3f8c0a6661
The C11 standard introduced a set of macros (CMPLX, CMPLXF, CMPLXL) that can be used to construct complex numbers from a pair of real and imaginary numbers. Unfortunately, they require some compiler support, which is why we only define them for Clang and GCC>=4.7. The cpack() function in libm performs the same task as CMPLX(), but cannot be used to generate compile-time constants. This means that all invocations of cpack() can safely be replaced by C11's CMPLX(). To keep the code building with GCC 4.2, provide copies of CMPLX() that can at least be used to generate run-time complex numbers. This makes it easier to build some of the functions outside of libm.
306 lines
11 KiB
C
306 lines
11 KiB
C
/* from: FreeBSD: head/lib/msun/ld80/s_expl.c 251343 2013-06-03 19:51:32Z kargl */
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/*-
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* Copyright (c) 2009-2013 Steven G. Kargl
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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 unmodified, this list of conditions, and the following
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* 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 ``AS IS'' AND ANY EXPRESS OR
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* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*
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* Optimized by Bruce D. Evans.
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*/
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/*
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* See s_expl.c for more comments about __k_expl().
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*
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* See ../src/e_exp.c and ../src/k_exp.h for precision-independent comments
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* about the secondary kernels.
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*/
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#define INTERVALS 128
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#define LOG2_INTERVALS 7
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#define BIAS (LDBL_MAX_EXP - 1)
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static const double
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/*
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* ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must
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* have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest
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* bits zero so that multiplication of it by n is exact.
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*/
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INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */
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L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */
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L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */
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/*
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* Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]:
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* |exp(x) - p(x)| < 2**-77.2
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* (0.002708 is ln2/(2*INTERVALS) rounded up a little).
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*/
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A2 = 0.5,
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A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
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A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
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A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
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A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
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/*
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* 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where
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* the first 53 bits of the significand are stored in hi and the next 53
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* bits are in lo. Tang's paper states that the trailing 6 bits of hi must
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* be zero for his algorithm in both single and double precision, because
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* the table is re-used in the implementation of expm1() where a floating
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* point addition involving hi must be exact. Here hi is double, so
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* converting it to long double gives 11 trailing zero bits.
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*/
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static const struct {
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double hi;
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double lo;
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} tbl[INTERVALS] = {
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0x1p+0, 0x0p+0,
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/*
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* XXX hi is rounded down, and the formatting is not quite normal.
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* But I rather like both. The 0x1.*p format is good for 4N+1
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* mantissa bits. Rounding down makes the lo terms positive,
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* so that the columnar formatting can be simpler.
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*/
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0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54,
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0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53,
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0x1.04315e86e7f84p+0, 0x1.7ae71f3441b49p-53,
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0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55,
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0x1.0706b29ddf6ddp+0, 0x1.8db880753b0f6p-53,
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0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57,
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0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54,
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0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54,
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0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54,
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0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b53p-59,
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0x1.0fb66affed31ap+0, 0x1.e464123bb1428p-53,
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0x1.11301d0125b50p+0, 0x1.49d77e35db263p-53,
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0x1.12abdc06c31cbp+0, 0x1.f72575a649ad2p-53,
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0x1.1429aaea92ddfp+0, 0x1.66820328764b1p-53,
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0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55,
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0x1.172b83c7d517ap+0, 0x1.b9bef918a1d63p-53,
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0x1.18af9388c8de9p+0, 0x1.777ee1734784ap-53,
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0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55,
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0x1.1bbe084045cd3p+0, 0x1.3563ce56884fcp-53,
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0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54,
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0x1.1ed5022fcd91cp+0, 0x1.71033fec2243ap-53,
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0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55,
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0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55,
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0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54,
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0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55,
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0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55,
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0x1.284dfe1f56380p+0, 0x1.2d9e2b9e07941p-53,
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0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55,
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0x1.2b87fd0dad98fp+0, 0x1.fbbd48ca71f95p-53,
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0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54,
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0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56,
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0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55,
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0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55,
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0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54,
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0x1.356c55f929ff0p+0, 0x1.928c468ec6e76p-53,
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0x1.371a7373aa9cap+0, 0x1.4e28aa05e8a8fp-53,
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0x1.38cae6d05d865p+0, 0x1.0b53961b37da2p-53,
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0x1.3a7db34e59ff6p+0, 0x1.d43792533c144p-53,
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0x1.3c32dc313a8e4p+0, 0x1.08003e4516b1ep-53,
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0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55,
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0x1.3fa4504ac801bp+0, 0x1.417ee03548306p-53,
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0x1.4160a21f72e29p+0, 0x1.f0864b71e7b6cp-53,
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0x1.431f5d950a896p+0, 0x1.b8e088728219ap-53,
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0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59,
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0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54,
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0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56,
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0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54,
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0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56,
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0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54,
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0x1.4f9b2769d2ca6p+0, 0x1.5a67b16d3540ep-53,
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0x1.516daa2cf6641p+0, 0x1.8225ea5909b04p-53,
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0x1.5342b569d4f81p+0, 0x1.be1507893b0d5p-53,
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0x1.551a4ca5d920ep+0, 0x1.8a5d8c4048699p-53,
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0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54,
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0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55,
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0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54,
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0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60,
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0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54,
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0x1.605e1b976dc08p+0, 0x1.60edeb25490dcp-53,
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0x1.6247eb03a5584p+0, 0x1.63e1f40dfa5b5p-53,
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0x1.6434634ccc31fp+0, 0x1.8edf0e2989db3p-53,
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0x1.6623882552224p+0, 0x1.224fb3c5371e6p-53,
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0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57,
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0x1.6a09e667f3bccp+0, 0x1.21165f626cdd5p-53,
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0x1.6c012750bdabep+0, 0x1.daed533001e9ep-53,
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0x1.6dfb23c651a2ep+0, 0x1.e441c597c3775p-53,
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0x1.6ff7df9519483p+0, 0x1.9f0fc369e7c42p-53,
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0x1.71f75e8ec5f73p+0, 0x1.ba46e1e5de15ap-53,
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0x1.73f9a48a58173p+0, 0x1.7ab9349cd1562p-53,
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0x1.75feb564267c8p+0, 0x1.7edd354674916p-53,
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0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54,
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0x1.7a11473eb0186p+0, 0x1.afaa2047ed9b4p-53,
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0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54,
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0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56,
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0x1.80427543e1a11p+0, 0x1.6c1bccec9346bp-53,
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0x1.82589994cce12p+0, 0x1.159f115f56694p-53,
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0x1.8471a4623c7acp+0, 0x1.9ca5ed72f8c81p-53,
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0x1.868d99b4492ecp+0, 0x1.01c83b21584a3p-53,
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0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54,
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0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54,
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0x1.8cf3216b5448bp+0, 0x1.de55439a2c38bp-53,
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0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55,
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0x1.9145b0b91ffc5p+0, 0x1.114c368d3ed6ep-53,
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0x1.93737b0cdc5e4p+0, 0x1.e8a0387e4a814p-53,
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0x1.95a44cbc8520ep+0, 0x1.d36906d2b41f9p-53,
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0x1.97d829fde4e4fp+0, 0x1.173d241f23d18p-53,
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0x1.9a0f170ca07b9p+0, 0x1.7462137188ce7p-53,
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0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56,
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0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56,
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0x1.a0c667b5de564p+0, 0x1.6535b51719567p-53,
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0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54,
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0x1.a5503b23e255cp+0, 0x1.1684892395f0fp-53,
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0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54,
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0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54,
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0x1.ac36bbfd3f379p+0, 0x1.81b72cd4624ccp-53,
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0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54,
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0x1.b0e07298db665p+0, 0x1.2108559bf8deep-53,
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0x1.b33a2b84f15fap+0, 0x1.ed7fa1cf7b290p-53,
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0x1.b59728de55939p+0, 0x1.1c7102222c90ep-53,
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0x1.b7f76f2fb5e46p+0, 0x1.d54f610356a79p-53,
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0x1.ba5b030a10649p+0, 0x1.0819678d5eb69p-53,
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0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55,
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0x1.bf2c25bd71e08p+0, 0x1.0811ae04a31c7p-53,
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0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55,
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0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54,
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0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54,
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0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56,
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0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56,
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0x1.cdf0b555dc3f9p+0, 0x1.889f12b1f58a3p-53,
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0x1.d072d4a07897bp+0, 0x1.1a1e45e4342b2p-53,
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0x1.d2f87080d89f1p+0, 0x1.15bc247313d44p-53,
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0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55,
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0x1.d80e316c98397p+0, 0x1.7709f3a09100cp-53,
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0x1.da9e603db3285p+0, 0x1.c2300696db532p-54,
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0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54,
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0x1.dfc97337b9b5ep+0, 0x1.72d195873da52p-53,
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0x1.e264614f5a128p+0, 0x1.424ec3f42f5b5p-53,
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0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55,
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0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54,
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0x1.ea4afa2a490d9p+0, 0x1.0b1ee7431ebb6p-53,
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0x1.ecf482d8e67f0p+0, 0x1.1b60625f7293ap-53,
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0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54,
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0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54,
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0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54,
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0x1.f7bfdad9cbe13p+0, 0x1.1227697fce57bp-53,
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0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55,
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0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57
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};
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/*
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* Kernel for expl(x). x must be finite and not tiny or huge.
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* "tiny" is anything that would make us underflow (|A6*x^6| < ~LDBL_MIN).
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* "huge" is anything that would make fn*L1 inexact (|x| > ~2**17*ln2).
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*/
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static inline void
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__k_expl(long double x, long double *hip, long double *lop, int *kp)
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{
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long double fn, q, r, r1, r2, t, z;
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int n, n2;
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/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
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/* Use a specialized rint() to get fn. Assume round-to-nearest. */
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fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
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r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */
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#if defined(HAVE_EFFICIENT_IRINTL)
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n = irintl(fn);
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#elif defined(HAVE_EFFICIENT_IRINT)
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n = irint(fn);
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#else
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n = (int)fn;
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#endif
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n2 = (unsigned)n % INTERVALS;
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/* Depend on the sign bit being propagated: */
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*kp = n >> LOG2_INTERVALS;
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r1 = x - fn * L1;
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r2 = fn * -L2;
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/* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */
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z = r * r;
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#if 0
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q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
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#else
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q = r2 + z * A2 + z * r * (A3 + r * A4 + z * (A5 + r * A6));
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#endif
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t = (long double)tbl[n2].lo + tbl[n2].hi;
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*hip = tbl[n2].hi;
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*lop = tbl[n2].lo + t * (q + r1);
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}
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static inline void
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k_hexpl(long double x, long double *hip, long double *lop)
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{
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float twopkm1;
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int k;
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__k_expl(x, hip, lop, &k);
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SET_FLOAT_WORD(twopkm1, 0x3f800000 + ((k - 1) << 23));
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*hip *= twopkm1;
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*lop *= twopkm1;
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}
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static inline long double
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hexpl(long double x)
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{
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long double hi, lo, twopkm2;
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int k;
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twopkm2 = 1;
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__k_expl(x, &hi, &lo, &k);
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SET_LDBL_EXPSIGN(twopkm2, BIAS + k - 2);
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return (lo + hi) * 2 * twopkm2;
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}
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#ifdef _COMPLEX_H
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/*
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* See ../src/k_exp.c for details.
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*/
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static inline long double complex
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__ldexp_cexpl(long double complex z, int expt)
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{
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long double exp_x, hi, lo;
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long double x, y, scale1, scale2;
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int half_expt, k;
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x = creall(z);
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y = cimagl(z);
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__k_expl(x, &hi, &lo, &k);
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exp_x = (lo + hi) * 0x1p16382;
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expt += k - 16382;
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scale1 = 1;
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half_expt = expt / 2;
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SET_LDBL_EXPSIGN(scale1, BIAS + half_expt);
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scale2 = 1;
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SET_LDBL_EXPSIGN(scale1, BIAS + expt - half_expt);
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return (CMPLXL(cos(y) * exp_x * scale1 * scale2,
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sinl(y) * exp_x * scale1 * scale2));
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}
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#endif /* _COMPLEX_H */
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