1 /*-
2 * Copyright (c) 2011 David Schultz
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice unmodified, this list of conditions, and the following
10 * disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
25 */
26
27 /*
28 * Hyperbolic tangent of a complex argument z = x + i y.
29 *
30 * The algorithm is from:
31 *
32 * W. Kahan. Branch Cuts for Complex Elementary Functions or Much
33 * Ado About Nothing's Sign Bit. In The State of the Art in
34 * Numerical Analysis, pp. 165 ff. Iserles and Powell, eds., 1987.
35 *
36 * Method:
37 *
38 * Let t = tan(x)
39 * beta = 1/cos^2(y)
40 * s = sinh(x)
41 * rho = cosh(x)
42 *
43 * We have:
44 *
45 * tanh(z) = sinh(z) / cosh(z)
46 *
47 * sinh(x) cos(y) + i cosh(x) sin(y)
48 * = ---------------------------------
49 * cosh(x) cos(y) + i sinh(x) sin(y)
50 *
51 * cosh(x) sinh(x) / cos^2(y) + i tan(y)
52 * = -------------------------------------
53 * 1 + sinh^2(x) / cos^2(y)
54 *
55 * beta rho s + i t
56 * = ----------------
57 * 1 + beta s^2
58 *
59 * Modifications:
60 *
61 * I omitted the original algorithm's handling of overflow in tan(x) after
62 * verifying with nearpi.c that this can't happen in IEEE single or double
63 * precision. I also handle large x differently.
64 */
65
66 #include <sys/cdefs.h>
67 __FBSDID("$FreeBSD$");
68
69 #include <complex.h>
70 #include <math.h>
71
72 #include "math_private.h"
73
74 double complex
ctanh(double complex z)75 ctanh(double complex z)
76 {
77 double x, y;
78 double t, beta, s, rho, denom;
79 uint32_t hx, ix, lx;
80
81 x = creal(z);
82 y = cimag(z);
83
84 EXTRACT_WORDS(hx, lx, x);
85 ix = hx & 0x7fffffff;
86
87 /*
88 * ctanh(NaN + i 0) = NaN + i 0
89 *
90 * ctanh(NaN + i y) = NaN + i NaN for y != 0
91 *
92 * The imaginary part has the sign of x*sin(2*y), but there's no
93 * special effort to get this right.
94 *
95 * ctanh(+-Inf +- i Inf) = +-1 +- 0
96 *
97 * ctanh(+-Inf + i y) = +-1 + 0 sin(2y) for y finite
98 *
99 * The imaginary part of the sign is unspecified. This special
100 * case is only needed to avoid a spurious invalid exception when
101 * y is infinite.
102 */
103 if (ix >= 0x7ff00000) {
104 if ((ix & 0xfffff) | lx) /* x is NaN */
105 return (cpack(x, (y == 0 ? y : x * y)));
106 SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */
107 return (cpack(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
108 }
109
110 /*
111 * ctanh(x + i NAN) = NaN + i NaN
112 * ctanh(x +- i Inf) = NaN + i NaN
113 */
114 if (!isfinite(y))
115 return (cpack(y - y, y - y));
116
117 /*
118 * ctanh(+-huge + i +-y) ~= +-1 +- i 2sin(2y)/exp(2x), using the
119 * approximation sinh^2(huge) ~= exp(2*huge) / 4.
120 * We use a modified formula to avoid spurious overflow.
121 */
122 if (ix >= 0x40360000) { /* x >= 22 */
123 double exp_mx = exp(-fabs(x));
124 return (cpack(copysign(1, x),
125 4 * sin(y) * cos(y) * exp_mx * exp_mx));
126 }
127
128 /* Kahan's algorithm */
129 t = tan(y);
130 beta = 1.0 + t * t; /* = 1 / cos^2(y) */
131 s = sinh(x);
132 rho = sqrt(1 + s * s); /* = cosh(x) */
133 denom = 1 + beta * s * s;
134 return (cpack((beta * rho * s) / denom, t / denom));
135 }
136
137 double complex
ctan(double complex z)138 ctan(double complex z)
139 {
140
141 /* ctan(z) = -I * ctanh(I * z) */
142 z = ctanh(cpack(-cimag(z), creal(z)));
143 return (cpack(cimag(z), -creal(z)));
144 }
145