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957c61cbbf
This change upgrades to GCC 12.3 and GNU binutils 2.42. The GNU linker appears to have changed things so that only a single de-duplicated str table is present in the binary, and it gets placed wherever the linker wants, regardless of what the linker script says. To cope with that we need to stop using .ident to embed licenses. As such, this change does significant work to revamp how third party licenses are defined in the codebase, using `.section .notice,"aR",@progbits`. This new GCC 12.3 toolchain has support for GNU indirect functions. It lets us support __target_clones__ for the first time. This is used for optimizing the performance of libc string functions such as strlen and friends so far on x86, by ensuring AVX systems favor a second codepath that uses VEX encoding. It shaves some latency off certain operations. It's a useful feature to have for scientific computing for the reasons explained by the test/libcxx/openmp_test.cc example which compiles for fifteen different microarchitectures. Thanks to the upgrades, it's now also possible to use newer instruction sets, such as AVX512FP16, VNNI. Cosmo now uses the %gs register on x86 by default for TLS. Doing it is helpful for any program that links `cosmo_dlopen()`. Such programs had to recompile their binaries at startup to change the TLS instructions. That's not great, since it means every page in the executable needs to be faulted. The work of rewriting TLS-related x86 opcodes, is moved to fixupobj.com instead. This is great news for MacOS x86 users, since we previously needed to morph the binary every time for that platform but now that's no longer necessary. The only platforms where we need fixup of TLS x86 opcodes at runtime are now Windows, OpenBSD, and NetBSD. On Windows we morph TLS to point deeper into the TIB, based on a TlsAlloc assignment, and on OpenBSD/NetBSD we morph %gs back into %fs since the kernels do not allow us to specify a value for the %gs register. OpenBSD users are now required to use APE Loader to run Cosmo binaries and assimilation is no longer possible. OpenBSD kernel needs to change to allow programs to specify a value for the %gs register, or it needs to stop marking executable pages loaded by the kernel as mimmutable(). This release fixes __constructor__, .ctor, .init_array, and lastly the .preinit_array so they behave the exact same way as glibc. We no longer use hex constants to define math.h symbols like M_PI.
657 lines
23 KiB
C
657 lines
23 KiB
C
/*-*- mode:c;indent-tabs-mode:t;c-basic-offset:8;tab-width:8;coding:utf-8 -*-│
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│ vi: set noet ft=c ts=8 sw=8 fenc=utf-8 :vi │
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╚──────────────────────────────────────────────────────────────────────────────╝
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│ │
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│ Musl Libc │
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│ Copyright © 2005-2014 Rich Felker, et al. │
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│ │
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│ Permission is hereby granted, free of charge, to any person obtaining │
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│ a copy of this software and associated documentation files (the │
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│ "Software"), to deal in the Software without restriction, including │
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│ without limitation the rights to use, copy, modify, merge, publish, │
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│ distribute, sublicense, and/or sell copies of the Software, and to │
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│ permit persons to whom the Software is furnished to do so, subject to │
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│ the following conditions: │
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│ │
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│ The above copyright notice and this permission notice shall be │
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│ included in all copies or substantial portions of the Software. │
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│ │
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│ THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, │
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│ EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF │
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│ MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. │
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│ IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY │
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│ CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, │
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│ TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE │
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│ SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. │
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│ │
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╚─────────────────────────────────────────────────────────────────────────────*/
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#include "libc/math.h"
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#include "libc/tinymath/internal.h"
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#include "libc/tinymath/ldshape.internal.h"
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__static_yoink("musl_libc_notice");
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__static_yoink("freebsd_libm_notice");
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#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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/* origin: FreeBSD /usr/src/lib/msun/ld80/s_exp2l.c and /usr/src/lib/msun/ld128/s_exp2l.c */
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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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#define TBLBITS 7
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#define TBLSIZE (1 << TBLBITS)
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static const double
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redux = 0x1.8p63 / TBLSIZE,
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P1 = 0x1.62e42fefa39efp-1,
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P2 = 0x1.ebfbdff82c58fp-3,
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P3 = 0x1.c6b08d7049fap-5,
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P4 = 0x1.3b2ab6fba4da5p-7,
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P5 = 0x1.5d8804780a736p-10,
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P6 = 0x1.430918835e33dp-13;
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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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* exp2l(x): compute the base 2 exponential of x
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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**-69.
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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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/**
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* Returns 2^𝑥.
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*/
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long double exp2l(long double x)
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{
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union ldshape u = {x};
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int e = u.i.se & 0x7fff;
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long double r, z;
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uint32_t i0;
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union {uint32_t u; int32_t i;} k;
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/* Filter out exceptional cases. */
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if (e >= 0x3fff + 13) { /* |x| >= 8192 or x is NaN */
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if (u.i.se >= 0x3fff + 14 && u.i.se >> 15 == 0)
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/* overflow */
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return x * 0x1p16383L;
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if (e == 0x7fff) /* -inf or -nan */
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return -1/x;
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if (x < -16382) {
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if (x <= -16446 || x - 0x1p63 + 0x1p63 != x)
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/* underflow */
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FORCE_EVAL((float)(-0x1p-149/x));
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if (x <= -16446)
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return 0;
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}
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} else if (e < 0x3fff - 64) {
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return 1 + x;
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}
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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.
|
||
* Then the low-order word of x + redux is 0x000abc12,
|
||
* We split this into k = 0xabc and i0 = 0x12 (adjusted to
|
||
* index into the table), then we compute z = 0x0.003456p0.
|
||
*/
|
||
u.f = x + redux;
|
||
i0 = u.i.m + TBLSIZE / 2;
|
||
k.u = i0 / TBLSIZE * TBLSIZE;
|
||
k.i /= TBLSIZE;
|
||
i0 %= TBLSIZE;
|
||
u.f -= redux;
|
||
z = x - u.f;
|
||
|
||
/* Compute r = exp2l(y) = exp2lt[i0] * p(z). */
|
||
long double t_hi = tbl[2*i0];
|
||
long double t_lo = tbl[2*i0 + 1];
|
||
/* XXX This gives > 1 ulp errors outside of FE_TONEAREST mode */
|
||
r = t_lo + (t_hi + t_lo) * z * (P1 + z * (P2 + z * (P3 + z * (P4
|
||
+ z * (P5 + z * P6))))) + t_hi;
|
||
|
||
return scalbnl(r, k.i);
|
||
}
|
||
|
||
#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
|
||
#define TBLBITS 7
|
||
#define TBLSIZE (1 << TBLBITS)
|
||
|
||
static const long double
|
||
P1 = 0x1.62e42fefa39ef35793c7673007e6p-1L,
|
||
P2 = 0x1.ebfbdff82c58ea86f16b06ec9736p-3L,
|
||
P3 = 0x1.c6b08d704a0bf8b33a762bad3459p-5L,
|
||
P4 = 0x1.3b2ab6fba4e7729ccbbe0b4f3fc2p-7L,
|
||
P5 = 0x1.5d87fe78a67311071dee13fd11d9p-10L,
|
||
P6 = 0x1.430912f86c7876f4b663b23c5fe5p-13L;
|
||
|
||
static const double
|
||
P7 = 0x1.ffcbfc588b041p-17,
|
||
P8 = 0x1.62c0223a5c7c7p-20,
|
||
P9 = 0x1.b52541ff59713p-24,
|
||
P10 = 0x1.e4cf56a391e22p-28,
|
||
redux = 0x1.8p112 / TBLSIZE;
|
||
|
||
static const long double tbl[TBLSIZE] = {
|
||
0x1.6a09e667f3bcc908b2fb1366dfeap-1L,
|
||
0x1.6c012750bdabeed76a99800f4edep-1L,
|
||
0x1.6dfb23c651a2ef220e2cbe1bc0d4p-1L,
|
||
0x1.6ff7df9519483cf87e1b4f3e1e98p-1L,
|
||
0x1.71f75e8ec5f73dd2370f2ef0b148p-1L,
|
||
0x1.73f9a48a58173bd5c9a4e68ab074p-1L,
|
||
0x1.75feb564267c8bf6e9aa33a489a8p-1L,
|
||
0x1.780694fde5d3f619ae02808592a4p-1L,
|
||
0x1.7a11473eb0186d7d51023f6ccb1ap-1L,
|
||
0x1.7c1ed0130c1327c49334459378dep-1L,
|
||
0x1.7e2f336cf4e62105d02ba1579756p-1L,
|
||
0x1.80427543e1a11b60de67649a3842p-1L,
|
||
0x1.82589994cce128acf88afab34928p-1L,
|
||
0x1.8471a4623c7acce52f6b97c6444cp-1L,
|
||
0x1.868d99b4492ec80e41d90ac2556ap-1L,
|
||
0x1.88ac7d98a669966530bcdf2d4cc0p-1L,
|
||
0x1.8ace5422aa0db5ba7c55a192c648p-1L,
|
||
0x1.8cf3216b5448bef2aa1cd161c57ap-1L,
|
||
0x1.8f1ae991577362b982745c72eddap-1L,
|
||
0x1.9145b0b91ffc588a61b469f6b6a0p-1L,
|
||
0x1.93737b0cdc5e4f4501c3f2540ae8p-1L,
|
||
0x1.95a44cbc8520ee9b483695a0e7fep-1L,
|
||
0x1.97d829fde4e4f8b9e920f91e8eb6p-1L,
|
||
0x1.9a0f170ca07b9ba3109b8c467844p-1L,
|
||
0x1.9c49182a3f0901c7c46b071f28dep-1L,
|
||
0x1.9e86319e323231824ca78e64c462p-1L,
|
||
0x1.a0c667b5de564b29ada8b8cabbacp-1L,
|
||
0x1.a309bec4a2d3358c171f770db1f4p-1L,
|
||
0x1.a5503b23e255c8b424491caf88ccp-1L,
|
||
0x1.a799e1330b3586f2dfb2b158f31ep-1L,
|
||
0x1.a9e6b5579fdbf43eb243bdff53a2p-1L,
|
||
0x1.ac36bbfd3f379c0db966a3126988p-1L,
|
||
0x1.ae89f995ad3ad5e8734d17731c80p-1L,
|
||
0x1.b0e07298db66590842acdfc6fb4ep-1L,
|
||
0x1.b33a2b84f15faf6bfd0e7bd941b0p-1L,
|
||
0x1.b59728de559398e3881111648738p-1L,
|
||
0x1.b7f76f2fb5e46eaa7b081ab53ff6p-1L,
|
||
0x1.ba5b030a10649840cb3c6af5b74cp-1L,
|
||
0x1.bcc1e904bc1d2247ba0f45b3d06cp-1L,
|
||
0x1.bf2c25bd71e088408d7025190cd0p-1L,
|
||
0x1.c199bdd85529c2220cb12a0916bap-1L,
|
||
0x1.c40ab5fffd07a6d14df820f17deap-1L,
|
||
0x1.c67f12e57d14b4a2137fd20f2a26p-1L,
|
||
0x1.c8f6d9406e7b511acbc48805c3f6p-1L,
|
||
0x1.cb720dcef90691503cbd1e949d0ap-1L,
|
||
0x1.cdf0b555dc3f9c44f8958fac4f12p-1L,
|
||
0x1.d072d4a07897b8d0f22f21a13792p-1L,
|
||
0x1.d2f87080d89f18ade123989ea50ep-1L,
|
||
0x1.d5818dcfba48725da05aeb66dff8p-1L,
|
||
0x1.d80e316c98397bb84f9d048807a0p-1L,
|
||
0x1.da9e603db3285708c01a5b6d480cp-1L,
|
||
0x1.dd321f301b4604b695de3c0630c0p-1L,
|
||
0x1.dfc97337b9b5eb968cac39ed284cp-1L,
|
||
0x1.e264614f5a128a12761fa17adc74p-1L,
|
||
0x1.e502ee78b3ff6273d130153992d0p-1L,
|
||
0x1.e7a51fbc74c834b548b2832378a4p-1L,
|
||
0x1.ea4afa2a490d9858f73a18f5dab4p-1L,
|
||
0x1.ecf482d8e67f08db0312fb949d50p-1L,
|
||
0x1.efa1bee615a27771fd21a92dabb6p-1L,
|
||
0x1.f252b376bba974e8696fc3638f24p-1L,
|
||
0x1.f50765b6e4540674f84b762861a6p-1L,
|
||
0x1.f7bfdad9cbe138913b4bfe72bd78p-1L,
|
||
0x1.fa7c1819e90d82e90a7e74b26360p-1L,
|
||
0x1.fd3c22b8f71f10975ba4b32bd006p-1L,
|
||
0x1.0000000000000000000000000000p+0L,
|
||
0x1.0163da9fb33356d84a66ae336e98p+0L,
|
||
0x1.02c9a3e778060ee6f7caca4f7a18p+0L,
|
||
0x1.04315e86e7f84bd738f9a20da442p+0L,
|
||
0x1.059b0d31585743ae7c548eb68c6ap+0L,
|
||
0x1.0706b29ddf6ddc6dc403a9d87b1ep+0L,
|
||
0x1.0874518759bc808c35f25d942856p+0L,
|
||
0x1.09e3ecac6f3834521e060c584d5cp+0L,
|
||
0x1.0b5586cf9890f6298b92b7184200p+0L,
|
||
0x1.0cc922b7247f7407b705b893dbdep+0L,
|
||
0x1.0e3ec32d3d1a2020742e4f8af794p+0L,
|
||
0x1.0fb66affed31af232091dd8a169ep+0L,
|
||
0x1.11301d0125b50a4ebbf1aed9321cp+0L,
|
||
0x1.12abdc06c31cbfb92bad324d6f84p+0L,
|
||
0x1.1429aaea92ddfb34101943b2588ep+0L,
|
||
0x1.15a98c8a58e512480d573dd562aep+0L,
|
||
0x1.172b83c7d517adcdf7c8c50eb162p+0L,
|
||
0x1.18af9388c8de9bbbf70b9a3c269cp+0L,
|
||
0x1.1a35beb6fcb753cb698f692d2038p+0L,
|
||
0x1.1bbe084045cd39ab1e72b442810ep+0L,
|
||
0x1.1d4873168b9aa7805b8028990be8p+0L,
|
||
0x1.1ed5022fcd91cb8819ff61121fbep+0L,
|
||
0x1.2063b88628cd63b8eeb0295093f6p+0L,
|
||
0x1.21f49917ddc962552fd29294bc20p+0L,
|
||
0x1.2387a6e75623866c1fadb1c159c0p+0L,
|
||
0x1.251ce4fb2a63f3582ab7de9e9562p+0L,
|
||
0x1.26b4565e27cdd257a673281d3068p+0L,
|
||
0x1.284dfe1f5638096cf15cf03c9fa0p+0L,
|
||
0x1.29e9df51fdee12c25d15f5a25022p+0L,
|
||
0x1.2b87fd0dad98ffddea46538fca24p+0L,
|
||
0x1.2d285a6e4030b40091d536d0733ep+0L,
|
||
0x1.2ecafa93e2f5611ca0f45d5239a4p+0L,
|
||
0x1.306fe0a31b7152de8d5a463063bep+0L,
|
||
0x1.32170fc4cd8313539cf1c3009330p+0L,
|
||
0x1.33c08b26416ff4c9c8610d96680ep+0L,
|
||
0x1.356c55f929ff0c94623476373be4p+0L,
|
||
0x1.371a7373aa9caa7145502f45452ap+0L,
|
||
0x1.38cae6d05d86585a9cb0d9bed530p+0L,
|
||
0x1.3a7db34e59ff6ea1bc9299e0a1fep+0L,
|
||
0x1.3c32dc313a8e484001f228b58cf0p+0L,
|
||
0x1.3dea64c12342235b41223e13d7eep+0L,
|
||
0x1.3fa4504ac801ba0bf701aa417b9cp+0L,
|
||
0x1.4160a21f72e29f84325b8f3dbacap+0L,
|
||
0x1.431f5d950a896dc704439410b628p+0L,
|
||
0x1.44e086061892d03136f409df0724p+0L,
|
||
0x1.46a41ed1d005772512f459229f0ap+0L,
|
||
0x1.486a2b5c13cd013c1a3b69062f26p+0L,
|
||
0x1.4a32af0d7d3de672d8bcf46f99b4p+0L,
|
||
0x1.4bfdad5362a271d4397afec42e36p+0L,
|
||
0x1.4dcb299fddd0d63b36ef1a9e19dep+0L,
|
||
0x1.4f9b2769d2ca6ad33d8b69aa0b8cp+0L,
|
||
0x1.516daa2cf6641c112f52c84d6066p+0L,
|
||
0x1.5342b569d4f81df0a83c49d86bf4p+0L,
|
||
0x1.551a4ca5d920ec52ec620243540cp+0L,
|
||
0x1.56f4736b527da66ecb004764e61ep+0L,
|
||
0x1.58d12d497c7fd252bc2b7343d554p+0L,
|
||
0x1.5ab07dd48542958c93015191e9a8p+0L,
|
||
0x1.5c9268a5946b701c4b1b81697ed4p+0L,
|
||
0x1.5e76f15ad21486e9be4c20399d12p+0L,
|
||
0x1.605e1b976dc08b076f592a487066p+0L,
|
||
0x1.6247eb03a5584b1f0fa06fd2d9eap+0L,
|
||
0x1.6434634ccc31fc76f8714c4ee122p+0L,
|
||
0x1.66238825522249127d9e29b92ea2p+0L,
|
||
0x1.68155d44ca973081c57227b9f69ep+0L,
|
||
};
|
||
|
||
static const float eps[TBLSIZE] = {
|
||
-0x1.5c50p-101,
|
||
-0x1.5d00p-106,
|
||
0x1.8e90p-102,
|
||
-0x1.5340p-103,
|
||
0x1.1bd0p-102,
|
||
-0x1.4600p-105,
|
||
-0x1.7a40p-104,
|
||
0x1.d590p-102,
|
||
-0x1.d590p-101,
|
||
0x1.b100p-103,
|
||
-0x1.0d80p-105,
|
||
0x1.6b00p-103,
|
||
-0x1.9f00p-105,
|
||
0x1.c400p-103,
|
||
0x1.e120p-103,
|
||
-0x1.c100p-104,
|
||
-0x1.9d20p-103,
|
||
0x1.a800p-108,
|
||
0x1.4c00p-106,
|
||
-0x1.9500p-106,
|
||
0x1.6900p-105,
|
||
-0x1.29d0p-100,
|
||
0x1.4c60p-103,
|
||
0x1.13a0p-102,
|
||
-0x1.5b60p-103,
|
||
-0x1.1c40p-103,
|
||
0x1.db80p-102,
|
||
0x1.91a0p-102,
|
||
0x1.dc00p-105,
|
||
0x1.44c0p-104,
|
||
0x1.9710p-102,
|
||
0x1.8760p-103,
|
||
-0x1.a720p-103,
|
||
0x1.ed20p-103,
|
||
-0x1.49c0p-102,
|
||
-0x1.e000p-111,
|
||
0x1.86a0p-103,
|
||
0x1.2b40p-103,
|
||
-0x1.b400p-108,
|
||
0x1.1280p-99,
|
||
-0x1.02d8p-102,
|
||
-0x1.e3d0p-103,
|
||
-0x1.b080p-105,
|
||
-0x1.f100p-107,
|
||
-0x1.16c0p-105,
|
||
-0x1.1190p-103,
|
||
-0x1.a7d2p-100,
|
||
0x1.3450p-103,
|
||
-0x1.67c0p-105,
|
||
0x1.4b80p-104,
|
||
-0x1.c4e0p-103,
|
||
0x1.6000p-108,
|
||
-0x1.3f60p-105,
|
||
0x1.93f0p-104,
|
||
0x1.5fe0p-105,
|
||
0x1.6f80p-107,
|
||
-0x1.7600p-106,
|
||
0x1.21e0p-106,
|
||
-0x1.3a40p-106,
|
||
-0x1.40c0p-104,
|
||
-0x1.9860p-105,
|
||
-0x1.5d40p-108,
|
||
-0x1.1d70p-106,
|
||
0x1.2760p-105,
|
||
0x0.0000p+0,
|
||
0x1.21e2p-104,
|
||
-0x1.9520p-108,
|
||
-0x1.5720p-106,
|
||
-0x1.4810p-106,
|
||
-0x1.be00p-109,
|
||
0x1.0080p-105,
|
||
-0x1.5780p-108,
|
||
-0x1.d460p-105,
|
||
-0x1.6140p-105,
|
||
0x1.4630p-104,
|
||
0x1.ad50p-103,
|
||
0x1.82e0p-105,
|
||
0x1.1d3cp-101,
|
||
0x1.6100p-107,
|
||
0x1.ec30p-104,
|
||
0x1.f200p-108,
|
||
0x1.0b40p-103,
|
||
0x1.3660p-102,
|
||
0x1.d9d0p-103,
|
||
-0x1.02d0p-102,
|
||
0x1.b070p-103,
|
||
0x1.b9c0p-104,
|
||
-0x1.01c0p-103,
|
||
-0x1.dfe0p-103,
|
||
0x1.1b60p-104,
|
||
-0x1.ae94p-101,
|
||
-0x1.3340p-104,
|
||
0x1.b3d8p-102,
|
||
-0x1.6e40p-105,
|
||
-0x1.3670p-103,
|
||
0x1.c140p-104,
|
||
0x1.1840p-101,
|
||
0x1.1ab0p-102,
|
||
-0x1.a400p-104,
|
||
0x1.1f00p-104,
|
||
-0x1.7180p-103,
|
||
0x1.4ce0p-102,
|
||
0x1.9200p-107,
|
||
-0x1.54c0p-103,
|
||
0x1.1b80p-105,
|
||
-0x1.1828p-101,
|
||
0x1.5720p-102,
|
||
-0x1.a060p-100,
|
||
0x1.9160p-102,
|
||
0x1.a280p-104,
|
||
0x1.3400p-107,
|
||
0x1.2b20p-102,
|
||
0x1.7800p-108,
|
||
0x1.cfd0p-101,
|
||
0x1.2ef0p-102,
|
||
-0x1.2760p-99,
|
||
0x1.b380p-104,
|
||
0x1.0048p-101,
|
||
-0x1.60b0p-102,
|
||
0x1.a1ccp-100,
|
||
-0x1.a640p-104,
|
||
-0x1.08a0p-101,
|
||
0x1.7e60p-102,
|
||
0x1.22c0p-103,
|
||
-0x1.7200p-106,
|
||
0x1.f0f0p-102,
|
||
0x1.eb4ep-99,
|
||
0x1.c6e0p-103,
|
||
};
|
||
|
||
/*
|
||
* exp2l(x): compute the base 2 exponential of x
|
||
*
|
||
* Accuracy: Peak error < 0.502 ulp.
|
||
*
|
||
* Method: (accurate tables)
|
||
*
|
||
* Reduce x:
|
||
* x = 2**k + y, for integer k and |y| <= 1/2.
|
||
* Thus we have exp2(x) = 2**k * exp2(y).
|
||
*
|
||
* Reduce y:
|
||
* y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
|
||
* Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
|
||
* with |z - eps[i]| <= 2**-8 + 2**-98 for the table used.
|
||
*
|
||
* We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
|
||
* a degree-10 minimax polynomial with maximum error under 2**-120.
|
||
* The values in exp2t[] and eps[] are chosen such that
|
||
* exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
|
||
* that exp2t[i] is accurate to 2**-122.
|
||
*
|
||
* Note that the range of i is +-TBLSIZE/2, so we actually index the tables
|
||
* by i0 = i + TBLSIZE/2.
|
||
*
|
||
* This method is due to Gal, with many details due to Gal and Bachelis:
|
||
*
|
||
* Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library
|
||
* for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991).
|
||
*/
|
||
|
||
/**
|
||
* Returns 2^𝑥.
|
||
*/
|
||
long double
|
||
exp2l(long double x)
|
||
{
|
||
union ldshape u = {x};
|
||
int e = u.i.se & 0x7fff;
|
||
long double r, z, t;
|
||
uint32_t i0;
|
||
union {uint32_t u; int32_t i;} k;
|
||
|
||
/* Filter out exceptional cases. */
|
||
if (e >= 0x3fff + 14) { /* |x| >= 16384 or x is NaN */
|
||
if (u.i.se >= 0x3fff + 15 && u.i.se >> 15 == 0)
|
||
/* overflow */
|
||
return x * 0x1p16383L;
|
||
if (e == 0x7fff) /* -inf or -nan */
|
||
return -1/x;
|
||
if (x < -16382) {
|
||
if (x <= -16495 || x - 0x1p112 + 0x1p112 != x)
|
||
/* underflow */
|
||
FORCE_EVAL((float)(-0x1p-149/x));
|
||
if (x <= -16446)
|
||
return 0;
|
||
}
|
||
} else if (e < 0x3fff - 114) {
|
||
return 1 + x;
|
||
}
|
||
|
||
/*
|
||
* Reduce x, computing z, i0, and k. The low bits of x + redux
|
||
* contain the 16-bit integer part of the exponent (k) followed by
|
||
* TBLBITS fractional bits (i0). We use bit tricks to extract these
|
||
* as integers, then set z to the remainder.
|
||
*
|
||
* Example: Suppose x is 0xabc.123456p0 and TBLBITS is 8.
|
||
* Then the low-order word of x + redux is 0x000abc12,
|
||
* We split this into k = 0xabc and i0 = 0x12 (adjusted to
|
||
* index into the table), then we compute z = 0x0.003456p0.
|
||
*/
|
||
u.f = x + redux;
|
||
i0 = u.i2.lo + TBLSIZE / 2;
|
||
k.u = i0 / TBLSIZE * TBLSIZE;
|
||
k.i /= TBLSIZE;
|
||
i0 %= TBLSIZE;
|
||
u.f -= redux;
|
||
z = x - u.f;
|
||
|
||
/* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
|
||
t = tbl[i0];
|
||
z -= eps[i0];
|
||
r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 + z * (P6
|
||
+ z * (P7 + z * (P8 + z * (P9 + z * P10)))))))));
|
||
|
||
return scalbnl(r, k.i);
|
||
}
|
||
|
||
#endif
|