s_exp2f.c revision 175468 
1/*-
2 * Copyright (c) 2005 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: head/lib/msun/src/s_exp2f.c 175468 2008-01-18 22:19:04Z das $");
29
30#include "math.h"
31#include "math_private.h"
32
33#define	TBLBITS	4
34#define	TBLSIZE	(1 << TBLBITS)
35
36static const float
37    huge    = 0x1p100f,
38    redux   = 0x1.8p23f / TBLSIZE,
39    P1	    = 0x1.62e430p-1f,
40    P2	    = 0x1.ebfbe0p-3f,
41    P3	    = 0x1.c6b348p-5f,
42    P4	    = 0x1.3b2c9cp-7f;
43
44static volatile float twom100 = 0x1p-100f;
45
46static const double exp2ft[TBLSIZE] = {
47	0x1.6a09e667f3bcdp-1,
48	0x1.7a11473eb0187p-1,
49	0x1.8ace5422aa0dbp-1,
50	0x1.9c49182a3f090p-1,
51	0x1.ae89f995ad3adp-1,
52	0x1.c199bdd85529cp-1,
53	0x1.d5818dcfba487p-1,
54	0x1.ea4afa2a490dap-1,
55	0x1.0000000000000p+0,
56	0x1.0b5586cf9890fp+0,
57	0x1.172b83c7d517bp+0,
58	0x1.2387a6e756238p+0,
59	0x1.306fe0a31b715p+0,
60	0x1.3dea64c123422p+0,
61	0x1.4bfdad5362a27p+0,
62	0x1.5ab07dd485429p+0,
63};
64
65/*
66 * exp2f(x): compute the base 2 exponential of x
67 *
68 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
69 *
70 * Method: (equally-spaced tables)
71 *
72 *   Reduce x:
73 *     x = 2**k + y, for integer k and |y| <= 1/2.
74 *     Thus we have exp2f(x) = 2**k * exp2(y).
75 *
76 *   Reduce y:
77 *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
78 *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
79 *     with |z| <= 2**-(TBLSIZE+1).
80 *
81 *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
82 *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
83 *   Using double precision in the final calculation avoids roundoff error.
84 *
85 *   This method is due to Tang, but I do not use his suggested parameters:
86 *
87 *	Tang, P.  Table-driven Implementation of the Exponential Function
88 *	in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
89 */
90float
91exp2f(float x)
92{
93	double tv;
94	float r, z;
95	volatile float t;	/* prevent gcc from using too much precision */
96	uint32_t hx, hr, ix, i0;
97	int32_t k;
98
99	/* Filter out exceptional cases. */
100	GET_FLOAT_WORD(hx,x);
101	ix = hx & 0x7fffffff;		/* high word of |x| */
102	if(ix >= 0x43000000) {			/* |x| >= 128 */
103		if(ix >= 0x7f800000) {
104			if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
105				return (x); 	/* x is NaN or +Inf */
106			else
107				return (0.0);	/* x is -Inf */
108		}
109		if(x >= 0x1.0p7f)
110			return (huge * huge);	/* overflow */
111		if(x <= -0x1.2cp7f)
112			return (twom100 * twom100); /* underflow */
113	} else if (ix <= 0x33000000) {		/* |x| <= 0x1p-25 */
114		return (1.0f + x);
115	}
116
117	/* Reduce x, computing z, i0, and k. */
118	t = x + redux;
119	GET_FLOAT_WORD(i0, t);
120	i0 += TBLSIZE / 2;
121	k = (i0 >> TBLBITS) << 23;
122	i0 &= TBLSIZE - 1;
123	t -= redux;
124	z = x - t;
125
126	/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
127	tv = exp2ft[i0];
128	r = tv + tv * (z * (P1 + z * (P2 + z * (P3 + z * P4))));
129
130	/* Scale by 2**(k>>23). */
131	if(k >= -125 << 23) {
132		if (k != 0) {
133			GET_FLOAT_WORD(hr, r);
134			SET_FLOAT_WORD(r, hr + k);
135		}
136		return (r);
137	} else {
138		GET_FLOAT_WORD(hr, r);
139		SET_FLOAT_WORD(r, hr + (k + (100 << 23)));
140		return (r * twom100);
141	}
142}
143