1*05a0b428SJohn Marino /* $OpenBSD: s_catan.c,v 1.6 2013/07/03 04:46:36 espie Exp $ */
2*05a0b428SJohn Marino /*
3*05a0b428SJohn Marino * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
4*05a0b428SJohn Marino *
5*05a0b428SJohn Marino * Permission to use, copy, modify, and distribute this software for any
6*05a0b428SJohn Marino * purpose with or without fee is hereby granted, provided that the above
7*05a0b428SJohn Marino * copyright notice and this permission notice appear in all copies.
8*05a0b428SJohn Marino *
9*05a0b428SJohn Marino * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
10*05a0b428SJohn Marino * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
11*05a0b428SJohn Marino * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
12*05a0b428SJohn Marino * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
13*05a0b428SJohn Marino * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
14*05a0b428SJohn Marino * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
15*05a0b428SJohn Marino * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
16*05a0b428SJohn Marino */
17*05a0b428SJohn Marino
18*05a0b428SJohn Marino /* catan()
19*05a0b428SJohn Marino *
20*05a0b428SJohn Marino * Complex circular arc tangent
21*05a0b428SJohn Marino *
22*05a0b428SJohn Marino *
23*05a0b428SJohn Marino *
24*05a0b428SJohn Marino * SYNOPSIS:
25*05a0b428SJohn Marino *
26*05a0b428SJohn Marino * double complex catan();
27*05a0b428SJohn Marino * double complex z, w;
28*05a0b428SJohn Marino *
29*05a0b428SJohn Marino * w = catan (z);
30*05a0b428SJohn Marino *
31*05a0b428SJohn Marino *
32*05a0b428SJohn Marino *
33*05a0b428SJohn Marino * DESCRIPTION:
34*05a0b428SJohn Marino *
35*05a0b428SJohn Marino * If
36*05a0b428SJohn Marino * z = x + iy,
37*05a0b428SJohn Marino *
38*05a0b428SJohn Marino * then
39*05a0b428SJohn Marino * 1 ( 2x )
40*05a0b428SJohn Marino * Re w = - arctan(-----------) + k PI
41*05a0b428SJohn Marino * 2 ( 2 2)
42*05a0b428SJohn Marino * (1 - x - y )
43*05a0b428SJohn Marino *
44*05a0b428SJohn Marino * ( 2 2)
45*05a0b428SJohn Marino * 1 (x + (y+1) )
46*05a0b428SJohn Marino * Im w = - log(------------)
47*05a0b428SJohn Marino * 4 ( 2 2)
48*05a0b428SJohn Marino * (x + (y-1) )
49*05a0b428SJohn Marino *
50*05a0b428SJohn Marino * Where k is an arbitrary integer.
51*05a0b428SJohn Marino *
52*05a0b428SJohn Marino * catan(z) = -i catanh(iz).
53*05a0b428SJohn Marino *
54*05a0b428SJohn Marino * ACCURACY:
55*05a0b428SJohn Marino *
56*05a0b428SJohn Marino * Relative error:
57*05a0b428SJohn Marino * arithmetic domain # trials peak rms
58*05a0b428SJohn Marino * DEC -10,+10 5900 1.3e-16 7.8e-18
59*05a0b428SJohn Marino * IEEE -10,+10 30000 2.3e-15 8.5e-17
60*05a0b428SJohn Marino * The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
61*05a0b428SJohn Marino * had peak relative error 1.5e-16, rms relative error
62*05a0b428SJohn Marino * 2.9e-17. See also clog().
63*05a0b428SJohn Marino */
64*05a0b428SJohn Marino
65*05a0b428SJohn Marino #include <complex.h>
66*05a0b428SJohn Marino #include <float.h>
67*05a0b428SJohn Marino #include <math.h>
68*05a0b428SJohn Marino
69*05a0b428SJohn Marino #define MAXNUM 1.0e308
70*05a0b428SJohn Marino
71*05a0b428SJohn Marino static const double DP1 = 3.14159265160560607910E0;
72*05a0b428SJohn Marino static const double DP2 = 1.98418714791870343106E-9;
73*05a0b428SJohn Marino static const double DP3 = 1.14423774522196636802E-17;
74*05a0b428SJohn Marino
75*05a0b428SJohn Marino static double
_redupi(double x)76*05a0b428SJohn Marino _redupi(double x)
77*05a0b428SJohn Marino {
78*05a0b428SJohn Marino double t;
79*05a0b428SJohn Marino long i;
80*05a0b428SJohn Marino
81*05a0b428SJohn Marino t = x/M_PI;
82*05a0b428SJohn Marino if(t >= 0.0)
83*05a0b428SJohn Marino t += 0.5;
84*05a0b428SJohn Marino else
85*05a0b428SJohn Marino t -= 0.5;
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 double complex
catan(double complex z)94*05a0b428SJohn Marino catan(double complex z)
95*05a0b428SJohn Marino {
96*05a0b428SJohn Marino double complex w;
97*05a0b428SJohn Marino double a, t, x, x2, y;
98*05a0b428SJohn Marino
99*05a0b428SJohn Marino x = creal (z);
100*05a0b428SJohn Marino y = cimag (z);
101*05a0b428SJohn Marino
102*05a0b428SJohn Marino if ((x == 0.0) && (y > 1.0))
103*05a0b428SJohn Marino goto ovrf;
104*05a0b428SJohn Marino
105*05a0b428SJohn Marino x2 = x * x;
106*05a0b428SJohn Marino a = 1.0 - x2 - (y * y);
107*05a0b428SJohn Marino if (a == 0.0)
108*05a0b428SJohn Marino goto ovrf;
109*05a0b428SJohn Marino
110*05a0b428SJohn Marino t = 0.5 * atan2 (2.0 * x, a);
111*05a0b428SJohn Marino w = _redupi (t);
112*05a0b428SJohn Marino
113*05a0b428SJohn Marino t = y - 1.0;
114*05a0b428SJohn Marino a = x2 + (t * t);
115*05a0b428SJohn Marino if (a == 0.0)
116*05a0b428SJohn Marino goto ovrf;
117*05a0b428SJohn Marino
118*05a0b428SJohn Marino t = y + 1.0;
119*05a0b428SJohn Marino a = (x2 + (t * t))/a;
120*05a0b428SJohn Marino w = w + (0.25 * log (a)) * I;
121*05a0b428SJohn Marino return (w);
122*05a0b428SJohn Marino
123*05a0b428SJohn Marino ovrf:
124*05a0b428SJohn Marino /*mtherr ("catan", OVERFLOW);*/
125*05a0b428SJohn Marino w = MAXNUM + MAXNUM * I;
126*05a0b428SJohn Marino return (w);
127*05a0b428SJohn Marino }
128*05a0b428SJohn Marino
129*05a0b428SJohn Marino #if LDBL_MANT_DIG == DBL_MANT_DIG
130*05a0b428SJohn Marino __strong_alias(catanl, catan);
131*05a0b428SJohn Marino #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */
132