145 lines
4.3 KiB
C
145 lines
4.3 KiB
C
/*-
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* Copyright (c) 2011 David Schultz
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice unmodified, this list of conditions, and the following
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* disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*/
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/*
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* Hyperbolic tangent of a complex argument z = x + i y.
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*
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* The algorithm is from:
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*
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* W. Kahan. Branch Cuts for Complex Elementary Functions or Much
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* Ado About Nothing's Sign Bit. In The State of the Art in
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* Numerical Analysis, pp. 165 ff. Iserles and Powell, eds., 1987.
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*
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* Method:
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*
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* Let t = tan(x)
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* beta = 1/cos^2(y)
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* s = sinh(x)
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* rho = cosh(x)
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*
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* We have:
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*
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* tanh(z) = sinh(z) / cosh(z)
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*
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* sinh(x) cos(y) + i cosh(x) sin(y)
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* = ---------------------------------
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* cosh(x) cos(y) + i sinh(x) sin(y)
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*
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* cosh(x) sinh(x) / cos^2(y) + i tan(y)
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* = -------------------------------------
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* 1 + sinh^2(x) / cos^2(y)
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*
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* beta rho s + i t
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* = ----------------
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* 1 + beta s^2
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*
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* Modifications:
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*
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* I omitted the original algorithm's handling of overflow in tan(x) after
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* verifying with nearpi.c that this can't happen in IEEE single or double
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* precision. I also handle large x differently.
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*/
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#include "cdefs-compat.h"
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//__FBSDID("$FreeBSD: src/lib/msun/src/s_ctanh.c,v 1.2 2011/10/21 06:30:16 das Exp $");
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#include <openlibm_complex.h>
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#include <openlibm_math.h>
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#include "math_private.h"
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OLM_DLLEXPORT double complex
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ctanh(double complex z)
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{
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double x, y;
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double t, beta, s, rho, denom;
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u_int32_t hx, ix, lx;
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x = creal(z);
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y = cimag(z);
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EXTRACT_WORDS(hx, lx, x);
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ix = hx & 0x7fffffff;
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/*
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* ctanh(NaN + i 0) = NaN + i 0
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*
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* ctanh(NaN + i y) = NaN + i NaN for y != 0
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*
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* The imaginary part has the sign of x*sin(2*y), but there's no
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* special effort to get this right.
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*
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* ctanh(+-Inf +- i Inf) = +-1 +- 0
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*
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* ctanh(+-Inf + i y) = +-1 + 0 sin(2y) for y finite
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*
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* The imaginary part of the sign is unspecified. This special
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* case is only needed to avoid a spurious invalid exception when
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* y is infinite.
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*/
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if (ix >= 0x7ff00000) {
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if ((ix & 0xfffff) | lx) /* x is NaN */
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return (CMPLX(x, (y == 0 ? y : x * y)));
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SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */
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return (CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
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}
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/*
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* ctanh(x + i NAN) = NaN + i NaN
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* ctanh(x +- i Inf) = NaN + i NaN
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*/
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if (!isfinite(y))
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return (CMPLX(y - y, y - y));
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/*
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* ctanh(+-huge + i +-y) ~= +-1 +- i 2sin(2y)/exp(2x), using the
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* approximation sinh^2(huge) ~= exp(2*huge) / 4.
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* We use a modified formula to avoid spurious overflow.
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*/
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if (ix >= 0x40360000) { /* x >= 22 */
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double exp_mx = exp(-fabs(x));
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return (CMPLX(copysign(1, x),
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4 * sin(y) * cos(y) * exp_mx * exp_mx));
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}
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/* Kahan's algorithm */
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t = tan(y);
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beta = 1.0 + t * t; /* = 1 / cos^2(y) */
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s = sinh(x);
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rho = sqrt(1 + s * s); /* = cosh(x) */
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denom = 1 + beta * s * s;
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return (CMPLX((beta * rho * s) / denom, t / denom));
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}
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OLM_DLLEXPORT double complex
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ctan(double complex z)
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{
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/* ctan(z) = -I * ctanh(I * z) */
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z = ctanh(CMPLX(-cimag(z), creal(z)));
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return (CMPLX(cimag(z), -creal(z)));
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}
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