src/complex/__cexp.c

/* origin: FreeBSD /usr/src/lib/msun/src/k_exp.c */
/*-
 * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
 * 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, 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 AND CONTRIBUTORS ``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 OR CONTRIBUTORS 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.
 */

#include "libm.h"

static const uint32_t k = 1799; /* constant for reduction */
static const double kln2 = 1246.97177782734161156; /* k * ln2 */

/*
 * Compute exp(x), scaled to avoid spurious overflow.  An exponent is
 * returned separately in 'expt'.
 *
 * Input:  ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
 * Output: 2**1023 <= y < 2**1024
 */
static double __frexp_exp(double x, int *expt)
{
	double exp_x;
	uint32_t hx;

	/*
	 * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to
	 * minimize |exp(kln2) - 2**k|.  We also scale the exponent of
	 * exp_x to MAX_EXP so that the result can be multiplied by
	 * a tiny number without losing accuracy due to denormalization.
	 */
	exp_x = exp(x - kln2);
	GET_HIGH_WORD(hx, exp_x);
	*expt = (hx >> 20) - (0x3ff + 1023) + k;
	SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20));
	return exp_x;
}

/*
 * __ldexp_cexp(x, expt) compute exp(x) * 2**expt.
 * It is intended for large arguments (real part >= ln(DBL_MAX))
 * where care is needed to avoid overflow.
 *
 * The present implementation is narrowly tailored for our hyperbolic and
 * exponential functions.  We assume expt is small (0 or -1), and the caller
 * has filtered out very large x, for which overflow would be inevitable.
 */
double complex __ldexp_cexp(double complex z, int expt)
{
	double x, y, exp_x, scale1, scale2;
	int ex_expt, half_expt;

	x = creal(z);
	y = cimag(z);
	exp_x = __frexp_exp(x, &ex_expt);
	expt += ex_expt;

	/*
	 * Arrange so that scale1 * scale2 == 2**expt.  We use this to
	 * compensate for scalbn being horrendously slow.
	 */
	half_expt = expt / 2;
	INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0);
	half_expt = expt - half_expt;
	INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0);

	return CMPLX(cos(y) * exp_x * scale1 * scale2, sin(y) * exp_x * scale1 * scale2);
}