1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause
3 *
4 * Copyright (c) 2011 David Schultz
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice unmodified, this list of conditions, and the following
12 * disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 * notice, this list of conditions and the following disclaimer in the
15 * documentation and/or other materials provided with the distribution.
16 *
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 */
28
29 /*
30 * Hyperbolic tangent of a complex argument z = x + I y.
31 *
32 * The algorithm is from:
33 *
34 * W. Kahan. Branch Cuts for Complex Elementary Functions or Much
35 * Ado About Nothing's Sign Bit. In The State of the Art in
36 * Numerical Analysis, pp. 165 ff. Iserles and Powell, eds., 1987.
37 *
38 * Method:
39 *
40 * Let t = tan(x)
41 * beta = 1/cos^2(y)
42 * s = sinh(x)
43 * rho = cosh(x)
44 *
45 * We have:
46 *
47 * tanh(z) = sinh(z) / cosh(z)
48 *
49 * sinh(x) cos(y) + I cosh(x) sin(y)
50 * = ---------------------------------
51 * cosh(x) cos(y) + I sinh(x) sin(y)
52 *
53 * cosh(x) sinh(x) / cos^2(y) + I tan(y)
54 * = -------------------------------------
55 * 1 + sinh^2(x) / cos^2(y)
56 *
57 * beta rho s + I t
58 * = ----------------
59 * 1 + beta s^2
60 *
61 * Modifications:
62 *
63 * I omitted the original algorithm's handling of overflow in tan(x) after
64 * verifying with nearpi.c that this can't happen in IEEE single or double
65 * precision. I also handle large x differently.
66 */
67
68 #include <complex.h>
69 #include <math.h>
70
71 #include "math_private.h"
72
73 double complex
ctanh(double complex z)74 ctanh(double complex z)
75 {
76 double x, y;
77 double t, beta, s, rho, denom;
78 uint32_t hx, ix, lx;
79
80 x = creal(z);
81 y = cimag(z);
82
83 EXTRACT_WORDS(hx, lx, x);
84 ix = hx & 0x7fffffff;
85
86 /*
87 * ctanh(NaN +- I 0) = d(NaN) +- I 0
88 *
89 * ctanh(NaN + I y) = d(NaN,y) + I d(NaN,y) for y != 0
90 *
91 * The imaginary part has the sign of x*sin(2*y), but there's no
92 * special effort to get this right.
93 *
94 * ctanh(+-Inf +- I Inf) = +-1 +- I 0
95 *
96 * ctanh(+-Inf + I y) = +-1 + I 0 sin(2y) for y finite
97 *
98 * The imaginary part of the sign is unspecified. This special
99 * case is only needed to avoid a spurious invalid exception when
100 * y is infinite.
101 */
102 if (ix >= 0x7ff00000) {
103 if ((ix & 0xfffff) | lx) /* x is NaN */
104 return (CMPLX(nan_mix(x, y),
105 y == 0 ? y : nan_mix(x, y)));
106 SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */
107 return (CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
108 }
109
110 /*
111 * ctanh(+-0 + i NAN) = +-0 + i NaN
112 * ctanh(+-0 +- i Inf) = +-0 + i NaN
113 * ctanh(x + i NAN) = NaN + i NaN
114 * ctanh(x +- i Inf) = NaN + i NaN
115 */
116 if (!isfinite(y))
117 return (CMPLX(x ? y - y : x, y - y));
118
119 /*
120 * ctanh(+-huge +- I y) ~= +-1 +- I 2sin(2y)/exp(2x), using the
121 * approximation sinh^2(huge) ~= exp(2*huge) / 4.
122 * We use a modified formula to avoid spurious overflow.
123 */
124 if (ix >= 0x40360000) { /* |x| >= 22 */
125 double exp_mx = exp(-fabs(x));
126 return (CMPLX(copysign(1, x),
127 4 * sin(y) * cos(y) * exp_mx * exp_mx));
128 }
129
130 /* Kahan's algorithm */
131 t = tan(y);
132 beta = 1.0 + t * t; /* = 1 / cos^2(y) */
133 s = sinh(x);
134 rho = sqrt(1 + s * s); /* = cosh(x) */
135 denom = 1 + beta * s * s;
136 return (CMPLX((beta * rho * s) / denom, t / denom));
137 }
138
139 double complex
ctan(double complex z)140 ctan(double complex z)
141 {
142
143 /* ctan(z) = -I * ctanh(I * z) = I * conj(ctanh(I * conj(z))) */
144 z = ctanh(CMPLX(cimag(z), creal(z)));
145 return (CMPLX(cimag(z), creal(z)));
146 }
147