| /* 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 "complex_impl.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 CMPLX(x, (y == 0 ? y : x * y)); |
| SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */ |
| return CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))); |
| } |
| |
| /* |
| * ctanh(+-0 + i NAN) = +-0 + i NaN |
| * ctanh(+-0 +- i Inf) = +-0 + i NaN |
| * ctanh(x + i NAN) = NaN + i NaN |
| * ctanh(x +- i Inf) = NaN + i NaN |
| */ |
| if (!isfinite(y)) |
| return CMPLX(x ? y - y : x, 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 CMPLX(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 CMPLX((beta * rho * s) / denom, t / denom); |
| } |