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