1*05a0b428SJohn Marino /* $OpenBSD: s_catanl.c,v 1.3 2011/07/20 21:02:51 martynas Exp $ */
2*05a0b428SJohn Marino
3*05a0b428SJohn Marino /*
4*05a0b428SJohn Marino * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5*05a0b428SJohn Marino *
6*05a0b428SJohn Marino * Permission to use, copy, modify, and distribute this software for any
7*05a0b428SJohn Marino * purpose with or without fee is hereby granted, provided that the above
8*05a0b428SJohn Marino * copyright notice and this permission notice appear in all copies.
9*05a0b428SJohn Marino *
10*05a0b428SJohn Marino * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11*05a0b428SJohn Marino * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12*05a0b428SJohn Marino * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13*05a0b428SJohn Marino * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14*05a0b428SJohn Marino * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15*05a0b428SJohn Marino * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16*05a0b428SJohn Marino * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17*05a0b428SJohn Marino */
18*05a0b428SJohn Marino
19*05a0b428SJohn Marino /* catanl()
20*05a0b428SJohn Marino *
21*05a0b428SJohn Marino * Complex circular arc tangent
22*05a0b428SJohn Marino *
23*05a0b428SJohn Marino *
24*05a0b428SJohn Marino *
25*05a0b428SJohn Marino * SYNOPSIS:
26*05a0b428SJohn Marino *
27*05a0b428SJohn Marino * long double complex catanl();
28*05a0b428SJohn Marino * long double complex z, w;
29*05a0b428SJohn Marino *
30*05a0b428SJohn Marino * w = catanl( z );
31*05a0b428SJohn Marino *
32*05a0b428SJohn Marino *
33*05a0b428SJohn Marino *
34*05a0b428SJohn Marino * DESCRIPTION:
35*05a0b428SJohn Marino *
36*05a0b428SJohn Marino * If
37*05a0b428SJohn Marino * z = x + iy,
38*05a0b428SJohn Marino *
39*05a0b428SJohn Marino * then
40*05a0b428SJohn Marino * 1 ( 2x )
41*05a0b428SJohn Marino * Re w = - arctan(-----------) + k PI
42*05a0b428SJohn Marino * 2 ( 2 2)
43*05a0b428SJohn Marino * (1 - x - y )
44*05a0b428SJohn Marino *
45*05a0b428SJohn Marino * ( 2 2)
46*05a0b428SJohn Marino * 1 (x + (y+1) )
47*05a0b428SJohn Marino * Im w = - log(------------)
48*05a0b428SJohn Marino * 4 ( 2 2)
49*05a0b428SJohn Marino * (x + (y-1) )
50*05a0b428SJohn Marino *
51*05a0b428SJohn Marino * Where k is an arbitrary integer.
52*05a0b428SJohn Marino *
53*05a0b428SJohn Marino *
54*05a0b428SJohn Marino *
55*05a0b428SJohn Marino * ACCURACY:
56*05a0b428SJohn Marino *
57*05a0b428SJohn Marino * Relative error:
58*05a0b428SJohn Marino * arithmetic domain # trials peak rms
59*05a0b428SJohn Marino * DEC -10,+10 5900 1.3e-16 7.8e-18
60*05a0b428SJohn Marino * IEEE -10,+10 30000 2.3e-15 8.5e-17
61*05a0b428SJohn Marino * The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
62*05a0b428SJohn Marino * had peak relative error 1.5e-16, rms relative error
63*05a0b428SJohn Marino * 2.9e-17. See also clog().
64*05a0b428SJohn Marino */
65*05a0b428SJohn Marino
66*05a0b428SJohn Marino #include <complex.h>
67*05a0b428SJohn Marino #include <float.h>
68*05a0b428SJohn Marino #include <math.h>
69*05a0b428SJohn Marino
70*05a0b428SJohn Marino static const long double PIL = 3.141592653589793238462643383279502884197169L;
71*05a0b428SJohn Marino static const long double DP1 = 3.14159265358979323829596852490908531763125L;
72*05a0b428SJohn Marino static const long double DP2 = 1.6667485837041756656403424829301998703007e-19L;
73*05a0b428SJohn Marino static const long double DP3 = 1.8830410776607851167459095484560349402753e-39L;
74*05a0b428SJohn Marino
75*05a0b428SJohn Marino static long double
redupil(long double x)76*05a0b428SJohn Marino redupil(long double x)
77*05a0b428SJohn Marino {
78*05a0b428SJohn Marino long double t;
79*05a0b428SJohn Marino long i;
80*05a0b428SJohn Marino
81*05a0b428SJohn Marino t = x / PIL;
82*05a0b428SJohn Marino if (t >= 0.0L)
83*05a0b428SJohn Marino t += 0.5L;
84*05a0b428SJohn Marino else
85*05a0b428SJohn Marino t -= 0.5L;
86*05a0b428SJohn Marino
87*05a0b428SJohn Marino i = t; /* the multiple */
88*05a0b428SJohn Marino t = i;
89*05a0b428SJohn Marino t = ((x - t * DP1) - t * DP2) - t * DP3;
90*05a0b428SJohn Marino return (t);
91*05a0b428SJohn Marino }
92*05a0b428SJohn Marino
93*05a0b428SJohn Marino long double complex
catanl(long double complex z)94*05a0b428SJohn Marino catanl(long double complex z)
95*05a0b428SJohn Marino {
96*05a0b428SJohn Marino long double complex w;
97*05a0b428SJohn Marino long double a, t, x, x2, y;
98*05a0b428SJohn Marino
99*05a0b428SJohn Marino x = creall(z);
100*05a0b428SJohn Marino y = cimagl(z);
101*05a0b428SJohn Marino
102*05a0b428SJohn Marino if ((x == 0.0L) && (y > 1.0L))
103*05a0b428SJohn Marino goto ovrf;
104*05a0b428SJohn Marino
105*05a0b428SJohn Marino x2 = x * x;
106*05a0b428SJohn Marino a = 1.0L - x2 - (y * y);
107*05a0b428SJohn Marino if (a == 0.0L)
108*05a0b428SJohn Marino goto ovrf;
109*05a0b428SJohn Marino
110*05a0b428SJohn Marino t = atan2l(2.0L * x, a) * 0.5L;
111*05a0b428SJohn Marino w = redupil(t);
112*05a0b428SJohn Marino
113*05a0b428SJohn Marino t = y - 1.0L;
114*05a0b428SJohn Marino a = x2 + (t * t);
115*05a0b428SJohn Marino if (a == 0.0L)
116*05a0b428SJohn Marino goto ovrf;
117*05a0b428SJohn Marino
118*05a0b428SJohn Marino t = y + 1.0L;
119*05a0b428SJohn Marino a = (x2 + (t * t)) / a;
120*05a0b428SJohn Marino w = w + (0.25L * logl(a)) * I;
121*05a0b428SJohn Marino return (w);
122*05a0b428SJohn Marino
123*05a0b428SJohn Marino ovrf:
124*05a0b428SJohn Marino /*mtherr( "catanl", OVERFLOW );*/
125*05a0b428SJohn Marino w = LDBL_MAX + LDBL_MAX * I;
126*05a0b428SJohn Marino return (w);
127*05a0b428SJohn Marino }
128