298 lines
8.5 KiB
Go
298 lines
8.5 KiB
Go
|
// Copyright 2010 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 cmplx
|
||
|
|
|
||
|
|
import (
|
||
|
|
"math"
|
||
|
|
"math/bits"
|
||
|
|
)
|
||
|
|
|
||
|
|
// The original C code, the long comment, and the constants
|
||
|
|
// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
|
||
|
|
// The go code is a simplified version of the original C.
|
||
|
|
//
|
||
|
|
// 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
|
||
|
|
|
||
|
|
// Complex circular tangent
|
||
|
|
//
|
||
|
|
// DESCRIPTION:
|
||
|
|
//
|
||
|
|
// If
|
||
|
|
// z = x + iy,
|
||
|
|
//
|
||
|
|
// then
|
||
|
|
//
|
||
|
|
// sin 2x + i sinh 2y
|
||
|
|
// w = --------------------.
|
||
|
|
// cos 2x + cosh 2y
|
||
|
|
//
|
||
|
|
// On the real axis the denominator is zero at odd multiples
|
||
|
|
// of PI/2. The denominator is evaluated by its Taylor
|
||
|
|
// series near these points.
|
||
|
|
//
|
||
|
|
// ctan(z) = -i ctanh(iz).
|
||
|
|
//
|
||
|
|
// ACCURACY:
|
||
|
|
//
|
||
|
|
// Relative error:
|
||
|
|
// arithmetic domain # trials peak rms
|
||
|
|
// DEC -10,+10 5200 7.1e-17 1.6e-17
|
||
|
|
// IEEE -10,+10 30000 7.2e-16 1.2e-16
|
||
|
|
// Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
|
||
|
|
|
||
|
|
// Tan returns the tangent of x.
|
||
|
|
func Tan(x complex128) complex128 {
|
||
|
|
switch re, im := real(x), imag(x); {
|
||
|
|
case math.IsInf(im, 0):
|
||
|
|
switch {
|
||
|
|
case math.IsInf(re, 0) || math.IsNaN(re):
|
||
|
|
return complex(math.Copysign(0, re), math.Copysign(1, im))
|
||
|
|
}
|
||
|
|
return complex(math.Copysign(0, math.Sin(2*re)), math.Copysign(1, im))
|
||
|
|
case re == 0 && math.IsNaN(im):
|
||
|
|
return x
|
||
|
|
}
|
||
|
|
d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
|
||
|
|
if math.Abs(d) < 0.25 {
|
||
|
|
d = tanSeries(x)
|
||
|
|
}
|
||
|
|
if d == 0 {
|
||
|
|
return Inf()
|
||
|
|
}
|
||
|
|
return complex(math.Sin(2*real(x))/d, math.Sinh(2*imag(x))/d)
|
||
|
|
}
|
||
|
|
|
||
|
|
// Complex hyperbolic tangent
|
||
|
|
//
|
||
|
|
// DESCRIPTION:
|
||
|
|
//
|
||
|
|
// tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) .
|
||
|
|
//
|
||
|
|
// ACCURACY:
|
||
|
|
//
|
||
|
|
// Relative error:
|
||
|
|
// arithmetic domain # trials peak rms
|
||
|
|
// IEEE -10,+10 30000 1.7e-14 2.4e-16
|
||
|
|
|
||
|
|
// Tanh returns the hyperbolic tangent of x.
|
||
|
|
func Tanh(x complex128) complex128 {
|
||
|
|
switch re, im := real(x), imag(x); {
|
||
|
|
case math.IsInf(re, 0):
|
||
|
|
switch {
|
||
|
|
case math.IsInf(im, 0) || math.IsNaN(im):
|
||
|
|
return complex(math.Copysign(1, re), math.Copysign(0, im))
|
||
|
|
}
|
||
|
|
return complex(math.Copysign(1, re), math.Copysign(0, math.Sin(2*im)))
|
||
|
|
case im == 0 && math.IsNaN(re):
|
||
|
|
return x
|
||
|
|
}
|
||
|
|
d := math.Cosh(2*real(x)) + math.Cos(2*imag(x))
|
||
|
|
if d == 0 {
|
||
|
|
return Inf()
|
||
|
|
}
|
||
|
|
return complex(math.Sinh(2*real(x))/d, math.Sin(2*imag(x))/d)
|
||
|
|
}
|
||
|
|
|
||
|
|
// reducePi reduces the input argument x to the range (-Pi/2, Pi/2].
|
||
|
|
// x must be greater than or equal to 0. For small arguments it
|
||
|
|
// uses Cody-Waite reduction in 3 float64 parts based on:
|
||
|
|
// "Elementary Function Evaluation: Algorithms and Implementation"
|
||
|
|
// Jean-Michel Muller, 1997.
|
||
|
|
// For very large arguments it uses Payne-Hanek range reduction based on:
|
||
|
|
// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit"
|
||
|
|
// K. C. Ng et al, March 24, 1992.
|
||
|
|
func reducePi(x float64) float64 {
|
||
|
|
// reduceThreshold is the maximum value of x where the reduction using
|
||
|
|
// Cody-Waite reduction still gives accurate results. This threshold
|
||
|
|
// is set by t*PIn being representable as a float64 without error
|
||
|
|
// where t is given by t = floor(x * (1 / Pi)) and PIn are the leading partial
|
||
|
|
// terms of Pi. Since the leading terms, PI1 and PI2 below, have 30 and 32
|
||
|
|
// trailing zero bits respectively, t should have less than 30 significant bits.
|
||
|
|
// t < 1<<30 -> floor(x*(1/Pi)+0.5) < 1<<30 -> x < (1<<30-1) * Pi - 0.5
|
||
|
|
// So, conservatively we can take x < 1<<30.
|
||
|
|
const reduceThreshold float64 = 1 << 30
|
||
|
|
if math.Abs(x) < reduceThreshold {
|
||
|
|
// Use Cody-Waite reduction in three parts.
|
||
|
|
const (
|
||
|
|
// PI1, PI2 and PI3 comprise an extended precision value of PI
|
||
|
|
// such that PI ~= PI1 + PI2 + PI3. The parts are chosen so
|
||
|
|
// that PI1 and PI2 have an approximately equal number of trailing
|
||
|
|
// zero bits. This ensures that t*PI1 and t*PI2 are exact for
|
||
|
|
// large integer values of t. The full precision PI3 ensures the
|
||
|
|
// approximation of PI is accurate to 102 bits to handle cancellation
|
||
|
|
// during subtraction.
|
||
|
|
PI1 = 3.141592502593994 // 0x400921fb40000000
|
||
|
|
PI2 = 1.5099578831723193e-07 // 0x3e84442d00000000
|
||
|
|
PI3 = 1.0780605716316238e-14 // 0x3d08469898cc5170
|
||
|
|
)
|
||
|
|
t := x / math.Pi
|
||
|
|
t += 0.5
|
||
|
|
t = float64(int64(t)) // int64(t) = the multiple
|
||
|
|
return ((x - t*PI1) - t*PI2) - t*PI3
|
||
|
|
}
|
||
|
|
// Must apply Payne-Hanek range reduction
|
||
|
|
const (
|
||
|
|
mask = 0x7FF
|
||
|
|
shift = 64 - 11 - 1
|
||
|
|
bias = 1023
|
||
|
|
fracMask = 1<<shift - 1
|
||
|
|
)
|
||
|
|
// Extract out the integer and exponent such that,
|
||
|
|
// x = ix * 2 ** exp.
|
||
|
|
ix := math.Float64bits(x)
|
||
|
|
exp := int(ix>>shift&mask) - bias - shift
|
||
|
|
ix &= fracMask
|
||
|
|
ix |= 1 << shift
|
||
|
|
|
||
|
|
// mPi is the binary digits of 1/Pi as a uint64 array,
|
||
|
|
// that is, 1/Pi = Sum mPi[i]*2^(-64*i).
|
||
|
|
// 19 64-bit digits give 1216 bits of precision
|
||
|
|
// to handle the largest possible float64 exponent.
|
||
|
|
var mPi = [...]uint64{
|
||
|
|
0x0000000000000000,
|
||
|
|
0x517cc1b727220a94,
|
||
|
|
0xfe13abe8fa9a6ee0,
|
||
|
|
0x6db14acc9e21c820,
|
||
|
|
0xff28b1d5ef5de2b0,
|
||
|
|
0xdb92371d2126e970,
|
||
|
|
0x0324977504e8c90e,
|
||
|
|
0x7f0ef58e5894d39f,
|
||
|
|
0x74411afa975da242,
|
||
|
|
0x74ce38135a2fbf20,
|
||
|
|
0x9cc8eb1cc1a99cfa,
|
||
|
|
0x4e422fc5defc941d,
|
||
|
|
0x8ffc4bffef02cc07,
|
||
|
|
0xf79788c5ad05368f,
|
||
|
|
0xb69b3f6793e584db,
|
||
|
|
0xa7a31fb34f2ff516,
|
||
|
|
0xba93dd63f5f2f8bd,
|
||
|
|
0x9e839cfbc5294975,
|
||
|
|
0x35fdafd88fc6ae84,
|
||
|
|
0x2b0198237e3db5d5,
|
||
|
|
}
|
||
|
|
// Use the exponent to extract the 3 appropriate uint64 digits from mPi,
|
||
|
|
// B ~ (z0, z1, z2), such that the product leading digit has the exponent -64.
|
||
|
|
// Note, exp >= 50 since x >= reduceThreshold and exp < 971 for maximum float64.
|
||
|
|
digit, bitshift := uint(exp+64)/64, uint(exp+64)%64
|
||
|
|
z0 := (mPi[digit] << bitshift) | (mPi[digit+1] >> (64 - bitshift))
|
||
|
|
z1 := (mPi[digit+1] << bitshift) | (mPi[digit+2] >> (64 - bitshift))
|
||
|
|
z2 := (mPi[digit+2] << bitshift) | (mPi[digit+3] >> (64 - bitshift))
|
||
|
|
// Multiply mantissa by the digits and extract the upper two digits (hi, lo).
|
||
|
|
z2hi, _ := bits.Mul64(z2, ix)
|
||
|
|
z1hi, z1lo := bits.Mul64(z1, ix)
|
||
|
|
z0lo := z0 * ix
|
||
|
|
lo, c := bits.Add64(z1lo, z2hi, 0)
|
||
|
|
hi, _ := bits.Add64(z0lo, z1hi, c)
|
||
|
|
// Find the magnitude of the fraction.
|
||
|
|
lz := uint(bits.LeadingZeros64(hi))
|
||
|
|
e := uint64(bias - (lz + 1))
|
||
|
|
// Clear implicit mantissa bit and shift into place.
|
||
|
|
hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1)))
|
||
|
|
hi >>= 64 - shift
|
||
|
|
// Include the exponent and convert to a float.
|
||
|
|
hi |= e << shift
|
||
|
|
x = math.Float64frombits(hi)
|
||
|
|
// map to (-Pi/2, Pi/2]
|
||
|
|
if x > 0.5 {
|
||
|
|
x--
|
||
|
|
}
|
||
|
|
return math.Pi * x
|
||
|
|
}
|
||
|
|
|
||
|
|
// Taylor series expansion for cosh(2y) - cos(2x)
|
||
|
|
func tanSeries(z complex128) float64 {
|
||
|
|
const MACHEP = 1.0 / (1 << 53)
|
||
|
|
x := math.Abs(2 * real(z))
|
||
|
|
y := math.Abs(2 * imag(z))
|
||
|
|
x = reducePi(x)
|
||
|
|
x = x * x
|
||
|
|
y = y * y
|
||
|
|
x2 := 1.0
|
||
|
|
y2 := 1.0
|
||
|
|
f := 1.0
|
||
|
|
rn := 0.0
|
||
|
|
d := 0.0
|
||
|
|
for {
|
||
|
|
rn++
|
||
|
|
f *= rn
|
||
|
|
rn++
|
||
|
|
f *= rn
|
||
|
|
x2 *= x
|
||
|
|
y2 *= y
|
||
|
|
t := y2 + x2
|
||
|
|
t /= f
|
||
|
|
d += t
|
||
|
|
|
||
|
|
rn++
|
||
|
|
f *= rn
|
||
|
|
rn++
|
||
|
|
f *= rn
|
||
|
|
x2 *= x
|
||
|
|
y2 *= y
|
||
|
|
t = y2 - x2
|
||
|
|
t /= f
|
||
|
|
d += t
|
||
|
|
if !(math.Abs(t/d) > MACHEP) {
|
||
|
|
// Caution: Use ! and > instead of <= for correct behavior if t/d is NaN.
|
||
|
|
// See issue 17577.
|
||
|
|
break
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return d
|
||
|
|
}
|
||
|
|
|
||
|
|
// Complex circular cotangent
|
||
|
|
//
|
||
|
|
// DESCRIPTION:
|
||
|
|
//
|
||
|
|
// If
|
||
|
|
// z = x + iy,
|
||
|
|
//
|
||
|
|
// then
|
||
|
|
//
|
||
|
|
// sin 2x - i sinh 2y
|
||
|
|
// w = --------------------.
|
||
|
|
// cosh 2y - cos 2x
|
||
|
|
//
|
||
|
|
// On the real axis, the denominator has zeros at even
|
||
|
|
// multiples of PI/2. Near these points it is evaluated
|
||
|
|
// by a Taylor series.
|
||
|
|
//
|
||
|
|
// ACCURACY:
|
||
|
|
//
|
||
|
|
// Relative error:
|
||
|
|
// arithmetic domain # trials peak rms
|
||
|
|
// DEC -10,+10 3000 6.5e-17 1.6e-17
|
||
|
|
// IEEE -10,+10 30000 9.2e-16 1.2e-16
|
||
|
|
// Also tested by ctan * ccot = 1 + i0.
|
||
|
|
|
||
|
|
// Cot returns the cotangent of x.
|
||
|
|
func Cot(x complex128) complex128 {
|
||
|
|
d := math.Cosh(2*imag(x)) - math.Cos(2*real(x))
|
||
|
|
if math.Abs(d) < 0.25 {
|
||
|
|
d = tanSeries(x)
|
||
|
|
}
|
||
|
|
if d == 0 {
|
||
|
|
return Inf()
|
||
|
|
}
|
||
|
|
return complex(math.Sin(2*real(x))/d, -math.Sinh(2*imag(x))/d)
|
||
|
|
}
|