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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 {
+ // Ensure that imag(r) has the same sign as imag(x) for imag(x) == signed zero.
+ if real(x) == 0 {
+ return complex(0, imag(x))
+ }
+ if real(x) < 0 {
+ return complex(0, math.Copysign(math.Sqrt(-real(x)), imag(x)))
+ }
+ return complex(math.Sqrt(real(x)), imag(x))
+ } else if math.IsInf(imag(x), 0) {
+ return complex(math.Inf(1.0), imag(x))
+ }
+ 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)
+}