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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)
}