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// 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 cmath
import "math"
// 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 {
d := math.Cos(2*real(x)) + math.Cosh(2*imag(x))
if math.Fabs(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 {
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)
}
// Program to subtract nearest integer multiple of PI
func reducePi(x float64) float64 {
const (
// extended precision value of PI:
DP1 = 3.14159265160560607910E0 // ?? 0x400921fb54000000
DP2 = 1.98418714791870343106E-9 // ?? 0x3e210b4610000000
DP3 = 1.14423774522196636802E-17 // ?? 0x3c6a62633145c06e
)
t := x / math.Pi
if t >= 0 {
t += 0.5
} else {
t -= 0.5
}
t = float64(int64(t)) // int64(t) = the multiple
return ((x - t*DP1) - t*DP2) - t*DP3
}
// Taylor series expansion for cosh(2y) - cos(2x)
func tanSeries(z complex128) float64 {
const MACHEP = 1.0 / (1 << 53)
x := math.Fabs(2 * real(z))
y := math.Fabs(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 += 1
f *= rn
rn += 1
f *= rn
x2 *= x
y2 *= y
t := y2 + x2
t /= f
d += t
rn += 1
f *= rn
rn += 1
f *= rn
x2 *= x
y2 *= y
t = y2 - x2
t /= f
d += t
if math.Fabs(t/d) <= MACHEP {
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.Fabs(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)
}
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