# Source file src/math/cmplx/sqrt.go

```     1  // Copyright 2010 The Go Authors. All rights reserved.
2  // Use of this source code is governed by a BSD-style
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
```

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