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 5 package cmplx
6 7 import "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 // moshier@na-net.ornl.gov
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).
58 func 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