1*05a0b428SJohn Marino /* $OpenBSD: s_ctan.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 /* ctan()
19*05a0b428SJohn Marino *
20*05a0b428SJohn Marino * Complex circular tangent
21*05a0b428SJohn Marino *
22*05a0b428SJohn Marino *
23*05a0b428SJohn Marino *
24*05a0b428SJohn Marino * SYNOPSIS:
25*05a0b428SJohn Marino *
26*05a0b428SJohn Marino * double complex ctan();
27*05a0b428SJohn Marino * double complex z, w;
28*05a0b428SJohn Marino *
29*05a0b428SJohn Marino * w = ctan (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 *
40*05a0b428SJohn Marino * sin 2x + i sinh 2y
41*05a0b428SJohn Marino * w = --------------------.
42*05a0b428SJohn Marino * cos 2x + cosh 2y
43*05a0b428SJohn Marino *
44*05a0b428SJohn Marino * On the real axis the denominator is zero at odd multiples
45*05a0b428SJohn Marino * of PI/2. The denominator is evaluated by its Taylor
46*05a0b428SJohn Marino * series near these points.
47*05a0b428SJohn Marino *
48*05a0b428SJohn Marino * ctan(z) = -i ctanh(iz).
49*05a0b428SJohn Marino *
50*05a0b428SJohn Marino * ACCURACY:
51*05a0b428SJohn Marino *
52*05a0b428SJohn Marino * Relative error:
53*05a0b428SJohn Marino * arithmetic domain # trials peak rms
54*05a0b428SJohn Marino * DEC -10,+10 5200 7.1e-17 1.6e-17
55*05a0b428SJohn Marino * IEEE -10,+10 30000 7.2e-16 1.2e-16
56*05a0b428SJohn Marino * Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
57*05a0b428SJohn Marino */
58*05a0b428SJohn Marino
59*05a0b428SJohn Marino #include <complex.h>
60*05a0b428SJohn Marino #include <float.h>
61*05a0b428SJohn Marino #include <math.h>
62*05a0b428SJohn Marino
63*05a0b428SJohn Marino #define MACHEP 1.1e-16
64*05a0b428SJohn Marino #define MAXNUM 1.0e308
65*05a0b428SJohn Marino
66*05a0b428SJohn Marino static const double DP1 = 3.14159265160560607910E0;
67*05a0b428SJohn Marino static const double DP2 = 1.98418714791870343106E-9;
68*05a0b428SJohn Marino static const double DP3 = 1.14423774522196636802E-17;
69*05a0b428SJohn Marino
70*05a0b428SJohn Marino static double
_redupi(double x)71*05a0b428SJohn Marino _redupi(double x)
72*05a0b428SJohn Marino {
73*05a0b428SJohn Marino double t;
74*05a0b428SJohn Marino long i;
75*05a0b428SJohn Marino
76*05a0b428SJohn Marino t = x/M_PI;
77*05a0b428SJohn Marino if (t >= 0.0)
78*05a0b428SJohn Marino t += 0.5;
79*05a0b428SJohn Marino else
80*05a0b428SJohn Marino t -= 0.5;
81*05a0b428SJohn Marino
82*05a0b428SJohn Marino i = t; /* the multiple */
83*05a0b428SJohn Marino t = i;
84*05a0b428SJohn Marino t = ((x - t * DP1) - t * DP2) - t * DP3;
85*05a0b428SJohn Marino return (t);
86*05a0b428SJohn Marino }
87*05a0b428SJohn Marino
88*05a0b428SJohn Marino /* Taylor series expansion for cosh(2y) - cos(2x) */
89*05a0b428SJohn Marino
90*05a0b428SJohn Marino static double
_ctans(double complex z)91*05a0b428SJohn Marino _ctans(double complex z)
92*05a0b428SJohn Marino {
93*05a0b428SJohn Marino double f, x, x2, y, y2, rn, t;
94*05a0b428SJohn Marino double d;
95*05a0b428SJohn Marino
96*05a0b428SJohn Marino x = fabs (2.0 * creal (z));
97*05a0b428SJohn Marino y = fabs (2.0 * cimag(z));
98*05a0b428SJohn Marino
99*05a0b428SJohn Marino x = _redupi(x);
100*05a0b428SJohn Marino
101*05a0b428SJohn Marino x = x * x;
102*05a0b428SJohn Marino y = y * y;
103*05a0b428SJohn Marino x2 = 1.0;
104*05a0b428SJohn Marino y2 = 1.0;
105*05a0b428SJohn Marino f = 1.0;
106*05a0b428SJohn Marino rn = 0.0;
107*05a0b428SJohn Marino d = 0.0;
108*05a0b428SJohn Marino do {
109*05a0b428SJohn Marino rn += 1.0;
110*05a0b428SJohn Marino f *= rn;
111*05a0b428SJohn Marino rn += 1.0;
112*05a0b428SJohn Marino f *= rn;
113*05a0b428SJohn Marino x2 *= x;
114*05a0b428SJohn Marino y2 *= y;
115*05a0b428SJohn Marino t = y2 + x2;
116*05a0b428SJohn Marino t /= f;
117*05a0b428SJohn Marino d += t;
118*05a0b428SJohn Marino
119*05a0b428SJohn Marino rn += 1.0;
120*05a0b428SJohn Marino f *= rn;
121*05a0b428SJohn Marino rn += 1.0;
122*05a0b428SJohn Marino f *= rn;
123*05a0b428SJohn Marino x2 *= x;
124*05a0b428SJohn Marino y2 *= y;
125*05a0b428SJohn Marino t = y2 - x2;
126*05a0b428SJohn Marino t /= f;
127*05a0b428SJohn Marino d += t;
128*05a0b428SJohn Marino }
129*05a0b428SJohn Marino while (fabs(t/d) > MACHEP)
130*05a0b428SJohn Marino ;
131*05a0b428SJohn Marino return (d);
132*05a0b428SJohn Marino }
133*05a0b428SJohn Marino
134*05a0b428SJohn Marino double complex
ctan(double complex z)135*05a0b428SJohn Marino ctan(double complex z)
136*05a0b428SJohn Marino {
137*05a0b428SJohn Marino double complex w;
138*05a0b428SJohn Marino double d;
139*05a0b428SJohn Marino
140*05a0b428SJohn Marino d = cos (2.0 * creal (z)) + cosh (2.0 * cimag (z));
141*05a0b428SJohn Marino
142*05a0b428SJohn Marino if (fabs(d) < 0.25)
143*05a0b428SJohn Marino d = _ctans (z);
144*05a0b428SJohn Marino
145*05a0b428SJohn Marino if (d == 0.0) {
146*05a0b428SJohn Marino /*mtherr ("ctan", OVERFLOW);*/
147*05a0b428SJohn Marino w = MAXNUM + MAXNUM * I;
148*05a0b428SJohn Marino return (w);
149*05a0b428SJohn Marino }
150*05a0b428SJohn Marino
151*05a0b428SJohn Marino w = sin (2.0 * creal(z)) / d + (sinh (2.0 * cimag(z)) / d) * I;
152*05a0b428SJohn Marino return (w);
153*05a0b428SJohn Marino }
154*05a0b428SJohn Marino
155*05a0b428SJohn Marino #if LDBL_MANT_DIG == DBL_MANT_DIG
156*05a0b428SJohn Marino __strong_alias(ctanl, ctan);
157*05a0b428SJohn Marino #endif /* LDBL_MANT_DIG == DBL_MANT_DIG */
158