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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 cmplx

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.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 {

        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.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 += 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.Abs(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.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)

}