2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
4 * Copyright (c) 2011 David Schultz <das@FreeBSD.ORG>
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
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.
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
29 #include <sys/cdefs.h>
30 __FBSDID("$FreeBSD$");
35 #include "math_private.h"
37 static const uint32_t k = 1799; /* constant for reduction */
38 static const double kln2 = 1246.97177782734161156; /* k * ln2 */
41 * Compute exp(x), scaled to avoid spurious overflow. An exponent is
42 * returned separately in 'expt'.
44 * Input: ln(DBL_MAX) <= x < ln(2 * DBL_MAX / DBL_MIN_DENORM) ~= 1454.91
45 * Output: 2**1023 <= y < 2**1024
48 __frexp_exp(double x, int *expt)
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.
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));
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.
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.
77 __ldexp_exp(double x, int expt)
82 exp_x = __frexp_exp(x, &ex_expt);
84 INSERT_WORDS(scale, (0x3ff + expt) << 20, 0);
85 return (exp_x * scale);
89 __ldexp_cexp(double complex z, int expt)
91 double x, y, exp_x, scale1, scale2;
92 int ex_expt, half_expt;
96 exp_x = __frexp_exp(x, &ex_expt);
100 * Arrange so that scale1 * scale2 == 2**expt. We use this to
101 * compensate for scalbn being horrendously slow.
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);
108 return (CMPLX(cos(y) * exp_x * scale1 * scale2,
109 sin(y) * exp_x * scale1 * scale2));