260ba0bff1
Mainly focus on files that use BSD 2-Clause license, however the tool I was using mis-identified many licenses so this was mostly a manual - error prone - task. The Software Package Data Exchange (SPDX) group provides a specification to make it easier for automated tools to detect and summarize well known opensource licenses. We are gradually adopting the specification, noting that the tags are considered only advisory and do not, in any way, superceed or replace the license texts.
291 lines
11 KiB
C
291 lines
11 KiB
C
/*-
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* SPDX-License-Identifier: BSD-2-Clause-FreeBSD
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*
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* Copyright (c) 2005-2008 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 <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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#include <float.h>
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#include <stdint.h>
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#ifdef __i386__
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#include <ieeefp.h>
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#endif
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#include "fpmath.h"
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#include "math.h"
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#include "math_private.h"
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#define TBLBITS 7
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#define TBLSIZE (1 << TBLBITS)
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#define BIAS (LDBL_MAX_EXP - 1)
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static volatile long double
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huge = 0x1p10000L,
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twom10000 = 0x1p-10000L;
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static const union IEEEl2bits
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P1 = LD80C(0xb17217f7d1cf79ac, -1, 6.93147180559945309429e-1L);
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static const double
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redux = 0x1.8p63 / TBLSIZE,
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/*
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* Domain [-0.00390625, 0.00390625], range ~[-1.7079e-23, 1.7079e-23]
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* |exp(x) - p(x)| < 2**-75.6
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*/
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P2 = 2.4022650695910072e-1, /* 0x1ebfbdff82c58f.0p-55 */
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P3 = 5.5504108664816879e-2, /* 0x1c6b08d7049e1a.0p-57 */
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P4 = 9.6181291055695180e-3, /* 0x13b2ab6fa8321a.0p-59 */
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P5 = 1.3333563089183052e-3, /* 0x15d8806f67f251.0p-62 */
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P6 = 1.5413361552277414e-4; /* 0x1433ddacff3441.0p-65 */
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static const double tbl[TBLSIZE * 2] = {
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0x1.6a09e667f3bcdp-1, -0x1.bdd3413b2648p-55,
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0x1.6c012750bdabfp-1, -0x1.2895667ff0cp-57,
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0x1.6dfb23c651a2fp-1, -0x1.bbe3a683c88p-58,
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0x1.6ff7df9519484p-1, -0x1.83c0f25860fp-56,
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0x1.71f75e8ec5f74p-1, -0x1.16e4786887bp-56,
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0x1.73f9a48a58174p-1, -0x1.0a8d96c65d5p-55,
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0x1.75feb564267c9p-1, -0x1.0245957316ep-55,
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0x1.780694fde5d3fp-1, 0x1.866b80a0216p-55,
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0x1.7a11473eb0187p-1, -0x1.41577ee0499p-56,
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0x1.7c1ed0130c132p-1, 0x1.f124cd1164ep-55,
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0x1.7e2f336cf4e62p-1, 0x1.05d02ba157ap-57,
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0x1.80427543e1a12p-1, -0x1.27c86626d97p-55,
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0x1.82589994cce13p-1, -0x1.d4c1dd41533p-55,
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0x1.8471a4623c7adp-1, -0x1.8d684a341cep-56,
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0x1.868d99b4492edp-1, -0x1.fc6f89bd4f68p-55,
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0x1.88ac7d98a6699p-1, 0x1.994c2f37cb5p-55,
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0x1.8ace5422aa0dbp-1, 0x1.6e9f156864bp-55,
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0x1.8cf3216b5448cp-1, -0x1.0d55e32e9e4p-57,
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0x1.8f1ae99157736p-1, 0x1.5cc13a2e397p-56,
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0x1.9145b0b91ffc6p-1, -0x1.dd6792e5825p-55,
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0x1.93737b0cdc5e5p-1, -0x1.75fc781b58p-58,
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0x1.95a44cbc8520fp-1, -0x1.64b7c96a5fp-57,
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0x1.97d829fde4e5p-1, -0x1.d185b7c1b86p-55,
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0x1.9a0f170ca07bap-1, -0x1.173bd91cee6p-55,
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0x1.9c49182a3f09p-1, 0x1.c7c46b071f2p-57,
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0x1.9e86319e32323p-1, 0x1.824ca78e64cp-57,
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0x1.a0c667b5de565p-1, -0x1.359495d1cd5p-55,
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0x1.a309bec4a2d33p-1, 0x1.6305c7ddc368p-55,
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0x1.a5503b23e255dp-1, -0x1.d2f6edb8d42p-55,
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0x1.a799e1330b358p-1, 0x1.bcb7ecac564p-55,
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0x1.a9e6b5579fdbfp-1, 0x1.0fac90ef7fdp-55,
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0x1.ac36bbfd3f37ap-1, -0x1.f9234cae76dp-56,
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0x1.ae89f995ad3adp-1, 0x1.7a1cd345dcc8p-55,
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0x1.b0e07298db666p-1, -0x1.bdef54c80e4p-55,
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0x1.b33a2b84f15fbp-1, -0x1.2805e3084d8p-58,
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0x1.b59728de5593ap-1, -0x1.c71dfbbba6ep-55,
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0x1.b7f76f2fb5e47p-1, -0x1.5584f7e54acp-57,
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0x1.ba5b030a1064ap-1, -0x1.efcd30e5429p-55,
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0x1.bcc1e904bc1d2p-1, 0x1.23dd07a2d9fp-56,
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0x1.bf2c25bd71e09p-1, -0x1.efdca3f6b9c8p-55,
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0x1.c199bdd85529cp-1, 0x1.11065895049p-56,
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0x1.c40ab5fffd07ap-1, 0x1.b4537e083c6p-55,
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0x1.c67f12e57d14bp-1, 0x1.2884dff483c8p-55,
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0x1.c8f6d9406e7b5p-1, 0x1.1acbc48805cp-57,
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0x1.cb720dcef9069p-1, 0x1.503cbd1e94ap-57,
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0x1.cdf0b555dc3fap-1, -0x1.dd83b53829dp-56,
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0x1.d072d4a07897cp-1, -0x1.cbc3743797a8p-55,
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0x1.d2f87080d89f2p-1, -0x1.d487b719d858p-55,
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0x1.d5818dcfba487p-1, 0x1.2ed02d75b37p-56,
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0x1.d80e316c98398p-1, -0x1.11ec18bedep-55,
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0x1.da9e603db3285p-1, 0x1.c2300696db5p-55,
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0x1.dd321f301b46p-1, 0x1.2da5778f019p-55,
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0x1.dfc97337b9b5fp-1, -0x1.1a5cd4f184b8p-55,
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0x1.e264614f5a129p-1, -0x1.7b627817a148p-55,
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0x1.e502ee78b3ff6p-1, 0x1.39e8980a9cdp-56,
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0x1.e7a51fbc74c83p-1, 0x1.2d522ca0c8ep-55,
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0x1.ea4afa2a490dap-1, -0x1.e9c23179c288p-55,
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0x1.ecf482d8e67f1p-1, -0x1.c93f3b411ad8p-55,
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0x1.efa1bee615a27p-1, 0x1.dc7f486a4b68p-55,
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0x1.f252b376bba97p-1, 0x1.3a1a5bf0d8e8p-55,
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0x1.f50765b6e454p-1, 0x1.9d3e12dd8a18p-55,
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0x1.f7bfdad9cbe14p-1, -0x1.dbb12d00635p-55,
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0x1.fa7c1819e90d8p-1, 0x1.74853f3a593p-56,
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0x1.fd3c22b8f71f1p-1, 0x1.2eb74966578p-58,
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0x1p+0, 0x0p+0,
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0x1.0163da9fb3335p+0, 0x1.b61299ab8cd8p-54,
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0x1.02c9a3e778061p+0, -0x1.19083535b08p-56,
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0x1.04315e86e7f85p+0, -0x1.0a31c1977c98p-54,
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0x1.059b0d3158574p+0, 0x1.d73e2a475b4p-55,
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0x1.0706b29ddf6dep+0, -0x1.c91dfe2b13cp-55,
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0x1.0874518759bc8p+0, 0x1.186be4bb284p-57,
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0x1.09e3ecac6f383p+0, 0x1.14878183161p-54,
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0x1.0b5586cf9890fp+0, 0x1.8a62e4adc61p-54,
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0x1.0cc922b7247f7p+0, 0x1.01edc16e24f8p-54,
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0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c58p-59,
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0x1.0fb66affed31bp+0, -0x1.b9bedc44ebcp-57,
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0x1.11301d0125b51p+0, -0x1.6c51039449bp-54,
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0x1.12abdc06c31ccp+0, -0x1.1b514b36ca8p-58,
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0x1.1429aaea92dep+0, -0x1.32fbf9af1368p-54,
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0x1.15a98c8a58e51p+0, 0x1.2406ab9eeabp-55,
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0x1.172b83c7d517bp+0, -0x1.19041b9d78ap-55,
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0x1.18af9388c8deap+0, -0x1.11023d1970f8p-54,
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0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4969p-55,
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0x1.1bbe084045cd4p+0, -0x1.95386352ef6p-54,
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0x1.1d4873168b9aap+0, 0x1.e016e00a264p-54,
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0x1.1ed5022fcd91dp+0, -0x1.1df98027bb78p-54,
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0x1.2063b88628cd6p+0, 0x1.dc775814a85p-55,
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0x1.21f49917ddc96p+0, 0x1.2a97e9494a6p-55,
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0x1.2387a6e756238p+0, 0x1.9b07eb6c7058p-54,
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0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f5p-55,
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0x1.26b4565e27cddp+0, 0x1.2bd339940eap-55,
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0x1.284dfe1f56381p+0, -0x1.a4c3a8c3f0d8p-54,
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0x1.29e9df51fdee1p+0, 0x1.612e8afad12p-55,
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0x1.2b87fd0dad99p+0, -0x1.10adcd6382p-59,
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0x1.2d285a6e4030bp+0, 0x1.0024754db42p-54,
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0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d524p-56,
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0x1.306fe0a31b715p+0, 0x1.6f46ad23183p-55,
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0x1.32170fc4cd831p+0, 0x1.a9ce78e1804p-55,
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0x1.33c08b26416ffp+0, 0x1.327218436598p-54,
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0x1.356c55f929ff1p+0, -0x1.b5cee5c4e46p-55,
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0x1.371a7373aa9cbp+0, -0x1.63aeabf42ebp-54,
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0x1.38cae6d05d866p+0, -0x1.e958d3c99048p-54,
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0x1.3a7db34e59ff7p+0, -0x1.5e436d661f6p-56,
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0x1.3c32dc313a8e5p+0, -0x1.efff8375d2ap-54,
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0x1.3dea64c123422p+0, 0x1.ada0911f09fp-55,
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0x1.3fa4504ac801cp+0, -0x1.7d023f956fap-54,
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0x1.4160a21f72e2ap+0, -0x1.ef3691c309p-58,
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0x1.431f5d950a897p+0, -0x1.1c7dde35f7ap-55,
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0x1.44e086061892dp+0, 0x1.89b7a04ef8p-59,
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0x1.46a41ed1d0057p+0, 0x1.c944bd1648a8p-54,
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0x1.486a2b5c13cdp+0, 0x1.3c1a3b69062p-56,
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0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be8p-54,
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0x1.4bfdad5362a27p+0, 0x1.d4397afec42p-56,
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0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a78p-54,
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0x1.4f9b2769d2ca7p+0, -0x1.4b309d25958p-54,
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0x1.516daa2cf6642p+0, -0x1.f768569bd94p-55,
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0x1.5342b569d4f82p+0, -0x1.07abe1db13dp-55,
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0x1.551a4ca5d920fp+0, -0x1.d689cefede6p-55,
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0x1.56f4736b527dap+0, 0x1.9bb2c011d938p-54,
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0x1.58d12d497c7fdp+0, 0x1.295e15b9a1ep-55,
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0x1.5ab07dd485429p+0, 0x1.6324c0546478p-54,
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0x1.5c9268a5946b7p+0, 0x1.c4b1b81698p-60,
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0x1.5e76f15ad2148p+0, 0x1.ba6f93080e68p-54,
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0x1.605e1b976dc09p+0, -0x1.3e2429b56de8p-54,
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0x1.6247eb03a5585p+0, -0x1.383c17e40b48p-54,
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0x1.6434634ccc32p+0, -0x1.c483c759d89p-55,
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0x1.6623882552225p+0, -0x1.bb60987591cp-54,
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0x1.68155d44ca973p+0, 0x1.038ae44f74p-57,
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};
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/**
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* Compute the base 2 exponential of x for Intel 80-bit format.
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*
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* Accuracy: Peak error < 0.511 ulp.
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*
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* Method: (equally-spaced tables)
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*
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* Reduce x:
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* x = 2**k + y, for integer k and |y| <= 1/2.
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* Thus we have exp2l(x) = 2**k * exp2(y).
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*
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* Reduce y:
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* y = i/TBLSIZE + z for integer i near y * TBLSIZE.
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* Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
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* with |z| <= 2**-(TBLBITS+1).
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*
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* We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
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* degree-6 minimax polynomial with maximum error under 2**-75.6.
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* The table entries each have 104 bits of accuracy, encoded as
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* a pair of double precision values.
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*/
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long double
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exp2l(long double x)
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{
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union IEEEl2bits u, v;
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long double r, twopk, twopkp10000, z;
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uint32_t hx, ix, i0;
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int k;
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/* Filter out exceptional cases. */
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u.e = x;
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hx = u.xbits.expsign;
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ix = hx & 0x7fff;
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if (ix >= BIAS + 14) { /* |x| >= 16384 or x is NaN */
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if (ix == BIAS + LDBL_MAX_EXP) {
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if (hx & 0x8000 && u.xbits.man == 1ULL << 63)
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return (0.0L); /* x is -Inf */
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return (x + x); /* x is +Inf, NaN or unsupported */
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}
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if (x >= 16384)
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return (huge * huge); /* overflow */
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if (x <= -16446)
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return (twom10000 * twom10000); /* underflow */
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} else if (ix <= BIAS - 66) { /* |x| < 0x1p-65 (includes pseudos) */
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return (1.0L + x); /* 1 with inexact */
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}
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ENTERI();
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/*
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* Reduce x, computing z, i0, and k. The low bits of x + redux
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* contain the 16-bit integer part of the exponent (k) followed by
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* TBLBITS fractional bits (i0). We use bit tricks to extract these
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* as integers, then set z to the remainder.
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*
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* Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8.
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* Then the low-order word of x + redux is 0x000abc12,
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* We split this into k = 0xabc and i0 = 0x12 (adjusted to
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* index into the table), then we compute z = 0x0.003456p0.
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*
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* XXX If the exponent is negative, the computation of k depends on
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* '>>' doing sign extension.
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*/
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u.e = x + redux;
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i0 = u.bits.manl + TBLSIZE / 2;
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k = (int)i0 >> TBLBITS;
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i0 = (i0 & (TBLSIZE - 1)) << 1;
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u.e -= redux;
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z = x - u.e;
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v.xbits.man = 1ULL << 63;
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if (k >= LDBL_MIN_EXP) {
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v.xbits.expsign = BIAS + k;
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twopk = v.e;
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} else {
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v.xbits.expsign = BIAS + k + 10000;
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twopkp10000 = v.e;
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}
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/* Compute r = exp2l(y) = exp2lt[i0] * p(z). */
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long double t_hi = tbl[i0];
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long double t_lo = tbl[i0 + 1];
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r = t_lo + (t_hi + t_lo) * z * (P1.e + z * (P2 + z * (P3 + z * (P4
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+ z * (P5 + z * P6))))) + t_hi;
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/* Scale by 2**k. */
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if (k >= LDBL_MIN_EXP) {
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if (k == LDBL_MAX_EXP)
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RETURNI(r * 2.0 * 0x1p16383L);
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RETURNI(r * twopk);
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} else {
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RETURNI(r * twopkp10000 * twom10000);
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}
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}
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