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      1 /*-
      2  * Copyright (c) 2005 Bruce D. Evans and Steven G. Kargl
      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 sine of a complex argument z = x + i y.
     29  *
     30  * sinh(z) = sinh(x+iy)
     31  *         = sinh(x) cos(y) + i cosh(x) sin(y).
     32  *
     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  */
     36 
     37 #include <sys/cdefs.h>
     38 __FBSDID("$FreeBSD$");
     39 
     40 #include <complex.h>
     41 #include <math.h>
     42 
     43 #include "math_private.h"
     44 
     45 static const double huge = 0x1p1023;
     46 
     47 double complex
     48 csinh(double complex z)
     49 {
     50 	double x, y, h;
     51 	int32_t hx, hy, ix, iy, lx, ly;
     52 
     53 	x = creal(z);
     54 	y = cimag(z);
     55 
     56 	EXTRACT_WORDS(hx, lx, x);
     57 	EXTRACT_WORDS(hy, ly, y);
     58 
     59 	ix = 0x7fffffff & hx;
     60 	iy = 0x7fffffff & hy;
     61 
     62 	/* Handle the nearly-non-exceptional cases where x and y are finite. */
     63 	if (ix < 0x7ff00000 && iy < 0x7ff00000) {
     64 		if ((iy | ly) == 0)
     65 			return (cpack(sinh(x), y));
     66 		if (ix < 0x40360000)	/* small x: normal case */
     67 			return (cpack(sinh(x) * cos(y), cosh(x) * sin(y)));
     68 
     69 		/* |x| >= 22, so cosh(x) ~= exp(|x|) */
     70 		if (ix < 0x40862e42) {
     71 			/* x < 710: exp(|x|) won't overflow */
     72 			h = exp(fabs(x)) * 0.5;
     73 			return (cpack(copysign(h, x) * cos(y), h * sin(y)));
     74 		} else if (ix < 0x4096bbaa) {
     75 			/* x < 1455: scale to avoid overflow */
     76 			z = __ldexp_cexp(cpack(fabs(x), y), -1);
     77 			return (cpack(creal(z) * copysign(1, x), cimag(z)));
     78 		} else {
     79 			/* x >= 1455: the result always overflows */
     80 			h = huge * x;
     81 			return (cpack(h * cos(y), h * h * sin(y)));
     82 		}
     83 	}
     84 
     85 	/*
     86 	 * sinh(+-0 +- I Inf) = sign(d(+-0, dNaN))0 + I dNaN.
     87 	 * The sign of 0 in the result is unspecified.  Choice = normally
     88 	 * the same as dNaN.  Raise the invalid floating-point exception.
     89 	 *
     90 	 * sinh(+-0 +- I NaN) = sign(d(+-0, NaN))0 + I d(NaN).
     91 	 * The sign of 0 in the result is unspecified.  Choice = normally
     92 	 * the same as d(NaN).
     93 	 */
     94 	if ((ix | lx) == 0 && iy >= 0x7ff00000)
     95 		return (cpack(copysign(0, x * (y - y)), y - y));
     96 
     97 	/*
     98 	 * sinh(+-Inf +- I 0) = +-Inf + I +-0.
     99 	 *
    100 	 * sinh(NaN +- I 0)   = d(NaN) + I +-0.
    101 	 */
    102 	if ((iy | ly) == 0 && ix >= 0x7ff00000) {
    103 		if (((hx & 0xfffff) | lx) == 0)
    104 			return (cpack(x, y));
    105 		return (cpack(x, copysign(0, y)));
    106 	}
    107 
    108 	/*
    109 	 * sinh(x +- I Inf) = dNaN + I dNaN.
    110 	 * Raise the invalid floating-point exception for finite nonzero x.
    111 	 *
    112 	 * sinh(x + I NaN) = d(NaN) + I d(NaN).
    113 	 * Optionally raises the invalid floating-point exception for finite
    114 	 * nonzero x.  Choice = don't raise (except for signaling NaNs).
    115 	 */
    116 	if (ix < 0x7ff00000 && iy >= 0x7ff00000)
    117 		return (cpack(y - y, x * (y - y)));
    118 
    119 	/*
    120 	 * sinh(+-Inf + I NaN)  = +-Inf + I d(NaN).
    121 	 * The sign of Inf in the result is unspecified.  Choice = normally
    122 	 * the same as d(NaN).
    123 	 *
    124 	 * sinh(+-Inf +- I Inf) = +Inf + I dNaN.
    125 	 * The sign of Inf in the result is unspecified.  Choice = always +.
    126 	 * Raise the invalid floating-point exception.
    127 	 *
    128 	 * sinh(+-Inf + I y)   = +-Inf cos(y) + I Inf sin(y)
    129 	 */
    130 	if (ix >= 0x7ff00000 && ((hx & 0xfffff) | lx) == 0) {
    131 		if (iy >= 0x7ff00000)
    132 			return (cpack(x * x, x * (y - y)));
    133 		return (cpack(x * cos(y), INFINITY * sin(y)));
    134 	}
    135 
    136 	/*
    137 	 * sinh(NaN + I NaN)  = d(NaN) + I d(NaN).
    138 	 *
    139 	 * sinh(NaN +- I Inf) = d(NaN) + I d(NaN).
    140 	 * Optionally raises the invalid floating-point exception.
    141 	 * Choice = raise.
    142 	 *
    143 	 * sinh(NaN + I y)    = d(NaN) + I d(NaN).
    144 	 * Optionally raises the invalid floating-point exception for finite
    145 	 * nonzero y.  Choice = don't raise (except for signaling NaNs).
    146 	 */
    147 	return (cpack((x * x) * (y - y), (x + x) * (y - y)));
    148 }
    149 
    150 double complex
    151 csin(double complex z)
    152 {
    153 
    154 	/* csin(z) = -I * csinh(I * z) */
    155 	z = csinh(cpack(-cimag(z), creal(z)));
    156 	return (cpack(cimag(z), -creal(z)));
    157 }
    158