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