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: n_exp2f.c,v 1.2 2014/03/23 15:26:47 martin 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 <stdint.h>
34 #include <float.h>
35 #include <string.h>
36
37 #include "math.h"
38
39 #define TBLBITS 4
40 #define TBLSIZE (1 << TBLBITS)
41
42 static const float
43 huge = 0x1p100f,
44 redux = 0x1.8p23f / TBLSIZE,
45 P1 = 0x1.62e430p-1f,
46 P2 = 0x1.ebfbe0p-3f,
47 P3 = 0x1.c6b348p-5f,
48 P4 = 0x1.3b2c9cp-7f;
49
50 static volatile float twom100 = 0x1p-100f;
51
52 static const double exp2ft[TBLSIZE] = {
53 0x1.6a09e667f3bcdp-1,
54 0x1.7a11473eb0187p-1,
55 0x1.8ace5422aa0dbp-1,
56 0x1.9c49182a3f090p-1,
57 0x1.ae89f995ad3adp-1,
58 0x1.c199bdd85529cp-1,
59 0x1.d5818dcfba487p-1,
60 0x1.ea4afa2a490dap-1,
61 0x1.0000000000000p+0,
62 0x1.0b5586cf9890fp+0,
63 0x1.172b83c7d517bp+0,
64 0x1.2387a6e756238p+0,
65 0x1.306fe0a31b715p+0,
66 0x1.3dea64c123422p+0,
67 0x1.4bfdad5362a27p+0,
68 0x1.5ab07dd485429p+0,
69 };
70
71 /*
72 * exp2f(x): compute the base 2 exponential of x
73 *
74 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
75 *
76 * Method: (equally-spaced tables)
77 *
78 * Reduce x:
79 * x = 2**k + y, for integer k and |y| <= 1/2.
80 * Thus we have exp2f(x) = 2**k * exp2(y).
81 *
82 * Reduce y:
83 * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
84 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
85 * with |z| <= 2**-(TBLSIZE+1).
86 *
87 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
88 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
89 * Using double precision for everything except the reduction makes
90 * roundoff error insignificant and simplifies the scaling step.
91 *
92 * This method is due to Tang, but I do not use his suggested parameters:
93 *
94 * Tang, P. Table-driven Implementation of the Exponential Function
95 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
96 */
97 float
exp2f(float x)98 exp2f(float x)
99 {
100 double tv, twopk, u, z;
101 float t;
102 uint32_t hx, ix, i0;
103 int32_t k, temp;
104
105 /* Filter out exceptional cases. */
106 memcpy(&hx, &x, sizeof(hx));
107 ix = hx & 0x7fffffff; /* high word of |x| */
108 if(ix >= 0x43000000) { /* |x| >= 128 */
109 if(x >= 0x1.0p7f)
110 return (huge * huge); /* overflow */
111 if(x <= -0x1.2cp7f)
112 return (twom100 * twom100); /* underflow */
113 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
114 return (1.0f + x);
115 }
116
117 /* Reduce x, computing z, i0, and k. */
118 i0 = x + redux;
119 memcpy(&t, &i0, sizeof(t));
120 i0 += TBLSIZE / 2;
121 k = (i0 >> TBLBITS) << 20;
122 i0 &= TBLSIZE - 1;
123 t -= redux;
124 z = x - t;
125 temp = 0x3ff00000+k;
126 twopk = 0.0;
127 memcpy(&twopk, &temp, sizeof(temp));
128
129 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
130 tv = exp2ft[i0];
131 u = tv * z;
132 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
133
134 /* Scale by 2**(k>>20). */
135 return (tv * twopk);
136 }
137