1 /*	$OpenBSD: s_catanl.c,v 1.3 2011/07/20 21:02:51 martynas Exp $	*/
2 
3 /*
4  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5  *
6  * Permission to use, copy, modify, and distribute this software for any
7  * purpose with or without fee is hereby granted, provided that the above
8  * copyright notice and this permission notice appear in all copies.
9  *
10  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17  */
18 
19 /*							catanl()
20  *
21  *	Complex circular arc tangent
22  *
23  *
24  *
25  * SYNOPSIS:
26  *
27  * long double complex catanl();
28  * long double complex z, w;
29  *
30  * w = catanl( z );
31  *
32  *
33  *
34  * DESCRIPTION:
35  *
36  * If
37  *     z = x + iy,
38  *
39  * then
40  *          1       (    2x     )
41  * Re w  =  - arctan(-----------)  +  k PI
42  *          2       (     2    2)
43  *                  (1 - x  - y )
44  *
45  *               ( 2         2)
46  *          1    (x  +  (y+1) )
47  * Im w  =  - log(------------)
48  *          4    ( 2         2)
49  *               (x  +  (y-1) )
50  *
51  * Where k is an arbitrary integer.
52  *
53  *
54  *
55  * ACCURACY:
56  *
57  *                      Relative error:
58  * arithmetic   domain     # trials      peak         rms
59  *    DEC       -10,+10      5900       1.3e-16     7.8e-18
60  *    IEEE      -10,+10     30000       2.3e-15     8.5e-17
61  * The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
62  * had peak relative error 1.5e-16, rms relative error
63  * 2.9e-17.  See also clog().
64  */
65 
66 #include <complex.h>
67 #include <float.h>
68 #include <math.h>
69 
70 static const long double PIL = 3.141592653589793238462643383279502884197169L;
71 static const long double DP1 = 3.14159265358979323829596852490908531763125L;
72 static const long double DP2 = 1.6667485837041756656403424829301998703007e-19L;
73 static const long double DP3 = 1.8830410776607851167459095484560349402753e-39L;
74 
75 static long double
76 redupil(long double x)
77 {
78 	long double t;
79 	long i;
80 
81 	t = x / PIL;
82 	if (t >= 0.0L)
83 		t += 0.5L;
84 	else
85 		t -= 0.5L;
86 
87 	i = t;	/* the multiple */
88 	t = i;
89 	t = ((x - t * DP1) - t * DP2) - t * DP3;
90 	return (t);
91 }
92 
93 long double complex
94 catanl(long double complex z)
95 {
96 	long double complex w;
97 	long double a, t, x, x2, y;
98 
99 	x = creall(z);
100 	y = cimagl(z);
101 
102 	if ((x == 0.0L) && (y > 1.0L))
103 		goto ovrf;
104 
105 	x2 = x * x;
106 	a = 1.0L - x2 - (y * y);
107 	if (a == 0.0L)
108 		goto ovrf;
109 
110 	t = atan2l(2.0L * x, a) * 0.5L;
111 	w = redupil(t);
112 
113 	t = y - 1.0L;
114 	a = x2 + (t * t);
115 	if (a == 0.0L)
116 		goto ovrf;
117 
118 	t = y + 1.0L;
119 	a = (x2 + (t * t)) / a;
120 	w = w + (0.25L * logl(a)) * I;
121 	return (w);
122 
123 ovrf:
124 	/*mtherr( "catanl", OVERFLOW );*/
125 	w = LDBL_MAX + LDBL_MAX * I;
126 	return (w);
127 }
128