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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.
// 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)
}
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