Sun Mar 23 15:26:47 2014 UTC ()
Avoid strict aliasing problems


(martin)
diff -r1.2 -r1.3 src/lib/libm/noieee_src/n_exp2.c
diff -r1.1 -r1.2 src/lib/libm/noieee_src/n_exp2f.c
cvs diff -r1.2 -r1.3 src/lib/libm/noieee_src/n_exp2.c (switch to unified diff)

--- src/lib/libm/noieee_src/n_exp2.c 2014/03/12 19:42:18 1.2
+++ src/lib/libm/noieee_src/n_exp2.c 2014/03/23 15:26:47 1.3
@@ -1,392 +1,395 @@ @@ -1,392 +1,395 @@
1/*- 1/*-
2 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> 2 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
3 * All rights reserved. 3 * All rights reserved.
4 * 4 *
5 * Redistribution and use in source and binary forms, with or without 5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions 6 * modification, are permitted provided that the following conditions
7 * are met: 7 * are met:
8 * 1. Redistributions of source code must retain the above copyright 8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer. 9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright 10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the 11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution. 12 * documentation and/or other materials provided with the distribution.
13 * 13 *
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 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 15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 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 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 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 23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
24 * SUCH DAMAGE. 24 * SUCH DAMAGE.
25 */ 25 */
26 26
27#include <sys/cdefs.h> 27#include <sys/cdefs.h>
28__RCSID("$NetBSD: n_exp2.c,v 1.2 2014/03/12 19:42:18 martin Exp $"); 28__RCSID("$NetBSD: n_exp2.c,v 1.3 2014/03/23 15:26:47 martin Exp $");
29#ifdef __FBSDID 29#ifdef __FBSDID
30__FBSDID("$FreeBSD: src/lib/msun/src/s_exp2.c,v 1.7 2008/02/22 02:27:34 das Exp $"); 30__FBSDID("$FreeBSD: src/lib/msun/src/s_exp2.c,v 1.7 2008/02/22 02:27:34 das Exp $");
31#endif 31#endif
32 32
33#include <stdint.h> 33#include <stdint.h>
34#include <float.h> 34#include <float.h>
 35#include <string.h>
35 36
36#include "math.h" 37#include "math.h"
37 38
38#define TBLBITS 8 39#define TBLBITS 8
39#define TBLSIZE (1 << TBLBITS) 40#define TBLSIZE (1 << TBLBITS)
40 41
41static const double 42static const double
42 huge = 0x1p126, 43 huge = 0x1p126,
43 redux = 0x1.8p52 / TBLSIZE, 44 redux = 0x1.8p52 / TBLSIZE,
44 P1 = 0x1.62e42fefa39efp-1, 45 P1 = 0x1.62e42fefa39efp-1,
45 P2 = 0x1.ebfbdff82c575p-3, 46 P2 = 0x1.ebfbdff82c575p-3,
46 P3 = 0x1.c6b08d704a0a6p-5, 47 P3 = 0x1.c6b08d704a0a6p-5,
47 P4 = 0x1.3b2ab88f70400p-7, 48 P4 = 0x1.3b2ab88f70400p-7,
48 P5 = 0x1.5d88003875c74p-10; 49 P5 = 0x1.5d88003875c74p-10;
49 50
50static volatile double twom1000 = 0x1p-100; 51static volatile double twom1000 = 0x1p-100;
51 52
52static const double tbl[TBLSIZE * 2] = { 53static const double tbl[TBLSIZE * 2] = {
53/* exp2(z + eps) eps */ 54/* exp2(z + eps) eps */
54 0x1.6a09e667f3d5dp-1, 0x1.9880p-44, 55 0x1.6a09e667f3d5dp-1, 0x1.9880p-44,
55 0x1.6b052fa751744p-1, 0x1.8000p-50, 56 0x1.6b052fa751744p-1, 0x1.8000p-50,
56 0x1.6c012750bd9fep-1, -0x1.8780p-45, 57 0x1.6c012750bd9fep-1, -0x1.8780p-45,
57 0x1.6cfdcddd476bfp-1, 0x1.ec00p-46, 58 0x1.6cfdcddd476bfp-1, 0x1.ec00p-46,
58 0x1.6dfb23c651a29p-1, -0x1.8000p-50, 59 0x1.6dfb23c651a29p-1, -0x1.8000p-50,
59 0x1.6ef9298593ae3p-1, -0x1.c000p-52, 60 0x1.6ef9298593ae3p-1, -0x1.c000p-52,
60 0x1.6ff7df9519386p-1, -0x1.fd80p-45, 61 0x1.6ff7df9519386p-1, -0x1.fd80p-45,
61 0x1.70f7466f42da3p-1, -0x1.c880p-45, 62 0x1.70f7466f42da3p-1, -0x1.c880p-45,
62 0x1.71f75e8ec5fc3p-1, 0x1.3c00p-46, 63 0x1.71f75e8ec5fc3p-1, 0x1.3c00p-46,
63 0x1.72f8286eacf05p-1, -0x1.8300p-44, 64 0x1.72f8286eacf05p-1, -0x1.8300p-44,
64 0x1.73f9a48a58152p-1, -0x1.0c00p-47, 65 0x1.73f9a48a58152p-1, -0x1.0c00p-47,
65 0x1.74fbd35d7ccfcp-1, 0x1.f880p-45, 66 0x1.74fbd35d7ccfcp-1, 0x1.f880p-45,
66 0x1.75feb564267f1p-1, 0x1.3e00p-47, 67 0x1.75feb564267f1p-1, 0x1.3e00p-47,
67 0x1.77024b1ab6d48p-1, -0x1.7d00p-45, 68 0x1.77024b1ab6d48p-1, -0x1.7d00p-45,
68 0x1.780694fde5d38p-1, -0x1.d000p-50, 69 0x1.780694fde5d38p-1, -0x1.d000p-50,
69 0x1.790b938ac1d00p-1, 0x1.3000p-49, 70 0x1.790b938ac1d00p-1, 0x1.3000p-49,
70 0x1.7a11473eb0178p-1, -0x1.d000p-49, 71 0x1.7a11473eb0178p-1, -0x1.d000p-49,
71 0x1.7b17b0976d060p-1, 0x1.0400p-45, 72 0x1.7b17b0976d060p-1, 0x1.0400p-45,
72 0x1.7c1ed0130c133p-1, 0x1.0000p-53, 73 0x1.7c1ed0130c133p-1, 0x1.0000p-53,
73 0x1.7d26a62ff8636p-1, -0x1.6900p-45, 74 0x1.7d26a62ff8636p-1, -0x1.6900p-45,
74 0x1.7e2f336cf4e3bp-1, -0x1.2e00p-47, 75 0x1.7e2f336cf4e3bp-1, -0x1.2e00p-47,
75 0x1.7f3878491c3e8p-1, -0x1.4580p-45, 76 0x1.7f3878491c3e8p-1, -0x1.4580p-45,
76 0x1.80427543e1b4ep-1, 0x1.3000p-44, 77 0x1.80427543e1b4ep-1, 0x1.3000p-44,
77 0x1.814d2add1071ap-1, 0x1.f000p-47, 78 0x1.814d2add1071ap-1, 0x1.f000p-47,
78 0x1.82589994ccd7ep-1, -0x1.1c00p-45, 79 0x1.82589994ccd7ep-1, -0x1.1c00p-45,
79 0x1.8364c1eb942d0p-1, 0x1.9d00p-45, 80 0x1.8364c1eb942d0p-1, 0x1.9d00p-45,
80 0x1.8471a4623cab5p-1, 0x1.7100p-43, 81 0x1.8471a4623cab5p-1, 0x1.7100p-43,
81 0x1.857f4179f5bbcp-1, 0x1.2600p-45, 82 0x1.857f4179f5bbcp-1, 0x1.2600p-45,
82 0x1.868d99b4491afp-1, -0x1.2c40p-44, 83 0x1.868d99b4491afp-1, -0x1.2c40p-44,
83 0x1.879cad931a395p-1, -0x1.3000p-45, 84 0x1.879cad931a395p-1, -0x1.3000p-45,
84 0x1.88ac7d98a65b8p-1, -0x1.a800p-45, 85 0x1.88ac7d98a65b8p-1, -0x1.a800p-45,
85 0x1.89bd0a4785800p-1, -0x1.d000p-49, 86 0x1.89bd0a4785800p-1, -0x1.d000p-49,
86 0x1.8ace5422aa223p-1, 0x1.3280p-44, 87 0x1.8ace5422aa223p-1, 0x1.3280p-44,
87 0x1.8be05bad619fap-1, 0x1.2b40p-43, 88 0x1.8be05bad619fap-1, 0x1.2b40p-43,
88 0x1.8cf3216b54383p-1, -0x1.ed00p-45, 89 0x1.8cf3216b54383p-1, -0x1.ed00p-45,
89 0x1.8e06a5e08664cp-1, -0x1.0500p-45, 90 0x1.8e06a5e08664cp-1, -0x1.0500p-45,
90 0x1.8f1ae99157807p-1, 0x1.8280p-45, 91 0x1.8f1ae99157807p-1, 0x1.8280p-45,
91 0x1.902fed0282c0ep-1, -0x1.cb00p-46, 92 0x1.902fed0282c0ep-1, -0x1.cb00p-46,
92 0x1.9145b0b91ff96p-1, -0x1.5e00p-47, 93 0x1.9145b0b91ff96p-1, -0x1.5e00p-47,
93 0x1.925c353aa2ff9p-1, 0x1.5400p-48, 94 0x1.925c353aa2ff9p-1, 0x1.5400p-48,
94 0x1.93737b0cdc64ap-1, 0x1.7200p-46, 95 0x1.93737b0cdc64ap-1, 0x1.7200p-46,
95 0x1.948b82b5f98aep-1, -0x1.9000p-47, 96 0x1.948b82b5f98aep-1, -0x1.9000p-47,
96 0x1.95a44cbc852cbp-1, 0x1.5680p-45, 97 0x1.95a44cbc852cbp-1, 0x1.5680p-45,
97 0x1.96bdd9a766f21p-1, -0x1.6d00p-44, 98 0x1.96bdd9a766f21p-1, -0x1.6d00p-44,
98 0x1.97d829fde4e2ap-1, -0x1.1000p-47, 99 0x1.97d829fde4e2ap-1, -0x1.1000p-47,
99 0x1.98f33e47a23a3p-1, 0x1.d000p-45, 100 0x1.98f33e47a23a3p-1, 0x1.d000p-45,
100 0x1.9a0f170ca0604p-1, -0x1.8a40p-44, 101 0x1.9a0f170ca0604p-1, -0x1.8a40p-44,
101 0x1.9b2bb4d53ff89p-1, 0x1.55c0p-44, 102 0x1.9b2bb4d53ff89p-1, 0x1.55c0p-44,
102 0x1.9c49182a3f15bp-1, 0x1.6b80p-45, 103 0x1.9c49182a3f15bp-1, 0x1.6b80p-45,
103 0x1.9d674194bb8c5p-1, -0x1.c000p-49, 104 0x1.9d674194bb8c5p-1, -0x1.c000p-49,
104 0x1.9e86319e3238ep-1, 0x1.7d00p-46, 105 0x1.9e86319e3238ep-1, 0x1.7d00p-46,
105 0x1.9fa5e8d07f302p-1, 0x1.6400p-46, 106 0x1.9fa5e8d07f302p-1, 0x1.6400p-46,
106 0x1.a0c667b5de54dp-1, -0x1.5000p-48, 107 0x1.a0c667b5de54dp-1, -0x1.5000p-48,
107 0x1.a1e7aed8eb8f6p-1, 0x1.9e00p-47, 108 0x1.a1e7aed8eb8f6p-1, 0x1.9e00p-47,
108 0x1.a309bec4a2e27p-1, 0x1.ad80p-45, 109 0x1.a309bec4a2e27p-1, 0x1.ad80p-45,
109 0x1.a42c980460a5dp-1, -0x1.af00p-46, 110 0x1.a42c980460a5dp-1, -0x1.af00p-46,
110 0x1.a5503b23e259bp-1, 0x1.b600p-47, 111 0x1.a5503b23e259bp-1, 0x1.b600p-47,
111 0x1.a674a8af46213p-1, 0x1.8880p-44, 112 0x1.a674a8af46213p-1, 0x1.8880p-44,
112 0x1.a799e1330b3a7p-1, 0x1.1200p-46, 113 0x1.a799e1330b3a7p-1, 0x1.1200p-46,
113 0x1.a8bfe53c12e8dp-1, 0x1.6c00p-47, 114 0x1.a8bfe53c12e8dp-1, 0x1.6c00p-47,
114 0x1.a9e6b5579fcd2p-1, -0x1.9b80p-45, 115 0x1.a9e6b5579fcd2p-1, -0x1.9b80p-45,
115 0x1.ab0e521356fb8p-1, 0x1.b700p-45, 116 0x1.ab0e521356fb8p-1, 0x1.b700p-45,
116 0x1.ac36bbfd3f381p-1, 0x1.9000p-50, 117 0x1.ac36bbfd3f381p-1, 0x1.9000p-50,
117 0x1.ad5ff3a3c2780p-1, 0x1.4000p-49, 118 0x1.ad5ff3a3c2780p-1, 0x1.4000p-49,
118 0x1.ae89f995ad2a3p-1, -0x1.c900p-45, 119 0x1.ae89f995ad2a3p-1, -0x1.c900p-45,
119 0x1.afb4ce622f367p-1, 0x1.6500p-46, 120 0x1.afb4ce622f367p-1, 0x1.6500p-46,
120 0x1.b0e07298db790p-1, 0x1.fd40p-45, 121 0x1.b0e07298db790p-1, 0x1.fd40p-45,
121 0x1.b20ce6c9a89a9p-1, 0x1.2700p-46, 122 0x1.b20ce6c9a89a9p-1, 0x1.2700p-46,
122 0x1.b33a2b84f1a4bp-1, 0x1.d470p-43, 123 0x1.b33a2b84f1a4bp-1, 0x1.d470p-43,
123 0x1.b468415b747e7p-1, -0x1.8380p-44, 124 0x1.b468415b747e7p-1, -0x1.8380p-44,
124 0x1.b59728de5593ap-1, 0x1.8000p-54, 125 0x1.b59728de5593ap-1, 0x1.8000p-54,
125 0x1.b6c6e29f1c56ap-1, 0x1.ad00p-47, 126 0x1.b6c6e29f1c56ap-1, 0x1.ad00p-47,
126 0x1.b7f76f2fb5e50p-1, 0x1.e800p-50, 127 0x1.b7f76f2fb5e50p-1, 0x1.e800p-50,
127 0x1.b928cf22749b2p-1, -0x1.4c00p-47, 128 0x1.b928cf22749b2p-1, -0x1.4c00p-47,
128 0x1.ba5b030a10603p-1, -0x1.d700p-47, 129 0x1.ba5b030a10603p-1, -0x1.d700p-47,
129 0x1.bb8e0b79a6f66p-1, 0x1.d900p-47, 130 0x1.bb8e0b79a6f66p-1, 0x1.d900p-47,
130 0x1.bcc1e904bc1ffp-1, 0x1.2a00p-47, 131 0x1.bcc1e904bc1ffp-1, 0x1.2a00p-47,
131 0x1.bdf69c3f3a16fp-1, -0x1.f780p-46, 132 0x1.bdf69c3f3a16fp-1, -0x1.f780p-46,
132 0x1.bf2c25bd71db8p-1, -0x1.0a00p-46, 133 0x1.bf2c25bd71db8p-1, -0x1.0a00p-46,
133 0x1.c06286141b2e9p-1, -0x1.1400p-46, 134 0x1.c06286141b2e9p-1, -0x1.1400p-46,
134 0x1.c199bdd8552e0p-1, 0x1.be00p-47, 135 0x1.c199bdd8552e0p-1, 0x1.be00p-47,
135 0x1.c2d1cd9fa64eep-1, -0x1.9400p-47, 136 0x1.c2d1cd9fa64eep-1, -0x1.9400p-47,
136 0x1.c40ab5fffd02fp-1, -0x1.ed00p-47, 137 0x1.c40ab5fffd02fp-1, -0x1.ed00p-47,
137 0x1.c544778fafd15p-1, 0x1.9660p-44, 138 0x1.c544778fafd15p-1, 0x1.9660p-44,
138 0x1.c67f12e57d0cbp-1, -0x1.a100p-46, 139 0x1.c67f12e57d0cbp-1, -0x1.a100p-46,
139 0x1.c7ba88988c1b6p-1, -0x1.8458p-42, 140 0x1.c7ba88988c1b6p-1, -0x1.8458p-42,
140 0x1.c8f6d9406e733p-1, -0x1.a480p-46, 141 0x1.c8f6d9406e733p-1, -0x1.a480p-46,
141 0x1.ca3405751c4dfp-1, 0x1.b000p-51, 142 0x1.ca3405751c4dfp-1, 0x1.b000p-51,
142 0x1.cb720dcef9094p-1, 0x1.1400p-47, 143 0x1.cb720dcef9094p-1, 0x1.1400p-47,
143 0x1.ccb0f2e6d1689p-1, 0x1.0200p-48, 144 0x1.ccb0f2e6d1689p-1, 0x1.0200p-48,
144 0x1.cdf0b555dc412p-1, 0x1.3600p-48, 145 0x1.cdf0b555dc412p-1, 0x1.3600p-48,
145 0x1.cf3155b5bab3bp-1, -0x1.6900p-47, 146 0x1.cf3155b5bab3bp-1, -0x1.6900p-47,
146 0x1.d072d4a0789bcp-1, 0x1.9a00p-47, 147 0x1.d072d4a0789bcp-1, 0x1.9a00p-47,
147 0x1.d1b532b08c8fap-1, -0x1.5e00p-46, 148 0x1.d1b532b08c8fap-1, -0x1.5e00p-46,
148 0x1.d2f87080d8a85p-1, 0x1.d280p-46, 149 0x1.d2f87080d8a85p-1, 0x1.d280p-46,
149 0x1.d43c8eacaa203p-1, 0x1.1a00p-47, 150 0x1.d43c8eacaa203p-1, 0x1.1a00p-47,
150 0x1.d5818dcfba491p-1, 0x1.f000p-50, 151 0x1.d5818dcfba491p-1, 0x1.f000p-50,
151 0x1.d6c76e862e6a1p-1, -0x1.3a00p-47, 152 0x1.d6c76e862e6a1p-1, -0x1.3a00p-47,
152 0x1.d80e316c9834ep-1, -0x1.cd80p-47, 153 0x1.d80e316c9834ep-1, -0x1.cd80p-47,
153 0x1.d955d71ff6090p-1, 0x1.4c00p-48, 154 0x1.d955d71ff6090p-1, 0x1.4c00p-48,
154 0x1.da9e603db32aep-1, 0x1.f900p-48, 155 0x1.da9e603db32aep-1, 0x1.f900p-48,
155 0x1.dbe7cd63a8325p-1, 0x1.9800p-49, 156 0x1.dbe7cd63a8325p-1, 0x1.9800p-49,
156 0x1.dd321f301b445p-1, -0x1.5200p-48, 157 0x1.dd321f301b445p-1, -0x1.5200p-48,
157 0x1.de7d5641c05bfp-1, -0x1.d700p-46, 158 0x1.de7d5641c05bfp-1, -0x1.d700p-46,
158 0x1.dfc97337b9aecp-1, -0x1.6140p-46, 159 0x1.dfc97337b9aecp-1, -0x1.6140p-46,
159 0x1.e11676b197d5ep-1, 0x1.b480p-47, 160 0x1.e11676b197d5ep-1, 0x1.b480p-47,
160 0x1.e264614f5a3e7p-1, 0x1.0ce0p-43, 161 0x1.e264614f5a3e7p-1, 0x1.0ce0p-43,
161 0x1.e3b333b16ee5cp-1, 0x1.c680p-47, 162 0x1.e3b333b16ee5cp-1, 0x1.c680p-47,
162 0x1.e502ee78b3fb4p-1, -0x1.9300p-47, 163 0x1.e502ee78b3fb4p-1, -0x1.9300p-47,
163 0x1.e653924676d68p-1, -0x1.5000p-49, 164 0x1.e653924676d68p-1, -0x1.5000p-49,
164 0x1.e7a51fbc74c44p-1, -0x1.7f80p-47, 165 0x1.e7a51fbc74c44p-1, -0x1.7f80p-47,
165 0x1.e8f7977cdb726p-1, -0x1.3700p-48, 166 0x1.e8f7977cdb726p-1, -0x1.3700p-48,
166 0x1.ea4afa2a490e8p-1, 0x1.5d00p-49, 167 0x1.ea4afa2a490e8p-1, 0x1.5d00p-49,
167 0x1.eb9f4867ccae4p-1, 0x1.61a0p-46, 168 0x1.eb9f4867ccae4p-1, 0x1.61a0p-46,
168 0x1.ecf482d8e680dp-1, 0x1.5500p-48, 169 0x1.ecf482d8e680dp-1, 0x1.5500p-48,
169 0x1.ee4aaa2188514p-1, 0x1.6400p-51, 170 0x1.ee4aaa2188514p-1, 0x1.6400p-51,
170 0x1.efa1bee615a13p-1, -0x1.e800p-49, 171 0x1.efa1bee615a13p-1, -0x1.e800p-49,
171 0x1.f0f9c1cb64106p-1, -0x1.a880p-48, 172 0x1.f0f9c1cb64106p-1, -0x1.a880p-48,
172 0x1.f252b376bb963p-1, -0x1.c900p-45, 173 0x1.f252b376bb963p-1, -0x1.c900p-45,
173 0x1.f3ac948dd7275p-1, 0x1.a000p-53, 174 0x1.f3ac948dd7275p-1, 0x1.a000p-53,
174 0x1.f50765b6e4524p-1, -0x1.4f00p-48, 175 0x1.f50765b6e4524p-1, -0x1.4f00p-48,
175 0x1.f6632798844fdp-1, 0x1.a800p-51, 176 0x1.f6632798844fdp-1, 0x1.a800p-51,
176 0x1.f7bfdad9cbe38p-1, 0x1.abc0p-48, 177 0x1.f7bfdad9cbe38p-1, 0x1.abc0p-48,
177 0x1.f91d802243c82p-1, -0x1.4600p-50, 178 0x1.f91d802243c82p-1, -0x1.4600p-50,
178 0x1.fa7c1819e908ep-1, -0x1.b0c0p-47, 179 0x1.fa7c1819e908ep-1, -0x1.b0c0p-47,
179 0x1.fbdba3692d511p-1, -0x1.0e00p-51, 180 0x1.fbdba3692d511p-1, -0x1.0e00p-51,
180 0x1.fd3c22b8f7194p-1, -0x1.0de8p-46, 181 0x1.fd3c22b8f7194p-1, -0x1.0de8p-46,
181 0x1.fe9d96b2a23eep-1, 0x1.e430p-49, 182 0x1.fe9d96b2a23eep-1, 0x1.e430p-49,
182 0x1.0000000000000p+0, 0x0.0000p+0, 183 0x1.0000000000000p+0, 0x0.0000p+0,
183 0x1.00b1afa5abcbep+0, -0x1.3400p-52, 184 0x1.00b1afa5abcbep+0, -0x1.3400p-52,
184 0x1.0163da9fb3303p+0, -0x1.2170p-46, 185 0x1.0163da9fb3303p+0, -0x1.2170p-46,
185 0x1.02168143b0282p+0, 0x1.a400p-52, 186 0x1.02168143b0282p+0, 0x1.a400p-52,
186 0x1.02c9a3e77806cp+0, 0x1.f980p-49, 187 0x1.02c9a3e77806cp+0, 0x1.f980p-49,
187 0x1.037d42e11bbcap+0, -0x1.7400p-51, 188 0x1.037d42e11bbcap+0, -0x1.7400p-51,
188 0x1.04315e86e7f89p+0, 0x1.8300p-50, 189 0x1.04315e86e7f89p+0, 0x1.8300p-50,
189 0x1.04e5f72f65467p+0, -0x1.a3f0p-46, 190 0x1.04e5f72f65467p+0, -0x1.a3f0p-46,
190 0x1.059b0d315855ap+0, -0x1.2840p-47, 191 0x1.059b0d315855ap+0, -0x1.2840p-47,
191 0x1.0650a0e3c1f95p+0, 0x1.1600p-48, 192 0x1.0650a0e3c1f95p+0, 0x1.1600p-48,
192 0x1.0706b29ddf71ap+0, 0x1.5240p-46, 193 0x1.0706b29ddf71ap+0, 0x1.5240p-46,
193 0x1.07bd42b72a82dp+0, -0x1.9a00p-49, 194 0x1.07bd42b72a82dp+0, -0x1.9a00p-49,
194 0x1.0874518759bd0p+0, 0x1.6400p-49, 195 0x1.0874518759bd0p+0, 0x1.6400p-49,
195 0x1.092bdf66607c8p+0, -0x1.0780p-47, 196 0x1.092bdf66607c8p+0, -0x1.0780p-47,
196 0x1.09e3ecac6f383p+0, -0x1.8000p-54, 197 0x1.09e3ecac6f383p+0, -0x1.8000p-54,
197 0x1.0a9c79b1f3930p+0, 0x1.fa00p-48, 198 0x1.0a9c79b1f3930p+0, 0x1.fa00p-48,
198 0x1.0b5586cf988fcp+0, -0x1.ac80p-48, 199 0x1.0b5586cf988fcp+0, -0x1.ac80p-48,
199 0x1.0c0f145e46c8ap+0, 0x1.9c00p-50, 200 0x1.0c0f145e46c8ap+0, 0x1.9c00p-50,
200 0x1.0cc922b724816p+0, 0x1.5200p-47, 201 0x1.0cc922b724816p+0, 0x1.5200p-47,
201 0x1.0d83b23395dd8p+0, -0x1.ad00p-48, 202 0x1.0d83b23395dd8p+0, -0x1.ad00p-48,
202 0x1.0e3ec32d3d1f3p+0, 0x1.bac0p-46, 203 0x1.0e3ec32d3d1f3p+0, 0x1.bac0p-46,
203 0x1.0efa55fdfa9a6p+0, -0x1.4e80p-47, 204 0x1.0efa55fdfa9a6p+0, -0x1.4e80p-47,
204 0x1.0fb66affed2f0p+0, -0x1.d300p-47, 205 0x1.0fb66affed2f0p+0, -0x1.d300p-47,
205 0x1.1073028d7234bp+0, 0x1.1500p-48, 206 0x1.1073028d7234bp+0, 0x1.1500p-48,
206 0x1.11301d0125b5bp+0, 0x1.c000p-49, 207 0x1.11301d0125b5bp+0, 0x1.c000p-49,
207 0x1.11edbab5e2af9p+0, 0x1.6bc0p-46, 208 0x1.11edbab5e2af9p+0, 0x1.6bc0p-46,
208 0x1.12abdc06c31d5p+0, 0x1.8400p-49, 209 0x1.12abdc06c31d5p+0, 0x1.8400p-49,
209 0x1.136a814f2047dp+0, -0x1.ed00p-47, 210 0x1.136a814f2047dp+0, -0x1.ed00p-47,
210 0x1.1429aaea92de9p+0, 0x1.8e00p-49, 211 0x1.1429aaea92de9p+0, 0x1.8e00p-49,
211 0x1.14e95934f3138p+0, 0x1.b400p-49, 212 0x1.14e95934f3138p+0, 0x1.b400p-49,
212 0x1.15a98c8a58e71p+0, 0x1.5300p-47, 213 0x1.15a98c8a58e71p+0, 0x1.5300p-47,
213 0x1.166a45471c3dfp+0, 0x1.3380p-47, 214 0x1.166a45471c3dfp+0, 0x1.3380p-47,
214 0x1.172b83c7d5211p+0, 0x1.8d40p-45, 215 0x1.172b83c7d5211p+0, 0x1.8d40p-45,
215 0x1.17ed48695bb9fp+0, -0x1.5d00p-47, 216 0x1.17ed48695bb9fp+0, -0x1.5d00p-47,
216 0x1.18af9388c8d93p+0, -0x1.c880p-46, 217 0x1.18af9388c8d93p+0, -0x1.c880p-46,
217 0x1.1972658375d66p+0, 0x1.1f00p-46, 218 0x1.1972658375d66p+0, 0x1.1f00p-46,
218 0x1.1a35beb6fcba7p+0, 0x1.0480p-46, 219 0x1.1a35beb6fcba7p+0, 0x1.0480p-46,
219 0x1.1af99f81387e3p+0, -0x1.7390p-43, 220 0x1.1af99f81387e3p+0, -0x1.7390p-43,
220 0x1.1bbe084045d54p+0, 0x1.4e40p-45, 221 0x1.1bbe084045d54p+0, 0x1.4e40p-45,
221 0x1.1c82f95281c43p+0, -0x1.a200p-47, 222 0x1.1c82f95281c43p+0, -0x1.a200p-47,
222 0x1.1d4873168b9b2p+0, 0x1.3800p-49, 223 0x1.1d4873168b9b2p+0, 0x1.3800p-49,
223 0x1.1e0e75eb44031p+0, 0x1.ac00p-49, 224 0x1.1e0e75eb44031p+0, 0x1.ac00p-49,
224 0x1.1ed5022fcd938p+0, 0x1.1900p-47, 225 0x1.1ed5022fcd938p+0, 0x1.1900p-47,
225 0x1.1f9c18438cdf7p+0, -0x1.b780p-46, 226 0x1.1f9c18438cdf7p+0, -0x1.b780p-46,
226 0x1.2063b88628d8fp+0, 0x1.d940p-45, 227 0x1.2063b88628d8fp+0, 0x1.d940p-45,
227 0x1.212be3578a81ep+0, 0x1.8000p-50, 228 0x1.212be3578a81ep+0, 0x1.8000p-50,
228 0x1.21f49917ddd41p+0, 0x1.b340p-45, 229 0x1.21f49917ddd41p+0, 0x1.b340p-45,
229 0x1.22bdda2791323p+0, 0x1.9f80p-46, 230 0x1.22bdda2791323p+0, 0x1.9f80p-46,
230 0x1.2387a6e7561e7p+0, -0x1.9c80p-46, 231 0x1.2387a6e7561e7p+0, -0x1.9c80p-46,
231 0x1.2451ffb821427p+0, 0x1.2300p-47, 232 0x1.2451ffb821427p+0, 0x1.2300p-47,
232 0x1.251ce4fb2a602p+0, -0x1.3480p-46, 233 0x1.251ce4fb2a602p+0, -0x1.3480p-46,
233 0x1.25e85711eceb0p+0, 0x1.2700p-46, 234 0x1.25e85711eceb0p+0, 0x1.2700p-46,
234 0x1.26b4565e27d16p+0, 0x1.1d00p-46, 235 0x1.26b4565e27d16p+0, 0x1.1d00p-46,
235 0x1.2780e341de00fp+0, 0x1.1ee0p-44, 236 0x1.2780e341de00fp+0, 0x1.1ee0p-44,
236 0x1.284dfe1f5633ep+0, -0x1.4c00p-46, 237 0x1.284dfe1f5633ep+0, -0x1.4c00p-46,
237 0x1.291ba7591bb30p+0, -0x1.3d80p-46, 238 0x1.291ba7591bb30p+0, -0x1.3d80p-46,
238 0x1.29e9df51fdf09p+0, 0x1.8b00p-47, 239 0x1.29e9df51fdf09p+0, 0x1.8b00p-47,
239 0x1.2ab8a66d10e9bp+0, -0x1.27c0p-45, 240 0x1.2ab8a66d10e9bp+0, -0x1.27c0p-45,
240 0x1.2b87fd0dada3ap+0, 0x1.a340p-45, 241 0x1.2b87fd0dada3ap+0, 0x1.a340p-45,
241 0x1.2c57e39771af9p+0, -0x1.0800p-46, 242 0x1.2c57e39771af9p+0, -0x1.0800p-46,
242 0x1.2d285a6e402d9p+0, -0x1.ed00p-47, 243 0x1.2d285a6e402d9p+0, -0x1.ed00p-47,
243 0x1.2df961f641579p+0, -0x1.4200p-48, 244 0x1.2df961f641579p+0, -0x1.4200p-48,
244 0x1.2ecafa93e2ecfp+0, -0x1.4980p-45, 245 0x1.2ecafa93e2ecfp+0, -0x1.4980p-45,
245 0x1.2f9d24abd8822p+0, -0x1.6300p-46, 246 0x1.2f9d24abd8822p+0, -0x1.6300p-46,
246 0x1.306fe0a31b625p+0, -0x1.2360p-44, 247 0x1.306fe0a31b625p+0, -0x1.2360p-44,
247 0x1.31432edeea50bp+0, -0x1.0df8p-40, 248 0x1.31432edeea50bp+0, -0x1.0df8p-40,
248 0x1.32170fc4cd7b8p+0, -0x1.2480p-45, 249 0x1.32170fc4cd7b8p+0, -0x1.2480p-45,
249 0x1.32eb83ba8e9a2p+0, -0x1.5980p-45, 250 0x1.32eb83ba8e9a2p+0, -0x1.5980p-45,
250 0x1.33c08b2641766p+0, 0x1.ed00p-46, 251 0x1.33c08b2641766p+0, 0x1.ed00p-46,
251 0x1.3496266e3fa27p+0, -0x1.c000p-50, 252 0x1.3496266e3fa27p+0, -0x1.c000p-50,
252 0x1.356c55f929f0fp+0, -0x1.0d80p-44, 253 0x1.356c55f929f0fp+0, -0x1.0d80p-44,
253 0x1.36431a2de88b9p+0, 0x1.2c80p-45, 254 0x1.36431a2de88b9p+0, 0x1.2c80p-45,
254 0x1.371a7373aaa39p+0, 0x1.0600p-45, 255 0x1.371a7373aaa39p+0, 0x1.0600p-45,
255 0x1.37f26231e74fep+0, -0x1.6600p-46, 256 0x1.37f26231e74fep+0, -0x1.6600p-46,
256 0x1.38cae6d05d838p+0, -0x1.ae00p-47, 257 0x1.38cae6d05d838p+0, -0x1.ae00p-47,
257 0x1.39a401b713ec3p+0, -0x1.4720p-43, 258 0x1.39a401b713ec3p+0, -0x1.4720p-43,
258 0x1.3a7db34e5a020p+0, 0x1.8200p-47, 259 0x1.3a7db34e5a020p+0, 0x1.8200p-47,
259 0x1.3b57fbfec6e95p+0, 0x1.e800p-44, 260 0x1.3b57fbfec6e95p+0, 0x1.e800p-44,
260 0x1.3c32dc313a8f2p+0, 0x1.f800p-49, 261 0x1.3c32dc313a8f2p+0, 0x1.f800p-49,
261 0x1.3d0e544ede122p+0, -0x1.7a00p-46, 262 0x1.3d0e544ede122p+0, -0x1.7a00p-46,
262 0x1.3dea64c1234bbp+0, 0x1.6300p-45, 263 0x1.3dea64c1234bbp+0, 0x1.6300p-45,
263 0x1.3ec70df1c4eccp+0, -0x1.8a60p-43, 264 0x1.3ec70df1c4eccp+0, -0x1.8a60p-43,
264 0x1.3fa4504ac7e8cp+0, -0x1.cdc0p-44, 265 0x1.3fa4504ac7e8cp+0, -0x1.cdc0p-44,
265 0x1.40822c367a0bbp+0, 0x1.5b80p-45, 266 0x1.40822c367a0bbp+0, 0x1.5b80p-45,
266 0x1.4160a21f72e95p+0, 0x1.ec00p-46, 267 0x1.4160a21f72e95p+0, 0x1.ec00p-46,
267 0x1.423fb27094646p+0, -0x1.3600p-46, 268 0x1.423fb27094646p+0, -0x1.3600p-46,
268 0x1.431f5d950a920p+0, 0x1.3980p-45, 269 0x1.431f5d950a920p+0, 0x1.3980p-45,
269 0x1.43ffa3f84b9ebp+0, 0x1.a000p-48, 270 0x1.43ffa3f84b9ebp+0, 0x1.a000p-48,
270 0x1.44e0860618919p+0, -0x1.6c00p-48, 271 0x1.44e0860618919p+0, -0x1.6c00p-48,
271 0x1.45c2042a7d201p+0, -0x1.bc00p-47, 272 0x1.45c2042a7d201p+0, -0x1.bc00p-47,
272 0x1.46a41ed1d0016p+0, -0x1.2800p-46, 273 0x1.46a41ed1d0016p+0, -0x1.2800p-46,
273 0x1.4786d668b3326p+0, 0x1.0e00p-44, 274 0x1.4786d668b3326p+0, 0x1.0e00p-44,
274 0x1.486a2b5c13c00p+0, -0x1.d400p-45, 275 0x1.486a2b5c13c00p+0, -0x1.d400p-45,
275 0x1.494e1e192af04p+0, 0x1.c200p-47, 276 0x1.494e1e192af04p+0, 0x1.c200p-47,
276 0x1.4a32af0d7d372p+0, -0x1.e500p-46, 277 0x1.4a32af0d7d372p+0, -0x1.e500p-46,
277 0x1.4b17dea6db801p+0, 0x1.7800p-47, 278 0x1.4b17dea6db801p+0, 0x1.7800p-47,
278 0x1.4bfdad53629e1p+0, -0x1.3800p-46, 279 0x1.4bfdad53629e1p+0, -0x1.3800p-46,
279 0x1.4ce41b817c132p+0, 0x1.0800p-47, 280 0x1.4ce41b817c132p+0, 0x1.0800p-47,
280 0x1.4dcb299fddddbp+0, 0x1.c700p-45, 281 0x1.4dcb299fddddbp+0, 0x1.c700p-45,
281 0x1.4eb2d81d8ab96p+0, -0x1.ce00p-46, 282 0x1.4eb2d81d8ab96p+0, -0x1.ce00p-46,
282 0x1.4f9b2769d2d02p+0, 0x1.9200p-46, 283 0x1.4f9b2769d2d02p+0, 0x1.9200p-46,
283 0x1.508417f4531c1p+0, -0x1.8c00p-47, 284 0x1.508417f4531c1p+0, -0x1.8c00p-47,
284 0x1.516daa2cf662ap+0, -0x1.a000p-48, 285 0x1.516daa2cf662ap+0, -0x1.a000p-48,
285 0x1.5257de83f51eap+0, 0x1.a080p-43, 286 0x1.5257de83f51eap+0, 0x1.a080p-43,
286 0x1.5342b569d4edap+0, -0x1.6d80p-45, 287 0x1.5342b569d4edap+0, -0x1.6d80p-45,
287 0x1.542e2f4f6ac1ap+0, -0x1.2440p-44, 288 0x1.542e2f4f6ac1ap+0, -0x1.2440p-44,
288 0x1.551a4ca5d94dbp+0, 0x1.83c0p-43, 289 0x1.551a4ca5d94dbp+0, 0x1.83c0p-43,
289 0x1.56070dde9116bp+0, 0x1.4b00p-45, 290 0x1.56070dde9116bp+0, 0x1.4b00p-45,
290 0x1.56f4736b529dep+0, 0x1.15a0p-43, 291 0x1.56f4736b529dep+0, 0x1.15a0p-43,
291 0x1.57e27dbe2c40ep+0, -0x1.9e00p-45, 292 0x1.57e27dbe2c40ep+0, -0x1.9e00p-45,
292 0x1.58d12d497c76fp+0, -0x1.3080p-45, 293 0x1.58d12d497c76fp+0, -0x1.3080p-45,
293 0x1.59c0827ff0b4cp+0, 0x1.dec0p-43, 294 0x1.59c0827ff0b4cp+0, 0x1.dec0p-43,
294 0x1.5ab07dd485427p+0, -0x1.4000p-51, 295 0x1.5ab07dd485427p+0, -0x1.4000p-51,
295 0x1.5ba11fba87af4p+0, 0x1.0080p-44, 296 0x1.5ba11fba87af4p+0, 0x1.0080p-44,
296 0x1.5c9268a59460bp+0, -0x1.6c80p-45, 297 0x1.5c9268a59460bp+0, -0x1.6c80p-45,
297 0x1.5d84590998e3fp+0, 0x1.69a0p-43, 298 0x1.5d84590998e3fp+0, 0x1.69a0p-43,
298 0x1.5e76f15ad20e1p+0, -0x1.b400p-46, 299 0x1.5e76f15ad20e1p+0, -0x1.b400p-46,
299 0x1.5f6a320dcebcap+0, 0x1.7700p-46, 300 0x1.5f6a320dcebcap+0, 0x1.7700p-46,
300 0x1.605e1b976dcb8p+0, 0x1.6f80p-45, 301 0x1.605e1b976dcb8p+0, 0x1.6f80p-45,
301 0x1.6152ae6cdf715p+0, 0x1.1000p-47, 302 0x1.6152ae6cdf715p+0, 0x1.1000p-47,
302 0x1.6247eb03a5531p+0, -0x1.5d00p-46, 303 0x1.6247eb03a5531p+0, -0x1.5d00p-46,
303 0x1.633dd1d1929b5p+0, -0x1.2d00p-46, 304 0x1.633dd1d1929b5p+0, -0x1.2d00p-46,
304 0x1.6434634ccc313p+0, -0x1.a800p-49, 305 0x1.6434634ccc313p+0, -0x1.a800p-49,
305 0x1.652b9febc8efap+0, -0x1.8600p-45, 306 0x1.652b9febc8efap+0, -0x1.8600p-45,
306 0x1.6623882553397p+0, 0x1.1fe0p-40, 307 0x1.6623882553397p+0, 0x1.1fe0p-40,
307 0x1.671c1c708328ep+0, -0x1.7200p-44, 308 0x1.671c1c708328ep+0, -0x1.7200p-44,
308 0x1.68155d44ca97ep+0, 0x1.6800p-49, 309 0x1.68155d44ca97ep+0, 0x1.6800p-49,
309 0x1.690f4b19e9471p+0, -0x1.9780p-45, 310 0x1.690f4b19e9471p+0, -0x1.9780p-45,
310}; 311};
311 312
312/* 313/*
313 * exp2(x): compute the base 2 exponential of x 314 * exp2(x): compute the base 2 exponential of x
314 * 315 *
315 * Accuracy: Peak error < 0.503 ulp for normalized results. 316 * Accuracy: Peak error < 0.503 ulp for normalized results.
316 * 317 *
317 * Method: (accurate tables) 318 * Method: (accurate tables)
318 * 319 *
319 * Reduce x: 320 * Reduce x:
320 * x = 2**k + y, for integer k and |y| <= 1/2. 321 * x = 2**k + y, for integer k and |y| <= 1/2.
321 * Thus we have exp2(x) = 2**k * exp2(y). 322 * Thus we have exp2(x) = 2**k * exp2(y).
322 * 323 *
323 * Reduce y: 324 * Reduce y:
324 * y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE. 325 * y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
325 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]), 326 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
326 * with |z - eps[i]| <= 2**-9 + 2**-39 for the table used. 327 * with |z - eps[i]| <= 2**-9 + 2**-39 for the table used.
327 * 328 *
328 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via 329 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
329 * a degree-5 minimax polynomial with maximum error under 1.3 * 2**-61. 330 * a degree-5 minimax polynomial with maximum error under 1.3 * 2**-61.
330 * The values in exp2t[] and eps[] are chosen such that 331 * The values in exp2t[] and eps[] are chosen such that
331 * exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such 332 * exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
332 * that exp2t[i] is accurate to 2**-64. 333 * that exp2t[i] is accurate to 2**-64.
333 * 334 *
334 * Note that the range of i is +-TBLSIZE/2, so we actually index the tables 335 * Note that the range of i is +-TBLSIZE/2, so we actually index the tables
335 * by i0 = i + TBLSIZE/2. For cache efficiency, exp2t[] and eps[] are 336 * by i0 = i + TBLSIZE/2. For cache efficiency, exp2t[] and eps[] are
336 * virtual tables, interleaved in the real table tbl[]. 337 * virtual tables, interleaved in the real table tbl[].
337 * 338 *
338 * This method is due to Gal, with many details due to Gal and Bachelis: 339 * This method is due to Gal, with many details due to Gal and Bachelis:
339 * 340 *
340 * Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library 341 * Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library
341 * for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991). 342 * for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991).
342 */ 343 */
343double 344double
344exp2(double x) 345exp2(double x)
345{ 346{
346 double r, t, twopk, z; 347 double r, t, twopk, z;
347 uint32_t hx, ix, i0; 348 uint32_t hx, ix, i0, temp;
348 int k, big; 349 int k, big;
349 350
350 /* Filter out exceptional cases. */ 351 /* Filter out exceptional cases. */
351 hx = ((uint32_t*)&x)[0]; 352 memcpy(&hx, &x, sizeof(hx));
352 ix = hx & 0x7fffffff; /* high word of |x| */ 353 ix = hx & 0x7fffffff; /* high word of |x| */
353 if(ix >= 0x40900000) { /* |x| >= 1024 */ 354 if(ix >= 0x40900000) { /* |x| >= 1024 */
354 if(x >= 0x1.0p10) 355 if(x >= 0x1.0p10)
355 return (huge * huge); /* overflow */ 356 return (huge * huge); /* overflow */
356 if(x <= -0x1.0ccp10) 357 if(x <= -0x1.0ccp10)
357 return (twom1000 * twom1000); /* underflow */ 358 return (twom1000 * twom1000); /* underflow */
358 } else if (ix < 0x3c900000) { /* |x| < 0x1p-54 */ 359 } else if (ix < 0x3c900000) { /* |x| < 0x1p-54 */
359 return (1.0 + x); 360 return (1.0 + x);
360 } 361 }
361 362
362 /* Reduce x, computing z, i0, and k. */ 363 /* Reduce x, computing z, i0, and k. */
363 (*(volatile double*)&t) = x + redux; 364 (*(volatile double*)&t) = x + redux;
364 i0 = ((uint32_t*)&t)[1]; 365 i0 = ((uint32_t*)&t)[1];
365 i0 += TBLSIZE / 2; 366 i0 += TBLSIZE / 2;
366 k = (i0 >> TBLBITS) << 20; 367 k = (i0 >> TBLBITS) << 20;
367 i0 = (i0 & (TBLSIZE - 1)) << 1; 368 i0 = (i0 & (TBLSIZE - 1)) << 1;
368 t -= redux; 369 t -= redux;
369 z = x - t; 370 z = x - t;
370 371
371 /* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */ 372 /* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
372 t = tbl[i0]; /* exp2t[i0] */ 373 t = tbl[i0]; /* exp2t[i0] */
373 z -= tbl[i0 + 1]; /* eps[i0] */ 374 z -= tbl[i0 + 1]; /* eps[i0] */
374 big = k >= -1021 << 20; 375 big = k >= -1021 << 20;
375 if (big) { 376 if (big) {
376 ((uint32_t*)&twopk)[0] = 0x3ff00000+k; 377 temp = 0x3ff00000+k;
377 ((uint32_t*)&twopk)[1] = 0; 378 twopk = 0.0;
 379 memcpy(&twopk, &temp, sizeof(temp));
378 } else { 380 } else {
379 ((uint32_t*)&twopk)[0] = 0x3ff00000+k + (1000 << 20); 381 temp = 0x3ff00000+k + (1000 << 20);
380 ((uint32_t*)&twopk)[1] = 0; 382 twopk = 0.0;
 383 memcpy(&twopk, &temp, sizeof(temp));
381 } 384 }
382 r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * P5)))); 385 r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * P5))));
383 386
384 /* Scale by 2**(k>>20). */ 387 /* Scale by 2**(k>>20). */
385 if (big) { 388 if (big) {
386 /*if (k == 1024 << 20) 389 /*if (k == 1024 << 20)
387 return (r * 2.0 * 0x1p1023); */ 390 return (r * 2.0 * 0x1p1023); */
388 return (r * twopk); 391 return (r * twopk);
389 } else { 392 } else {
390 return (r * twopk * twom1000); 393 return (r * twopk * twom1000);
391 } 394 }
392} 395}
cvs diff -r1.1 -r1.2 src/lib/libm/noieee_src/n_exp2f.c (switch to unified diff)

--- src/lib/libm/noieee_src/n_exp2f.c 2014/03/06 10:55:57 1.1
+++ src/lib/libm/noieee_src/n_exp2f.c 2014/03/23 15:26:47 1.2
@@ -1,134 +1,136 @@ @@ -1,134 +1,136 @@
1/*- 1/*-
2 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> 2 * Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
3 * All rights reserved. 3 * All rights reserved.
4 * 4 *
5 * Redistribution and use in source and binary forms, with or without 5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions 6 * modification, are permitted provided that the following conditions
7 * are met: 7 * are met:
8 * 1. Redistributions of source code must retain the above copyright 8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer. 9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright 10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the 11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution. 12 * documentation and/or other materials provided with the distribution.
13 * 13 *
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 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 15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 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 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 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 23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
24 * SUCH DAMAGE. 24 * SUCH DAMAGE.
25 */ 25 */
26 26
27#include <sys/cdefs.h> 27#include <sys/cdefs.h>
28__RCSID("$NetBSD: n_exp2f.c,v 1.1 2014/03/06 10:55:57 martin Exp $"); 28__RCSID("$NetBSD: n_exp2f.c,v 1.2 2014/03/23 15:26:47 martin Exp $");
29#ifdef __FBSDID 29#ifdef __FBSDID
30__FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.9 2008/02/22 02:27:34 das Exp $"); 30__FBSDID("$FreeBSD: src/lib/msun/src/s_exp2f.c,v 1.9 2008/02/22 02:27:34 das Exp $");
31#endif 31#endif
32 32
33#include <stdint.h> 33#include <stdint.h>
34#include <float.h> 34#include <float.h>
 35#include <string.h>
35 36
36#include "math.h" 37#include "math.h"
37 38
38#define TBLBITS 4 39#define TBLBITS 4
39#define TBLSIZE (1 << TBLBITS) 40#define TBLSIZE (1 << TBLBITS)
40 41
41static const float 42static const float
42 huge = 0x1p100f, 43 huge = 0x1p100f,
43 redux = 0x1.8p23f / TBLSIZE, 44 redux = 0x1.8p23f / TBLSIZE,
44 P1 = 0x1.62e430p-1f, 45 P1 = 0x1.62e430p-1f,
45 P2 = 0x1.ebfbe0p-3f, 46 P2 = 0x1.ebfbe0p-3f,
46 P3 = 0x1.c6b348p-5f, 47 P3 = 0x1.c6b348p-5f,
47 P4 = 0x1.3b2c9cp-7f; 48 P4 = 0x1.3b2c9cp-7f;
48 49
49static volatile float twom100 = 0x1p-100f; 50static volatile float twom100 = 0x1p-100f;
50 51
51static const double exp2ft[TBLSIZE] = { 52static const double exp2ft[TBLSIZE] = {
52 0x1.6a09e667f3bcdp-1, 53 0x1.6a09e667f3bcdp-1,
53 0x1.7a11473eb0187p-1, 54 0x1.7a11473eb0187p-1,
54 0x1.8ace5422aa0dbp-1, 55 0x1.8ace5422aa0dbp-1,
55 0x1.9c49182a3f090p-1, 56 0x1.9c49182a3f090p-1,
56 0x1.ae89f995ad3adp-1, 57 0x1.ae89f995ad3adp-1,
57 0x1.c199bdd85529cp-1, 58 0x1.c199bdd85529cp-1,
58 0x1.d5818dcfba487p-1, 59 0x1.d5818dcfba487p-1,
59 0x1.ea4afa2a490dap-1, 60 0x1.ea4afa2a490dap-1,
60 0x1.0000000000000p+0, 61 0x1.0000000000000p+0,
61 0x1.0b5586cf9890fp+0, 62 0x1.0b5586cf9890fp+0,
62 0x1.172b83c7d517bp+0, 63 0x1.172b83c7d517bp+0,
63 0x1.2387a6e756238p+0, 64 0x1.2387a6e756238p+0,
64 0x1.306fe0a31b715p+0, 65 0x1.306fe0a31b715p+0,
65 0x1.3dea64c123422p+0, 66 0x1.3dea64c123422p+0,
66 0x1.4bfdad5362a27p+0, 67 0x1.4bfdad5362a27p+0,
67 0x1.5ab07dd485429p+0, 68 0x1.5ab07dd485429p+0,
68}; 69};
69  70
70/* 71/*
71 * exp2f(x): compute the base 2 exponential of x 72 * exp2f(x): compute the base 2 exponential of x
72 * 73 *
73 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927. 74 * Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
74 * 75 *
75 * Method: (equally-spaced tables) 76 * Method: (equally-spaced tables)
76 * 77 *
77 * Reduce x: 78 * Reduce x:
78 * x = 2**k + y, for integer k and |y| <= 1/2. 79 * x = 2**k + y, for integer k and |y| <= 1/2.
79 * Thus we have exp2f(x) = 2**k * exp2(y). 80 * Thus we have exp2f(x) = 2**k * exp2(y).
80 * 81 *
81 * Reduce y: 82 * Reduce y:
82 * y = i/TBLSIZE + z for integer i near y * TBLSIZE. 83 * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
83 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z), 84 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
84 * with |z| <= 2**-(TBLSIZE+1). 85 * with |z| <= 2**-(TBLSIZE+1).
85 * 86 *
86 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a 87 * 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 * degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
88 * Using double precision for everything except the reduction makes 89 * Using double precision for everything except the reduction makes
89 * roundoff error insignificant and simplifies the scaling step. 90 * roundoff error insignificant and simplifies the scaling step.
90 * 91 *
91 * This method is due to Tang, but I do not use his suggested parameters: 92 * This method is due to Tang, but I do not use his suggested parameters:
92 * 93 *
93 * Tang, P. Table-driven Implementation of the Exponential Function 94 * Tang, P. Table-driven Implementation of the Exponential Function
94 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989). 95 * in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
95 */ 96 */
96float 97float
97exp2f(float x) 98exp2f(float x)
98{ 99{
99 double tv, twopk, u, z; 100 double tv, twopk, u, z;
100 float t; 101 float t;
101 uint32_t hx, ix, i0; 102 uint32_t hx, ix, i0;
102 int32_t k; 103 int32_t k, temp;
103 104
104 /* Filter out exceptional cases. */ 105 /* Filter out exceptional cases. */
105 hx = *((uint32_t*)&x); 106 memcpy(&hx, &x, sizeof(hx));
106 ix = hx & 0x7fffffff; /* high word of |x| */ 107 ix = hx & 0x7fffffff; /* high word of |x| */
107 if(ix >= 0x43000000) { /* |x| >= 128 */ 108 if(ix >= 0x43000000) { /* |x| >= 128 */
108 if(x >= 0x1.0p7f) 109 if(x >= 0x1.0p7f)
109 return (huge * huge); /* overflow */ 110 return (huge * huge); /* overflow */
110 if(x <= -0x1.2cp7f) 111 if(x <= -0x1.2cp7f)
111 return (twom100 * twom100); /* underflow */ 112 return (twom100 * twom100); /* underflow */
112 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */ 113 } else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
113 return (1.0f + x); 114 return (1.0f + x);
114 } 115 }
115 116
116 /* Reduce x, computing z, i0, and k. */ 117 /* Reduce x, computing z, i0, and k. */
117 *((volatile float*)&t) = x + redux; 118 i0 = x + redux;
118 i0 = *((uint32_t*)&t); 119 memcpy(&t, &i0, sizeof(t));
119 i0 += TBLSIZE / 2; 120 i0 += TBLSIZE / 2;
120 k = (i0 >> TBLBITS) << 20; 121 k = (i0 >> TBLBITS) << 20;
121 i0 &= TBLSIZE - 1; 122 i0 &= TBLSIZE - 1;
122 t -= redux; 123 t -= redux;
123 z = x - t; 124 z = x - t;
124 ((uint32_t*)&twopk)[0] = 0x3ff00000+k; 125 temp = 0x3ff00000+k;
125 ((uint32_t*)&twopk)[1] = 0; 126 twopk = 0.0;
 127 memcpy(&twopk, &temp, sizeof(temp));
126 128
127 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */ 129 /* Compute r = exp2(y) = exp2ft[i0] * p(z). */
128 tv = exp2ft[i0]; 130 tv = exp2ft[i0];
129 u = tv * z; 131 u = tv * z;
130 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4); 132 tv = tv + u * (P1 + z * P2) + u * (z * z) * (P3 + z * P4);
131 133
132 /* Scale by 2**(k>>20). */ 134 /* Scale by 2**(k>>20). */
133 return (tv * twopk); 135 return (tv * twopk);
134} 136}