1 /*-
2 * Copyright (c) 2005-2008 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
28
29 #define TBLBITS 7
30 #define TBLSIZE (1 << TBLBITS)
31
32 #define BIAS (LDBL_MAX_EXP - 1)
33 #define EXPMASK (BIAS + LDBL_MAX_EXP)
34
35 static const long double huge = 0x1p10000L;
36 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */
37 static const long double twom10000 = 0x1p-10000L;
38 #else
39 static volatile long double twom10000 = 0x1p-10000L;
40 #endif
41
42 static const double
43 redux = 0x1.8p63 / TBLSIZE,
44 P1 = 0x1.62e42fefa39efp-1,
45 P2 = 0x1.ebfbdff82c58fp-3,
46 P3 = 0x1.c6b08d7049fap-5,
47 P4 = 0x1.3b2ab6fba4da5p-7,
48 P5 = 0x1.5d8804780a736p-10,
49 P6 = 0x1.430918835e33dp-13;
50
51 static const double tbl[TBLSIZE * 2] = {
52 0x1.6a09e667f3bcdp-1, -0x1.bdd3413b2648p-55,
53 0x1.6c012750bdabfp-1, -0x1.2895667ff0cp-57,
54 0x1.6dfb23c651a2fp-1, -0x1.bbe3a683c88p-58,
55 0x1.6ff7df9519484p-1, -0x1.83c0f25860fp-56,
56 0x1.71f75e8ec5f74p-1, -0x1.16e4786887bp-56,
57 0x1.73f9a48a58174p-1, -0x1.0a8d96c65d5p-55,
58 0x1.75feb564267c9p-1, -0x1.0245957316ep-55,
59 0x1.780694fde5d3fp-1, 0x1.866b80a0216p-55,
60 0x1.7a11473eb0187p-1, -0x1.41577ee0499p-56,
61 0x1.7c1ed0130c132p-1, 0x1.f124cd1164ep-55,
62 0x1.7e2f336cf4e62p-1, 0x1.05d02ba157ap-57,
63 0x1.80427543e1a12p-1, -0x1.27c86626d97p-55,
64 0x1.82589994cce13p-1, -0x1.d4c1dd41533p-55,
65 0x1.8471a4623c7adp-1, -0x1.8d684a341cep-56,
66 0x1.868d99b4492edp-1, -0x1.fc6f89bd4f68p-55,
67 0x1.88ac7d98a6699p-1, 0x1.994c2f37cb5p-55,
68 0x1.8ace5422aa0dbp-1, 0x1.6e9f156864bp-55,
69 0x1.8cf3216b5448cp-1, -0x1.0d55e32e9e4p-57,
70 0x1.8f1ae99157736p-1, 0x1.5cc13a2e397p-56,
71 0x1.9145b0b91ffc6p-1, -0x1.dd6792e5825p-55,
72 0x1.93737b0cdc5e5p-1, -0x1.75fc781b58p-58,
73 0x1.95a44cbc8520fp-1, -0x1.64b7c96a5fp-57,
74 0x1.97d829fde4e5p-1, -0x1.d185b7c1b86p-55,
75 0x1.9a0f170ca07bap-1, -0x1.173bd91cee6p-55,
76 0x1.9c49182a3f09p-1, 0x1.c7c46b071f2p-57,
77 0x1.9e86319e32323p-1, 0x1.824ca78e64cp-57,
78 0x1.a0c667b5de565p-1, -0x1.359495d1cd5p-55,
79 0x1.a309bec4a2d33p-1, 0x1.6305c7ddc368p-55,
80 0x1.a5503b23e255dp-1, -0x1.d2f6edb8d42p-55,
81 0x1.a799e1330b358p-1, 0x1.bcb7ecac564p-55,
82 0x1.a9e6b5579fdbfp-1, 0x1.0fac90ef7fdp-55,
83 0x1.ac36bbfd3f37ap-1, -0x1.f9234cae76dp-56,
84 0x1.ae89f995ad3adp-1, 0x1.7a1cd345dcc8p-55,
85 0x1.b0e07298db666p-1, -0x1.bdef54c80e4p-55,
86 0x1.b33a2b84f15fbp-1, -0x1.2805e3084d8p-58,
87 0x1.b59728de5593ap-1, -0x1.c71dfbbba6ep-55,
88 0x1.b7f76f2fb5e47p-1, -0x1.5584f7e54acp-57,
89 0x1.ba5b030a1064ap-1, -0x1.efcd30e5429p-55,
90 0x1.bcc1e904bc1d2p-1, 0x1.23dd07a2d9fp-56,
91 0x1.bf2c25bd71e09p-1, -0x1.efdca3f6b9c8p-55,
92 0x1.c199bdd85529cp-1, 0x1.11065895049p-56,
93 0x1.c40ab5fffd07ap-1, 0x1.b4537e083c6p-55,
94 0x1.c67f12e57d14bp-1, 0x1.2884dff483c8p-55,
95 0x1.c8f6d9406e7b5p-1, 0x1.1acbc48805cp-57,
96 0x1.cb720dcef9069p-1, 0x1.503cbd1e94ap-57,
97 0x1.cdf0b555dc3fap-1, -0x1.dd83b53829dp-56,
98 0x1.d072d4a07897cp-1, -0x1.cbc3743797a8p-55,
99 0x1.d2f87080d89f2p-1, -0x1.d487b719d858p-55,
100 0x1.d5818dcfba487p-1, 0x1.2ed02d75b37p-56,
101 0x1.d80e316c98398p-1, -0x1.11ec18bedep-55,
102 0x1.da9e603db3285p-1, 0x1.c2300696db5p-55,
103 0x1.dd321f301b46p-1, 0x1.2da5778f019p-55,
104 0x1.dfc97337b9b5fp-1, -0x1.1a5cd4f184b8p-55,
105 0x1.e264614f5a129p-1, -0x1.7b627817a148p-55,
106 0x1.e502ee78b3ff6p-1, 0x1.39e8980a9cdp-56,
107 0x1.e7a51fbc74c83p-1, 0x1.2d522ca0c8ep-55,
108 0x1.ea4afa2a490dap-1, -0x1.e9c23179c288p-55,
109 0x1.ecf482d8e67f1p-1, -0x1.c93f3b411ad8p-55,
110 0x1.efa1bee615a27p-1, 0x1.dc7f486a4b68p-55,
111 0x1.f252b376bba97p-1, 0x1.3a1a5bf0d8e8p-55,
112 0x1.f50765b6e454p-1, 0x1.9d3e12dd8a18p-55,
113 0x1.f7bfdad9cbe14p-1, -0x1.dbb12d00635p-55,
114 0x1.fa7c1819e90d8p-1, 0x1.74853f3a593p-56,
115 0x1.fd3c22b8f71f1p-1, 0x1.2eb74966578p-58,
116 0x1p+0, 0x0p+0,
117 0x1.0163da9fb3335p+0, 0x1.b61299ab8cd8p-54,
118 0x1.02c9a3e778061p+0, -0x1.19083535b08p-56,
119 0x1.04315e86e7f85p+0, -0x1.0a31c1977c98p-54,
120 0x1.059b0d3158574p+0, 0x1.d73e2a475b4p-55,
121 0x1.0706b29ddf6dep+0, -0x1.c91dfe2b13cp-55,
122 0x1.0874518759bc8p+0, 0x1.186be4bb284p-57,
123 0x1.09e3ecac6f383p+0, 0x1.14878183161p-54,
124 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc61p-54,
125 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f8p-54,
126 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c58p-59,
127 0x1.0fb66affed31bp+0, -0x1.b9bedc44ebcp-57,
128 0x1.11301d0125b51p+0, -0x1.6c51039449bp-54,
129 0x1.12abdc06c31ccp+0, -0x1.1b514b36ca8p-58,
130 0x1.1429aaea92dep+0, -0x1.32fbf9af1368p-54,
131 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeabp-55,
132 0x1.172b83c7d517bp+0, -0x1.19041b9d78ap-55,
133 0x1.18af9388c8deap+0, -0x1.11023d1970f8p-54,
134 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4969p-55,
135 0x1.1bbe084045cd4p+0, -0x1.95386352ef6p-54,
136 0x1.1d4873168b9aap+0, 0x1.e016e00a264p-54,
137 0x1.1ed5022fcd91dp+0, -0x1.1df98027bb78p-54,
138 0x1.2063b88628cd6p+0, 0x1.dc775814a85p-55,
139 0x1.21f49917ddc96p+0, 0x1.2a97e9494a6p-55,
140 0x1.2387a6e756238p+0, 0x1.9b07eb6c7058p-54,
141 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f5p-55,
142 0x1.26b4565e27cddp+0, 0x1.2bd339940eap-55,
143 0x1.284dfe1f56381p+0, -0x1.a4c3a8c3f0d8p-54,
144 0x1.29e9df51fdee1p+0, 0x1.612e8afad12p-55,
145 0x1.2b87fd0dad99p+0, -0x1.10adcd6382p-59,
146 0x1.2d285a6e4030bp+0, 0x1.0024754db42p-54,
147 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d524p-56,
148 0x1.306fe0a31b715p+0, 0x1.6f46ad23183p-55,
149 0x1.32170fc4cd831p+0, 0x1.a9ce78e1804p-55,
150 0x1.33c08b26416ffp+0, 0x1.327218436598p-54,
151 0x1.356c55f929ff1p+0, -0x1.b5cee5c4e46p-55,
152 0x1.371a7373aa9cbp+0, -0x1.63aeabf42ebp-54,
153 0x1.38cae6d05d866p+0, -0x1.e958d3c99048p-54,
154 0x1.3a7db34e59ff7p+0, -0x1.5e436d661f6p-56,
155 0x1.3c32dc313a8e5p+0, -0x1.efff8375d2ap-54,
156 0x1.3dea64c123422p+0, 0x1.ada0911f09fp-55,
157 0x1.3fa4504ac801cp+0, -0x1.7d023f956fap-54,
158 0x1.4160a21f72e2ap+0, -0x1.ef3691c309p-58,
159 0x1.431f5d950a897p+0, -0x1.1c7dde35f7ap-55,
160 0x1.44e086061892dp+0, 0x1.89b7a04ef8p-59,
161 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a8p-54,
162 0x1.486a2b5c13cdp+0, 0x1.3c1a3b69062p-56,
163 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be8p-54,
164 0x1.4bfdad5362a27p+0, 0x1.d4397afec42p-56,
165 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a78p-54,
166 0x1.4f9b2769d2ca7p+0, -0x1.4b309d25958p-54,
167 0x1.516daa2cf6642p+0, -0x1.f768569bd94p-55,
168 0x1.5342b569d4f82p+0, -0x1.07abe1db13dp-55,
169 0x1.551a4ca5d920fp+0, -0x1.d689cefede6p-55,
170 0x1.56f4736b527dap+0, 0x1.9bb2c011d938p-54,
171 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1ep-55,
172 0x1.5ab07dd485429p+0, 0x1.6324c0546478p-54,
173 0x1.5c9268a5946b7p+0, 0x1.c4b1b81698p-60,
174 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e68p-54,
175 0x1.605e1b976dc09p+0, -0x1.3e2429b56de8p-54,
176 0x1.6247eb03a5585p+0, -0x1.383c17e40b48p-54,
177 0x1.6434634ccc32p+0, -0x1.c483c759d89p-55,
178 0x1.6623882552225p+0, -0x1.bb60987591cp-54,
179 0x1.68155d44ca973p+0, 0x1.038ae44f74p-57,
180 };
181
182 /*
183 * exp2l(x): compute the base 2 exponential of x
184 *
185 * Accuracy: Peak error < 0.511 ulp.
186 *
187 * Method: (equally-spaced tables)
188 *
189 * Reduce x:
190 * x = 2**k + y, for integer k and |y| <= 1/2.
191 * Thus we have exp2l(x) = 2**k * exp2(y).
192 *
193 * Reduce y:
194 * y = i/TBLSIZE + z for integer i near y * TBLSIZE.
195 * Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
196 * with |z| <= 2**-(TBLBITS+1).
197 *
198 * We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
199 * degree-6 minimax polynomial with maximum error under 2**-69.
200 * The table entries each have 104 bits of accuracy, encoded as
201 * a pair of double precision values.
202 */
203 long double
exp2l(long double x)204 exp2l(long double x)
205 {
206 union IEEEl2bits u, v;
207 long double r, twopk, twopkp10000, z;
208 uint32_t hx, ix, i0;
209 int k;
210
211 /* Filter out exceptional cases. */
212 u.e = x;
213 hx = u.xbits.expsign;
214 ix = hx & EXPMASK;
215 if (ix >= BIAS + 14) { /* |x| >= 16384 or x is NaN */
216 if (ix == BIAS + LDBL_MAX_EXP) {
217 if (u.xbits.man != (unsigned long long) LDBL_NBIT_INF << 32 || (hx & 0x8000) == 0)
218 return (x + x); /* x is +Inf or NaN */
219 else
220 return (0.0l); /* x is -Inf */
221 }
222 if (x >= 16384)
223 return __math_oflowl(0); /* overflow */
224 if (x <= -16446)
225 return __math_uflowl(0); /* underflow */
226 } else if (ix <= BIAS - 66) { /* |x| < 0x1p-66 */
227 return (1.0l + x);
228 }
229
230 /*
231 * Reduce x, computing z, i0, and k. The low bits of x + redux
232 * contain the 16-bit integer part of the exponent (k) followed by
233 * TBLBITS fractional bits (i0). We use bit tricks to extract these
234 * as integers, then set z to the remainder.
235 *
236 * Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8.
237 * Then the low-order word of x + redux is 0x000abc12,
238 * We split this into k = 0xabc and i0 = 0x12 (adjusted to
239 * index into the table), then we compute z = 0x0.003456p0.
240 *
241 * XXX If the exponent is negative, the computation of k depends on
242 * '>>' doing sign extension.
243 */
244 u.e = x + (long double)redux;
245 i0 = u.bits.manl + TBLSIZE / 2;
246 k = (int)i0 >> TBLBITS;
247 i0 = (i0 & (TBLSIZE - 1)) << 1;
248 u.e -= (long double)redux;
249 z = x - u.e;
250 v.xbits.man = 1ULL << 63;
251 if (k >= LDBL_MIN_EXP) {
252 v.xbits.expsign = LDBL_MAX_EXP - 1 + k;
253 twopk = v.e;
254 } else {
255 v.xbits.expsign = LDBL_MAX_EXP - 1 + k + 10000;
256 twopkp10000 = v.e;
257 }
258
259 /* Compute r = exp2l(y) = exp2lt[i0] * p(z). */
260 long double t_hi = (long double)tbl[i0];
261 long double t_lo = (long double)tbl[i0 + 1];
262 /* XXX This gives > 1 ulp errors outside of FE_TONEAREST mode */
263 r = t_lo + (t_hi + t_lo) * z * ((long double)P1 + z * ((long double)P2 + z * ((long double)P3 + z * ((long double)P4
264 + z * ((long double)P5 + z * (long double)P6))))) + t_hi;
265
266 /* Scale by 2**k. */
267 if (k >= LDBL_MIN_EXP) {
268 if (k == LDBL_MAX_EXP)
269 return (r * 2.0l * 0x1p16383L);
270 return (r * twopk);
271 } else {
272 return (r * twopkp10000 * twom10000);
273 }
274 }
275
