xref: /freebsd/lib/msun/src/s_exp2f.c (revision b134ea7211146d57edd96001d5fb63ff60a882da)
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 
38 static const float
39     huge    = 0x1p100f,
40     redux   = 0x1.8p23f / TBLSIZE,
41     P1	    = 0x1.62e430p-1f,
42     P2	    = 0x1.ebfbe0p-3f,
43     P3	    = 0x1.c6b348p-5f,
44     P4	    = 0x1.3b2c9cp-7f;
45 
46 static volatile float twom100 = 0x1p-100f;
47 
48 static const double exp2ft[TBLSIZE] = {
49 	0x1.6a09e667f3bcdp-1,
50 	0x1.7a11473eb0187p-1,
51 	0x1.8ace5422aa0dbp-1,
52 	0x1.9c49182a3f090p-1,
53 	0x1.ae89f995ad3adp-1,
54 	0x1.c199bdd85529cp-1,
55 	0x1.d5818dcfba487p-1,
56 	0x1.ea4afa2a490dap-1,
57 	0x1.0000000000000p+0,
58 	0x1.0b5586cf9890fp+0,
59 	0x1.172b83c7d517bp+0,
60 	0x1.2387a6e756238p+0,
61 	0x1.306fe0a31b715p+0,
62 	0x1.3dea64c123422p+0,
63 	0x1.4bfdad5362a27p+0,
64 	0x1.5ab07dd485429p+0,
65 };
66 
67 /*
68  * exp2f(x): compute the base 2 exponential of x
69  *
70  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
71  *
72  * Method: (equally-spaced tables)
73  *
74  *   Reduce x:
75  *     x = 2**k + y, for integer k and |y| <= 1/2.
76  *     Thus we have exp2f(x) = 2**k * exp2(y).
77  *
78  *   Reduce y:
79  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
80  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
81  *     with |z| <= 2**-(TBLSIZE+1).
82  *
83  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
84  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
85  *   Using double precision in the final calculation avoids roundoff error.
86  *
87  *   This method is due to Tang, but I do not use his suggested parameters:
88  *
89  *	Tang, P.  Table-driven Implementation of the Exponential Function
90  *	in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
91  */
92 float
93 exp2f(float x)
94 {
95 	double tv;
96 	float t, z;
97 	uint32_t hx, htv, ix, i0;
98 	int32_t k;
99 
100 	/* Filter out exceptional cases. */
101 	GET_FLOAT_WORD(hx, x);
102 	ix = hx & 0x7fffffff;		/* high word of |x| */
103 	if(ix >= 0x43000000) {			/* |x| >= 128 */
104 		if(ix >= 0x7f800000) {
105 			if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
106 				return (x); 	/* x is NaN or +Inf */
107 			else
108 				return (0.0);	/* x is -Inf */
109 		}
110 		if(x >= 0x1.0p7f)
111 			return (huge * huge);	/* overflow */
112 		if(x <= -0x1.2cp7f)
113 			return (twom100 * twom100); /* underflow */
114 	} else if (ix <= 0x33000000) {		/* |x| <= 0x1p-25 */
115 		return (1.0f + x);
116 	}
117 
118 	/* Reduce x, computing z, i0, and k. */
119 	STRICT_ASSIGN(float, t, x + redux);
120 	GET_FLOAT_WORD(i0, t);
121 	i0 += TBLSIZE / 2;
122 	k = (i0 >> TBLBITS) << 20;
123 	i0 &= TBLSIZE - 1;
124 	t -= redux;
125 	z = x - t;
126 
127 	/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
128 	tv = exp2ft[i0];
129 	tv = tv + tv * (z * (P1 + z * (P2 + z * (P3 + z * P4))));
130 
131 	/* Scale by 2**(k>>20). */
132 	if (k != 0) {
133 		GET_HIGH_WORD(htv, tv);
134 		SET_HIGH_WORD(tv, htv + k);
135 	}
136 	return (tv);
137 }
138