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/* $OpenBSD: s_exp2.c,v 1.9 2016/09/12 19:47:02 guenther Exp $ */ |
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/*- |
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* Copyright (c) 2005 David Schultz <das@FreeBSD.ORG> |
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* All rights reserved. |
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* |
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* Redistribution and use in source and binary forms, with or without |
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* modification, are permitted provided that the following conditions |
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* are met: |
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* 1. Redistributions of source code must retain the above copyright |
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* notice, this list of conditions and the following disclaimer. |
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* 2. Redistributions in binary form must reproduce the above copyright |
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* notice, this list of conditions and the following disclaimer in the |
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* documentation and/or other materials provided with the distribution. |
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* |
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* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND |
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE |
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE |
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* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE |
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL |
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS |
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) |
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT |
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY |
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
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* SUCH DAMAGE. |
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*/ |
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#include <float.h> |
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#include <math.h> |
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#include "math_private.h" |
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#define TBLBITS 8 |
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#define TBLSIZE (1 << TBLBITS) |
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static const double |
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huge = 0x1p1000, |
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redux = 0x1.8p52 / TBLSIZE, |
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P1 = 0x1.62e42fefa39efp-1, |
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P2 = 0x1.ebfbdff82c575p-3, |
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P3 = 0x1.c6b08d704a0a6p-5, |
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P4 = 0x1.3b2ab88f70400p-7, |
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P5 = 0x1.5d88003875c74p-10; |
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static volatile double twom1000 = 0x1p-1000; |
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static const double tbl[TBLSIZE * 2] = { |
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/* exp2(z + eps) eps */ |
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0x1.6a09e667f3d5dp-1, 0x1.9880p-44, |
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0x1.6b052fa751744p-1, 0x1.8000p-50, |
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0x1.6c012750bd9fep-1, -0x1.8780p-45, |
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0x1.6cfdcddd476bfp-1, 0x1.ec00p-46, |
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0x1.6dfb23c651a29p-1, -0x1.8000p-50, |
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0x1.6ef9298593ae3p-1, -0x1.c000p-52, |
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0x1.6ff7df9519386p-1, -0x1.fd80p-45, |
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0x1.70f7466f42da3p-1, -0x1.c880p-45, |
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0x1.71f75e8ec5fc3p-1, 0x1.3c00p-46, |
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0x1.72f8286eacf05p-1, -0x1.8300p-44, |
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0x1.73f9a48a58152p-1, -0x1.0c00p-47, |
60 |
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0x1.74fbd35d7ccfcp-1, 0x1.f880p-45, |
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0x1.75feb564267f1p-1, 0x1.3e00p-47, |
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0x1.77024b1ab6d48p-1, -0x1.7d00p-45, |
63 |
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0x1.780694fde5d38p-1, -0x1.d000p-50, |
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0x1.790b938ac1d00p-1, 0x1.3000p-49, |
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0x1.7a11473eb0178p-1, -0x1.d000p-49, |
66 |
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0x1.7b17b0976d060p-1, 0x1.0400p-45, |
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0x1.7c1ed0130c133p-1, 0x1.0000p-53, |
68 |
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0x1.7d26a62ff8636p-1, -0x1.6900p-45, |
69 |
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0x1.7e2f336cf4e3bp-1, -0x1.2e00p-47, |
70 |
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0x1.7f3878491c3e8p-1, -0x1.4580p-45, |
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0x1.80427543e1b4ep-1, 0x1.3000p-44, |
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0x1.814d2add1071ap-1, 0x1.f000p-47, |
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0x1.82589994ccd7ep-1, -0x1.1c00p-45, |
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0x1.8364c1eb942d0p-1, 0x1.9d00p-45, |
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0x1.8471a4623cab5p-1, 0x1.7100p-43, |
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0x1.857f4179f5bbcp-1, 0x1.2600p-45, |
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0x1.868d99b4491afp-1, -0x1.2c40p-44, |
78 |
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0x1.879cad931a395p-1, -0x1.3000p-45, |
79 |
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0x1.88ac7d98a65b8p-1, -0x1.a800p-45, |
80 |
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0x1.89bd0a4785800p-1, -0x1.d000p-49, |
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0x1.8ace5422aa223p-1, 0x1.3280p-44, |
82 |
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0x1.8be05bad619fap-1, 0x1.2b40p-43, |
83 |
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0x1.8cf3216b54383p-1, -0x1.ed00p-45, |
84 |
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0x1.8e06a5e08664cp-1, -0x1.0500p-45, |
85 |
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0x1.8f1ae99157807p-1, 0x1.8280p-45, |
86 |
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0x1.902fed0282c0ep-1, -0x1.cb00p-46, |
87 |
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0x1.9145b0b91ff96p-1, -0x1.5e00p-47, |
88 |
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0x1.925c353aa2ff9p-1, 0x1.5400p-48, |
89 |
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0x1.93737b0cdc64ap-1, 0x1.7200p-46, |
90 |
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0x1.948b82b5f98aep-1, -0x1.9000p-47, |
91 |
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0x1.95a44cbc852cbp-1, 0x1.5680p-45, |
92 |
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0x1.96bdd9a766f21p-1, -0x1.6d00p-44, |
93 |
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0x1.97d829fde4e2ap-1, -0x1.1000p-47, |
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0x1.98f33e47a23a3p-1, 0x1.d000p-45, |
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0x1.9a0f170ca0604p-1, -0x1.8a40p-44, |
96 |
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0x1.9b2bb4d53ff89p-1, 0x1.55c0p-44, |
97 |
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0x1.9c49182a3f15bp-1, 0x1.6b80p-45, |
98 |
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0x1.9d674194bb8c5p-1, -0x1.c000p-49, |
99 |
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0x1.9e86319e3238ep-1, 0x1.7d00p-46, |
100 |
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0x1.9fa5e8d07f302p-1, 0x1.6400p-46, |
101 |
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0x1.a0c667b5de54dp-1, -0x1.5000p-48, |
102 |
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0x1.a1e7aed8eb8f6p-1, 0x1.9e00p-47, |
103 |
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0x1.a309bec4a2e27p-1, 0x1.ad80p-45, |
104 |
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0x1.a42c980460a5dp-1, -0x1.af00p-46, |
105 |
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0x1.a5503b23e259bp-1, 0x1.b600p-47, |
106 |
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0x1.a674a8af46213p-1, 0x1.8880p-44, |
107 |
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0x1.a799e1330b3a7p-1, 0x1.1200p-46, |
108 |
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0x1.a8bfe53c12e8dp-1, 0x1.6c00p-47, |
109 |
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0x1.a9e6b5579fcd2p-1, -0x1.9b80p-45, |
110 |
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0x1.ab0e521356fb8p-1, 0x1.b700p-45, |
111 |
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0x1.ac36bbfd3f381p-1, 0x1.9000p-50, |
112 |
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0x1.ad5ff3a3c2780p-1, 0x1.4000p-49, |
113 |
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0x1.ae89f995ad2a3p-1, -0x1.c900p-45, |
114 |
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0x1.afb4ce622f367p-1, 0x1.6500p-46, |
115 |
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0x1.b0e07298db790p-1, 0x1.fd40p-45, |
116 |
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0x1.b20ce6c9a89a9p-1, 0x1.2700p-46, |
117 |
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0x1.b33a2b84f1a4bp-1, 0x1.d470p-43, |
118 |
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0x1.b468415b747e7p-1, -0x1.8380p-44, |
119 |
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0x1.b59728de5593ap-1, 0x1.8000p-54, |
120 |
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0x1.b6c6e29f1c56ap-1, 0x1.ad00p-47, |
121 |
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0x1.b7f76f2fb5e50p-1, 0x1.e800p-50, |
122 |
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0x1.b928cf22749b2p-1, -0x1.4c00p-47, |
123 |
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0x1.ba5b030a10603p-1, -0x1.d700p-47, |
124 |
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0x1.bb8e0b79a6f66p-1, 0x1.d900p-47, |
125 |
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0x1.bcc1e904bc1ffp-1, 0x1.2a00p-47, |
126 |
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0x1.bdf69c3f3a16fp-1, -0x1.f780p-46, |
127 |
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0x1.bf2c25bd71db8p-1, -0x1.0a00p-46, |
128 |
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0x1.c06286141b2e9p-1, -0x1.1400p-46, |
129 |
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0x1.c199bdd8552e0p-1, 0x1.be00p-47, |
130 |
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0x1.c2d1cd9fa64eep-1, -0x1.9400p-47, |
131 |
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0x1.c40ab5fffd02fp-1, -0x1.ed00p-47, |
132 |
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0x1.c544778fafd15p-1, 0x1.9660p-44, |
133 |
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0x1.c67f12e57d0cbp-1, -0x1.a100p-46, |
134 |
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0x1.c7ba88988c1b6p-1, -0x1.8458p-42, |
135 |
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0x1.c8f6d9406e733p-1, -0x1.a480p-46, |
136 |
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0x1.ca3405751c4dfp-1, 0x1.b000p-51, |
137 |
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0x1.cb720dcef9094p-1, 0x1.1400p-47, |
138 |
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0x1.ccb0f2e6d1689p-1, 0x1.0200p-48, |
139 |
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0x1.cdf0b555dc412p-1, 0x1.3600p-48, |
140 |
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0x1.cf3155b5bab3bp-1, -0x1.6900p-47, |
141 |
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0x1.d072d4a0789bcp-1, 0x1.9a00p-47, |
142 |
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0x1.d1b532b08c8fap-1, -0x1.5e00p-46, |
143 |
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0x1.d2f87080d8a85p-1, 0x1.d280p-46, |
144 |
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0x1.d43c8eacaa203p-1, 0x1.1a00p-47, |
145 |
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0x1.d5818dcfba491p-1, 0x1.f000p-50, |
146 |
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0x1.d6c76e862e6a1p-1, -0x1.3a00p-47, |
147 |
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0x1.d80e316c9834ep-1, -0x1.cd80p-47, |
148 |
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0x1.d955d71ff6090p-1, 0x1.4c00p-48, |
149 |
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0x1.da9e603db32aep-1, 0x1.f900p-48, |
150 |
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0x1.dbe7cd63a8325p-1, 0x1.9800p-49, |
151 |
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0x1.dd321f301b445p-1, -0x1.5200p-48, |
152 |
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0x1.de7d5641c05bfp-1, -0x1.d700p-46, |
153 |
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0x1.dfc97337b9aecp-1, -0x1.6140p-46, |
154 |
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0x1.e11676b197d5ep-1, 0x1.b480p-47, |
155 |
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0x1.e264614f5a3e7p-1, 0x1.0ce0p-43, |
156 |
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0x1.e3b333b16ee5cp-1, 0x1.c680p-47, |
157 |
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0x1.e502ee78b3fb4p-1, -0x1.9300p-47, |
158 |
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0x1.e653924676d68p-1, -0x1.5000p-49, |
159 |
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0x1.e7a51fbc74c44p-1, -0x1.7f80p-47, |
160 |
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0x1.e8f7977cdb726p-1, -0x1.3700p-48, |
161 |
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0x1.ea4afa2a490e8p-1, 0x1.5d00p-49, |
162 |
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0x1.eb9f4867ccae4p-1, 0x1.61a0p-46, |
163 |
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0x1.ecf482d8e680dp-1, 0x1.5500p-48, |
164 |
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0x1.ee4aaa2188514p-1, 0x1.6400p-51, |
165 |
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0x1.efa1bee615a13p-1, -0x1.e800p-49, |
166 |
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0x1.f0f9c1cb64106p-1, -0x1.a880p-48, |
167 |
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0x1.f252b376bb963p-1, -0x1.c900p-45, |
168 |
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0x1.f3ac948dd7275p-1, 0x1.a000p-53, |
169 |
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0x1.f50765b6e4524p-1, -0x1.4f00p-48, |
170 |
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0x1.f6632798844fdp-1, 0x1.a800p-51, |
171 |
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0x1.f7bfdad9cbe38p-1, 0x1.abc0p-48, |
172 |
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0x1.f91d802243c82p-1, -0x1.4600p-50, |
173 |
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0x1.fa7c1819e908ep-1, -0x1.b0c0p-47, |
174 |
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0x1.fbdba3692d511p-1, -0x1.0e00p-51, |
175 |
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0x1.fd3c22b8f7194p-1, -0x1.0de8p-46, |
176 |
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0x1.fe9d96b2a23eep-1, 0x1.e430p-49, |
177 |
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0x1.0000000000000p+0, 0x0.0000p+0, |
178 |
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0x1.00b1afa5abcbep+0, -0x1.3400p-52, |
179 |
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0x1.0163da9fb3303p+0, -0x1.2170p-46, |
180 |
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0x1.02168143b0282p+0, 0x1.a400p-52, |
181 |
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0x1.02c9a3e77806cp+0, 0x1.f980p-49, |
182 |
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0x1.037d42e11bbcap+0, -0x1.7400p-51, |
183 |
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0x1.04315e86e7f89p+0, 0x1.8300p-50, |
184 |
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0x1.04e5f72f65467p+0, -0x1.a3f0p-46, |
185 |
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0x1.059b0d315855ap+0, -0x1.2840p-47, |
186 |
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0x1.0650a0e3c1f95p+0, 0x1.1600p-48, |
187 |
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0x1.0706b29ddf71ap+0, 0x1.5240p-46, |
188 |
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0x1.07bd42b72a82dp+0, -0x1.9a00p-49, |
189 |
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0x1.0874518759bd0p+0, 0x1.6400p-49, |
190 |
|
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0x1.092bdf66607c8p+0, -0x1.0780p-47, |
191 |
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0x1.09e3ecac6f383p+0, -0x1.8000p-54, |
192 |
|
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0x1.0a9c79b1f3930p+0, 0x1.fa00p-48, |
193 |
|
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0x1.0b5586cf988fcp+0, -0x1.ac80p-48, |
194 |
|
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0x1.0c0f145e46c8ap+0, 0x1.9c00p-50, |
195 |
|
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0x1.0cc922b724816p+0, 0x1.5200p-47, |
196 |
|
|
0x1.0d83b23395dd8p+0, -0x1.ad00p-48, |
197 |
|
|
0x1.0e3ec32d3d1f3p+0, 0x1.bac0p-46, |
198 |
|
|
0x1.0efa55fdfa9a6p+0, -0x1.4e80p-47, |
199 |
|
|
0x1.0fb66affed2f0p+0, -0x1.d300p-47, |
200 |
|
|
0x1.1073028d7234bp+0, 0x1.1500p-48, |
201 |
|
|
0x1.11301d0125b5bp+0, 0x1.c000p-49, |
202 |
|
|
0x1.11edbab5e2af9p+0, 0x1.6bc0p-46, |
203 |
|
|
0x1.12abdc06c31d5p+0, 0x1.8400p-49, |
204 |
|
|
0x1.136a814f2047dp+0, -0x1.ed00p-47, |
205 |
|
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0x1.1429aaea92de9p+0, 0x1.8e00p-49, |
206 |
|
|
0x1.14e95934f3138p+0, 0x1.b400p-49, |
207 |
|
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0x1.15a98c8a58e71p+0, 0x1.5300p-47, |
208 |
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0x1.166a45471c3dfp+0, 0x1.3380p-47, |
209 |
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0x1.172b83c7d5211p+0, 0x1.8d40p-45, |
210 |
|
|
0x1.17ed48695bb9fp+0, -0x1.5d00p-47, |
211 |
|
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0x1.18af9388c8d93p+0, -0x1.c880p-46, |
212 |
|
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0x1.1972658375d66p+0, 0x1.1f00p-46, |
213 |
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0x1.1a35beb6fcba7p+0, 0x1.0480p-46, |
214 |
|
|
0x1.1af99f81387e3p+0, -0x1.7390p-43, |
215 |
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0x1.1bbe084045d54p+0, 0x1.4e40p-45, |
216 |
|
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0x1.1c82f95281c43p+0, -0x1.a200p-47, |
217 |
|
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0x1.1d4873168b9b2p+0, 0x1.3800p-49, |
218 |
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0x1.1e0e75eb44031p+0, 0x1.ac00p-49, |
219 |
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0x1.1ed5022fcd938p+0, 0x1.1900p-47, |
220 |
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0x1.1f9c18438cdf7p+0, -0x1.b780p-46, |
221 |
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0x1.2063b88628d8fp+0, 0x1.d940p-45, |
222 |
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0x1.212be3578a81ep+0, 0x1.8000p-50, |
223 |
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0x1.21f49917ddd41p+0, 0x1.b340p-45, |
224 |
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0x1.22bdda2791323p+0, 0x1.9f80p-46, |
225 |
|
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0x1.2387a6e7561e7p+0, -0x1.9c80p-46, |
226 |
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0x1.2451ffb821427p+0, 0x1.2300p-47, |
227 |
|
|
0x1.251ce4fb2a602p+0, -0x1.3480p-46, |
228 |
|
|
0x1.25e85711eceb0p+0, 0x1.2700p-46, |
229 |
|
|
0x1.26b4565e27d16p+0, 0x1.1d00p-46, |
230 |
|
|
0x1.2780e341de00fp+0, 0x1.1ee0p-44, |
231 |
|
|
0x1.284dfe1f5633ep+0, -0x1.4c00p-46, |
232 |
|
|
0x1.291ba7591bb30p+0, -0x1.3d80p-46, |
233 |
|
|
0x1.29e9df51fdf09p+0, 0x1.8b00p-47, |
234 |
|
|
0x1.2ab8a66d10e9bp+0, -0x1.27c0p-45, |
235 |
|
|
0x1.2b87fd0dada3ap+0, 0x1.a340p-45, |
236 |
|
|
0x1.2c57e39771af9p+0, -0x1.0800p-46, |
237 |
|
|
0x1.2d285a6e402d9p+0, -0x1.ed00p-47, |
238 |
|
|
0x1.2df961f641579p+0, -0x1.4200p-48, |
239 |
|
|
0x1.2ecafa93e2ecfp+0, -0x1.4980p-45, |
240 |
|
|
0x1.2f9d24abd8822p+0, -0x1.6300p-46, |
241 |
|
|
0x1.306fe0a31b625p+0, -0x1.2360p-44, |
242 |
|
|
0x1.31432edeea50bp+0, -0x1.0df8p-40, |
243 |
|
|
0x1.32170fc4cd7b8p+0, -0x1.2480p-45, |
244 |
|
|
0x1.32eb83ba8e9a2p+0, -0x1.5980p-45, |
245 |
|
|
0x1.33c08b2641766p+0, 0x1.ed00p-46, |
246 |
|
|
0x1.3496266e3fa27p+0, -0x1.c000p-50, |
247 |
|
|
0x1.356c55f929f0fp+0, -0x1.0d80p-44, |
248 |
|
|
0x1.36431a2de88b9p+0, 0x1.2c80p-45, |
249 |
|
|
0x1.371a7373aaa39p+0, 0x1.0600p-45, |
250 |
|
|
0x1.37f26231e74fep+0, -0x1.6600p-46, |
251 |
|
|
0x1.38cae6d05d838p+0, -0x1.ae00p-47, |
252 |
|
|
0x1.39a401b713ec3p+0, -0x1.4720p-43, |
253 |
|
|
0x1.3a7db34e5a020p+0, 0x1.8200p-47, |
254 |
|
|
0x1.3b57fbfec6e95p+0, 0x1.e800p-44, |
255 |
|
|
0x1.3c32dc313a8f2p+0, 0x1.f800p-49, |
256 |
|
|
0x1.3d0e544ede122p+0, -0x1.7a00p-46, |
257 |
|
|
0x1.3dea64c1234bbp+0, 0x1.6300p-45, |
258 |
|
|
0x1.3ec70df1c4eccp+0, -0x1.8a60p-43, |
259 |
|
|
0x1.3fa4504ac7e8cp+0, -0x1.cdc0p-44, |
260 |
|
|
0x1.40822c367a0bbp+0, 0x1.5b80p-45, |
261 |
|
|
0x1.4160a21f72e95p+0, 0x1.ec00p-46, |
262 |
|
|
0x1.423fb27094646p+0, -0x1.3600p-46, |
263 |
|
|
0x1.431f5d950a920p+0, 0x1.3980p-45, |
264 |
|
|
0x1.43ffa3f84b9ebp+0, 0x1.a000p-48, |
265 |
|
|
0x1.44e0860618919p+0, -0x1.6c00p-48, |
266 |
|
|
0x1.45c2042a7d201p+0, -0x1.bc00p-47, |
267 |
|
|
0x1.46a41ed1d0016p+0, -0x1.2800p-46, |
268 |
|
|
0x1.4786d668b3326p+0, 0x1.0e00p-44, |
269 |
|
|
0x1.486a2b5c13c00p+0, -0x1.d400p-45, |
270 |
|
|
0x1.494e1e192af04p+0, 0x1.c200p-47, |
271 |
|
|
0x1.4a32af0d7d372p+0, -0x1.e500p-46, |
272 |
|
|
0x1.4b17dea6db801p+0, 0x1.7800p-47, |
273 |
|
|
0x1.4bfdad53629e1p+0, -0x1.3800p-46, |
274 |
|
|
0x1.4ce41b817c132p+0, 0x1.0800p-47, |
275 |
|
|
0x1.4dcb299fddddbp+0, 0x1.c700p-45, |
276 |
|
|
0x1.4eb2d81d8ab96p+0, -0x1.ce00p-46, |
277 |
|
|
0x1.4f9b2769d2d02p+0, 0x1.9200p-46, |
278 |
|
|
0x1.508417f4531c1p+0, -0x1.8c00p-47, |
279 |
|
|
0x1.516daa2cf662ap+0, -0x1.a000p-48, |
280 |
|
|
0x1.5257de83f51eap+0, 0x1.a080p-43, |
281 |
|
|
0x1.5342b569d4edap+0, -0x1.6d80p-45, |
282 |
|
|
0x1.542e2f4f6ac1ap+0, -0x1.2440p-44, |
283 |
|
|
0x1.551a4ca5d94dbp+0, 0x1.83c0p-43, |
284 |
|
|
0x1.56070dde9116bp+0, 0x1.4b00p-45, |
285 |
|
|
0x1.56f4736b529dep+0, 0x1.15a0p-43, |
286 |
|
|
0x1.57e27dbe2c40ep+0, -0x1.9e00p-45, |
287 |
|
|
0x1.58d12d497c76fp+0, -0x1.3080p-45, |
288 |
|
|
0x1.59c0827ff0b4cp+0, 0x1.dec0p-43, |
289 |
|
|
0x1.5ab07dd485427p+0, -0x1.4000p-51, |
290 |
|
|
0x1.5ba11fba87af4p+0, 0x1.0080p-44, |
291 |
|
|
0x1.5c9268a59460bp+0, -0x1.6c80p-45, |
292 |
|
|
0x1.5d84590998e3fp+0, 0x1.69a0p-43, |
293 |
|
|
0x1.5e76f15ad20e1p+0, -0x1.b400p-46, |
294 |
|
|
0x1.5f6a320dcebcap+0, 0x1.7700p-46, |
295 |
|
|
0x1.605e1b976dcb8p+0, 0x1.6f80p-45, |
296 |
|
|
0x1.6152ae6cdf715p+0, 0x1.1000p-47, |
297 |
|
|
0x1.6247eb03a5531p+0, -0x1.5d00p-46, |
298 |
|
|
0x1.633dd1d1929b5p+0, -0x1.2d00p-46, |
299 |
|
|
0x1.6434634ccc313p+0, -0x1.a800p-49, |
300 |
|
|
0x1.652b9febc8efap+0, -0x1.8600p-45, |
301 |
|
|
0x1.6623882553397p+0, 0x1.1fe0p-40, |
302 |
|
|
0x1.671c1c708328ep+0, -0x1.7200p-44, |
303 |
|
|
0x1.68155d44ca97ep+0, 0x1.6800p-49, |
304 |
|
|
0x1.690f4b19e9471p+0, -0x1.9780p-45, |
305 |
|
|
}; |
306 |
|
|
|
307 |
|
|
/* |
308 |
|
|
* exp2(x): compute the base 2 exponential of x |
309 |
|
|
* |
310 |
|
|
* Accuracy: Peak error < 0.503 ulp for normalized results. |
311 |
|
|
* |
312 |
|
|
* Method: (accurate tables) |
313 |
|
|
* |
314 |
|
|
* Reduce x: |
315 |
|
|
* x = 2**k + y, for integer k and |y| <= 1/2. |
316 |
|
|
* Thus we have exp2(x) = 2**k * exp2(y). |
317 |
|
|
* |
318 |
|
|
* Reduce y: |
319 |
|
|
* y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE. |
320 |
|
|
* Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]), |
321 |
|
|
* with |z - eps[i]| <= 2**-9 + 2**-39 for the table used. |
322 |
|
|
* |
323 |
|
|
* We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via |
324 |
|
|
* a degree-5 minimax polynomial with maximum error under 1.3 * 2**-61. |
325 |
|
|
* The values in exp2t[] and eps[] are chosen such that |
326 |
|
|
* exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such |
327 |
|
|
* that exp2t[i] is accurate to 2**-64. |
328 |
|
|
* |
329 |
|
|
* Note that the range of i is +-TBLSIZE/2, so we actually index the tables |
330 |
|
|
* by i0 = i + TBLSIZE/2. For cache efficiency, exp2t[] and eps[] are |
331 |
|
|
* virtual tables, interleaved in the real table tbl[]. |
332 |
|
|
* |
333 |
|
|
* This method is due to Gal, with many details due to Gal and Bachelis: |
334 |
|
|
* |
335 |
|
|
* Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library |
336 |
|
|
* for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991). |
337 |
|
|
*/ |
338 |
|
|
double |
339 |
|
|
exp2(double x) |
340 |
|
|
{ |
341 |
|
|
double r, t, twopk, twopkp1000, z; |
342 |
|
|
uint32_t hx, ix, lx, i0; |
343 |
|
|
int k; |
344 |
|
|
|
345 |
|
|
/* Filter out exceptional cases. */ |
346 |
|
17604 |
GET_HIGH_WORD(hx,x); |
347 |
|
8802 |
ix = hx & 0x7fffffff; /* high word of |x| */ |
348 |
✗✓ |
8802 |
if(ix >= 0x40900000) { /* |x| >= 1024 */ |
349 |
|
|
if(ix >= 0x7ff00000) { |
350 |
|
|
GET_LOW_WORD(lx,x); |
351 |
|
|
if(((ix & 0xfffff) | lx) != 0 || (hx & 0x80000000) == 0) |
352 |
|
|
return (x + x); /* x is NaN or +Inf */ |
353 |
|
|
else |
354 |
|
|
return (0.0); /* x is -Inf */ |
355 |
|
|
} |
356 |
|
|
if(x >= 0x1.0p10) |
357 |
|
|
return (huge * huge); /* overflow */ |
358 |
|
|
if(x <= -0x1.0ccp10) |
359 |
|
|
return (twom1000 * twom1000); /* underflow */ |
360 |
✓✓ |
8802 |
} else if (ix < 0x3c900000) { /* |x| < 0x1p-54 */ |
361 |
|
18 |
return (1.0 + x); |
362 |
|
|
} |
363 |
|
|
|
364 |
|
|
/* Reduce x, computing z, i0, and k. */ |
365 |
|
8784 |
STRICT_ASSIGN(double, t, x + redux); |
366 |
|
8784 |
GET_LOW_WORD(i0, t); |
367 |
|
8784 |
i0 += TBLSIZE / 2; |
368 |
|
8784 |
k = (i0 >> TBLBITS) << 20; |
369 |
|
8784 |
i0 = (i0 & (TBLSIZE - 1)) << 1; |
370 |
|
8784 |
t -= redux; |
371 |
|
8784 |
z = x - t; |
372 |
|
|
|
373 |
|
|
/* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */ |
374 |
|
8784 |
t = tbl[i0]; /* exp2t[i0] */ |
375 |
|
8784 |
z -= tbl[i0 + 1]; /* eps[i0] */ |
376 |
✓✗ |
8784 |
if (k >= -(1021 << 20)) |
377 |
|
8784 |
INSERT_WORDS(twopk, 0x3ff00000 + k, 0); |
378 |
|
|
else |
379 |
|
|
INSERT_WORDS(twopkp1000, 0x3ff00000 + k + (1000 << 20), 0); |
380 |
|
8784 |
r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * P5)))); |
381 |
|
|
|
382 |
|
|
/* Scale by 2**(k>>20). */ |
383 |
✓✗ |
8784 |
if(k >= -(1021 << 20)) { |
384 |
✗✓ |
8784 |
if (k == 1024 << 20) |
385 |
|
|
return (r * 2.0 * 0x1p1023); |
386 |
|
8784 |
return (r * twopk); |
387 |
|
|
} else { |
388 |
|
|
return (r * twopkp1000 * twom1000); |
389 |
|
|
} |
390 |
|
8802 |
} |
391 |
|
|
DEF_STD(exp2); |
392 |
|
|
LDBL_MAYBE_UNUSED_CLONE(exp2); |