1// Copyright 2010 The Go Authors. All rights reserved.
2// Use of this source code is governed by a BSD-style
3// license that can be found in the LICENSE file.
4
5package cmplx
6
7import "math"
8
9// The original C code, the long comment, and the constants
10// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c.
11// The go code is a simplified version of the original C.
12//
13// Cephes Math Library Release 2.8:  June, 2000
14// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
15//
16// The readme file at http://netlib.sandia.gov/cephes/ says:
17//    Some software in this archive may be from the book _Methods and
18// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
19// International, 1989) or from the Cephes Mathematical Library, a
20// commercial product. In either event, it is copyrighted by the author.
21// What you see here may be used freely but it comes with no support or
22// guarantee.
23//
24//   The two known misprints in the book are repaired here in the
25// source listings for the gamma function and the incomplete beta
26// integral.
27//
28//   Stephen L. Moshier
29//   [email protected]
30
31// Complex square root
32//
33// DESCRIPTION:
34//
35// If z = x + iy,  r = |z|, then
36//
37//                       1/2
38// Re w  =  [ (r + x)/2 ]   ,
39//
40//                       1/2
41// Im w  =  [ (r - x)/2 ]   .
42//
43// Cancellation error in r-x or r+x is avoided by using the
44// identity  2 Re w Im w  =  y.
45//
46// Note that -w is also a square root of z. The root chosen
47// is always in the right half plane and Im w has the same sign as y.
48//
49// ACCURACY:
50//
51//                      Relative error:
52// arithmetic   domain     # trials      peak         rms
53//    DEC       -10,+10     25000       3.2e-17     9.6e-18
54//    IEEE      -10,+10   1,000,000     2.9e-16     6.1e-17
55
56// Sqrt returns the square root of x.
57// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x).
58func Sqrt(x complex128) complex128 {
59	if imag(x) == 0 {
60		// Ensure that imag(r) has the same sign as imag(x) for imag(x) == signed zero.
61		if real(x) == 0 {
62			return complex(0, imag(x))
63		}
64		if real(x) < 0 {
65			return complex(0, math.Copysign(math.Sqrt(-real(x)), imag(x)))
66		}
67		return complex(math.Sqrt(real(x)), imag(x))
68	} else if math.IsInf(imag(x), 0) {
69		return complex(math.Inf(1.0), imag(x))
70	}
71	if real(x) == 0 {
72		if imag(x) < 0 {
73			r := math.Sqrt(-0.5 * imag(x))
74			return complex(r, -r)
75		}
76		r := math.Sqrt(0.5 * imag(x))
77		return complex(r, r)
78	}
79	a := real(x)
80	b := imag(x)
81	var scale float64
82	// Rescale to avoid internal overflow or underflow.
83	if math.Abs(a) > 4 || math.Abs(b) > 4 {
84		a *= 0.25
85		b *= 0.25
86		scale = 2
87	} else {
88		a *= 1.8014398509481984e16 // 2**54
89		b *= 1.8014398509481984e16
90		scale = 7.450580596923828125e-9 // 2**-27
91	}
92	r := math.Hypot(a, b)
93	var t float64
94	if a > 0 {
95		t = math.Sqrt(0.5*r + 0.5*a)
96		r = scale * math.Abs((0.5*b)/t)
97		t *= scale
98	} else {
99		r = math.Sqrt(0.5*r - 0.5*a)
100		t = scale * math.Abs((0.5*b)/r)
101		r *= scale
102	}
103	if b < 0 {
104		return complex(t, -r)
105	}
106	return complex(t, r)
107}
108