{ + 1.0, + 0h4003A5D7C2BD619C, + 0h40010725A42B18F5, + 0h3FE89DFBE45050AF, + 0h3FBAAE55D6537C88, + 0h3F6A5ABB57D0CF61, + } + @static lgamW := [?]f64{ + 0h3FDACFE390C97D69, + 0h3FB555555555553B, + 0hBF66C16C16B02E5C, + 0h3F4A019F98CF38B6, + 0hBF4380CB8C0FE741, + 0h3F4B67BA4CDAD5D1, + 0hBF5AB89D0B9E43E4, + } + + + Y_MIN :: 1.461632144968362245 + TWO_52 :: 0h4330000000000000 // ~4.5036e+15 + TWO_53 :: 0h4340000000000000 // ~9.0072e+15 + TWO_58 :: 0h4390000000000000 // ~2.8823e+17 + TINY :: 0h3b90000000000000 // ~8.47033e-22 + Tc :: 0h3FF762D86356BE3F + Tf :: 0hBFBF19B9BCC38A42 + Tt :: 0hBC50C7CAA48A971F + + // special cases + sign = 1 + switch { + case is_nan(x): + lgamma = x + return + case is_inf(x): + lgamma = x + return + case x == 0: + lgamma = inf_f64(1) + return + } + + x := x + neg := false + if x < 0 { + x = -x + neg = true + } + + if x < TINY { // if |x| < 2**-70, return -log(|x|) + if neg { + sign = -1 + } + lgamma = -ln(x) + return + } + nadj: f64 + if neg { + if x >= TWO_52 { // |x| >= 2**52, must be -integer + lgamma = inf_f64(1) + return + } + t := sin_pi(x) + if t == 0 { + lgamma = inf_f64(1) // -integer + return + } + nadj = ln(PI / abs(t*x)) + if t < 0 { + sign = -1 + } + } + + switch { + case x == 1 || x == 2: // purge off 1 and 2 + lgamma = 0 + return + case x < 2: // use lgamma(x) = lgamma(x+1) - log(x) + y: f64 + i: int + if x <= 0.9 { + lgamma = -ln(x) + switch { + case x >= (Y_MIN - 1 + 0.27): // 0.7316 <= x <= 0.9 + y = 1 - x + i = 0 + case x >= (Y_MIN - 1 - 0.27): // 0.2316 <= x < 0.7316 + y = x - (Tc - 1) + i = 1 + case: // 0 < x < 0.2316 + y = x + i = 2 + } + } else { + lgamma = 0 + switch { + case x >= (Y_MIN + 0.27): // 1.7316 <= x < 2 + y = 2 - x + i = 0 + case x >= (Y_MIN - 0.27): // 1.2316 <= x < 1.7316 + y = x - Tc + i = 1 + case: // 0.9 < x < 1.2316 + y = x - 1 + i = 2 + } + } + switch i { + case 0: + z := y * y + p1 := lgamA[0] + z*(lgamA[2]+z*(lgamA[4]+z*(lgamA[6]+z*(lgamA[8]+z*lgamA[10])))) + p2 := z * (lgamA[1] + z*(+lgamA[3]+z*(lgamA[5]+z*(lgamA[7]+z*(lgamA[9]+z*lgamA[11]))))) + p := y*p1 + p2 + lgamma += (p - 0.5*y) + case 1: + z := y * y + w := z * y + p1 := lgamT[0] + w*(lgamT[3]+w*(lgamT[6]+w*(lgamT[9]+w*lgamT[12]))) // parallel comp + p2 := lgamT[1] + w*(lgamT[4]+w*(lgamT[7]+w*(lgamT[10]+w*lgamT[13]))) + p3 := lgamT[2] + w*(lgamT[5]+w*(lgamT[8]+w*(lgamT[11]+w*lgamT[14]))) + p := z*p1 - (Tt - w*(p2+y*p3)) + lgamma += (Tf + p) + case 2: + p1 := y * (lgamU[0] + y*(lgamU[1]+y*(lgamU[2]+y*(lgamU[3]+y*(lgamU[4]+y*lgamU[5]))))) + p2 := 1 + y*(lgamV[1]+y*(lgamV[2]+y*(lgamV[3]+y*(lgamV[4]+y*lgamV[5])))) + lgamma += (-0.5*y + p1/p2) + } + case x < 8: // 2 <= x < 8 + i := int(x) + y := x - f64(i) + p := y * (lgamS[0] + y*(lgamS[1]+y*(lgamS[2]+y*(lgamS[3]+y*(lgamS[4]+y*(lgamS[5]+y*lgamS[6])))))) + q := 1 + y*(lgamR[1]+y*(lgamR[2]+y*(lgamR[3]+y*(lgamR[4]+y*(lgamR[5]+y*lgamR[6]))))) + lgamma = 0.5*y + p/q + z := 1.0 // lgamma(1+s) = ln(s) + lgamma(s) + switch i { + case 7: + z *= (y + 6) + fallthrough + case 6: + z *= (y + 5) + fallthrough + case 5: + z *= (y + 4) + fallthrough + case 4: + z *= (y + 3) + fallthrough + case 3: + z *= (y + 2) + lgamma += ln(z) + } + case x < TWO_58: // 8 <= x < 2**58 + t := ln(x) + z := 1 / x + y := z * z + w := lgamW[0] + z*(lgamW[1]+y*(lgamW[2]+y*(lgamW[3]+y*(lgamW[4]+y*(lgamW[5]+y*lgamW[6]))))) + lgamma = (x-0.5)*(t-1) + w + case: // 2**58 <= x <= Inf + lgamma = x * (ln(x) - 1) + } + if neg { + lgamma = nadj - lgamma + } + return +} + + +lgamma_f16 :: proc "contextless" (x: f16) -> (lgamma: f16, sign: int) { r, s := lgamma_f64(f64(x)); return f16(r), s } +lgamma_f32 :: proc "contextless" (x: f32) -> (lgamma: f32, sign: int) { r, s := lgamma_f64(f64(x)); return f32(r), s } +lgamma_f16le :: proc "contextless" (x: f16le) -> (lgamma: f16le, sign: int) { r, s := lgamma_f64(f64(x)); return f16le(r), s } +lgamma_f16be :: proc "contextless" (x: f16be) -> (lgamma: f16be, sign: int) { r, s := lgamma_f64(f64(x)); return f16be(r), s } +lgamma_f32le :: proc "contextless" (x: f32le) -> (lgamma: f32le, sign: int) { r, s := lgamma_f64(f64(x)); return f32le(r), s } +lgamma_f32be :: proc "contextless" (x: f32be) -> (lgamma: f32be, sign: int) { r, s := lgamma_f64(f64(x)); return f32be(r), s } +lgamma_f64le :: proc "contextless" (x: f64le) -> (lgamma: f64le, sign: int) { r, s := lgamma_f64(f64(x)); return f64le(r), s } +lgamma_f64be :: proc "contextless" (x: f64be) -> (lgamma: f64be, sign: int) { r, s := lgamma_f64(f64(x)); return f64be(r), s } + +lgamma :: proc{ + lgamma_f16, lgamma_f16le, lgamma_f16be, + lgamma_f32, lgamma_f32le, lgamma_f32be, + lgamma_f64, lgamma_f64le, lgamma_f64be, +} \ No newline at end of file