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: s_exp2.c,v 1.5 2014/03/18 21:52:19 dsl Exp $");
29 #ifdef __FBSDID
30 __FBSDID("$FreeBSD: src/lib/msun/src/s_exp2.c,v 1.7 2008/02/22 02:27:34 das Exp $");
31 #endif
32
33 #include <float.h>
34
35 #include "math.h"
36 #include "math_private.h"
37
38 #define TBLBITS 8
39 #define TBLSIZE (1 << TBLBITS)
40
41 static const double
42 redux = 0x1.8p52 / TBLSIZE,
43 P1 = 0x1.62e42fefa39efp-1,
44 P2 = 0x1.ebfbdff82c575p-3,
45 P3 = 0x1.c6b08d704a0a6p-5,
46 P4 = 0x1.3b2ab88f70400p-7,
47 P5 = 0x1.5d88003875c74p-10;
48
49 #ifdef __i386__
50 /* The return value is in %st(0) - we need a 'long double' error. */
51 static volatile long double huge = 0x1p+16000L;
52 static volatile long double tiny = 0x1p-16000L;
53 #else
54 static volatile double huge = 0x1p+1000;
55 static volatile double tiny = 0x1p-1000;
56 #endif
57
58 static const struct {
59 double exp2; /* exp2(z + eps) */
60 double eps; /* eps */
61 } tbl[TBLSIZE] = {
62 /* exp2(z + eps) eps */
63 { 0x1.6a09e667f3d5dp-1, 0x1.9880p-44 },
64 { 0x1.6b052fa751744p-1, 0x1.8000p-50 },
65 { 0x1.6c012750bd9fep-1, -0x1.8780p-45 },
66 { 0x1.6cfdcddd476bfp-1, 0x1.ec00p-46 },
67 { 0x1.6dfb23c651a29p-1, -0x1.8000p-50 },
68 { 0x1.6ef9298593ae3p-1, -0x1.c000p-52 },
69 { 0x1.6ff7df9519386p-1, -0x1.fd80p-45 },
70 { 0x1.70f7466f42da3p-1, -0x1.c880p-45 },
71 { 0x1.71f75e8ec5fc3p-1, 0x1.3c00p-46 },
72 { 0x1.72f8286eacf05p-1, -0x1.8300p-44 },
73 { 0x1.73f9a48a58152p-1, -0x1.0c00p-47 },
74 { 0x1.74fbd35d7ccfcp-1, 0x1.f880p-45 },
75 { 0x1.75feb564267f1p-1, 0x1.3e00p-47 },
76 { 0x1.77024b1ab6d48p-1, -0x1.7d00p-45 },
77 { 0x1.780694fde5d38p-1, -0x1.d000p-50 },
78 { 0x1.790b938ac1d00p-1, 0x1.3000p-49 },
79 { 0x1.7a11473eb0178p-1, -0x1.d000p-49 },
80 { 0x1.7b17b0976d060p-1, 0x1.0400p-45 },
81 { 0x1.7c1ed0130c133p-1, 0x1.0000p-53 },
82 { 0x1.7d26a62ff8636p-1, -0x1.6900p-45 },
83 { 0x1.7e2f336cf4e3bp-1, -0x1.2e00p-47 },
84 { 0x1.7f3878491c3e8p-1, -0x1.4580p-45 },
85 { 0x1.80427543e1b4ep-1, 0x1.3000p-44 },
86 { 0x1.814d2add1071ap-1, 0x1.f000p-47 },
87 { 0x1.82589994ccd7ep-1, -0x1.1c00p-45 },
88 { 0x1.8364c1eb942d0p-1, 0x1.9d00p-45 },
89 { 0x1.8471a4623cab5p-1, 0x1.7100p-43 },
90 { 0x1.857f4179f5bbcp-1, 0x1.2600p-45 },
91 { 0x1.868d99b4491afp-1, -0x1.2c40p-44 },
92 { 0x1.879cad931a395p-1, -0x1.3000p-45 },
93 { 0x1.88ac7d98a65b8p-1, -0x1.a800p-45 },
94 { 0x1.89bd0a4785800p-1, -0x1.d000p-49 },
95 { 0x1.8ace5422aa223p-1, 0x1.3280p-44 },
96 { 0x1.8be05bad619fap-1, 0x1.2b40p-43 },
97 { 0x1.8cf3216b54383p-1, -0x1.ed00p-45 },
98 { 0x1.8e06a5e08664cp-1, -0x1.0500p-45 },
99 { 0x1.8f1ae99157807p-1, 0x1.8280p-45 },
100 { 0x1.902fed0282c0ep-1, -0x1.cb00p-46 },
101 { 0x1.9145b0b91ff96p-1, -0x1.5e00p-47 },
102 { 0x1.925c353aa2ff9p-1, 0x1.5400p-48 },
103 { 0x1.93737b0cdc64ap-1, 0x1.7200p-46 },
104 { 0x1.948b82b5f98aep-1, -0x1.9000p-47 },
105 { 0x1.95a44cbc852cbp-1, 0x1.5680p-45 },
106 { 0x1.96bdd9a766f21p-1, -0x1.6d00p-44 },
107 { 0x1.97d829fde4e2ap-1, -0x1.1000p-47 },
108 { 0x1.98f33e47a23a3p-1, 0x1.d000p-45 },
109 { 0x1.9a0f170ca0604p-1, -0x1.8a40p-44 },
110 { 0x1.9b2bb4d53ff89p-1, 0x1.55c0p-44 },
111 { 0x1.9c49182a3f15bp-1, 0x1.6b80p-45 },
112 { 0x1.9d674194bb8c5p-1, -0x1.c000p-49 },
113 { 0x1.9e86319e3238ep-1, 0x1.7d00p-46 },
114 { 0x1.9fa5e8d07f302p-1, 0x1.6400p-46 },
115 { 0x1.a0c667b5de54dp-1, -0x1.5000p-48 },
116 { 0x1.a1e7aed8eb8f6p-1, 0x1.9e00p-47 },
117 { 0x1.a309bec4a2e27p-1, 0x1.ad80p-45 },
118 { 0x1.a42c980460a5dp-1, -0x1.af00p-46 },
119 { 0x1.a5503b23e259bp-1, 0x1.b600p-47 },
120 { 0x1.a674a8af46213p-1, 0x1.8880p-44 },
121 { 0x1.a799e1330b3a7p-1, 0x1.1200p-46 },
122 { 0x1.a8bfe53c12e8dp-1, 0x1.6c00p-47 },
123 { 0x1.a9e6b5579fcd2p-1, -0x1.9b80p-45 },
124 { 0x1.ab0e521356fb8p-1, 0x1.b700p-45 },
125 { 0x1.ac36bbfd3f381p-1, 0x1.9000p-50 },
126 { 0x1.ad5ff3a3c2780p-1, 0x1.4000p-49 },
127 { 0x1.ae89f995ad2a3p-1, -0x1.c900p-45 },
128 { 0x1.afb4ce622f367p-1, 0x1.6500p-46 },
129 { 0x1.b0e07298db790p-1, 0x1.fd40p-45 },
130 { 0x1.b20ce6c9a89a9p-1, 0x1.2700p-46 },
131 { 0x1.b33a2b84f1a4bp-1, 0x1.d470p-43 },
132 { 0x1.b468415b747e7p-1, -0x1.8380p-44 },
133 { 0x1.b59728de5593ap-1, 0x1.8000p-54 },
134 { 0x1.b6c6e29f1c56ap-1, 0x1.ad00p-47 },
135 { 0x1.b7f76f2fb5e50p-1, 0x1.e800p-50 },
136 { 0x1.b928cf22749b2p-1, -0x1.4c00p-47 },
137 { 0x1.ba5b030a10603p-1, -0x1.d700p-47 },
138 { 0x1.bb8e0b79a6f66p-1, 0x1.d900p-47 },
139 { 0x1.bcc1e904bc1ffp-1, 0x1.2a00p-47 },
140 { 0x1.bdf69c3f3a16fp-1, -0x1.f780p-46 },
141 { 0x1.bf2c25bd71db8p-1, -0x1.0a00p-46 },
142 { 0x1.c06286141b2e9p-1, -0x1.1400p-46 },
143 { 0x1.c199bdd8552e0p-1, 0x1.be00p-47 },
144 { 0x1.c2d1cd9fa64eep-1, -0x1.9400p-47 },
145 { 0x1.c40ab5fffd02fp-1, -0x1.ed00p-47 },
146 { 0x1.c544778fafd15p-1, 0x1.9660p-44 },
147 { 0x1.c67f12e57d0cbp-1, -0x1.a100p-46 },
148 { 0x1.c7ba88988c1b6p-1, -0x1.8458p-42 },
149 { 0x1.c8f6d9406e733p-1, -0x1.a480p-46 },
150 { 0x1.ca3405751c4dfp-1, 0x1.b000p-51 },
151 { 0x1.cb720dcef9094p-1, 0x1.1400p-47 },
152 { 0x1.ccb0f2e6d1689p-1, 0x1.0200p-48 },
153 { 0x1.cdf0b555dc412p-1, 0x1.3600p-48 },
154 { 0x1.cf3155b5bab3bp-1, -0x1.6900p-47 },
155 { 0x1.d072d4a0789bcp-1, 0x1.9a00p-47 },
156 { 0x1.d1b532b08c8fap-1, -0x1.5e00p-46 },
157 { 0x1.d2f87080d8a85p-1, 0x1.d280p-46 },
158 { 0x1.d43c8eacaa203p-1, 0x1.1a00p-47 },
159 { 0x1.d5818dcfba491p-1, 0x1.f000p-50 },
160 { 0x1.d6c76e862e6a1p-1, -0x1.3a00p-47 },
161 { 0x1.d80e316c9834ep-1, -0x1.cd80p-47 },
162 { 0x1.d955d71ff6090p-1, 0x1.4c00p-48 },
163 { 0x1.da9e603db32aep-1, 0x1.f900p-48 },
164 { 0x1.dbe7cd63a8325p-1, 0x1.9800p-49 },
165 { 0x1.dd321f301b445p-1, -0x1.5200p-48 },
166 { 0x1.de7d5641c05bfp-1, -0x1.d700p-46 },
167 { 0x1.dfc97337b9aecp-1, -0x1.6140p-46 },
168 { 0x1.e11676b197d5ep-1, 0x1.b480p-47 },
169 { 0x1.e264614f5a3e7p-1, 0x1.0ce0p-43 },
170 { 0x1.e3b333b16ee5cp-1, 0x1.c680p-47 },
171 { 0x1.e502ee78b3fb4p-1, -0x1.9300p-47 },
172 { 0x1.e653924676d68p-1, -0x1.5000p-49 },
173 { 0x1.e7a51fbc74c44p-1, -0x1.7f80p-47 },
174 { 0x1.e8f7977cdb726p-1, -0x1.3700p-48 },
175 { 0x1.ea4afa2a490e8p-1, 0x1.5d00p-49 },
176 { 0x1.eb9f4867ccae4p-1, 0x1.61a0p-46 },
177 { 0x1.ecf482d8e680dp-1, 0x1.5500p-48 },
178 { 0x1.ee4aaa2188514p-1, 0x1.6400p-51 },
179 { 0x1.efa1bee615a13p-1, -0x1.e800p-49 },
180 { 0x1.f0f9c1cb64106p-1, -0x1.a880p-48 },
181 { 0x1.f252b376bb963p-1, -0x1.c900p-45 },
182 { 0x1.f3ac948dd7275p-1, 0x1.a000p-53 },
183 { 0x1.f50765b6e4524p-1, -0x1.4f00p-48 },
184 { 0x1.f6632798844fdp-1, 0x1.a800p-51 },
185 { 0x1.f7bfdad9cbe38p-1, 0x1.abc0p-48 },
186 { 0x1.f91d802243c82p-1, -0x1.4600p-50 },
187 { 0x1.fa7c1819e908ep-1, -0x1.b0c0p-47 },
188 { 0x1.fbdba3692d511p-1, -0x1.0e00p-51 },
189 { 0x1.fd3c22b8f7194p-1, -0x1.0de8p-46 },
190 { 0x1.fe9d96b2a23eep-1, 0x1.e430p-49 },
191 { 0x1.0000000000000p+0, 0x0.0000p+0 },
192 { 0x1.00b1afa5abcbep+0, -0x1.3400p-52 },
193 { 0x1.0163da9fb3303p+0, -0x1.2170p-46 },
194 { 0x1.02168143b0282p+0, 0x1.a400p-52 },
195 { 0x1.02c9a3e77806cp+0, 0x1.f980p-49 },
196 { 0x1.037d42e11bbcap+0, -0x1.7400p-51 },
197 { 0x1.04315e86e7f89p+0, 0x1.8300p-50 },
198 { 0x1.04e5f72f65467p+0, -0x1.a3f0p-46 },
199 { 0x1.059b0d315855ap+0, -0x1.2840p-47 },
200 { 0x1.0650a0e3c1f95p+0, 0x1.1600p-48 },
201 { 0x1.0706b29ddf71ap+0, 0x1.5240p-46 },
202 { 0x1.07bd42b72a82dp+0, -0x1.9a00p-49 },
203 { 0x1.0874518759bd0p+0, 0x1.6400p-49 },
204 { 0x1.092bdf66607c8p+0, -0x1.0780p-47 },
205 { 0x1.09e3ecac6f383p+0, -0x1.8000p-54 },
206 { 0x1.0a9c79b1f3930p+0, 0x1.fa00p-48 },
207 { 0x1.0b5586cf988fcp+0, -0x1.ac80p-48 },
208 { 0x1.0c0f145e46c8ap+0, 0x1.9c00p-50 },
209 { 0x1.0cc922b724816p+0, 0x1.5200p-47 },
210 { 0x1.0d83b23395dd8p+0, -0x1.ad00p-48 },
211 { 0x1.0e3ec32d3d1f3p+0, 0x1.bac0p-46 },
212 { 0x1.0efa55fdfa9a6p+0, -0x1.4e80p-47 },
213 { 0x1.0fb66affed2f0p+0, -0x1.d300p-47 },
214 { 0x1.1073028d7234bp+0, 0x1.1500p-48 },
215 { 0x1.11301d0125b5bp+0, 0x1.c000p-49 },
216 { 0x1.11edbab5e2af9p+0, 0x1.6bc0p-46 },
217 { 0x1.12abdc06c31d5p+0, 0x1.8400p-49 },
218 { 0x1.136a814f2047dp+0, -0x1.ed00p-47 },
219 { 0x1.1429aaea92de9p+0, 0x1.8e00p-49 },
220 { 0x1.14e95934f3138p+0, 0x1.b400p-49 },
221 { 0x1.15a98c8a58e71p+0, 0x1.5300p-47 },
222 { 0x1.166a45471c3dfp+0, 0x1.3380p-47 },
223 { 0x1.172b83c7d5211p+0, 0x1.8d40p-45 },
224 { 0x1.17ed48695bb9fp+0, -0x1.5d00p-47 },
225 { 0x1.18af9388c8d93p+0, -0x1.c880p-46 },
226 { 0x1.1972658375d66p+0, 0x1.1f00p-46 },
227 { 0x1.1a35beb6fcba7p+0, 0x1.0480p-46 },
228 { 0x1.1af99f81387e3p+0, -0x1.7390p-43 },
229 { 0x1.1bbe084045d54p+0, 0x1.4e40p-45 },
230 { 0x1.1c82f95281c43p+0, -0x1.a200p-47 },
231 { 0x1.1d4873168b9b2p+0, 0x1.3800p-49 },
232 { 0x1.1e0e75eb44031p+0, 0x1.ac00p-49 },
233 { 0x1.1ed5022fcd938p+0, 0x1.1900p-47 },
234 { 0x1.1f9c18438cdf7p+0, -0x1.b780p-46 },
235 { 0x1.2063b88628d8fp+0, 0x1.d940p-45 },
236 { 0x1.212be3578a81ep+0, 0x1.8000p-50 },
237 { 0x1.21f49917ddd41p+0, 0x1.b340p-45 },
238 { 0x1.22bdda2791323p+0, 0x1.9f80p-46 },
239 { 0x1.2387a6e7561e7p+0, -0x1.9c80p-46 },
240 { 0x1.2451ffb821427p+0, 0x1.2300p-47 },
241 { 0x1.251ce4fb2a602p+0, -0x1.3480p-46 },
242 { 0x1.25e85711eceb0p+0, 0x1.2700p-46 },
243 { 0x1.26b4565e27d16p+0, 0x1.1d00p-46 },
244 { 0x1.2780e341de00fp+0, 0x1.1ee0p-44 },
245 { 0x1.284dfe1f5633ep+0, -0x1.4c00p-46 },
246 { 0x1.291ba7591bb30p+0, -0x1.3d80p-46 },
247 { 0x1.29e9df51fdf09p+0, 0x1.8b00p-47 },
248 { 0x1.2ab8a66d10e9bp+0, -0x1.27c0p-45 },
249 { 0x1.2b87fd0dada3ap+0, 0x1.a340p-45 },
250 { 0x1.2c57e39771af9p+0, -0x1.0800p-46 },
251 { 0x1.2d285a6e402d9p+0, -0x1.ed00p-47 },
252 { 0x1.2df961f641579p+0, -0x1.4200p-48 },
253 { 0x1.2ecafa93e2ecfp+0, -0x1.4980p-45 },
254 { 0x1.2f9d24abd8822p+0, -0x1.6300p-46 },
255 { 0x1.306fe0a31b625p+0, -0x1.2360p-44 },
256 { 0x1.31432edeea50bp+0, -0x1.0df8p-40 },
257 { 0x1.32170fc4cd7b8p+0, -0x1.2480p-45 },
258 { 0x1.32eb83ba8e9a2p+0, -0x1.5980p-45 },
259 { 0x1.33c08b2641766p+0, 0x1.ed00p-46 },
260 { 0x1.3496266e3fa27p+0, -0x1.c000p-50 },
261 { 0x1.356c55f929f0fp+0, -0x1.0d80p-44 },
262 { 0x1.36431a2de88b9p+0, 0x1.2c80p-45 },
263 { 0x1.371a7373aaa39p+0, 0x1.0600p-45 },
264 { 0x1.37f26231e74fep+0, -0x1.6600p-46 },
265 { 0x1.38cae6d05d838p+0, -0x1.ae00p-47 },
266 { 0x1.39a401b713ec3p+0, -0x1.4720p-43 },
267 { 0x1.3a7db34e5a020p+0, 0x1.8200p-47 },
268 { 0x1.3b57fbfec6e95p+0, 0x1.e800p-44 },
269 { 0x1.3c32dc313a8f2p+0, 0x1.f800p-49 },
270 { 0x1.3d0e544ede122p+0, -0x1.7a00p-46 },
271 { 0x1.3dea64c1234bbp+0, 0x1.6300p-45 },
272 { 0x1.3ec70df1c4eccp+0, -0x1.8a60p-43 },
273 { 0x1.3fa4504ac7e8cp+0, -0x1.cdc0p-44 },
274 { 0x1.40822c367a0bbp+0, 0x1.5b80p-45 },
275 { 0x1.4160a21f72e95p+0, 0x1.ec00p-46 },
276 { 0x1.423fb27094646p+0, -0x1.3600p-46 },
277 { 0x1.431f5d950a920p+0, 0x1.3980p-45 },
278 { 0x1.43ffa3f84b9ebp+0, 0x1.a000p-48 },
279 { 0x1.44e0860618919p+0, -0x1.6c00p-48 },
280 { 0x1.45c2042a7d201p+0, -0x1.bc00p-47 },
281 { 0x1.46a41ed1d0016p+0, -0x1.2800p-46 },
282 { 0x1.4786d668b3326p+0, 0x1.0e00p-44 },
283 { 0x1.486a2b5c13c00p+0, -0x1.d400p-45 },
284 { 0x1.494e1e192af04p+0, 0x1.c200p-47 },
285 { 0x1.4a32af0d7d372p+0, -0x1.e500p-46 },
286 { 0x1.4b17dea6db801p+0, 0x1.7800p-47 },
287 { 0x1.4bfdad53629e1p+0, -0x1.3800p-46 },
288 { 0x1.4ce41b817c132p+0, 0x1.0800p-47 },
289 { 0x1.4dcb299fddddbp+0, 0x1.c700p-45 },
290 { 0x1.4eb2d81d8ab96p+0, -0x1.ce00p-46 },
291 { 0x1.4f9b2769d2d02p+0, 0x1.9200p-46 },
292 { 0x1.508417f4531c1p+0, -0x1.8c00p-47 },
293 { 0x1.516daa2cf662ap+0, -0x1.a000p-48 },
294 { 0x1.5257de83f51eap+0, 0x1.a080p-43 },
295 { 0x1.5342b569d4edap+0, -0x1.6d80p-45 },
296 { 0x1.542e2f4f6ac1ap+0, -0x1.2440p-44 },
297 { 0x1.551a4ca5d94dbp+0, 0x1.83c0p-43 },
298 { 0x1.56070dde9116bp+0, 0x1.4b00p-45 },
299 { 0x1.56f4736b529dep+0, 0x1.15a0p-43 },
300 { 0x1.57e27dbe2c40ep+0, -0x1.9e00p-45 },
301 { 0x1.58d12d497c76fp+0, -0x1.3080p-45 },
302 { 0x1.59c0827ff0b4cp+0, 0x1.dec0p-43 },
303 { 0x1.5ab07dd485427p+0, -0x1.4000p-51 },
304 { 0x1.5ba11fba87af4p+0, 0x1.0080p-44 },
305 { 0x1.5c9268a59460bp+0, -0x1.6c80p-45 },
306 { 0x1.5d84590998e3fp+0, 0x1.69a0p-43 },
307 { 0x1.5e76f15ad20e1p+0, -0x1.b400p-46 },
308 { 0x1.5f6a320dcebcap+0, 0x1.7700p-46 },
309 { 0x1.605e1b976dcb8p+0, 0x1.6f80p-45 },
310 { 0x1.6152ae6cdf715p+0, 0x1.1000p-47 },
311 { 0x1.6247eb03a5531p+0, -0x1.5d00p-46 },
312 { 0x1.633dd1d1929b5p+0, -0x1.2d00p-46 },
313 { 0x1.6434634ccc313p+0, -0x1.a800p-49 },
314 { 0x1.652b9febc8efap+0, -0x1.8600p-45 },
315 { 0x1.6623882553397p+0, 0x1.1fe0p-40 },
316 { 0x1.671c1c708328ep+0, -0x1.7200p-44 },
317 { 0x1.68155d44ca97ep+0, 0x1.6800p-49 },
318 { 0x1.690f4b19e9471p+0, -0x1.9780p-45 },
319 };
320
321 /*
322 * exp2(x): compute the base 2 exponential of x
323 *
324 * Accuracy: Peak error < 0.503 ulp for normalized results.
325 *
326 * Method: (accurate tables)
327 *
328 * Reduce x:
329 * x = 2**k + y, for integer k and |y| <= 1/2.
330 * Thus we have exp2(x) = 2**k * exp2(y).
331 *
332 * Reduce y:
333 * y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
334 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
335 * with |z - eps[i]| <= 2**-9 + 2**-39 for the table used.
336 *
337 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
338 * a degree-5 minimax polynomial with maximum error under 1.3 * 2**-61.
339 * The values in exp2t[] and eps[] are chosen such that
340 * exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
341 * that exp2t[i] is accurate to 2**-64.
342 *
343 * Note that the range of i is +-TBLSIZE/2, so we actually index the tables
344 * by i0 = i + TBLSIZE/2.
345 *
346 * This method is due to Gal, with many details due to Gal and Bachelis:
347 *
348 * Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library
349 * for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991).
350 */
351 double
exp2(double x)352 exp2(double x)
353 {
354 volatile ieee_double_shape_type x_p_redux;
355 double r, t, twopk, z;
356 uint32_t hx, ix, lx;
357 int32_t i0;
358 int k;
359
360 /* Filter out exceptional cases. */
361 GET_HIGH_WORD(hx,x);
362 ix = hx & 0x7fffffff; /* high word of |x| */
363 if(ix >= 0x40900000) { /* |x| >= 1024 */
364 if(ix >= 0x7ff00000) {
365 /* x is infinity or NaN */
366 GET_LOW_WORD(lx,x);
367 if(((ix & 0xfffff) | lx) != 0 || (hx & 0x80000000) == 0)
368 return (x + x); /* x is NaN or +Inf */
369 else
370 return (0.0); /* x is -Inf */
371 }
372 if(x >= 0x1.0p10)
373 return (huge * huge); /* overflow */
374 if(x <= -0x1.0ccp10)
375 return (tiny * tiny); /* underflow */
376 } else if (ix < 0x3c900000) { /* |x| < 0x1p-54 */
377 return (1.0 + x);
378 }
379
380 /* Reduce x, computing z, i0, and k. */
381 x_p_redux.value = x + redux;
382 i0 = x_p_redux.parts.lsw;
383
384 i0 += TBLSIZE / 2;
385 k = i0 >> TBLBITS;
386 i0 &= TBLSIZE - 1;
387 z = x - (x_p_redux.value - redux);
388
389 /* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
390 t = tbl[i0].exp2;
391 z -= tbl[i0].eps;
392 r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * P5))));
393
394 /*
395 * Scale by 2**k.
396 * For 'normal' values of 'k' we just multply be the correct value,
397 * but at the extremes this doesn't work and the multiply must
398 * be split.
399 */
400 if (k >= -1021) {
401 if (k == 1024)
402 /* Overflow */
403 return (r * 2.0 * 0x1p1023);
404 } else {
405 /* result might be subnormal */
406 r *= 0x1p-1000;
407 k += 1000;
408 }
409 INSERT_WORDS(twopk, 0x3ff00000 + (k << 20), 0);
410 return (r * twopk);
411 }
412
413 #ifdef notyet
414 #if (LDBL_MANT_DIG == 53)
415 __weak_reference(exp2, exp2l);
416 #endif
417 #endif
418