2 * Copyright (c) 2005 Bruce D. Evans and Steven G. Kargl
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice unmodified, this list of conditions, and the following
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.
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17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
28 * Hyperbolic sine of a complex argument z = x + i y.
30 * sinh(z) = sinh(x+iy)
31 * = sinh(x) cos(y) + i cosh(x) sin(y).
33 * Exceptional values are noted in the comments within the source code.
34 * These values and the return value were taken from n1124.pdf.
35 * The sign of the result for some exceptional values is unspecified but
36 * must satisfy both sinh(conj(z)) == conj(sinh(z)) and sinh(-z) == -sinh(z).
39 #include <sys/cdefs.h>
40 __FBSDID("$FreeBSD$");
45 #include "math_private.h"
47 static const double huge = 0x1p1023;
50 csinh(double complex z)
53 int32_t hx, hy, ix, iy, lx, ly;
58 EXTRACT_WORDS(hx, lx, x);
59 EXTRACT_WORDS(hy, ly, y);
64 /* Handle the nearly-non-exceptional cases where x and y are finite. */
65 if (ix < 0x7ff00000 && iy < 0x7ff00000) {
67 return (CMPLX(sinh(x), y));
68 if (ix < 0x40360000) /* |x| < 22: normal case */
69 return (CMPLX(sinh(x) * cos(y), cosh(x) * sin(y)));
71 /* |x| >= 22, so cosh(x) ~= exp(|x|) */
72 if (ix < 0x40862e42) {
73 /* x < 710: exp(|x|) won't overflow */
74 h = exp(fabs(x)) * 0.5;
75 return (CMPLX(copysign(h, x) * cos(y), h * sin(y)));
76 } else if (ix < 0x4096bbaa) {
77 /* x < 1455: scale to avoid overflow */
78 z = __ldexp_cexp(CMPLX(fabs(x), y), -1);
79 return (CMPLX(creal(z) * copysign(1, x), cimag(z)));
81 /* x >= 1455: the result always overflows */
83 return (CMPLX(h * cos(y), h * h * sin(y)));
88 * sinh(+-0 +- I Inf) = +-0 + I dNaN.
89 * The sign of 0 in the result is unspecified. Choice = same sign
90 * as the argument. Raise the invalid floating-point exception.
92 * sinh(+-0 +- I NaN) = +-0 + I d(NaN).
93 * The sign of 0 in the result is unspecified. Choice = same sign
96 if ((ix | lx) == 0) /* && iy >= 0x7ff00000 */
97 return (CMPLX(x, y - y));
100 * sinh(+-Inf +- I 0) = +-Inf + I +-0.
102 * sinh(NaN +- I 0) = d(NaN) + I +-0.
104 if ((iy | ly) == 0) /* && ix >= 0x7ff00000 */
105 return (CMPLX(x + x, y));
108 * sinh(x +- I Inf) = dNaN + I dNaN.
109 * Raise the invalid floating-point exception for finite nonzero x.
111 * sinh(x + I NaN) = d(NaN) + I d(NaN).
112 * Optionally raises the invalid floating-point exception for finite
113 * nonzero x. Choice = don't raise (except for signaling NaNs).
115 if (ix < 0x7ff00000) /* && iy >= 0x7ff00000 */
116 return (CMPLX(y - y, y - y));
119 * sinh(+-Inf + I NaN) = +-Inf + I d(NaN).
120 * The sign of Inf in the result is unspecified. Choice = same sign
123 * sinh(+-Inf +- I Inf) = +-Inf + I dNaN.
124 * The sign of Inf in the result is unspecified. Choice = same sign
125 * as the argument. Raise the invalid floating-point exception.
127 * sinh(+-Inf + I y) = +-Inf cos(y) + I Inf sin(y)
129 if (ix == 0x7ff00000 && lx == 0) {
130 if (iy >= 0x7ff00000)
131 return (CMPLX(x, y - y));
132 return (CMPLX(x * cos(y), INFINITY * sin(y)));
136 * sinh(NaN1 + I NaN2) = d(NaN1, NaN2) + I d(NaN1, NaN2).
138 * sinh(NaN +- I Inf) = d(NaN, dNaN) + I d(NaN, dNaN).
139 * Optionally raises the invalid floating-point exception.
142 * sinh(NaN + I y) = d(NaN) + I d(NaN).
143 * Optionally raises the invalid floating-point exception for finite
144 * nonzero y. Choice = don't raise (except for signaling NaNs).
146 return (CMPLX((x + x) * (y - y), (x * x) * (y - y)));
150 csin(double complex z)
153 /* csin(z) = -I * csinh(I * z) = I * conj(csinh(I * conj(z))). */
154 z = csinh(CMPLX(cimag(z), creal(z)));
155 return (CMPLX(cimag(z), creal(z)));