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237 lines
6.4 KiB
Go
237 lines
6.4 KiB
Go
// 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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package math
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/*
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Floating-point sine and cosine.
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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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// 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 the radian argument x.
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//
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// Special cases are:
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// Cos(±Inf) = NaN
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// Cos(NaN) = NaN
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//extern cos
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func libc_cos(float64) float64
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func Cos(x float64) float64 {
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return libc_cos(x)
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}
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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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// special cases
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switch {
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case 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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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++
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y++
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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 the radian argument x.
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//
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// Special cases 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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//extern sin
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func libc_sin(float64) float64
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func Sin(x float64) float64 {
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return libc_sin(x)
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}
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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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// special cases
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switch {
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case x == 0 || IsNaN(x):
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return x // return ±0 || NaN()
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case 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++
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y++
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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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