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