1 /* $OpenBSD: s_exp2f.c,v 1.2 2012/12/05 23:20:04 deraadt Exp $ */
2 /*-
3 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
4 * All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
25 * SUCH DAMAGE.
26 */
27
28 #include <float.h>
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 redux = 0x1.8p23f / TBLSIZE,
39 P1 = 0x1.62e430p-1f,
40 P2 = 0x1.ebfbe0p-3f,
41 P3 = 0x1.c6b348p-5f,
42 P4 = 0x1.3b2c9cp-7f;
43
44 static volatile float twom100 = 0x1p-100f;
45
46 static const double exp2ft[TBLSIZE] = {
47 0x1.6a09e667f3bcdp-1,
48 0x1.7a11473eb0187p-1,
49 0x1.8ace5422aa0dbp-1,
50 0x1.9c49182a3f090p-1,
51 0x1.ae89f995ad3adp-1,
52 0x1.c199bdd85529cp-1,
53 0x1.d5818dcfba487p-1,
54 0x1.ea4afa2a490dap-1,
55 0x1.0000000000000p+0,
56 0x1.0b5586cf9890fp+0,
57 0x1.172b83c7d517bp+0,
58 0x1.2387a6e756238p+0,
59 0x1.306fe0a31b715p+0,
60 0x1.3dea64c123422p+0,
61 0x1.4bfdad5362a27p+0,
62 0x1.5ab07dd485429p+0,
63 };
64
65 /*
66 * exp2f(x): compute the base 2 exponential of x
67 *
68 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
69 *
70 * Method: (equally-spaced tables)
71 *
72 * Reduce x:
73 * x = 2**k + y, for integer k and |y| <= 1/2.
74 * Thus we have exp2f(x) = 2**k * exp2(y).
75 *
76 * Reduce y:
77 * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
78 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
79 * with |z| <= 2**-(TBLSIZE+1).
80 *
81 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
82 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
83 * Using double precision for everything except the reduction makes
84 * roundoff error insignificant and simplifies the scaling step.
85 *
86 * This method is due to Tang, but I do not use his suggested parameters:
87 *
88 * Tang, P. Table-driven Implementation of the Exponential Function
89 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
90 */
91 float
exp2f(float x)92 exp2f(float x)
93 {
94 double tv, twopk, u, z;
95 float t;
96 uint32_t hx, ix, i0;
97 int32_t k;
98
99 /* Filter out exceptional cases. */
100 GET_FLOAT_WORD(hx, x);
101 ix = hx & 0x7fffffff; /* high word of |x| */
102 if(ix >= 0x43000000) { /* |x| >= 128 */
103 if(ix >= 0x7f800000) {
104 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
105 return (x + x); /* x is NaN or +Inf */
106 else
107 return (0.0); /* x is -Inf */
108 }
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 STRICT_ASSIGN(float, t, x + redux);
119 GET_FLOAT_WORD(i0, t);
120 i0 += TBLSIZE / 2;
121 k = (i0 >> TBLBITS) << 20;
122 i0 &= TBLSIZE - 1;
123 t -= redux;
124 z = x - t;
125 INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
126
127 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
128 tv = exp2ft[i0];
129 u = tv * z;
130 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
131
132 /* Scale by 2**(k>>20). */
133 return (tv * twopk);
134 }
135