k_exp.c revision 330897 
1/*-
2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
3 *
4 * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 *    notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 *    notice, this list of conditions and the following disclaimer in the
14 *    documentation and/or other materials provided with the distribution.
15 *
16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
17 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
18 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
19 * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
20 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
21 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
22 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
23 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
24 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
25 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
26 * SUCH DAMAGE.
27 */
28
29#include <sys/cdefs.h>
30__FBSDID("$FreeBSD: stable/11/lib/msun/src/k_exp.c 330897 2018-03-14 03:19:51Z eadler $");
31
32#include <complex.h>
33
34#include "math.h"
35#include "math_private.h"
36
37static const uint32_t k = 1799;		/* constant for reduction */
38static const double kln2 =  1246.97177782734161156;	/* k * ln2 */
39
40/*
41 * Compute exp(x), scaled to avoid spurious overflow.  An exponent is
42 * returned separately in 'expt'.
43 *
44 * Input:  ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
45 * Output: 2**1023 <= y < 2**1024
46 */
47static double
48__frexp_exp(double x, int *expt)
49{
50	double exp_x;
51	uint32_t hx;
52
53	/*
54	 * We use exp(x) = exp(x - kln2) * 2**k, carefully chosen to
55	 * minimize |exp(kln2) - 2**k|.  We also scale the exponent of
56	 * exp_x to MAX_EXP so that the result can be multiplied by
57	 * a tiny number without losing accuracy due to denormalization.
58	 */
59	exp_x = exp(x - kln2);
60	GET_HIGH_WORD(hx, exp_x);
61	*expt = (hx >> 20) - (0x3ff + 1023) + k;
62	SET_HIGH_WORD(exp_x, (hx & 0xfffff) | ((0x3ff + 1023) << 20));
63	return (exp_x);
64}
65
66/*
67 * __ldexp_exp(x, expt) and __ldexp_cexp(x, expt) compute exp(x) * 2**expt.
68 * They are intended for large arguments (real part >= ln(DBL_MAX))
69 * where care is needed to avoid overflow.
70 *
71 * The present implementation is narrowly tailored for our hyperbolic and
72 * exponential functions.  We assume expt is small (0 or -1), and the caller
73 * has filtered out very large x, for which overflow would be inevitable.
74 */
75
76double
77__ldexp_exp(double x, int expt)
78{
79	double exp_x, scale;
80	int ex_expt;
81
82	exp_x = __frexp_exp(x, &ex_expt);
83	expt += ex_expt;
84	INSERT_WORDS(scale, (0x3ff + expt) << 20, 0);
85	return (exp_x * scale);
86}
87
88double complex
89__ldexp_cexp(double complex z, int expt)
90{
91	double x, y, exp_x, scale1, scale2;
92	int ex_expt, half_expt;
93
94	x = creal(z);
95	y = cimag(z);
96	exp_x = __frexp_exp(x, &ex_expt);
97	expt += ex_expt;
98
99	/*
100	 * Arrange so that scale1 * scale2 == 2**expt.  We use this to
101	 * compensate for scalbn being horrendously slow.
102	 */
103	half_expt = expt / 2;
104	INSERT_WORDS(scale1, (0x3ff + half_expt) << 20, 0);
105	half_expt = expt - half_expt;
106	INSERT_WORDS(scale2, (0x3ff + half_expt) << 20, 0);
107
108	return (CMPLX(cos(y) * exp_x * scale1 * scale2,
109	    sin(y) * exp_x * scale1 * scale2));
110}
111