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math: Improved accuracy for Sin and Cos.
Fixes #1564. R=rsc, dchest CC=golang-dev https://golang.org/cl/5320056
This commit is contained in:
Charles L. Dorian
Russ Cox
parent
48c75c5f9c
commit
739c442e42
@ -1709,7 +1709,7 @@ func TestCopysign(t *testing.T) {
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func TestCos(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f := Cos(vf[i]); !close(cos[i], f) {
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if f := Cos(vf[i]); !veryclose(cos[i], f) {
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t.Errorf("Cos(%g) = %g, want %g", vf[i], f, cos[i])
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}
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}
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@ -2192,7 +2192,7 @@ func TestSignbit(t *testing.T) {
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}
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func TestSin(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if f := Sin(vf[i]); !close(sin[i], f) {
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if f := Sin(vf[i]); !veryclose(sin[i], f) {
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t.Errorf("Sin(%g) = %g, want %g", vf[i], f, sin[i])
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}
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}
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@ -2205,7 +2205,7 @@ func TestSin(t *testing.T) {
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func TestSincos(t *testing.T) {
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for i := 0; i < len(vf); i++ {
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if s, c := Sincos(vf[i]); !close(sin[i], s) || !close(cos[i], c) {
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if s, c := Sincos(vf[i]); !veryclose(sin[i], s) || !veryclose(cos[i], c) {
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t.Errorf("Sincos(%g) = %g, %g want %g, %g", vf[i], s, c, sin[i], cos[i])
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}
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}
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@ -1,4 +1,4 @@
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// Copyright 2009 The Go Authors. All rights reserved.
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// Copyright 2011 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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@ -6,60 +6,218 @@ package math
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/*
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Floating-point sine and cosine.
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Coefficients are #5077 from Hart & Cheney. (18.80D)
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*/
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func sinus(x float64, quad int) float64 {
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const (
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P0 = .1357884097877375669092680e8
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P1 = -.4942908100902844161158627e7
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P2 = .4401030535375266501944918e6
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P3 = -.1384727249982452873054457e5
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P4 = .1459688406665768722226959e3
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Q0 = .8644558652922534429915149e7
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Q1 = .4081792252343299749395779e6
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Q2 = .9463096101538208180571257e4
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Q3 = .1326534908786136358911494e3
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)
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if x < 0 {
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x = -x
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quad = quad + 2
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}
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x = x * (2 / Pi) /* underflow? */
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var y float64
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if x > 32764 {
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var e float64
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e, y = Modf(x)
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e = e + float64(quad)
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f, _ := Modf(0.25 * e)
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quad = int(e - 4*f)
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} else {
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k := int32(x)
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y = x - float64(k)
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quad = (quad + int(k)) & 3
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}
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// The original C code, the long comment, and the constants
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// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
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// available from http://www.netlib.org/cephes/cmath.tgz.
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// The go code is a simplified version of the original C.
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//
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// sin.c
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//
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// Circular sine
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//
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// SYNOPSIS:
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//
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// double x, y, sin();
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// y = sin( x );
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//
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// DESCRIPTION:
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//
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// Range reduction is into intervals of pi/4. The reduction error is nearly
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// eliminated by contriving an extended precision modular arithmetic.
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//
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// Two polynomial approximating functions are employed.
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// Between 0 and pi/4 the sine is approximated by
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// x + x**3 P(x**2).
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// Between pi/4 and pi/2 the cosine is represented as
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// 1 - x**2 Q(x**2).
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//
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// ACCURACY:
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//
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// Relative error:
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// arithmetic domain # trials peak rms
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// DEC 0, 10 150000 3.0e-17 7.8e-18
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// IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
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//
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// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9. The loss
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// is not gradual, but jumps suddenly to about 1 part in 10e7. Results may
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// be meaningless for x > 2**49 = 5.6e14.
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//
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// cos.c
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//
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// Circular cosine
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//
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// SYNOPSIS:
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//
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// double x, y, cos();
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// y = cos( x );
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//
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// DESCRIPTION:
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//
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// Range reduction is into intervals of pi/4. The reduction error is nearly
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// eliminated by contriving an extended precision modular arithmetic.
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//
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// Two polynomial approximating functions are employed.
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// Between 0 and pi/4 the cosine is approximated by
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// 1 - x**2 Q(x**2).
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// Between pi/4 and pi/2 the sine is represented as
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// x + x**3 P(x**2).
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//
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// ACCURACY:
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//
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// Relative error:
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// arithmetic domain # trials peak rms
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// IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
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// DEC 0,+1.07e9 17000 3.0e-17 7.2e-18
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//
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// Cephes Math Library Release 2.8: June, 2000
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// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
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//
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// The readme file at http://netlib.sandia.gov/cephes/ says:
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// Some software in this archive may be from the book _Methods and
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// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
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// International, 1989) or from the Cephes Mathematical Library, a
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// commercial product. In either event, it is copyrighted by the author.
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// What you see here may be used freely but it comes with no support or
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// guarantee.
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//
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// The two known misprints in the book are repaired here in the
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// source listings for the gamma function and the incomplete beta
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// integral.
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//
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// Stephen L. Moshier
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// moshier@na-net.ornl.gov
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if quad&1 != 0 {
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y = 1 - y
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}
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if quad > 1 {
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y = -y
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}
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yy := y * y
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temp1 := ((((P4*yy+P3)*yy+P2)*yy+P1)*yy + P0) * y
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temp2 := ((((yy+Q3)*yy+Q2)*yy+Q1)*yy + Q0)
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return temp1 / temp2
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// sin coefficients
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var _sin = [...]float64{
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1.58962301576546568060E-10, // 0x3de5d8fd1fd19ccd
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-2.50507477628578072866E-8, // 0xbe5ae5e5a9291f5d
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2.75573136213857245213E-6, // 0x3ec71de3567d48a1
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-1.98412698295895385996E-4, // 0xbf2a01a019bfdf03
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8.33333333332211858878E-3, // 0x3f8111111110f7d0
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-1.66666666666666307295E-1, // 0xbfc5555555555548
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}
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// cos coefficients
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var _cos = [...]float64{
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-1.13585365213876817300E-11, // 0xbda8fa49a0861a9b
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2.08757008419747316778E-9, // 0x3e21ee9d7b4e3f05
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-2.75573141792967388112E-7, // 0xbe927e4f7eac4bc6
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2.48015872888517045348E-5, // 0x3efa01a019c844f5
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-1.38888888888730564116E-3, // 0xbf56c16c16c14f91
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4.16666666666665929218E-2, // 0x3fa555555555554b
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}
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// Cos returns the cosine of x.
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//
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// Special conditions are:
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// Cos(±Inf) = NaN
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// Cos(NaN) = NaN
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func Cos(x float64) float64 {
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const (
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PI4A = 7.85398125648498535156E-1 // 0x3fe921fb40000000, Pi/4 split into three parts
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PI4B = 3.77489470793079817668E-8 // 0x3e64442d00000000,
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PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170,
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M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
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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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switch {
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case x != x || x < -MaxFloat64 || x > MaxFloat64: // IsNaN(x) || IsInf(x, 0):
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return NaN()
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}
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// make argument positive
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sign := false
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if x < 0 {
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x = -x
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}
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return sinus(x, 1)
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j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
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y := float64(j) // integer part of x/(Pi/4), as float
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// map zeros to origin
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if j&1 == 1 {
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j += 1
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y += 1
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}
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j &= 7 // octant modulo 2Pi radians (360 degrees)
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if j > 3 {
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j -= 4
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sign = !sign
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}
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if j > 1 {
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sign = !sign
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}
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z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
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zz := z * z
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if j == 1 || j == 2 {
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y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
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} else {
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y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
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}
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if sign {
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y = -y
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}
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return y
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}
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// Sin returns the sine of x.
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func Sin(x float64) float64 { return sinus(x, 0) }
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//
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// Special conditions are:
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// Sin(±0) = ±0
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// Sin(±Inf) = NaN
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// Sin(NaN) = NaN
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func Sin(x float64) float64 {
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const (
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PI4A = 7.85398125648498535156E-1 // 0x3fe921fb40000000, Pi/4 split into three parts
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PI4B = 3.77489470793079817668E-8 // 0x3e64442d00000000,
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PI4C = 2.69515142907905952645E-15 // 0x3ce8469898cc5170,
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M4PI = 1.273239544735162542821171882678754627704620361328125 // 4/pi
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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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switch {
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case x == 0 || x != x: // x == 0 || IsNaN():
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return x // return ±0 || NaN()
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case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
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return NaN()
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}
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// make argument positive but save the sign
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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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j := int64(x * M4PI) // integer part of x/(Pi/4), as integer for tests on the phase angle
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y := float64(j) // integer part of x/(Pi/4), as float
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// map zeros to origin
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if j&1 == 1 {
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j += 1
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y += 1
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}
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j &= 7 // octant modulo 2Pi radians (360 degrees)
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// reflect in x axis
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if j > 3 {
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sign = !sign
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j -= 4
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}
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z := ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
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zz := z * z
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if j == 1 || j == 2 {
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y = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
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} else {
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y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
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}
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if sign {
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y = -y
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}
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return y
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}
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