1/*-
2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
3 *
4 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 *    notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 *    notice, this list of conditions and the following disclaimer in the
14 *    documentation and/or other materials provided with the distribution.
15 *
16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
17 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
18 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
19 * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
20 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
21 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
22 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
23 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
24 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
25 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
26 * SUCH DAMAGE.
27 */
28
29#include <sys/cdefs.h>
30__FBSDID("$FreeBSD$");
31
32#include <float.h>
33
34#include "math.h"
35#include "math_private.h"
36
37#define	TBLBITS	4
38#define	TBLSIZE	(1 << TBLBITS)
39
40static const float
41    redux   = 0x1.8p23f / TBLSIZE,
42    P1	    = 0x1.62e430p-1f,
43    P2	    = 0x1.ebfbe0p-3f,
44    P3	    = 0x1.c6b348p-5f,
45    P4	    = 0x1.3b2c9cp-7f;
46
47static volatile float
48    huge    = 0x1p100f,
49    twom100 = 0x1p-100f;
50
51static const double exp2ft[TBLSIZE] = {
52	0x1.6a09e667f3bcdp-1,
53	0x1.7a11473eb0187p-1,
54	0x1.8ace5422aa0dbp-1,
55	0x1.9c49182a3f090p-1,
56	0x1.ae89f995ad3adp-1,
57	0x1.c199bdd85529cp-1,
58	0x1.d5818dcfba487p-1,
59	0x1.ea4afa2a490dap-1,
60	0x1.0000000000000p+0,
61	0x1.0b5586cf9890fp+0,
62	0x1.172b83c7d517bp+0,
63	0x1.2387a6e756238p+0,
64	0x1.306fe0a31b715p+0,
65	0x1.3dea64c123422p+0,
66	0x1.4bfdad5362a27p+0,
67	0x1.5ab07dd485429p+0,
68};
69
70/*
71 * exp2f(x): compute the base 2 exponential of x
72 *
73 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
74 *
75 * Method: (equally-spaced tables)
76 *
77 *   Reduce x:
78 *     x = 2**k + y, for integer k and |y| <= 1/2.
79 *     Thus we have exp2f(x) = 2**k * exp2(y).
80 *
81 *   Reduce y:
82 *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
83 *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
84 *     with |z| <= 2**-(TBLSIZE+1).
85 *
86 *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
87 *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
88 *   Using double precision for everything except the reduction makes
89 *   roundoff error insignificant and simplifies the scaling step.
90 *
91 *   This method is due to Tang, but I do not use his suggested parameters:
92 *
93 *	Tang, P.  Table-driven Implementation of the Exponential Function
94 *	in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
95 */
96float
97exp2f(float x)
98{
99	double tv, twopk, u, z;
100	float t;
101	uint32_t hx, ix, i0;
102	int32_t k;
103
104	/* Filter out exceptional cases. */
105	GET_FLOAT_WORD(hx, x);
106	ix = hx & 0x7fffffff;		/* high word of |x| */
107	if(ix >= 0x43000000) {			/* |x| >= 128 */
108		if(ix >= 0x7f800000) {
109			if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
110				return (x + x);	/* x is NaN or +Inf */
111			else
112				return (0.0);	/* x is -Inf */
113		}
114		if(x >= 0x1.0p7f)
115			return (huge * huge);	/* overflow */
116		if(x <= -0x1.2cp7f)
117			return (twom100 * twom100); /* underflow */
118	} else if (ix <= 0x33000000) {		/* |x| <= 0x1p-25 */
119		return (1.0f + x);
120	}
121
122	/* Reduce x, computing z, i0, and k. */
123	STRICT_ASSIGN(float, t, x + redux);
124	GET_FLOAT_WORD(i0, t);
125	i0 += TBLSIZE / 2;
126	k = (i0 >> TBLBITS) << 20;
127	i0 &= TBLSIZE - 1;
128	t -= redux;
129	z = x - t;
130	INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
131
132	/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
133	tv = exp2ft[i0];
134	u = tv * z;
135	tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
136
137	/* Scale by 2**(k>>20). */
138	return (tv * twopk);
139}
140