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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 square root

//

// DESCRIPTION:

//

// If z = x + iy,  r = |z|, then

//

//                       1/2

// Re w  =  [ (r + x)/2 ]   ,

//

//                       1/2

// Im w  =  [ (r - x)/2 ]   .

//

// Cancellation error in r-x or r+x is avoided by using the

// identity  2 Re w Im w  =  y.

//

// Note that -w is also a square root of z.  The root chosen

// is always in the right half plane and Im w has the same sign as y.

//

// ACCURACY:

//

//                      Relative error:

// arithmetic   domain     # trials      peak         rms

//    DEC       -10,+10     25000       3.2e-17     9.6e-18

//    IEEE      -10,+10   1,000,000     2.9e-16     6.1e-17

// Sqrt returns the square root of x.

func Sqrt(x complex128) complex128 {

        if imag(x) == 0 {

                if real(x) == 0 {

                        return complex(0, 0)

                }

                if real(x) < 0 {

                        return complex(0, math.Sqrt(-real(x)))

                }

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

        }

        if real(x) == 0 {

                if imag(x) < 0 {

                        r := math.Sqrt(-0.5 * imag(x))

                        return complex(r, -r)

                }

                r := math.Sqrt(0.5 * imag(x))

                return complex(r, r)

        }

        a := real(x)

        b := imag(x)

        var scale float64

        // Rescale to avoid internal overflow or underflow.

        if math.Abs(a) > 4 || math.Abs(b) > 4 {

                a *= 0.25

                b *= 0.25

                scale = 2

        } else {

                a *= 1.8014398509481984e16 // 2**54

                b *= 1.8014398509481984e16

                scale = 7.450580596923828125e-9 // 2**-27

        }

        r := math.Hypot(a, b)

        var t float64

        if a > 0 {

                t = math.Sqrt(0.5*r + 0.5*a)

                r = scale * math.Abs((0.5*b)/t)

                t *= scale

        } else {

                r = math.Sqrt(0.5*r - 0.5*a)

                t = scale * math.Abs((0.5*b)/r)

                r *= scale

        }

        if b < 0 {

                return complex(t, -r)

        }

        return complex(t, r)

}