xref: /freebsd/lib/msun/src/s_exp2f.c (revision 1e413cf93298b5b97441a21d9a50fdcd0ee9945e)
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 "math.h"
31 #include "math_private.h"
32 
33 #define	TBLBITS	4
34 #define	TBLSIZE	(1 << TBLBITS)
35 
36 static const float
37     huge    = 0x1p100f,
38     twom100 = 0x1p-100f,
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 
45 static const double exp2ft[TBLSIZE] = {
46 	0x1.6a09e667f3bcdp-1,
47 	0x1.7a11473eb0187p-1,
48 	0x1.8ace5422aa0dbp-1,
49 	0x1.9c49182a3f090p-1,
50 	0x1.ae89f995ad3adp-1,
51 	0x1.c199bdd85529cp-1,
52 	0x1.d5818dcfba487p-1,
53 	0x1.ea4afa2a490dap-1,
54 	0x1.0000000000000p+0,
55 	0x1.0b5586cf9890fp+0,
56 	0x1.172b83c7d517bp+0,
57 	0x1.2387a6e756238p+0,
58 	0x1.306fe0a31b715p+0,
59 	0x1.3dea64c123422p+0,
60 	0x1.4bfdad5362a27p+0,
61 	0x1.5ab07dd485429p+0,
62 };
63 
64 /*
65  * exp2f(x): compute the base 2 exponential of x
66  *
67  * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
68  *
69  * Method: (equally-spaced tables)
70  *
71  *   Reduce x:
72  *     x = 2**k + y, for integer k and |y| <= 1/2.
73  *     Thus we have exp2f(x) = 2**k * exp2(y).
74  *
75  *   Reduce y:
76  *     y = i/TBLSIZE + z for integer i near y * TBLSIZE.
77  *     Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
78  *     with |z| <= 2**-(TBLSIZE+1).
79  *
80  *   We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
81  *   degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
82  *   Using double precision in the final calculation avoids roundoff error.
83  *
84  *   This method is due to Tang, but I do not use his suggested parameters:
85  *
86  *	Tang, P.  Table-driven Implementation of the Exponential Function
87  *	in IEEE Floating-Point Arithmetic.  TOMS 15(2), 144-157 (1989).
88  */
89 float
90 exp2f(float x)
91 {
92 	double tv;
93 	float r, z;
94 	volatile float t;	/* prevent gcc from using too much precision */
95 	uint32_t hx, hr, ix, i0;
96 	int32_t k;
97 
98 	/* Filter out exceptional cases. */
99 	GET_FLOAT_WORD(hx,x);
100 	ix = hx & 0x7fffffff;		/* high word of |x| */
101 	if(ix >= 0x43000000) {			/* |x| >= 128 */
102 		if(ix >= 0x7f800000) {
103 			if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
104 				return (x); 	/* x is NaN or +Inf */
105 			else
106 				return (0.0);	/* x is -Inf */
107 		}
108 		if(x >= 0x1.0p7f)
109 			return (huge * huge);	/* overflow */
110 		if(x <= -0x1.2cp7f)
111 			return (twom100 * twom100); /* underflow */
112 	} else if (ix <= 0x33000000) {		/* |x| <= 0x1p-25 */
113 		return (1.0f + x);
114 	}
115 
116 	/* Reduce x, computing z, i0, and k. */
117 	t = x + redux;
118 	GET_FLOAT_WORD(i0, t);
119 	i0 += TBLSIZE / 2;
120 	k = (i0 >> TBLBITS) << 23;
121 	i0 &= TBLSIZE - 1;
122 	t -= redux;
123 	z = x - t;
124 
125 	/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
126 	tv = exp2ft[i0];
127 	r = tv + tv * (z * (P1 + z * (P2 + z * (P3 + z * P4))));
128 
129 	/* Scale by 2**(k>>23). */
130 	if(k >= -125 << 23) {
131 		if (k != 0) {
132 			GET_FLOAT_WORD(hr, r);
133 			SET_FLOAT_WORD(r, hr + k);
134 		}
135 		return (r);
136 	} else {
137 		GET_FLOAT_WORD(hr, r);
138 		SET_FLOAT_WORD(r, hr + (k + (100 << 23)));
139 		return (r * twom100);
140 	}
141 }
142