| /*- |
| * 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 <sys/cdefs.h> |
| __FBSDID("$FreeBSD: head/lib/msun/src/s_ctanh.c 284427 2015-06-15 20:40:44Z tijl $"); |
| |
| #include <complex.h> |
| #include <math.h> |
| |
| #include "math_private.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) = d(NaN) +- I 0 |
| * |
| * ctanh(NaN + I y) = d(NaN,y) + I d(NaN,y) 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 +- I 0 |
| * |
| * ctanh(+-Inf + I y) = +-1 + I 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 + 0) * (y + 0), |
| y == 0 ? y : (x + 0) * (y + 0))); |
| SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */ |
| return (CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y)))); |
| } |
| |
| /* |
| * ctanh(x + I NaN) = d(NaN) + I d(NaN) |
| * ctanh(x +- I Inf) = dNaN + I dNaN |
| */ |
| if (!isfinite(y)) |
| return (CMPLX(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 (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)); |
| } |
| |
| double complex |
| ctan(double complex z) |
| { |
| |
| /* ctan(z) = -I * ctanh(I * z) = I * conj(ctanh(I * conj(z))) */ |
| z = ctanh(CMPLX(cimag(z), creal(z))); |
| return (CMPLX(cimag(z), creal(z))); |
| } |