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("$FreeBSD$");
29
30#include <float.h>
31
32#include "math.h"
33#include "math_private.h"
34
35#define	TBLBITS	4
36#define	TBLSIZE	(1 << TBLBITS)
37
38static const float
39    redux   = 0x1.8p23f / TBLSIZE,
40    P1	    = 0x1.62e430p-1f,
41    P2	    = 0x1.ebfbe0p-3f,
42    P3	    = 0x1.c6b348p-5f,
43    P4	    = 0x1.3b2c9cp-7f;
44
45static volatile float
46    huge    = 0x1p100f,
47    twom100 = 0x1p-100f;
48
49static const double exp2ft[TBLSIZE] = {
50	0x1.6a09e667f3bcdp-1,
51	0x1.7a11473eb0187p-1,
52	0x1.8ace5422aa0dbp-1,
53	0x1.9c49182a3f090p-1,
54	0x1.ae89f995ad3adp-1,
55	0x1.c199bdd85529cp-1,
56	0x1.d5818dcfba487p-1,
57	0x1.ea4afa2a490dap-1,
58	0x1.0000000000000p+0,
59	0x1.0b5586cf9890fp+0,
60	0x1.172b83c7d517bp+0,
61	0x1.2387a6e756238p+0,
62	0x1.306fe0a31b715p+0,
63	0x1.3dea64c123422p+0,
64	0x1.4bfdad5362a27p+0,
65	0x1.5ab07dd485429p+0,
66};
67
68/*
69 * exp2f(x): compute the base 2 exponential of x
70 *
71 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
72 *
73 * Method: (equally-spaced tables)
74 *
75 *   Reduce x:
76 *     x = 2**k + y, for integer k and |y| <= 1/2.
77 *     Thus we have exp2f(x) = 2**k * exp2(y).
78 *
79 *   Reduce y:
80 *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
81 *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
82 *     with |z| <= 2**-(TBLSIZE+1).
83 *
84 *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
85 *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
86 *   Using double precision for everything except the reduction makes
87 *   roundoff error insignificant and simplifies the scaling step.
88 *
89 *   This method is due to Tang, but I do not use his suggested parameters:
90 *
91 *	Tang, P.  Table-driven Implementation of the Exponential Function
92 *	in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
93 */
94float
95exp2f(float x)
96{
97	double tv, twopk, u, z;
98	float t;
99	uint32_t hx, ix, i0;
100	int32_t k;
101
102	/* Filter out exceptional cases. */
103	GET_FLOAT_WORD(hx, x);
104	ix = hx & 0x7fffffff;		/* high word of |x| */
105	if(ix >= 0x43000000) {			/* |x| >= 128 */
106		if(ix >= 0x7f800000) {
107			if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
108				return (x + x);	/* x is NaN or +Inf */
109			else
110				return (0.0);	/* x is -Inf */
111		}
112		if(x >= 0x1.0p7f)
113			return (huge * huge);	/* overflow */
114		if(x <= -0x1.2cp7f)
115			return (twom100 * twom100); /* underflow */
116	} else if (ix <= 0x33000000) {		/* |x| <= 0x1p-25 */
117		return (1.0f + x);
118	}
119
120	/* Reduce x, computing z, i0, and k. */
121	STRICT_ASSIGN(float, t, x + redux);
122	GET_FLOAT_WORD(i0, t);
123	i0 += TBLSIZE / 2;
124	k = (i0 >> TBLBITS) << 20;
125	i0 &= TBLSIZE - 1;
126	t -= redux;
127	z = x - t;
128	INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
129
130	/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
131	tv = exp2ft[i0];
132	u = tv * z;
133	tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
134
135	/* Scale by 2**(k>>20). */
136	return (tv * twopk);
137}
138