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math: add lgamma; in-line special cases of acosh, nextafter
Added lgamma.go, tests and special cases. R=rsc CC=golang-dev https://golang.org/cl/217060
This commit is contained in:
Charles L. Dorian
Russ Cox
parent
2ac3df210b
commit
78e918ccba
@ -52,6 +52,7 @@ ALLGOFILES=\
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fmod.go\
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frexp.go\
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hypot.go\
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lgamma.go\
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ldexp.go\
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log.go\
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log1p.go\
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@ -44,8 +44,11 @@ func Acosh(x float64) float64 {
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Ln2 = 6.93147180559945286227e-01 // 0x3FE62E42FEFA39EF
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Large = 1 << 28 // 2^28
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)
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// TODO(rsc): Remove manual inlining of IsNaN
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// when compiler does it for us
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// first case is special case
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switch {
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case x < 1 || IsNaN(x):
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case x < 1 || x != x: // x < 1 || IsNaN(x):
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return NaN()
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case x == 1:
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return 0
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@ -286,7 +286,18 @@ var frexp = []fi{
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fi{9.1265404584042750000e-01, 1},
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fi{-5.4287029803597508250e-01, 4},
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}
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var lgamma = []fi{
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fi{3.146492141244545774319734e+00, 1},
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fi{8.003414490659126375852113e+00, 1},
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fi{1.517575735509779707488106e+00, -1},
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fi{-2.588480028182145853558748e-01, 1},
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fi{1.1989897050205555002007985e+01, 1},
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fi{6.262899811091257519386906e-01, 1},
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fi{3.5287924899091566764846037e+00, 1},
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fi{4.5725644770161182299423372e-01, 1},
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fi{-6.363667087767961257654854e-02, 1},
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fi{-1.077385130910300066425564e+01, -1},
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}
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var log = []float64{
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1.605231462693062999102599e+00,
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2.0462560018708770653153909e+00,
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@ -736,6 +747,21 @@ var hypotSC = []float64{
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NaN(),
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}
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var vflgammaSC = []float64{
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Inf(-1),
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-3,
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0,
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Inf(1),
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NaN(),
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}
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var lgammaSC = []fi{
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fi{Inf(-1), 1},
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fi{Inf(1), 1},
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fi{Inf(1), 1},
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fi{Inf(1), 1},
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fi{NaN(), 1},
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}
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var vflogSC = []float64{
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Inf(-1),
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-Pi,
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@ -1229,6 +1255,19 @@ func TestLdexp(t *testing.T) {
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}
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}
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func TestLgamma(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f, s := Lgamma(vf[i]); !close(lgamma[i].f, f) || lgamma[i].i != s {
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t.Errorf("Lgamma(%g) = %g, %d, want %g, %d\n", vf[i], f, s, lgamma[i].f, lgamma[i].i)
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}
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}
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for i := 0; i < len(vflgammaSC); i++ {
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if f, s := Lgamma(vflgammaSC[i]); !alike(lgammaSC[i].f, f) || lgammaSC[i].i != s {
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t.Errorf("Lgamma(%g) = %g, %d, want %g, %d\n", vflgammaSC[i], f, s, lgammaSC[i].f, lgammaSC[i].i)
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}
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}
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}
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func TestLog(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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a := Fabs(vf[i])
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@ -1632,6 +1671,12 @@ func BenchmarkLdexp(b *testing.B) {
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}
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}
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func BenchmarkLgamma(b *testing.B) {
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for i := 0; i < b.N; i++ {
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Lgamma(2.5)
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}
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}
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func BenchmarkLog(b *testing.B) {
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for i := 0; i < b.N; i++ {
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Log(.5)
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350
src/pkg/math/lgamma.go
Normal file
350
src/pkg/math/lgamma.go
Normal file
@ -0,0 +1,350 @@
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// Copyright 2010 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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/*
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Floating-point logarithm of the Gamma function.
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*/
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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/e_lgamma_r.c and
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// came with this notice. The go code is a simplified
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// version of the original C.
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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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// __ieee754_lgamma_r(x, signgamp)
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// Reentrant version of the logarithm of the Gamma function
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// with user provided pointer for the sign of Gamma(x).
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//
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// Method:
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// 1. Argument Reduction for 0 < x <= 8
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// Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
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// reduce x to a number in [1.5,2.5] by
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// lgamma(1+s) = log(s) + lgamma(s)
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// for example,
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// lgamma(7.3) = log(6.3) + lgamma(6.3)
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// = log(6.3*5.3) + lgamma(5.3)
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// = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
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// 2. Polynomial approximation of lgamma around its
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// minimum (ymin=1.461632144968362245) to maintain monotonicity.
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// On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
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// Let z = x-ymin;
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// lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
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// poly(z) is a 14 degree polynomial.
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// 2. Rational approximation in the primary interval [2,3]
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// We use the following approximation:
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// s = x-2.0;
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// lgamma(x) = 0.5*s + s*P(s)/Q(s)
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// with accuracy
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// |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
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// Our algorithms are based on the following observation
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//
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// zeta(2)-1 2 zeta(3)-1 3
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// lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
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// 2 3
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//
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// where Euler = 0.5772156649... is the Euler constant, which
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// is very close to 0.5.
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//
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// 3. For x>=8, we have
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// lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
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// (better formula:
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// lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
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// Let z = 1/x, then we approximation
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// f(z) = lgamma(x) - (x-0.5)(log(x)-1)
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// by
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// 3 5 11
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// w = w0 + w1*z + w2*z + w3*z + ... + w6*z
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// where
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// |w - f(z)| < 2**-58.74
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//
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// 4. For negative x, since (G is gamma function)
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// -x*G(-x)*G(x) = pi/sin(pi*x),
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// we have
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// G(x) = pi/(sin(pi*x)*(-x)*G(-x))
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// since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
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// Hence, for x<0, signgam = sign(sin(pi*x)) and
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// lgamma(x) = log(|Gamma(x)|)
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// = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
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// Note: one should avoid computing pi*(-x) directly in the
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// computation of sin(pi*(-x)).
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//
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// 5. Special Cases
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// lgamma(2+s) ~ s*(1-Euler) for tiny s
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// lgamma(1)=lgamma(2)=0
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// lgamma(x) ~ -log(x) for tiny x
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// lgamma(0) = lgamma(inf) = inf
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// lgamma(-integer) = +-inf
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//
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//
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// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
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//
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// Special cases are:
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// Lgamma(+Inf) = +Inf
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// Lgamma(0) = +Inf
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// Lgamma(-integer) = +Inf
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// Lgamma(-Inf) = -Inf
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// Lgamma(NaN) = NaN
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func Lgamma(x float64) (lgamma float64, sign int) {
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const (
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Ymin = 1.461632144968362245
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Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
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Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
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Two58 = 1 << 58 // 0x4390000000000000 ~2.8823e+17
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Tiny = 1.0 / (1 << 70) // 0x3b90000000000000 ~8.47033e-22
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A0 = 7.72156649015328655494e-02 // 0x3FB3C467E37DB0C8
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A1 = 3.22467033424113591611e-01 // 0x3FD4A34CC4A60FAD
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A2 = 6.73523010531292681824e-02 // 0x3FB13E001A5562A7
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A3 = 2.05808084325167332806e-02 // 0x3F951322AC92547B
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A4 = 7.38555086081402883957e-03 // 0x3F7E404FB68FEFE8
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A5 = 2.89051383673415629091e-03 // 0x3F67ADD8CCB7926B
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A6 = 1.19270763183362067845e-03 // 0x3F538A94116F3F5D
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A7 = 5.10069792153511336608e-04 // 0x3F40B6C689B99C00
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A8 = 2.20862790713908385557e-04 // 0x3F2CF2ECED10E54D
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A9 = 1.08011567247583939954e-04 // 0x3F1C5088987DFB07
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A10 = 2.52144565451257326939e-05 // 0x3EFA7074428CFA52
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A11 = 4.48640949618915160150e-05 // 0x3F07858E90A45837
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Tc = 1.46163214496836224576e+00 // 0x3FF762D86356BE3F
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Tf = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
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// Tt = -(tail of Tf)
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Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
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T0 = 4.83836122723810047042e-01 // 0x3FDEF72BC8EE38A2
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T1 = -1.47587722994593911752e-01 // 0xBFC2E4278DC6C509
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T2 = 6.46249402391333854778e-02 // 0x3FB08B4294D5419B
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T3 = -3.27885410759859649565e-02 // 0xBFA0C9A8DF35B713
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T4 = 1.79706750811820387126e-02 // 0x3F9266E7970AF9EC
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T5 = -1.03142241298341437450e-02 // 0xBF851F9FBA91EC6A
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T6 = 6.10053870246291332635e-03 // 0x3F78FCE0E370E344
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T7 = -3.68452016781138256760e-03 // 0xBF6E2EFFB3E914D7
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T8 = 2.25964780900612472250e-03 // 0x3F6282D32E15C915
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T9 = -1.40346469989232843813e-03 // 0xBF56FE8EBF2D1AF1
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T10 = 8.81081882437654011382e-04 // 0x3F4CDF0CEF61A8E9
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T11 = -5.38595305356740546715e-04 // 0xBF41A6109C73E0EC
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T12 = 3.15632070903625950361e-04 // 0x3F34AF6D6C0EBBF7
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T13 = -3.12754168375120860518e-04 // 0xBF347F24ECC38C38
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T14 = 3.35529192635519073543e-04 // 0x3F35FD3EE8C2D3F4
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U0 = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
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U1 = 6.32827064025093366517e-01 // 0x3FE4401E8B005DFF
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U2 = 1.45492250137234768737e+00 // 0x3FF7475CD119BD6F
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U3 = 9.77717527963372745603e-01 // 0x3FEF497644EA8450
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U4 = 2.28963728064692451092e-01 // 0x3FCD4EAEF6010924
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U5 = 1.33810918536787660377e-02 // 0x3F8B678BBF2BAB09
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V1 = 2.45597793713041134822e+00 // 0x4003A5D7C2BD619C
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V2 = 2.12848976379893395361e+00 // 0x40010725A42B18F5
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V3 = 7.69285150456672783825e-01 // 0x3FE89DFBE45050AF
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V4 = 1.04222645593369134254e-01 // 0x3FBAAE55D6537C88
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V5 = 3.21709242282423911810e-03 // 0x3F6A5ABB57D0CF61
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S0 = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
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S1 = 2.14982415960608852501e-01 // 0x3FCB848B36E20878
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S2 = 3.25778796408930981787e-01 // 0x3FD4D98F4F139F59
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S3 = 1.46350472652464452805e-01 // 0x3FC2BB9CBEE5F2F7
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S4 = 2.66422703033638609560e-02 // 0x3F9B481C7E939961
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S5 = 1.84028451407337715652e-03 // 0x3F5E26B67368F239
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S6 = 3.19475326584100867617e-05 // 0x3F00BFECDD17E945
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R1 = 1.39200533467621045958e+00 // 0x3FF645A762C4AB74
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R2 = 7.21935547567138069525e-01 // 0x3FE71A1893D3DCDC
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R3 = 1.71933865632803078993e-01 // 0x3FC601EDCCFBDF27
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R4 = 1.86459191715652901344e-02 // 0x3F9317EA742ED475
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R5 = 7.77942496381893596434e-04 // 0x3F497DDACA41A95B
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R6 = 7.32668430744625636189e-06 // 0x3EDEBAF7A5B38140
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W0 = 4.18938533204672725052e-01 // 0x3FDACFE390C97D69
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W1 = 8.33333333333329678849e-02 // 0x3FB555555555553B
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W2 = -2.77777777728775536470e-03 // 0xBF66C16C16B02E5C
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W3 = 7.93650558643019558500e-04 // 0x3F4A019F98CF38B6
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W4 = -5.95187557450339963135e-04 // 0xBF4380CB8C0FE741
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W5 = 8.36339918996282139126e-04 // 0x3F4B67BA4CDAD5D1
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W6 = -1.63092934096575273989e-03 // 0xBF5AB89D0B9E43E4
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)
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// TODO(rsc): Remove manual inlining of IsNaN, IsInf
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// when compiler does it for us
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// special cases
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sign = 1
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switch {
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case x != x: // IsNaN(x):
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lgamma = x
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return
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case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
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lgamma = x
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return
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case x == 0:
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lgamma = Inf(1)
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return
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}
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neg := false
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if x < 0 {
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x = -x
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neg = true
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}
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if x < Tiny { // if |x| < 2**-70, return -log(|x|)
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if neg {
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sign = -1
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}
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lgamma = -Log(x)
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return
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}
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var nadj float64
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if neg {
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if x >= Two52 { // |x| >= 2**52, must be -integer
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lgamma = Inf(1)
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return
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}
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t := sinPi(x)
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if t == 0 {
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lgamma = Inf(1) // -integer
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return
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}
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nadj = Log(Pi / Fabs(t*x))
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if t < 0 {
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sign = -1
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}
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}
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switch {
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case x == 1 || x == 2: // purge off 1 and 2
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lgamma = 0
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return
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case x < 2: // use lgamma(x) = lgamma(x+1) - log(x)
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var y float64
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var i int
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if x <= 0.9 {
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lgamma = -Log(x)
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switch {
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case x >= (Ymin - 1 + 0.27): // 0.7316 <= x <= 0.9
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y = 1 - x
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i = 0
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case x >= (Ymin - 1 - 0.27): // 0.2316 <= x < 0.7316
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y = x - (Tc - 1)
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i = 1
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default: // 0 < x < 0.2316
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y = x
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i = 2
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}
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} else {
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lgamma = 0
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switch {
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case x >= (Ymin + 0.27): // 1.7316 <= x < 2
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y = 2 - x
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i = 0
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case x >= (Ymin - 0.27): // 1.2316 <= x < 1.7316
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y = x - Tc
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i = 1
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default: // 0.9 < x < 1.2316
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y = x - 1
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i = 2
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}
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}
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switch i {
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case 0:
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z := y * y
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p1 := A0 + z*(A2+z*(A4+z*(A6+z*(A8+z*A10))))
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p2 := z * (A1 + z*(A3+z*(A5+z*(A7+z*(A9+z*A11)))))
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p := y*p1 + p2
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lgamma += (p - 0.5*y)
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case 1:
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z := y * y
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w := z * y
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p1 := T0 + w*(T3+w*(T6+w*(T9+w*T12))) // parallel comp
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p2 := T1 + w*(T4+w*(T7+w*(T10+w*T13)))
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p3 := T2 + w*(T5+w*(T8+w*(T11+w*T14)))
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p := z*p1 - (Tt - w*(p2+y*p3))
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lgamma += (Tf + p)
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case 2:
|
||||
p1 := y * (U0 + y*(U1+y*(U2+y*(U3+y*(U4+y*U5)))))
|
||||
p2 := 1 + y*(V1+y*(V2+y*(V3+y*(V4+y*V5))))
|
||||
lgamma += (-0.5*y + p1/p2)
|
||||
}
|
||||
case x < 8: // 2 <= x < 8
|
||||
i := int(x)
|
||||
y := x - float64(i)
|
||||
p := y * (S0 + y*(S1+y*(S2+y*(S3+y*(S4+y*(S5+y*S6))))))
|
||||
q := 1 + y*(R1+y*(R2+y*(R3+y*(R4+y*(R5+y*R6)))))
|
||||
lgamma = 0.5*y + p/q
|
||||
z := float64(1) // Lgamma(1+s) = Log(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 += Log(z)
|
||||
}
|
||||
case x < Two58: // 8 <= x < 2**58
|
||||
t := Log(x)
|
||||
z := 1 / x
|
||||
y := z * z
|
||||
w := W0 + z*(W1+y*(W2+y*(W3+y*(W4+y*(W5+y*W6)))))
|
||||
lgamma = (x-0.5)*(t-1) + w
|
||||
default: // 2**58 <= x <= Inf
|
||||
lgamma = x * (Log(x) - 1)
|
||||
}
|
||||
if neg {
|
||||
lgamma = nadj - lgamma
|
||||
}
|
||||
return
|
||||
}
|
||||
|
||||
// sinPi(x) is a helper function for negative x
|
||||
func sinPi(x float64) float64 {
|
||||
const (
|
||||
Two52 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
|
||||
Two53 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
|
||||
)
|
||||
if x < 0.25 {
|
||||
return -Sin(Pi * x)
|
||||
}
|
||||
|
||||
// argument reduction
|
||||
z := Floor(x)
|
||||
var n int
|
||||
if z != x { // inexact
|
||||
x = Fmod(x, 2)
|
||||
n = int(x * 4)
|
||||
} else {
|
||||
if x >= Two53 { // x must be even
|
||||
x = 0
|
||||
n = 0
|
||||
} else {
|
||||
if x < Two52 {
|
||||
z = x + Two52 // exact
|
||||
}
|
||||
n = int(1 & Float64bits(z))
|
||||
x = float64(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))
|
||||
default:
|
||||
x = Sin(Pi * (x - 2))
|
||||
}
|
||||
return -x
|
||||
}
|
||||
@ -11,8 +11,10 @@ package math
|
||||
// Nextafter(NaN, y) = NaN
|
||||
// Nextafter(x, NaN) = NaN
|
||||
func Nextafter(x, y float64) (r float64) {
|
||||
// TODO(rsc): Remove manual inlining of IsNaN
|
||||
// when compiler does it for us
|
||||
switch {
|
||||
case IsNaN(x) || IsNaN(y): // special case
|
||||
case x != x || y != y: // IsNaN(x) || IsNaN(y): // special case
|
||||
r = NaN()
|
||||
case x == y:
|
||||
r = x
|
||||
|
||||
Loading…
Reference in New Issue
Block a user