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Add log1p, erf, erfc, ilogb logb (implemented based of FreeBSD's)
This commit is contained in:
gingerBill
@@ -0,0 +1,410 @@
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package math
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// The original C code and the long comment below are
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// from FreeBSD's /usr/src/lib/msun/src/s_erf.c and
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// came with this notice.
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//
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// ====================================================
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// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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//
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// Developed at SunPro, a Sun Microsystems, Inc. business.
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// Permission to use, copy, modify, and distribute this
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// software is freely granted, provided that this notice
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// is preserved.
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// ====================================================
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//
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//
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// double erf(double x)
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// double erfc(double x)
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// x
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// 2 |\
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// erf(x) = --------- | exp(-t*t)dt
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// sqrt(pi) \|
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// 0
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//
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// erfc(x) = 1-erf(x)
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// Note that
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// erf(-x) = -erf(x)
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// erfc(-x) = 2 - erfc(x)
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//
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// Method:
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// 1. For |x| in [0, 0.84375]
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// erf(x) = x + x*R(x**2)
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// erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
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// = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
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// where R = P/Q where P is an odd poly of degree 8 and
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// Q is an odd poly of degree 10.
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// -57.90
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// | R - (erf(x)-x)/x | <= 2
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//
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//
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// Remark. The formula is derived by noting
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// erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....)
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// and that
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// 2/sqrt(pi) = 1.128379167095512573896158903121545171688
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// is close to one. The interval is chosen because the fix
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// point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
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// near 0.6174), and by some experiment, 0.84375 is chosen to
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// guarantee the error is less than one ulp for erf.
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//
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// 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
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// c = 0.84506291151 rounded to single (24 bits)
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// erf(x) = sign(x) * (c + P1(s)/Q1(s))
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// erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
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// 1+(c+P1(s)/Q1(s)) if x < 0
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// |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
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// Remark: here we use the taylor series expansion at x=1.
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// erf(1+s) = erf(1) + s*Poly(s)
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// = 0.845.. + P1(s)/Q1(s)
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// That is, we use rational approximation to approximate
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// erf(1+s) - (c = (single)0.84506291151)
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// Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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// where
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// P1(s) = degree 6 poly in s
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// Q1(s) = degree 6 poly in s
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//
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// 3. For x in [1.25,1/0.35(~2.857143)],
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// erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
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// erf(x) = 1 - erfc(x)
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// where
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// R1(z) = degree 7 poly in z, (z=1/x**2)
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// S1(z) = degree 8 poly in z
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//
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// 4. For x in [1/0.35,28]
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// erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
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// = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
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// = 2.0 - tiny (if x <= -6)
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// erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
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// erf(x) = sign(x)*(1.0 - tiny)
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// where
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// R2(z) = degree 6 poly in z, (z=1/x**2)
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// S2(z) = degree 7 poly in z
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//
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// Note1:
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// To compute exp(-x*x-0.5625+R/S), let s be a single
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// precision number and s := x; then
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// -x*x = -s*s + (s-x)*(s+x)
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// exp(-x*x-0.5626+R/S) =
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// exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
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// Note2:
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// Here 4 and 5 make use of the asymptotic series
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// exp(-x*x)
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// erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) )
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// x*sqrt(pi)
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// We use rational approximation to approximate
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// g(s)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625
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// Here is the error bound for R1/S1 and R2/S2
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// |R1/S1 - f(x)| < 2**(-62.57)
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// |R2/S2 - f(x)| < 2**(-61.52)
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//
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// 5. For inf > x >= 28
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// erf(x) = sign(x) *(1 - tiny) (raise inexact)
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// erfc(x) = tiny*tiny (raise underflow) if x > 0
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// = 2 - tiny if x<0
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//
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// 7. Special case:
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// erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
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// erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
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// erfc/erf(NaN) is NaN
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erf :: proc{
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erf_f16,
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erf_f16le,
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erf_f16be,
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erf_f32,
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erf_f32le,
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erf_f32be,
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erf_f64,
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}
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erf_f16 :: proc "contextless" (x: f16) -> f16 { return f16(erf_f64(f64(x))) }
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erf_f16le :: proc "contextless" (x: f16le) -> f16le { return f16le(erf_f64(f64(x))) }
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erf_f16be :: proc "contextless" (x: f16be) -> f16be { return f16be(erf_f64(f64(x))) }
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erf_f32 :: proc "contextless" (x: f32) -> f32 { return f32(erf_f64(f64(x))) }
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erf_f32le :: proc "contextless" (x: f32le) -> f32le { return f32le(erf_f64(f64(x))) }
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erf_f32be :: proc "contextless" (x: f32be) -> f32be { return f32be(erf_f64(f64(x))) }
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erf_f64 :: proc "contextless" (x: f64) -> f64 {
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erx :: 0h3FEB0AC160000000
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// Coefficients for approximation to erf in [0, 0.84375]
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efx :: 0h3FC06EBA8214DB69
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efx8 :: 0h3FF06EBA8214DB69
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pp0 :: 0h3FC06EBA8214DB68
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pp1 :: 0hBFD4CD7D691CB913
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pp2 :: 0hBF9D2A51DBD7194F
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pp3 :: 0hBF77A291236668E4
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pp4 :: 0hBEF8EAD6120016AC
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qq1 :: 0h3FD97779CDDADC09
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qq2 :: 0h3FB0A54C5536CEBA
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qq3 :: 0h3F74D022C4D36B0F
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qq4 :: 0h3F215DC9221C1A10
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qq5 :: 0hBED09C4342A26120
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// Coefficients for approximation to erf in [0.84375, 1.25]
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pa0 :: 0hBF6359B8BEF77538
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pa1 :: 0h3FDA8D00AD92B34D
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pa2 :: 0hBFD7D240FBB8C3F1
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pa3 :: 0h3FD45FCA805120E4
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pa4 :: 0hBFBC63983D3E28EC
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pa5 :: 0h3FA22A36599795EB
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pa6 :: 0hBF61BF380A96073F
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qa1 :: 0h3FBB3E6618EEE323
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qa2 :: 0h3FE14AF092EB6F33
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qa3 :: 0h3FB2635CD99FE9A7
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qa4 :: 0h3FC02660E763351F
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qa5 :: 0h3F8BEDC26B51DD1C
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qa6 :: 0h3F888B545735151D
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// Coefficients for approximation to erfc in [1.25, 1/0.35]
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ra0 :: 0hBF843412600D6435
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ra1 :: 0hBFE63416E4BA7360
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ra2 :: 0hC0251E0441B0E726
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ra3 :: 0hC04F300AE4CBA38D
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ra4 :: 0hC0644CB184282266
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ra5 :: 0hC067135CEBCCABB2
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ra6 :: 0hC054526557E4D2F2
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ra7 :: 0hC023A0EFC69AC25C
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sa1 :: 0h4033A6B9BD707687
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sa2 :: 0h4061350C526AE721
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sa3 :: 0h407B290DD58A1A71
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sa4 :: 0h40842B1921EC2868
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sa5 :: 0h407AD02157700314
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sa6 :: 0h405B28A3EE48AE2C
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sa7 :: 0h401A47EF8E484A93
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sa8 :: 0hBFAEEFF2EE749A62
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// Coefficients for approximation to erfc in [1/.35, 28]
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rb0 :: 0hBF84341239E86F4A
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rb1 :: 0hBFE993BA70C285DE
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rb2 :: 0hC031C209555F995A
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rb3 :: 0hC064145D43C5ED98
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rb4 :: 0hC083EC881375F228
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rb5 :: 0hC09004616A2E5992
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rb6 :: 0hC07E384E9BDC383F
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sb1 :: 0h403E568B261D5190
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sb2 :: 0h40745CAE221B9F0A
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sb3 :: 0h409802EB189D5118
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sb4 :: 0h40A8FFB7688C246A
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sb5 :: 0h40A3F219CEDF3BE6
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sb6 :: 0h407DA874E79FE763
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sb7 :: 0hC03670E242712D62
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VERY_TINY :: 0h0080000000000000
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SMALL :: 1.0 / (1 << 28) // 2**-28
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// special cases
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switch {
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case is_nan(x):
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return nan_f64()
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case is_inf(x, 1):
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return 1
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case is_inf(x, -1):
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return -1
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}
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x := x
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sign := false
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if x < 0 {
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x = -x
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sign = true
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}
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if x < 0.84375 { // |x| < 0.84375
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temp: f64
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if x < SMALL { // |x| < 2**-28
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if x < VERY_TINY {
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temp = 0.125 * (8.0*x + efx8*x) // avoid underflow
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} else {
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temp = x + efx*x
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}
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} else {
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z := x * x
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r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
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s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
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y := r / s
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temp = x + x*y
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}
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if sign {
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return -temp
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}
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return temp
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}
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if x < 1.25 { // 0.84375 <= |x| < 1.25
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s := x - 1
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P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
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Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
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if sign {
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return -erx - P/Q
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}
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return erx + P/Q
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}
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if x >= 6 { // inf > |x| >= 6
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if sign {
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return -1
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}
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return 1
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}
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s := 1 / (x * x)
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R, S: f64
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if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
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R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
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S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
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} else { // |x| >= 1 / 0.35 ~ 2.857143
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R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
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S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
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}
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z := transmute(f64)(0xffffffff00000000 & transmute(u64)x) // pseudo-single (20-bit) precision x
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r := exp(-z*z-0.5625) * exp((z-x)*(z+x)+R/S)
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if sign {
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return r/x - 1
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}
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return 1 - r/x
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}
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erfc :: proc{
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erfc_f16,
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erfc_f16le,
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erfc_f16be,
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erfc_f32,
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erfc_f32le,
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erfc_f32be,
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erfc_f64,
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}
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erfc_f16 :: proc "contextless" (x: f16) -> f16 { return f16(erfc_f64(f64(x))) }
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erfc_f16le :: proc "contextless" (x: f16le) -> f16le { return f16le(erfc_f64(f64(x))) }
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erfc_f16be :: proc "contextless" (x: f16be) -> f16be { return f16be(erfc_f64(f64(x))) }
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erfc_f32 :: proc "contextless" (x: f32) -> f32 { return f32(erfc_f64(f64(x))) }
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erfc_f32le :: proc "contextless" (x: f32le) -> f32le { return f32le(erfc_f64(f64(x))) }
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erfc_f32be :: proc "contextless" (x: f32be) -> f32be { return f32be(erfc_f64(f64(x))) }
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erfc_f64 :: proc "contextless" (x: f64) -> f64 {
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erx :: 0h3FEB0AC160000000
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// Coefficients for approximation to erf in [0, 0.84375]
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efx :: 0h3FC06EBA8214DB69
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efx8 :: 0h3FF06EBA8214DB69
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pp0 :: 0h3FC06EBA8214DB68
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pp1 :: 0hBFD4CD7D691CB913
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pp2 :: 0hBF9D2A51DBD7194F
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pp3 :: 0hBF77A291236668E4
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pp4 :: 0hBEF8EAD6120016AC
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qq1 :: 0h3FD97779CDDADC09
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qq2 :: 0h3FB0A54C5536CEBA
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qq3 :: 0h3F74D022C4D36B0F
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qq4 :: 0h3F215DC9221C1A10
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qq5 :: 0hBED09C4342A26120
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// Coefficients for approximation to erf in [0.84375, 1.25]
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pa0 :: 0hBF6359B8BEF77538
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pa1 :: 0h3FDA8D00AD92B34D
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pa2 :: 0hBFD7D240FBB8C3F1
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pa3 :: 0h3FD45FCA805120E4
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pa4 :: 0hBFBC63983D3E28EC
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pa5 :: 0h3FA22A36599795EB
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pa6 :: 0hBF61BF380A96073F
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qa1 :: 0h3FBB3E6618EEE323
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qa2 :: 0h3FE14AF092EB6F33
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qa3 :: 0h3FB2635CD99FE9A7
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qa4 :: 0h3FC02660E763351F
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qa5 :: 0h3F8BEDC26B51DD1C
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qa6 :: 0h3F888B545735151D
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// Coefficients for approximation to erfc in [1.25, 1/0.35]
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ra0 :: 0hBF843412600D6435
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ra1 :: 0hBFE63416E4BA7360
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ra2 :: 0hC0251E0441B0E726
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ra3 :: 0hC04F300AE4CBA38D
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ra4 :: 0hC0644CB184282266
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ra5 :: 0hC067135CEBCCABB2
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ra6 :: 0hC054526557E4D2F2
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ra7 :: 0hC023A0EFC69AC25C
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sa1 :: 0h4033A6B9BD707687
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sa2 :: 0h4061350C526AE721
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sa3 :: 0h407B290DD58A1A71
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sa4 :: 0h40842B1921EC2868
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sa5 :: 0h407AD02157700314
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sa6 :: 0h405B28A3EE48AE2C
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sa7 :: 0h401A47EF8E484A93
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sa8 :: 0hBFAEEFF2EE749A62
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// Coefficients for approximation to erfc in [1/.35, 28]
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rb0 :: 0hBF84341239E86F4A
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rb1 :: 0hBFE993BA70C285DE
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rb2 :: 0hC031C209555F995A
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rb3 :: 0hC064145D43C5ED98
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rb4 :: 0hC083EC881375F228
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rb5 :: 0hC09004616A2E5992
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rb6 :: 0hC07E384E9BDC383F
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sb1 :: 0h403E568B261D5190
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sb2 :: 0h40745CAE221B9F0A
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sb3 :: 0h409802EB189D5118
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sb4 :: 0h40A8FFB7688C246A
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sb5 :: 0h40A3F219CEDF3BE6
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sb6 :: 0h407DA874E79FE763
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sb7 :: 0hC03670E242712D62
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TINY :: 1.0 / (1 << 56) // 2**-56
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// special cases
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switch {
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case is_nan(x):
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return nan_f64()
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case is_inf(x, 1):
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return 0
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case is_inf(x, -1):
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return 2
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}
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x := x
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sign := false
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if x < 0 {
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x = -x
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sign = true
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}
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if x < 0.84375 { // |x| < 0.84375
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temp: f64
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if x < TINY { // |x| < 2**-56
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temp = x
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} else {
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z := x * x
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r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
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s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
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y := r / s
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if x < 0.25 { // |x| < 1/4
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temp = x + x*y
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} else {
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temp = 0.5 + (x*y + (x - 0.5))
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}
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}
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if sign {
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return 1 + temp
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}
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return 1 - temp
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}
|
||||
if x < 1.25 { // 0.84375 <= |x| < 1.25
|
||||
s := x - 1
|
||||
P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
|
||||
Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
|
||||
if sign {
|
||||
return 1 + erx + P/Q
|
||||
}
|
||||
return 1 - erx - P/Q
|
||||
|
||||
}
|
||||
if x < 28 { // |x| < 28
|
||||
s := 1 / (x * x)
|
||||
R, S: f64
|
||||
if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
|
||||
R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
|
||||
S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
|
||||
} else { // |x| >= 1 / 0.35 ~ 2.857143
|
||||
if sign && x > 6 {
|
||||
return 2 // x < -6
|
||||
}
|
||||
R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
|
||||
S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
|
||||
}
|
||||
z := transmute(f64)(0xffffffff00000000 & transmute(u64)x) // pseudo-single (20-bit) precision x
|
||||
r := exp(-z*z-0.5625) * exp((z-x)*(z+x)+R/S)
|
||||
if sign {
|
||||
return 2 - r/x
|
||||
}
|
||||
return r / x
|
||||
}
|
||||
if sign {
|
||||
return 2
|
||||
}
|
||||
return 0
|
||||
}
|
||||
Reference in New Issue
Block a user