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Diffstat (limited to 'src/complex/ctanh.c')
| -rw-r--r-- | src/complex/ctanh.c | 127 |
1 files changed, 127 insertions, 0 deletions
diff --git a/src/complex/ctanh.c b/src/complex/ctanh.c new file mode 100644 index 00000000..dd569fc3 --- /dev/null +++ b/src/complex/ctanh.c @@ -0,0 +1,127 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/s_ctanh.c */ +/*- + * 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 "libm.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); +} |