2012-07-23 19:13:55 +00:00
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/*-
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* Copyright (c) 2009-2012 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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* Compute the exponential of x for Intel 80-bit format. This is based on:
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*
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* PTP Tang, "Table-driven implementation of the exponential function
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* in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 15,
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* 144-157 (1989).
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*
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* where the 32 table entries have been expanded to NUM (see below).
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*/
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#include <float.h>
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#ifdef __i386__
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#include <ieeefp.h>
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#endif
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2012-07-26 03:59:33 +00:00
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#include "fpmath.h"
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2012-07-23 19:13:55 +00:00
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#include "math.h"
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#include "math_private.h"
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#define BIAS (LDBL_MAX_EXP - 1)
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static const long double
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huge = 0x1p10000L,
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twom10000 = 0x1p-10000L;
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/* XXX Prevent gcc from erroneously constant folding this: */
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static volatile const long double tiny = 0x1p-10000L;
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static const union IEEEl2bits
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/* log(2**16384 - 0.5) rounded towards zero: */
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o_threshold = LD80C(0xb17217f7d1cf79ab, 13, 0, 11356.5234062941439488L),
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/* log(2**(-16381-64-1)) rounded towards zero: */
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u_threshold = LD80C(0xb21dfe7f09e2baa9, 13, 1, -11399.4985314888605581L);
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static const double __aligned(64)
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/*
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* ln2/NUM = L1+L2 (hi+lo decomposition for multiplication). L1 must have
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* at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(NUM)) lowest bits zero
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* so that multiplication of it by n is exact.
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*/
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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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INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */
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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*NUM) rounded up a little).
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*/
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P2 = 0.5,
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P3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
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P4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
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P5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
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P6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
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/*
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* 2^(i/NUM) for i in [0,NUM] is represented by two values where the
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* first 47 (?!) bits of the significand is stored in hi and the next 53
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* bits are in lo.
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*/
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#define NUM 128
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static const struct {
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double hi;
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double lo;
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} s[NUM] __aligned(16) = {
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0x1p+0, 0x0p+0,
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0x1.0163da9fb330p+0, 0x1.ab6c25335719bp-47,
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0x1.02c9a3e77804p+0, 0x1.07737be56527cp-47,
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0x1.04315e86e7f8p+0, 0x1.2f5ce3e688369p-50,
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0x1.059b0d315854p+0, 0x1.a1d73e2a475b4p-47,
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0x1.0706b29ddf6cp+0, 0x1.dc6dc403a9d88p-48,
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0x1.0874518759bcp+0, 0x1.01186be4bb285p-49,
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0x1.09e3ecac6f38p+0, 0x1.a290f03062c27p-51,
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0x1.0b5586cf9890p+0, 0x1.ec5317256e308p-49,
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0x1.0cc922b7247cp+0, 0x1.ba03db82dc49fp-47,
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0x1.0e3ec32d3d18p+0, 0x1.10103a1727c58p-47,
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0x1.0fb66affed30p+0, 0x1.af232091dd8a1p-48,
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0x1.11301d0125b4p+0, 0x1.0a4ebbf1aed93p-48,
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0x1.12abdc06c31cp+0, 0x1.7f72575a649adp-49,
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0x1.1429aaea92dcp+0, 0x1.fb34101943b26p-48,
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0x1.15a98c8a58e4p+0, 0x1.12480d573dd56p-48,
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0x1.172b83c7d514p+0, 0x1.d6e6fbe462876p-47,
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0x1.18af9388c8dcp+0, 0x1.4dddfb85cd1e1p-47,
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0x1.1a35beb6fcb4p+0, 0x1.a9e5b4c7b4969p-47,
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0x1.1bbe084045ccp+0, 0x1.39ab1e72b4428p-48,
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0x1.1d4873168b98p+0, 0x1.53c02dc0144c8p-47,
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0x1.1ed5022fcd90p+0, 0x1.cb8819ff61122p-48,
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0x1.2063b88628ccp+0, 0x1.63b8eeb029509p-48,
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0x1.21f49917ddc8p+0, 0x1.62552fd29294cp-48,
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0x1.2387a6e75620p+0, 0x1.c3360fd6d8e0bp-47,
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0x1.251ce4fb2a60p+0, 0x1.f9ac155bef4f5p-47,
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0x1.26b4565e27ccp+0, 0x1.d257a673281d4p-48,
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0x1.284dfe1f5638p+0, 0x1.2d9e2b9e07941p-53,
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0x1.29e9df51fdecp+0, 0x1.09612e8afad12p-47,
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0x1.2b87fd0dad98p+0, 0x1.ffbbd48ca71f9p-49,
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0x1.2d285a6e4030p+0, 0x1.680123aa6da0fp-49,
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0x1.2ecafa93e2f4p+0, 0x1.611ca0f45d524p-48,
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0x1.306fe0a31b70p+0, 0x1.52de8d5a46306p-48,
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0x1.32170fc4cd80p+0, 0x1.89a9ce78e1804p-47,
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0x1.33c08b26416cp+0, 0x1.fa64e43086cb3p-47,
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0x1.356c55f929fcp+0, 0x1.864a311a3b1bap-47,
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0x1.371a7373aa9cp+0, 0x1.54e28aa05e8a9p-49,
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0x1.38cae6d05d84p+0, 0x1.2c2d4e586cdf7p-47,
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0x1.3a7db34e59fcp+0, 0x1.b750de494cf05p-47,
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0x1.3c32dc313a8cp+0, 0x1.242000f9145acp-47,
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0x1.3dea64c12340p+0, 0x1.11ada0911f09fp-47,
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0x1.3fa4504ac800p+0, 0x1.ba0bf701aa418p-48,
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0x1.4160a21f72e0p+0, 0x1.4fc2192dc79eep-47,
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0x1.431f5d950a88p+0, 0x1.6dc704439410dp-48,
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0x1.44e086061890p+0, 0x1.68189b7a04ef8p-47,
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0x1.46a41ed1d004p+0, 0x1.772512f45922ap-48,
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0x1.486a2b5c13ccp+0, 0x1.013c1a3b69063p-48,
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0x1.4a32af0d7d3cp+0, 0x1.e672d8bcf46f9p-48,
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0x1.4bfdad5362a0p+0, 0x1.38ea1cbd7f621p-47,
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0x1.4dcb299fddd0p+0, 0x1.ac766dde353c2p-49,
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0x1.4f9b2769d2c8p+0, 0x1.35699ec5b4d50p-47,
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0x1.516daa2cf664p+0, 0x1.c112f52c84d82p-52,
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0x1.5342b569d4f8p+0, 0x1.df0a83c49d86ap-52,
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0x1.551a4ca5d920p+0, 0x1.d8a5d8c40486ap-49,
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0x1.56f4736b527cp+0, 0x1.a66ecb004764fp-48,
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0x1.58d12d497c7cp+0, 0x1.e9295e15b9a1ep-47,
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0x1.5ab07dd48540p+0, 0x1.4ac64980a8c8fp-47,
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0x1.5c9268a59468p+0, 0x1.b80e258dc0b4cp-47,
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0x1.5e76f15ad214p+0, 0x1.0dd37c9840733p-49,
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0x1.605e1b976dc0p+0, 0x1.160edeb25490ep-49,
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0x1.6247eb03a558p+0, 0x1.2c7c3e81bf4b7p-50,
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0x1.6434634ccc30p+0, 0x1.fc76f8714c4eep-48,
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0x1.662388255220p+0, 0x1.24893ecf14dc8p-47,
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0x1.68155d44ca94p+0, 0x1.9840e2b913dd0p-47,
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0x1.6a09e667f3bcp+0, 0x1.921165f626cddp-49,
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0x1.6c012750bda8p+0, 0x1.f76bb54cc007ap-47,
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0x1.6dfb23c651a0p+0, 0x1.779107165f0dep-47,
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0x1.6ff7df951948p+0, 0x1.e7c3f0da79f11p-51,
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0x1.71f75e8ec5f4p+0, 0x1.9ee91b8797785p-47,
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0x1.73f9a48a5814p+0, 0x1.9deae4d273456p-47,
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0x1.75feb564267cp+0, 0x1.17edd35467491p-49,
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0x1.780694fde5d0p+0, 0x1.fb0cd7014042cp-47,
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0x1.7a11473eb018p+0, 0x1.b5f54408fdb37p-50,
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0x1.7c1ed0130c10p+0, 0x1.93e2499a22c9cp-47,
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0x1.7e2f336cf4e4p+0, 0x1.1082e815d0abdp-47,
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0x1.80427543e1a0p+0, 0x1.1b60de67649a3p-48,
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0x1.82589994cce0p+0, 0x1.28acf88afab35p-48,
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0x1.8471a4623c78p+0, 0x1.667297b5cbe32p-47,
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0x1.868d99b4492cp+0, 0x1.640720ec85613p-47,
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0x1.88ac7d98a668p+0, 0x1.966530bcdf2d5p-48,
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0x1.8ace5422aa0cp+0, 0x1.b5ba7c55a192dp-48,
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0x1.8cf3216b5448p+0, 0x1.7de55439a2c39p-49,
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0x1.8f1ae9915770p+0, 0x1.b15cc13a2e397p-47,
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0x1.9145b0b91ffcp+0, 0x1.622986d1a7daep-50,
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0x1.93737b0cdc5cp+0, 0x1.27a280e1f92a0p-47,
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0x1.95a44cbc8520p+0, 0x1.dd36906d2b420p-49,
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0x1.97d829fde4e4p+0, 0x1.f173d241f23d1p-49,
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0x1.9a0f170ca078p+0, 0x1.cdd1884dc6234p-47,
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0x1.9c49182a3f08p+0, 0x1.01c7c46b071f3p-48,
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0x1.9e86319e3230p+0, 0x1.18c12653c7326p-47,
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0x1.a0c667b5de54p+0, 0x1.2594d6d45c656p-47,
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0x1.a309bec4a2d0p+0, 0x1.9ac60b8fbb86dp-47,
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0x1.a5503b23e254p+0, 0x1.c8b424491caf8p-48,
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0x1.a799e1330b34p+0, 0x1.86f2dfb2b158fp-48,
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0x1.a9e6b5579fd8p+0, 0x1.fa1f5921deffap-47,
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0x1.ac36bbfd3f34p+0, 0x1.ce06dcb351893p-47,
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0x1.ae89f995ad38p+0, 0x1.6af439a68bb99p-47,
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0x1.b0e07298db64p+0, 0x1.2c8421566fe38p-47,
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0x1.b33a2b84f15cp+0, 0x1.d7b5fe873decap-47,
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0x1.b59728de5590p+0, 0x1.cc71c40888b24p-47,
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0x1.b7f76f2fb5e4p+0, 0x1.baa9ec206ad4fp-50,
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0x1.ba5b030a1064p+0, 0x1.30819678d5eb7p-49,
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0x1.bcc1e904bc1cp+0, 0x1.2247ba0f45b3dp-48,
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0x1.bf2c25bd71e0p+0, 0x1.10811ae04a31cp-49,
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0x1.c199bdd85528p+0, 0x1.c2220cb12a092p-48,
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0x1.c40ab5fffd04p+0, 0x1.d368a6fc1078cp-47,
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0x1.c67f12e57d14p+0, 0x1.694426ffa41e5p-49,
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0x1.c8f6d9406e78p+0, 0x1.a88d65e24402ep-47,
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0x1.cb720dcef904p+0, 0x1.48a81e5e8f4a5p-47,
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0x1.cdf0b555dc3cp+0, 0x1.ce227c4ac7d63p-47,
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0x1.d072d4a07894p+0, 0x1.dc68791790d0bp-47,
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0x1.d2f87080d89cp+0, 0x1.8c56f091cc4f5p-47,
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0x1.d5818dcfba48p+0, 0x1.c976816bad9b8p-50,
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0x1.d80e316c9838p+0, 0x1.7bb84f9d04880p-48,
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0x1.da9e603db328p+0, 0x1.5c2300696db53p-50,
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|
0x1.dd321f301b44p+0, 0x1.025b4aef1e032p-47,
|
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|
|
|
0x1.dfc97337b9b4p+0, 0x1.eb968cac39ed3p-48,
|
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|
|
|
0x1.e264614f5a10p+0, 0x1.45093b0fd0bd7p-47,
|
|
|
|
|
0x1.e502ee78b3fcp+0, 0x1.b139e8980a9cdp-47,
|
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|
|
|
0x1.e7a51fbc74c8p+0, 0x1.a5aa4594191bcp-51,
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|
0x1.ea4afa2a490cp+0, 0x1.9858f73a18f5ep-48,
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0x1.ecf482d8e67cp+0, 0x1.846d81897dca5p-47,
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0x1.efa1bee615a0p+0, 0x1.3bb8fe90d496dp-47,
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|
0x1.f252b376bba8p+0, 0x1.74e8696fc3639p-48,
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0x1.f50765b6e454p+0, 0x1.9d3e12dd8a18bp-54,
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0x1.f7bfdad9cbe0p+0, 0x1.38913b4bfe72cp-48,
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0x1.fa7c1819e90cp+0, 0x1.82e90a7e74b26p-48,
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|
0x1.fd3c22b8f71cp+0, 0x1.884badd25995ep-47
|
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|
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|
};
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|
|
|
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|
|
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|
long double
|
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|
|
expl(long double x)
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|
|
|
|
{
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union IEEEl2bits u, v;
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|
long double fn, r, r1, r2, q, t, t23, t45, twopk, twopkp10000, z;
|
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|
|
|
int k, n, n2;
|
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|
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|
uint16_t hx, ix;
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|
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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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|
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|
ix = hx & 0x7fff;
|
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|
|
|
if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
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|
|
|
if (ix == BIAS + LDBL_MAX_EXP) {
|
|
|
|
|
if (hx & 0x8000 && u.xbits.man == 1ULL << 63)
|
|
|
|
|
return (0.0L); /* x is -Inf */
|
|
|
|
|
return (x + x); /* x is +Inf, NaN or unsupported */
|
|
|
|
|
}
|
|
|
|
|
if (x > o_threshold.e)
|
|
|
|
|
return (huge * huge);
|
|
|
|
|
if (x < u_threshold.e)
|
|
|
|
|
return (tiny * tiny);
|
|
|
|
|
} else if (ix <= BIAS - 34) { /* |x| < 0x1p-33 */
|
|
|
|
|
/* includes pseudo-denormals */
|
|
|
|
|
if (huge + x > 1.0L) /* trigger inexact iff x != 0 */
|
|
|
|
|
return (1.0L + x);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
ENTERI();
|
|
|
|
|
|
|
|
|
|
/* Reduce x to (k*ln2 + midpoint[n2] + r1 + r2). */
|
|
|
|
|
/* Use a specialized rint() to get fn. Assume round-to-nearest. */
|
|
|
|
|
fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
|
|
|
|
|
r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */
|
|
|
|
|
#if defined(HAVE_EFFICIENT_IRINTL)
|
|
|
|
|
n = irintl(fn);
|
|
|
|
|
#elif defined(HAVE_EFFICIENT_IRINT)
|
|
|
|
|
n = irint(fn);
|
|
|
|
|
#else
|
|
|
|
|
n = (int)fn;
|
|
|
|
|
#endif
|
|
|
|
|
n2 = (unsigned)n % NUM; /* Tang's j. */
|
|
|
|
|
k = (n - n2) / NUM;
|
|
|
|
|
r1 = x - fn * L1;
|
|
|
|
|
r2 = -fn * L2;
|
|
|
|
|
|
|
|
|
|
/* Prepare scale factors. */
|
|
|
|
|
v.xbits.man = 1ULL << 63;
|
|
|
|
|
if (k >= LDBL_MIN_EXP) {
|
|
|
|
|
v.xbits.expsign = BIAS + k;
|
|
|
|
|
twopk = v.e;
|
|
|
|
|
} else {
|
|
|
|
|
v.xbits.expsign = BIAS + k + 10000;
|
|
|
|
|
twopkp10000 = v.e;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* Evaluate expl(midpoint[n2] + r1 + r2) = s[n2] * expl(r1 + r2). */
|
|
|
|
|
/* Here q = q(r), not q(r1), since r1 is lopped like L1. */
|
|
|
|
|
t45 = r * P5 + P4;
|
|
|
|
|
z = r * r;
|
|
|
|
|
t23 = r * P3 + P2;
|
|
|
|
|
q = r2 + z * t23 + z * z * t45 + z * z * z * P6;
|
|
|
|
|
t = (long double)s[n2].lo + s[n2].hi;
|
|
|
|
|
t = s[n2].lo + t * (q + r1) + s[n2].hi;
|
|
|
|
|
|
|
|
|
|
/* Scale by 2**k. */
|
|
|
|
|
if (k >= LDBL_MIN_EXP) {
|
|
|
|
|
if (k == LDBL_MAX_EXP)
|
|
|
|
|
RETURNI(t * 2.0L * 0x1p16383L);
|
|
|
|
|
RETURNI(t * twopk);
|
|
|
|
|
} else {
|
|
|
|
|
RETURNI(t * twopkp10000 * twom10000);
|
|
|
|
|
}
|
|
|
|
|
}
|
