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: head/lib/msun/src/k_exp.c 326219 2017-11-26 02:00:33Z pfg $");
31
32 #include <complex.h>
33
34 #include "math.h"
35 #include "math_private.h"
36
37 static const uint32_t k = 1799; /* constant for reduction */
38 static 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 */
47 static double
__frexp_exp(double x,int * expt)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
76 double
__ldexp_exp(double x,int expt)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
88 double complex
__ldexp_cexp(double complex z,int expt)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