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__RCSID("$NetBSD: s_exp2f.c,v 1.1 2010/01/11 16:28:39 christos Exp $");
29#ifdef __FBSDID
30__FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.9 2008/02/22 02:27:34 das Exp $");
31#endif
32
33#include <float.h>
34
35#include "math.h"
36#include "math_private.h"
37
38#define	TBLBITS	4
39#define	TBLSIZE	(1 << TBLBITS)
40
41static const float
42    redux   = 0x1.8p23f / TBLSIZE,
43    P1	    = 0x1.62e430p-1f,
44    P2	    = 0x1.ebfbe0p-3f,
45    P3	    = 0x1.c6b348p-5f,
46    P4	    = 0x1.3b2c9cp-7f;
47
48/*
49 * For out of range values we need to generate the appropriate
50 * underflow or overflow trap as well as generating infinity or zero.
51 * This means we have to get the fpu to execute an instruction that
52 * will generate the trap (and not have the compiler optimise it away).
53 * This is normally done by calculating 'huge * huge' or 'tiny * tiny'.
54 *
55 * i386 is particularly problematic.
56 * The 'float' result is returned on the x87 stack, so is 'long double'.
57 * If we just multiply two 'float' values the caller will see 0x1p+/-200
58 * (not 0 or infinity).
59 * If we use 'double' the compiler does a store-load which will convert the
60 * value and generate the required exception.
61 */
62#ifdef __i386__
63static volatile double overflow = 0x1p+1000;
64static volatile double underflow = 0x1p-1000;
65#else
66static volatile float huge = 0x1p+100;
67static volatile float tiny = 0x1p-100;
68#define overflow (huge * huge)
69#define underflow (tiny * tiny)
70#endif
71
72static const double exp2ft[TBLSIZE] = {
73	0x1.6a09e667f3bcdp-1,
74	0x1.7a11473eb0187p-1,
75	0x1.8ace5422aa0dbp-1,
76	0x1.9c49182a3f090p-1,
77	0x1.ae89f995ad3adp-1,
78	0x1.c199bdd85529cp-1,
79	0x1.d5818dcfba487p-1,
80	0x1.ea4afa2a490dap-1,
81	0x1.0000000000000p+0,
82	0x1.0b5586cf9890fp+0,
83	0x1.172b83c7d517bp+0,
84	0x1.2387a6e756238p+0,
85	0x1.306fe0a31b715p+0,
86	0x1.3dea64c123422p+0,
87	0x1.4bfdad5362a27p+0,
88	0x1.5ab07dd485429p+0,
89};
90
91/*
92 * exp2f(x): compute the base 2 exponential of x
93 *
94 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
95 *
96 * Method: (equally-spaced tables)
97 *
98 *   Reduce x:
99 *     x = 2**k + y, for integer k and |y| <= 1/2.
100 *     Thus we have exp2f(x) = 2**k * exp2(y).
101 *
102 *   Reduce y:
103 *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
104 *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
105 *     with |z| <= 2**-(TBLSIZE+1).
106 *
107 *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
108 *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
109 *   Using double precision for everything except the reduction makes
110 *   roundoff error insignificant and simplifies the scaling step.
111 *
112 *   This method is due to Tang, but I do not use his suggested parameters:
113 *
114 *	Tang, P.  Table-driven Implementation of the Exponential Function
115 *	in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
116 */
117float
118exp2f(float x)
119{
120	double tv, twopk, u, z;
121	float t;
122	uint32_t hx, ix, i0;
123	int32_t k;
124
125	/* Filter out exceptional cases. */
126	GET_FLOAT_WORD(hx, x);
127	ix = hx & 0x7fffffff;		/* high word of |x| */
128	if(ix >= 0x43000000) {			/* |x| >= 128 */
129		if(ix >= 0x7f800000) {
130			if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
131				return (x + x);	/* x is NaN or +Inf */
132			else
133				return (0.0);	/* x is -Inf */
134		}
135		if(x >= 0x1.0p7f)
136			return overflow;	/* +infinity with overflow */
137		if(x <= -0x1.2cp7f)
138			return underflow;	/* zero with underflow */
139	} else if (ix <= 0x33000000) {		/* |x| <= 0x1p-25 */
140		return (1.0f + x);
141	}
142
143	/* Reduce x, computing z, i0, and k. */
144	STRICT_ASSIGN(float, t, x + redux);
145	GET_FLOAT_WORD(i0, t);
146	i0 += TBLSIZE / 2;
147	k = (i0 >> TBLBITS) << 20;
148	i0 &= TBLSIZE - 1;
149	t -= redux;
150	z = x - t;
151	INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
152
153	/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
154	tv = exp2ft[i0];
155	u = tv * z;
156	tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
157
158	/* Scale by 2**(k>>20). */
159	return (tv * twopk);
160}
161