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/*-
* Copyright (c) 2011 David Schultz
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice unmodified, this list of conditions, and the following
* disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
/*
* Hyperbolic tangent of a complex argument z = x + i y.
*
* The algorithm is from:
*
* W. Kahan. Branch Cuts for Complex Elementary Functions or Much
* Ado About Nothing's Sign Bit. In The State of the Art in
* Numerical Analysis, pp. 165 ff. Iserles and Powell, eds., 1987.
*
* Method:
*
* Let t = tan(x)
* beta = 1/cos^2(y)
* s = sinh(x)
* rho = cosh(x)
*
* We have:
*
* tanh(z) = sinh(z) / cosh(z)
*
* sinh(x) cos(y) + i cosh(x) sin(y)
* = ---------------------------------
* cosh(x) cos(y) + i sinh(x) sin(y)
*
* cosh(x) sinh(x) / cos^2(y) + i tan(y)
* = -------------------------------------
* 1 + sinh^2(x) / cos^2(y)
*
* beta rho s + i t
* = ----------------
* 1 + beta s^2
*
* Modifications:
*
* I omitted the original algorithm's handling of overflow in tan(x) after
* verifying with nearpi.c that this can't happen in IEEE single or double
* precision. I also handle large x differently.
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
#include <complex.h>
#include <math.h>
#include "math_private.h"
double complex
ctanh(double complex z)
{
double x, y;
double t, beta, s, rho, denom;
uint32_t hx, ix, lx;
x = creal(z);
y = cimag(z);
EXTRACT_WORDS(hx, lx, x);
ix = hx & 0x7fffffff;
/*
* ctanh(NaN + i 0) = NaN + i 0
*
* ctanh(NaN + i y) = NaN + i NaN for y != 0
*
* The imaginary part has the sign of x*sin(2*y), but there's no
* special effort to get this right.
*
* ctanh(+-Inf +- i Inf) = +-1 +- 0
*
* ctanh(+-Inf + i y) = +-1 + 0 sin(2y) for y finite
*
* The imaginary part of the sign is unspecified. This special
* case is only needed to avoid a spurious invalid exception when
* y is infinite.
*/
if (ix >= 0x7ff00000) {
if ((ix & 0xfffff) | lx) /* x is NaN */
return (cpack(x, (y == 0 ? y : x * y)));
SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */
return (cpack(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
}
/*
* ctanh(x + i NAN) = NaN + i NaN
* ctanh(x +- i Inf) = NaN + i NaN
*/
if (!isfinite(y))
return (cpack(y - y, y - y));
/*
* ctanh(+-huge + i +-y) ~= +-1 +- i 2sin(2y)/exp(2x), using the
* approximation sinh^2(huge) ~= exp(2*huge) / 4.
* We use a modified formula to avoid spurious overflow.
*/
if (ix >= 0x40360000) { /* x >= 22 */
double exp_mx = exp(-fabs(x));
return (cpack(copysign(1, x),
4 * sin(y) * cos(y) * exp_mx * exp_mx));
}
/* Kahan's algorithm */
t = tan(y);
beta = 1.0 + t * t; /* = 1 / cos^2(y) */
s = sinh(x);
rho = sqrt(1 + s * s); /* = cosh(x) */
denom = 1 + beta * s * s;
return (cpack((beta * rho * s) / denom, t / denom));
}
double complex
ctan(double complex z)
{
/* ctan(z) = -I * ctanh(I * z) */
z = ctanh(cpack(-cimag(z), creal(z)));
return (cpack(cimag(z), -creal(z)));
}
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