1 /* $OpenBSD: s_csqrt.c,v 1.6 2013/07/03 04:46:36 espie Exp $ */
2 /*
3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
4 *
5 * Permission to use, copy, modify, and distribute this software for any
6 * purpose with or without fee is hereby granted, provided that the above
7 * copyright notice and this permission notice appear in all copies.
8 *
9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
16 */
17
18 /* csqrt()
19 *
20 * Complex square root
21 *
22 *
23 *
24 * SYNOPSIS:
25 *
26 * double complex csqrt();
27 * double complex z, w;
28 *
29 * w = csqrt (z);
30 *
31 *
32 *
33 * DESCRIPTION:
34 *
35 *
36 * If z = x + iy, r = |z|, then
37 *
38 * 1/2
39 * Re w = [ (r + x)/2 ] ,
40 *
41 * 1/2
42 * Im w = [ (r - x)/2 ] .
43 *
44 * Cancellation error in r-x or r+x is avoided by using the
45 * identity 2 Re w Im w = y.
46 *
47 * Note that -w is also a square root of z. The root chosen
48 * is always in the right half plane and Im w has the same sign as y.
49 *
50 *
51 *
52 * ACCURACY:
53 *
54 * Relative error:
55 * arithmetic domain # trials peak rms
56 * DEC -10,+10 25000 3.2e-17 9.6e-18
57 * IEEE -10,+10 1,000,000 2.9e-16 6.1e-17
58 *
59 */
60
61 #include <complex.h>
62 #include <float.h>
63 #include <math.h>
64
65 double complex
csqrt(double complex z)66 csqrt(double complex z)
67 {
68 double complex w;
69 double x, y, r, t, scale;
70
71 x = creal (z);
72 y = cimag (z);
73
74 if (y == 0.0) {
75 if (x == 0.0) {
76 w = 0.0 + y * I;
77 }
78 else {
79 r = fabs (x);
80 r = sqrt (r);
81 if (x < 0.0) {
82 w = 0.0 + r * I;
83 }
84 else {
85 w = r + y * I;
86 }
87 }
88 return (w);
89 }
90 if (x == 0.0) {
91 r = fabs (y);
92 r = sqrt (0.5*r);
93 if (y > 0)
94 w = r + r * I;
95 else
96 w = r - r * I;
97 return (w);
98 }
99 /* Rescale to avoid internal overflow or underflow. */
100 if ((fabs(x) > 4.0) || (fabs(y) > 4.0)) {
101 x *= 0.25;
102 y *= 0.25;
103 scale = 2.0;
104 }
105 else {
106 x *= 1.8014398509481984e16; /* 2^54 */
107 y *= 1.8014398509481984e16;
108 scale = 7.450580596923828125e-9; /* 2^-27 */
109 #if 0
110 x *= 4.0;
111 y *= 4.0;
112 scale = 0.5;
113 #endif
114 }
115 w = x + y * I;
116 r = cabs(w);
117 if (x > 0) {
118 t = sqrt(0.5 * r + 0.5 * x);
119 r = scale * fabs((0.5 * y) / t);
120 t *= scale;
121 }
122 else {
123 r = sqrt( 0.5 * r - 0.5 * x );
124 t = scale * fabs( (0.5 * y) / r );
125 r *= scale;
126 }
127 if (y < 0)
128 w = t - r * I;
129 else
130 w = t + r * I;
131 return (w);
132 }
133
134 #if LDBL_MANT_DIG == DBL_MANT_DIG
135 __strong_alias(csqrtl, csqrt);
136 #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */
137