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: head/lib/msun/src/s_ctanh.c 284427 2015-06-15 20:40:44Z tijl $");
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) = d(NaN) +- I 0
89 *
90 * ctanh(NaN + I y) = d(NaN,y) + I d(NaN,y) 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 +- I 0
96 *
97 * ctanh(+-Inf + I y) = +-1 + I 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 (CMPLX((x + 0) * (y + 0),
106 y == 0 ? y : (x + 0) * (y + 0)));
107 SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */
108 return (CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
109 }
110
111 /*
112 * ctanh(x + I NaN) = d(NaN) + I d(NaN)
113 * ctanh(x +- I Inf) = dNaN + I dNaN
114 */
115 if (!isfinite(y))
116 return (CMPLX(y - y, y - y));
117
118 /*
119 * ctanh(+-huge +- I y) ~= +-1 +- I 2sin(2y)/exp(2x), using the
120 * approximation sinh^2(huge) ~= exp(2*huge) / 4.
121 * We use a modified formula to avoid spurious overflow.
122 */
123 if (ix >= 0x40360000) { /* |x| >= 22 */
124 double exp_mx = exp(-fabs(x));
125 return (CMPLX(copysign(1, x),
126 4 * sin(y) * cos(y) * exp_mx * exp_mx));
127 }
128
129 /* Kahan's algorithm */
130 t = tan(y);
131 beta = 1.0 + t * t; /* = 1 / cos^2(y) */
132 s = sinh(x);
133 rho = sqrt(1 + s * s); /* = cosh(x) */
134 denom = 1 + beta * s * s;
135 return (CMPLX((beta * rho * s) / denom, t / denom));
136 }
137
138 double complex
ctan(double complex z)139 ctan(double complex z)
140 {
141
142 /* ctan(z) = -I * ctanh(I * z) = I * conj(ctanh(I * conj(z))) */
143 z = ctanh(CMPLX(cimag(z), creal(z)));
144 return (CMPLX(cimag(z), creal(z)));
145 }
146