1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause-FreeBSD
3 *
4 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in the
14 * documentation and/or other materials provided with the distribution.
15 *
16 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
17 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
18 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
19 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
20 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
21 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
22 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
23 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
24 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
25 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
26 * SUCH DAMAGE.
27 */
28
29 #include <sys/cdefs.h>
30 __FBSDID("$FreeBSD$");
31
32 #include <float.h>
33
34 #include "math.h"
35 #include "math_private.h"
36
37 #define TBLBITS 4
38 #define TBLSIZE (1 << TBLBITS)
39
40 static const float
41 redux = 0x1.8p23f / TBLSIZE,
42 P1 = 0x1.62e430p-1f,
43 P2 = 0x1.ebfbe0p-3f,
44 P3 = 0x1.c6b348p-5f,
45 P4 = 0x1.3b2c9cp-7f;
46
47 static volatile float
48 huge = 0x1p100f,
49 twom100 = 0x1p-100f;
50
51 static const double exp2ft[TBLSIZE] = {
52 0x1.6a09e667f3bcdp-1,
53 0x1.7a11473eb0187p-1,
54 0x1.8ace5422aa0dbp-1,
55 0x1.9c49182a3f090p-1,
56 0x1.ae89f995ad3adp-1,
57 0x1.c199bdd85529cp-1,
58 0x1.d5818dcfba487p-1,
59 0x1.ea4afa2a490dap-1,
60 0x1.0000000000000p+0,
61 0x1.0b5586cf9890fp+0,
62 0x1.172b83c7d517bp+0,
63 0x1.2387a6e756238p+0,
64 0x1.306fe0a31b715p+0,
65 0x1.3dea64c123422p+0,
66 0x1.4bfdad5362a27p+0,
67 0x1.5ab07dd485429p+0,
68 };
69
70 /*
71 * exp2f(x): compute the base 2 exponential of x
72 *
73 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
74 *
75 * Method: (equally-spaced tables)
76 *
77 * Reduce x:
78 * x = 2**k + y, for integer k and |y| <= 1/2.
79 * Thus we have exp2f(x) = 2**k * exp2(y).
80 *
81 * Reduce y:
82 * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
83 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
84 * with |z| <= 2**-(TBLSIZE+1).
85 *
86 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
87 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
88 * Using double precision for everything except the reduction makes
89 * roundoff error insignificant and simplifies the scaling step.
90 *
91 * This method is due to Tang, but I do not use his suggested parameters:
92 *
93 * Tang, P. Table-driven Implementation of the Exponential Function
94 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
95 */
96 float
exp2f(float x)97 exp2f(float x)
98 {
99 double tv, twopk, u, z;
100 float t;
101 uint32_t hx, ix, i0;
102 int32_t k;
103
104 /* Filter out exceptional cases. */
105 GET_FLOAT_WORD(hx, x);
106 ix = hx & 0x7fffffff; /* high word of |x| */
107 if(ix >= 0x43000000) { /* |x| >= 128 */
108 if(ix >= 0x7f800000) {
109 if ((ix & 0x7fffff) != 0 || (hx & 0x80000000) == 0)
110 return (x + x); /* x is NaN or +Inf */
111 else
112 return (0.0); /* x is -Inf */
113 }
114 if(x >= 0x1.0p7f)
115 return (huge * huge); /* overflow */
116 if(x <= -0x1.2cp7f)
117 return (twom100 * twom100); /* underflow */
118 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
119 return (1.0f + x);
120 }
121
122 /* Reduce x, computing z, i0, and k. */
123 STRICT_ASSIGN(float, t, x + redux);
124 GET_FLOAT_WORD(i0, t);
125 i0 += TBLSIZE / 2;
126 k = (i0 >> TBLBITS) << 20;
127 i0 &= TBLSIZE - 1;
128 t -= redux;
129 z = x - t;
130 INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
131
132 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
133 tv = exp2ft[i0];
134 u = tv * z;
135 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
136
137 /* Scale by 2**(k>>20). */
138 return (tv * twopk);
139 }
140