1/* origin: FreeBSD /usr/src/lib/msun/src/s_ctanh.c */ 2/*- 3 * Copyright (c) 2011 David Schultz 4 * All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice unmodified, this list of conditions, and the following 11 * disclaimer. 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in the 14 * documentation and/or other materials provided with the distribution. 15 * 16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR 17 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES 18 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. 19 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, 20 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 21 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, 22 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY 23 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT 24 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF 25 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 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 "libm.h" 67 68double complex ctanh(double complex z) 69{ 70 double x, y; 71 double t, beta, s, rho, denom; 72 uint32_t hx, ix, lx; 73 74 x = creal(z); 75 y = cimag(z); 76 77 EXTRACT_WORDS(hx, lx, x); 78 ix = hx & 0x7fffffff; 79 80 /* 81 * ctanh(NaN + i 0) = NaN + i 0 82 * 83 * ctanh(NaN + i y) = NaN + i NaN for y != 0 84 * 85 * The imaginary part has the sign of x*sin(2*y), but there's no 86 * special effort to get this right. 87 * 88 * ctanh(+-Inf +- i Inf) = +-1 +- 0 89 * 90 * ctanh(+-Inf + i y) = +-1 + 0 sin(2y) for y finite 91 * 92 * The imaginary part of the sign is unspecified. This special 93 * case is only needed to avoid a spurious invalid exception when 94 * y is infinite. 95 */ 96 if (ix >= 0x7ff00000) { 97 if ((ix & 0xfffff) | lx) /* x is NaN */ 98 return CMPLX(x, (y == 0 ? y : x * y)); 99 SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */ 100 return CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))); 101 } 102 103 /* 104 * ctanh(+-0 + i NAN) = +-0 + i NaN 105 * ctanh(+-0 +- i Inf) = +-0 + i NaN 106 * ctanh(x + i NAN) = NaN + i NaN 107 * ctanh(x +- i Inf) = NaN + i NaN 108 */ 109 if (!isfinite(y)) 110 return CMPLX(x ? y - y : x, y - y); 111 112 /* 113 * ctanh(+-huge + i +-y) ~= +-1 +- i 2sin(2y)/exp(2x), using the 114 * approximation sinh^2(huge) ~= exp(2*huge) / 4. 115 * We use a modified formula to avoid spurious overflow. 116 */ 117 if (ix >= 0x40360000) { /* x >= 22 */ 118 double exp_mx = exp(-fabs(x)); 119 return CMPLX(copysign(1, x), 4 * sin(y) * cos(y) * exp_mx * exp_mx); 120 } 121 122 /* Kahan's algorithm */ 123 t = tan(y); 124 beta = 1.0 + t * t; /* = 1 / cos^2(y) */ 125 s = sinh(x); 126 rho = sqrt(1 + s * s); /* = cosh(x) */ 127 denom = 1 + beta * s * s; 128 return CMPLX((beta * rho * s) / denom, t / denom); 129} 130