Elliott Hughes | a0ee078 | 2013-01-30 19:06:37 -0800 | [diff] [blame] | 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 | } |