sqrt.go

// 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 ]   .
//
// Cancelation 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.
// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(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)
}