1/* origin: OpenBSD /usr/src/lib/libm/src/s_catan.c */
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 * Complex circular arc tangent
19 *
20 *
21 * SYNOPSIS:
22 *
23 * double complex catan();
24 * double complex z, w;
25 *
26 * w = catan (z);
27 *
28 *
29 * DESCRIPTION:
30 *
31 * If
32 * z = x + iy,
33 *
34 * then
35 * 1 ( 2x )
36 * Re w = - arctan(-----------) + k PI
37 * 2 ( 2 2)
38 * (1 - x - y )
39 *
40 * ( 2 2)
41 * 1 (x + (y+1) )
42 * Im w = - log(------------)
43 * 4 ( 2 2)
44 * (x + (y-1) )
45 *
46 * Where k is an arbitrary integer.
47 *
48 * catan(z) = -i catanh(iz).
49 *
50 * ACCURACY:
51 *
52 * Relative error:
53 * arithmetic domain # trials peak rms
54 * DEC -10,+10 5900 1.3e-16 7.8e-18
55 * IEEE -10,+10 30000 2.3e-15 8.5e-17
56 * The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
57 * had peak relative error 1.5e-16, rms relative error
58 * 2.9e-17. See also clog().
59 */
60
61#include "complex_impl.h"
62
63#define MAXNUM 1.0e308
64
65static const double DP1 = 3.14159265160560607910E0;
66static const double DP2 = 1.98418714791870343106E-9;
67static const double DP3 = 1.14423774522196636802E-17;
68
69static double _redupi(double x)
70{
71 double t;
72 long i;
73
74 t = x/M_PI;
75 if (t >= 0.0)
76 t += 0.5;
77 else
78 t -= 0.5;
79
80 i = t; /* the multiple */
81 t = i;
82 t = ((x - t * DP1) - t * DP2) - t * DP3;
83 return t;
84}
85
86double complex catan(double complex z)
87{
88 double complex w;
89 double a, t, x, x2, y;
90
91 x = creal(z);
92 y = cimag(z);
93
94 x2 = x * x;
95 a = 1.0 - x2 - (y * y);
96
97 t = 0.5 * atan2(2.0 * x, a);
98 w = _redupi(t);
99
100 t = y - 1.0;
101 a = x2 + (t * t);
102
103 t = y + 1.0;
104 a = (x2 + t * t)/a;
105 w = CMPLX(w, 0.25 * log(a));
106 return w;
107}