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      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
     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
    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