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// Copyright 2011 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package math
/*
Floating-point sine and cosine.
*/
// The original C code, the long comment, and the constants
// below were from http://netlib.sandia.gov/cephes/cmath/sin.c,
// available from http://www.netlib.org/cephes/cmath.tgz.
// The go code is a simplified version of the original C.
//
// sin.c
//
// Circular sine
//
// SYNOPSIS:
//
// double x, y, sin();
// y = sin( x );
//
// DESCRIPTION:
//
// Range reduction is into intervals of pi/4. The reduction error is nearly
// eliminated by contriving an extended precision modular arithmetic.
//
// Two polynomial approximating functions are employed.
// Between 0 and pi/4 the sine is approximated by
// x + x**3 P(x**2).
// Between pi/4 and pi/2 the cosine is represented as
// 1 - x**2 Q(x**2).
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC 0, 10 150000 3.0e-17 7.8e-18
// IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
//
// Partial loss of accuracy begins to occur at x = 2**30 = 1.074e9. The loss
// is not gradual, but jumps suddenly to about 1 part in 10e7. Results may
// be meaningless for x > 2**49 = 5.6e14.
//
// cos.c
//
// Circular cosine
//
// SYNOPSIS:
//
// double x, y, cos();
// y = cos( x );
//
// DESCRIPTION:
//
// Range reduction is into intervals of pi/4. The reduction error is nearly
// eliminated by contriving an extended precision modular arithmetic.
//
// Two polynomial approximating functions are employed.
// Between 0 and pi/4 the cosine is approximated by
// 1 - x**2 Q(x**2).
// Between pi/4 and pi/2 the sine is represented as
// x + x**3 P(x**2).
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
// DEC 0,+1.07e9 17000 3.0e-17 7.2e-18
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
// sin coefficients
var _sin = [...]float64{
1.58962301576546568060e-10, // 0x3de5d8fd1fd19ccd
-2.50507477628578072866e-8, // 0xbe5ae5e5a9291f5d
2.75573136213857245213e-6, // 0x3ec71de3567d48a1
-1.98412698295895385996e-4, // 0xbf2a01a019bfdf03
8.33333333332211858878e-3, // 0x3f8111111110f7d0
-1.66666666666666307295e-1, // 0xbfc5555555555548
}
// cos coefficients
var _cos = [...]float64{
-1.13585365213876817300e-11, // 0xbda8fa49a0861a9b
2.08757008419747316778e-9, // 0x3e21ee9d7b4e3f05
-2.75573141792967388112e-7, // 0xbe927e4f7eac4bc6
2.48015872888517045348e-5, // 0x3efa01a019c844f5
-1.38888888888730564116e-3, // 0xbf56c16c16c14f91
4.16666666666665929218e-2, // 0x3fa555555555554b
}
// Cos returns the cosine of the radian argument x.
//
// Special cases are:
// Cos(±Inf) = NaN
// Cos(NaN) = NaN
func Cos(x float64) float64
func cos(x float64) float64 {
const (
PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts
PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000,
PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
)
// special cases
switch {
case IsNaN(x) || IsInf(x, 0):
return NaN()
}
// make argument positive
sign := false
x = Abs(x)
var j uint64
var y, z float64
if x >= reduceThreshold {
j, z = trigReduce(x)
} else {
j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
y = float64(j) // integer part of x/(Pi/4), as float
// map zeros to origin
if j&1 == 1 {
j++
y++
}
j &= 7 // octant modulo 2Pi radians (360 degrees)
z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
}
if j > 3 {
j -= 4
sign = !sign
}
if j > 1 {
sign = !sign
}
zz := z * z
if j == 1 || j == 2 {
y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
} else {
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])
}
if sign {
y = -y
}
return y
}
// Sin returns the sine of the radian argument x.
//
// Special cases are:
// Sin(±0) = ±0
// Sin(±Inf) = NaN
// Sin(NaN) = NaN
func Sin(x float64) float64
func sin(x float64) float64 {
const (
PI4A = 7.85398125648498535156e-1 // 0x3fe921fb40000000, Pi/4 split into three parts
PI4B = 3.77489470793079817668e-8 // 0x3e64442d00000000,
PI4C = 2.69515142907905952645e-15 // 0x3ce8469898cc5170,
)
// special cases
switch {
case x == 0 || IsNaN(x):
return x // return ±0 || NaN()
case IsInf(x, 0):
return NaN()
}
// make argument positive but save the sign
sign := false
if x < 0 {
x = -x
sign = true
}
var j uint64
var y, z float64
if x >= reduceThreshold {
j, z = trigReduce(x)
} else {
j = uint64(x * (4 / Pi)) // integer part of x/(Pi/4), as integer for tests on the phase angle
y = float64(j) // integer part of x/(Pi/4), as float
// map zeros to origin
if j&1 == 1 {
j++
y++
}
j &= 7 // octant modulo 2Pi radians (360 degrees)
z = ((x - y*PI4A) - y*PI4B) - y*PI4C // Extended precision modular arithmetic
}
// reflect in x axis
if j > 3 {
sign = !sign
j -= 4
}
zz := z * z
if j == 1 || j == 2 {
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])
} else {
y = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
}
if sign {
y = -y
}
return y
}
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