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sincos.go
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sincos.go
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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 math32
import "math/bits"
/*
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 = [...]float32{
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 = [...]float32{
-1.13585365213876817300e-11, // 0xbda8fa49a0861a9b
2.08757008419747316778e-9, // 0x3e21ee9d7b4e3f05
-2.75573141792967388112e-7, // 0xbe927e4f7eac4bc6
2.48015872888517045348e-5, // 0x3efa01a019c844f5
-1.38888888888730564116e-3, // 0xbf56c16c16c14f91
4.16666666666665929218e-2, // 0x3fa555555555554b
}
// Sincos returns Sin(x), Cos(x).
//
// Special cases are:
// Sincos(±0) = ±0, 1
// Sincos(±Inf) = NaN, NaN
// Sincos(NaN) = NaN, NaN
func Sincos(x float32) (sin, cos float32) {
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:
return x, 1 // return ±0.0, 1.0
case IsNaN(x) || IsInf(x, 0):
return NaN(), NaN()
}
// make argument positive
sinSign, cosSign := false, false
if x < 0 {
x = -x
sinSign = true
}
var j uint64
var y, z float32
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 = float32(j) // integer part of x/(Pi/4), as float
if j&1 == 1 { // map zeros to origin
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 { // reflect in x axis
j -= 4
sinSign, cosSign = !sinSign, !cosSign
}
if j > 1 {
cosSign = !cosSign
}
zz := z * z
cos = 1.0 - 0.5*zz + zz*zz*((((((_cos[0]*zz)+_cos[1])*zz+_cos[2])*zz+_cos[3])*zz+_cos[4])*zz+_cos[5])
sin = z + z*zz*((((((_sin[0]*zz)+_sin[1])*zz+_sin[2])*zz+_sin[3])*zz+_sin[4])*zz+_sin[5])
if j == 1 || j == 2 {
sin, cos = cos, sin
}
if cosSign {
cos = -cos
}
if sinSign {
sin = -sin
}
return
}
// Sin returns the sine of the radian argument x.
//
// Special cases are:
// Sin(±0) = ±0
// Sin(±Inf) = NaN
// Sin(NaN) = NaN
func Sin(x float32) float32 {
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 float32
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 = float32(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
}
// Cos returns the cosine of the radian argument x.
//
// Special cases are:
// Cos(±Inf) = NaN
// Cos(NaN) = NaN
func Cos(x float32) float32 {
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 float32
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 = float32(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
}
// reduceThreshold is the maximum value of x where the reduction using Pi/4
// in 3 float64 parts still gives accurate results. This threshold
// is set by y*C being representable as a float64 without error
// where y is given by y = floor(x * (4 / Pi)) and C is the leading partial
// terms of 4/Pi. Since the leading terms (PI4A and PI4B in sin.go) have 30
// and 32 trailing zero bits, y should have less than 30 significant bits.
// y < 1<<30 -> floor(x*4/Pi) < 1<<30 -> x < (1<<30 - 1) * Pi/4
// So, conservatively we can take x < 1<<29.
// Above this threshold Payne-Hanek range reduction must be used.
const reduceThreshold = 1 << 29
// trigReduce implements Payne-Hanek range reduction by Pi/4
// for x > 0. It returns the integer part mod 8 (j) and
// the fractional part (z) of x / (Pi/4).
// The implementation is based on:
// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
// K. C. Ng et al, March 24, 1992
// The simulated multi-precision calculation of x*B uses 64-bit integer arithmetic.
func trigReduce(x float32) (j uint64, z float32) {
const PI4 = Pi / 4
if x < PI4 {
return 0, x
}
// Extract out the integer and exponent such that,
// x = ix * 2 ** exp.
ix := Float32bits(x)
exp := int(ix>>shift&mask) - bias - shift
ix &^= mask << shift
ix |= 1 << shift
// Use the exponent to extract the 3 appropriate uint64 digits from mPi4,
// B ~ (z0, z1, z2), such that the product leading digit has the exponent -61.
// Note, exp >= -53 since x >= PI4 and exp < 971 for maximum float64.
const floatingbits = 32 - 3
digit, bitshift := uint(exp+floatingbits)/32, uint(exp+floatingbits)%32
z0 := (mPi4[digit] << bitshift) | (mPi4[digit+1] >> (32 - bitshift))
z1 := (mPi4[digit+1] << bitshift) | (mPi4[digit+2] >> (32 - bitshift))
z2 := (mPi4[digit+2] << bitshift) | (mPi4[digit+3] >> (32 - bitshift))
// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
z2hi, _ := bits.Mul64(z2, uint64(ix))
z1hi, z1lo := bits.Mul64(z1, uint64(ix))
z0lo := z0 * uint64(ix)
lo, c := bits.Add64(z1lo, z2hi, 0)
hi, _ := bits.Add64(z0lo, z1hi, c)
// The top 3 bits are j.
j = hi >> floatingbits
// Extract the fraction and find its magnitude.
hi = hi<<3 | lo>>floatingbits
lz := uint(bits.LeadingZeros64(hi))
e := uint64(bias - (lz + 1))
// Clear implicit mantissa bit and shift into place.
hi = (hi << (lz + 1)) | (lo >> (32 - (lz + 1)))
hi >>= 43 - shift
// Include the exponent and convert to a float.
hi |= e << shift
z = Float32frombits(uint32(hi))
// Map zeros to origin.
if j&1 == 1 {
j++
j &= 7
z--
}
// Multiply the fractional part by pi/4.
return j, z * PI4
}
// mPi4 is the binary digits of 4/pi as a uint64 array,
// that is, 4/pi = Sum mPi4[i]*2^(-64*i)
// 19 64-bit digits and the leading one bit give 1217 bits
// of precision to handle the largest possible float64 exponent.
var mPi4 = [...]uint64{
0x0000000000000001,
0x45f306dc9c882a53,
0xf84eafa3ea69bb81,
0xb6c52b3278872083,
0xfca2c757bd778ac3,
0x6e48dc74849ba5c0,
0x0c925dd413a32439,
0xfc3bd63962534e7d,
0xd1046bea5d768909,
0xd338e04d68befc82,
0x7323ac7306a673e9,
0x3908bf177bf25076,
0x3ff12fffbc0b301f,
0xde5e2316b414da3e,
0xda6cfd9e4f96136e,
0x9e8c7ecd3cbfd45a,
0xea4f758fd7cbe2f6,
0x7a0e73ef14a525d4,
0xd7f6bf623f1aba10,
0xac06608df8f6d757,
}