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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 arc sine

//

// DESCRIPTION:

//

// Inverse complex sine:

//                               2

// w = -i clog( iz + csqrt( 1 - z ) ).

//

// casin(z) = -i casinh(iz)

//

// ACCURACY:

//

//                      Relative error:

// arithmetic   domain     # trials      peak         rms

//    DEC       -10,+10     10100       2.1e-15     3.4e-16

//    IEEE      -10,+10     30000       2.2e-14     2.7e-15

// Larger relative error can be observed for z near zero.

// Also tested by csin(casin(z)) = z.

// Asin returns the inverse sine of x.

func Asin(x complex128) complex128 {

        if imag(x) == 0 {

                if math.Abs(real(x)) > 1 {

                        return complex(math.Pi/2, 0) // DOMAIN error

                }

                return complex(math.Asin(real(x)), 0)

        }

        ct := complex(-imag(x), real(x)) // i * x

        xx := x * x

        x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x

        x2 := Sqrt(x1)                       // x2 = sqrt(1 - x*x)

        w := Log(ct + x2)

        return complex(imag(w), -real(w)) // -i * w

}

// Asinh returns the inverse hyperbolic sine of x.

func Asinh(x complex128) complex128 {

        // TODO check range

        if imag(x) == 0 {

                if math.Abs(real(x)) > 1 {

                        return complex(math.Pi/2, 0) // DOMAIN error

                }

                return complex(math.Asinh(real(x)), 0)

        }

        xx := x * x

        x1 := complex(1+real(xx), imag(xx)) // 1 + x*x

        return Log(x + Sqrt(x1))            // log(x + sqrt(1 + x*x))

}

// Complex circular arc cosine

//

// DESCRIPTION:

//

// w = arccos z  =  PI/2 - arcsin z.

//

// ACCURACY:

//

//                      Relative error:

// arithmetic   domain     # trials      peak         rms

//    DEC       -10,+10      5200      1.6e-15      2.8e-16

//    IEEE      -10,+10     30000      1.8e-14      2.2e-15

// Acos returns the inverse cosine of x.

func Acos(x complex128) complex128 {

        w := Asin(x)

        return complex(math.Pi/2-real(w), -imag(w))

}

// Acosh returns the inverse hyperbolic cosine of x.

func Acosh(x complex128) complex128 {

        w := Acos(x)

        if imag(w) <= 0 {

                return complex(-imag(w), real(w)) // i * w

        }

        return complex(imag(w), -real(w)) // -i * w

}

// Complex circular arc tangent

//

// DESCRIPTION:

//

// If

//     z = x + iy,

//

// then

//          1       (    2x     )

// Re w  =  - arctan(-----------)  +  k PI

//          2       (     2    2)

//                  (1 - x  - y )

//

//               ( 2         2)

//          1    (x  +  (y+1) )

// Im w  =  - log(------------)

//          4    ( 2         2)

//               (x  +  (y-1) )

//

// Where k is an arbitrary integer.

//

// catan(z) = -i catanh(iz).

//

// ACCURACY:

//

//                      Relative error:

// arithmetic   domain     # trials      peak         rms

//    DEC       -10,+10      5900       1.3e-16     7.8e-18

//    IEEE      -10,+10     30000       2.3e-15     8.5e-17

// The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,

// had peak relative error 1.5e-16, rms relative error

// 2.9e-17.  See also clog().

// Atan returns the inverse tangent of x.

func Atan(x complex128) complex128 {

        if real(x) == 0 && imag(x) > 1 {

                return NaN()

        }

        x2 := real(x) * real(x)

        a := 1 - x2 - imag(x)*imag(x)

        if a == 0 {

                return NaN()

        }

        t := 0.5 * math.Atan2(2*real(x), a)

        w := reducePi(t)

        t = imag(x) - 1

        b := x2 + t*t

        if b == 0 {

                return NaN()

        }

        t = imag(x) + 1

        c := (x2 + t*t) / b

        return complex(w, 0.25*math.Log(c))

}

// Atanh returns the inverse hyperbolic tangent of x.

func Atanh(x complex128) complex128 {

        z := complex(-imag(x), real(x)) // z = i * x

        z = Atan(z)

        return complex(imag(z), -real(z)) // z = -i * z

}