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Improves __ieee754_exp(x) performance by 18-37% when |x| < 1.0397
Adds a fast path to e_exp.c when |x| < 1.03972053527832. When values are tested in isolation, reduction in execution time is: aarch 30%, sparc 18%, x86 37%. When comparing benchtests/bench.out which includes values outside that range, the gains are: aarch 8%, sparc 5%, x86 9%. make check is clean (no increase in ulp for any math test). Testing 20M values for each rounding mode in that range shows approximately one in 200 values is off by 1 ulp. No value tested for exp(x) changed by 2 or more ulp. No observed change in performance or accuracy for x outside fast path range. These changes will be active for all platforms that don't provide their own exp() routines. They will also be active for ieee754 versions of ccos, ccosh, cosh, csin, csinh, sinh, exp10, gamma, and erf.
This commit is contained in:
Patrick McGehearty
parent
a700e7cb37
commit
a14d8acd32
@ -1,3 +1,8 @@
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2018-04-15 Patrick McGehearty <patrick.mcgehearty@oracle.com>
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* sysdeps/ieee754/dbl-64/e_exp.c: faster __ieee754_exp()
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* sysdeps/ieee754/dbl-64/eexp.tbl: New file for e_exp.c
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2018-04-12 DJ Delorie <dj@redhat.com>
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* sysdeps/unix/sysv/linux/syscall-names.list: Update kernel
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@ -40,6 +40,7 @@
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#include <math_private.h>
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#include <fenv.h>
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#include <float.h>
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#include "eexp.tbl"
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#ifndef SECTION
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# define SECTION
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@ -50,8 +51,10 @@ SECTION
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__ieee754_exp (double x)
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{
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double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
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double z;
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mynumber junk1, junk2, binexp = {{0, 0}};
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int4 i, j, m, n, ex;
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int4 k;
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double retval;
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{
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@ -61,7 +64,42 @@ __ieee754_exp (double x)
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m = junk1.i[HIGH_HALF];
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n = m & hugeint;
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if (n > smallint && n < bigint)
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if (n < 0x3ff0a2b2) /* |x| < 1.03972053527832 */
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{
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if (n < 0x3f862e42) /* |x| < 3/2 ln 2 */
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{
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if (n < 0x3ed00000) /* |x| < 1/64 ln 2 */
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{
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if (n < 0x3e300000) /* |x| < 2^18 */
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{
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retval = one + junk1.x;
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goto ret;
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}
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retval = one + junk1.x * (one + half * junk1.x);
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goto ret;
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}
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t = junk1.x * junk1.x;
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retval = junk1.x + (t * (half + junk1.x * t2) +
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(t * t) * (t3 + junk1.x * t4 + t * t5));
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retval = one + retval;
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goto ret;
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}
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/* Find the multiple of 2^-6 nearest x. */
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k = n >> 20;
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j = (0x00100000 | (n & 0x000fffff)) >> (0x40c - k);
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j = (j - 1) & ~1;
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if (m < 0)
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j += 134;
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z = junk1.x - TBL2[j];
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t = z * z;
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retval = z + (t * (half + (z * t2))
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+ (t * t) * (t3 + z * t4 + t * t5));
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retval = TBL2[j + 1] + TBL2[j + 1] * retval;
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goto ret;
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}
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if (n < bigint) /* && |x| >= 1.03972053527832 */
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{
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y = x * log2e.x + three51.x;
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bexp = y - three51.x; /* multiply the result by 2**bexp */
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@ -94,12 +132,6 @@ __ieee754_exp (double x)
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goto ret;
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}
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if (n <= smallint)
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{
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retval = 1.0;
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goto ret;
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}
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if (n >= badint)
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{
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if (n > infint)
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172
sysdeps/ieee754/dbl-64/eexp.tbl
Normal file
172
sysdeps/ieee754/dbl-64/eexp.tbl
Normal file
@ -0,0 +1,172 @@
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/* EXP function tables - for use in computing double precision exponential
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Copyright (C) 2018 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<http://www.gnu.org/licenses/>. */
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/* For i = 0, ..., 66,
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TBL2[2*i] is a double precision number near (i+1)*2^-6, and
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TBL2[2*i+1] = exp(TBL2[2*i]) to within a relative error less
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than 2^-60.
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For i = 67, ..., 133,
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TBL2[2*i] is a double precision number near -(i+1)*2^-6, and
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TBL2[2*i+1] = exp(TBL2[2*i]) to within a relative error less
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than 2^-60. */
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static const double TBL2[268] = {
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0x1.ffffffffffc82p-7, 0x1.04080ab55de32p+0,
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0x1.fffffffffffdbp-6, 0x1.08205601127ecp+0,
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0x1.80000000000a0p-5, 0x1.0c49236829e91p+0,
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0x1.fffffffffff79p-5, 0x1.1082b577d34e9p+0,
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0x1.3fffffffffffcp-4, 0x1.14cd4fc989cd6p+0,
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0x1.8000000000060p-4, 0x1.192937074e0d4p+0,
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0x1.c000000000061p-4, 0x1.1d96b0eff0e80p+0,
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0x1.fffffffffffd6p-4, 0x1.2216045b6f5cap+0,
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0x1.1ffffffffff58p-3, 0x1.26a7793f6014cp+0,
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0x1.3ffffffffff75p-3, 0x1.2b4b58b372c65p+0,
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0x1.5ffffffffff00p-3, 0x1.3001ecf601ad1p+0,
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0x1.8000000000020p-3, 0x1.34cb8170b583ap+0,
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0x1.9ffffffffa629p-3, 0x1.39a862bd3b344p+0,
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0x1.c00000000000fp-3, 0x1.3e98deaa11dcep+0,
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0x1.e00000000007fp-3, 0x1.439d443f5f16dp+0,
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0x1.0000000000072p-2, 0x1.48b5e3c3e81abp+0,
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0x1.0fffffffffecap-2, 0x1.4de30ec211dfbp+0,
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0x1.1ffffffffff8fp-2, 0x1.5325180cfacd2p+0,
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0x1.300000000003bp-2, 0x1.587c53c5a7b04p+0,
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0x1.4000000000034p-2, 0x1.5de9176046007p+0,
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0x1.4ffffffffff89p-2, 0x1.636bb9a98322fp+0,
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0x1.5ffffffffffe7p-2, 0x1.690492cbf942ap+0,
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0x1.6ffffffffff78p-2, 0x1.6eb3fc55b1e45p+0,
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0x1.7ffffffffff65p-2, 0x1.747a513dbef32p+0,
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0x1.8ffffffffffd5p-2, 0x1.7a57ede9ea22ep+0,
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0x1.9ffffffffff6ep-2, 0x1.804d30347b50fp+0,
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0x1.affffffffffc3p-2, 0x1.865a7772164aep+0,
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0x1.c000000000053p-2, 0x1.8c802477b0030p+0,
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0x1.d00000000004dp-2, 0x1.92be99a09bf1ep+0,
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0x1.e000000000096p-2, 0x1.99163ad4b1e08p+0,
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0x1.efffffffffefap-2, 0x1.9f876d8e8c4fcp+0,
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0x1.fffffffffffd0p-2, 0x1.a61298e1e0688p+0,
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0x1.0800000000002p-1, 0x1.acb82581eee56p+0,
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0x1.100000000001fp-1, 0x1.b3787dc80f979p+0,
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0x1.17ffffffffff8p-1, 0x1.ba540dba56e4fp+0,
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0x1.1fffffffffffap-1, 0x1.c14b431256441p+0,
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0x1.27fffffffffc4p-1, 0x1.c85e8d43f7c9bp+0,
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0x1.2fffffffffffdp-1, 0x1.cf8e5d84758a6p+0,
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0x1.380000000001fp-1, 0x1.d6db26d16cd84p+0,
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0x1.3ffffffffffd8p-1, 0x1.de455df80e39bp+0,
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0x1.4800000000052p-1, 0x1.e5cd799c6a59cp+0,
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0x1.4ffffffffffc8p-1, 0x1.ed73f240dc10cp+0,
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0x1.5800000000013p-1, 0x1.f539424d90f71p+0,
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0x1.5ffffffffffbcp-1, 0x1.fd1de6182f885p+0,
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0x1.680000000002dp-1, 0x1.02912df5ce741p+1,
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0x1.7000000000040p-1, 0x1.06a39207f0a2ap+1,
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0x1.780000000004fp-1, 0x1.0ac660691652ap+1,
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0x1.7ffffffffff6fp-1, 0x1.0ef9db467dcabp+1,
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0x1.87fffffffffe5p-1, 0x1.133e45d82e943p+1,
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0x1.9000000000035p-1, 0x1.1793e4652cc6dp+1,
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0x1.97fffffffffb3p-1, 0x1.1bfafc47bda48p+1,
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0x1.a000000000000p-1, 0x1.2073d3f1bd518p+1,
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0x1.a80000000004ap-1, 0x1.24feb2f105ce2p+1,
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0x1.affffffffffedp-1, 0x1.299be1f3e7f11p+1,
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0x1.b7ffffffffffbp-1, 0x1.2e4baacdb6611p+1,
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0x1.c00000000001dp-1, 0x1.330e587b62b39p+1,
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0x1.c800000000079p-1, 0x1.37e437282d538p+1,
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0x1.cffffffffff51p-1, 0x1.3ccd943268248p+1,
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0x1.d7fffffffff74p-1, 0x1.41cabe304cadcp+1,
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0x1.e000000000011p-1, 0x1.46dc04f4e5343p+1,
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0x1.e80000000001ep-1, 0x1.4c01b9950a124p+1,
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0x1.effffffffff9ep-1, 0x1.513c2e6c73196p+1,
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0x1.f7fffffffffedp-1, 0x1.568bb722dd586p+1,
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0x1.0000000000034p+0, 0x1.5bf0a8b1457b0p+1,
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0x1.03fffffffffe2p+0, 0x1.616b5967376dfp+1,
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0x1.07fffffffff4bp+0, 0x1.66fc20f0337a9p+1,
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0x1.0bffffffffffdp+0, 0x1.6ca35859290f5p+1,
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-0x1.fffffffffffe4p-7, 0x1.f80feabfeefa5p-1,
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-0x1.ffffffffffb0bp-6, 0x1.f03f56a88b5fep-1,
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-0x1.7ffffffffffa7p-5, 0x1.e88dc6afecfc5p-1,
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-0x1.ffffffffffea8p-5, 0x1.e0fabfbc702b8p-1,
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-0x1.3ffffffffffb3p-4, 0x1.d985c89d041acp-1,
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-0x1.7ffffffffffe3p-4, 0x1.d22e6a0197c06p-1,
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-0x1.bffffffffff9ap-4, 0x1.caf42e73a4c89p-1,
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-0x1.fffffffffff98p-4, 0x1.c3d6a24ed822dp-1,
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-0x1.1ffffffffffe9p-3, 0x1.bcd553b9d7b67p-1,
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-0x1.3ffffffffffe0p-3, 0x1.b5efd29f24c2dp-1,
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-0x1.5fffffffff553p-3, 0x1.af25b0a61a9f4p-1,
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-0x1.7ffffffffff8bp-3, 0x1.a876812c08794p-1,
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-0x1.9fffffffffe51p-3, 0x1.a1e1d93d68828p-1,
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-0x1.bffffffffff6ep-3, 0x1.9b674f8f2f3f5p-1,
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-0x1.dffffffffff7fp-3, 0x1.95067c7837a0cp-1,
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-0x1.fffffffffff7ap-3, 0x1.8ebef9eac8225p-1,
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-0x1.0fffffffffffep-2, 0x1.8890636e31f55p-1,
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-0x1.1ffffffffff41p-2, 0x1.827a56188975ep-1,
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-0x1.2ffffffffffbap-2, 0x1.7c7c708877656p-1,
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-0x1.3fffffffffff8p-2, 0x1.769652df22f81p-1,
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-0x1.4ffffffffff90p-2, 0x1.70c79eba33c2fp-1,
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-0x1.5ffffffffffdbp-2, 0x1.6b0ff72deb8aap-1,
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-0x1.6ffffffffff9ap-2, 0x1.656f00bf5798ep-1,
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-0x1.7ffffffffff9fp-2, 0x1.5fe4615e98eb0p-1,
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-0x1.8ffffffffffeep-2, 0x1.5a6fc061433cep-1,
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-0x1.9fffffffffc4ap-2, 0x1.5510c67cd26cdp-1,
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-0x1.affffffffff30p-2, 0x1.4fc71dc13566bp-1,
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-0x1.bfffffffffff0p-2, 0x1.4a9271936fd0ep-1,
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-0x1.cfffffffffff3p-2, 0x1.45726ea84fb8cp-1,
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-0x1.dfffffffffff3p-2, 0x1.4066c2ff3912bp-1,
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-0x1.effffffffff80p-2, 0x1.3b6f1ddd05ab9p-1,
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-0x1.fffffffffffdfp-2, 0x1.368b2fc6f9614p-1,
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-0x1.0800000000000p-1, 0x1.31baaa7dca843p-1,
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-0x1.0ffffffffffa4p-1, 0x1.2cfd40f8bdce4p-1,
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-0x1.17fffffffff0ap-1, 0x1.2852a760d5ce7p-1,
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-0x1.2000000000000p-1, 0x1.23ba930c1568bp-1,
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-0x1.27fffffffffbbp-1, 0x1.1f34ba78d568dp-1,
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-0x1.2fffffffffe32p-1, 0x1.1ac0d5492c1dbp-1,
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-0x1.37ffffffff042p-1, 0x1.165e9c3e67ef2p-1,
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-0x1.3ffffffffff77p-1, 0x1.120dc93499431p-1,
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-0x1.47fffffffff6bp-1, 0x1.0dce171e34ecep-1,
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-0x1.4fffffffffff1p-1, 0x1.099f41ffbe588p-1,
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-0x1.57ffffffffe02p-1, 0x1.058106eb8a7aep-1,
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-0x1.5ffffffffffe5p-1, 0x1.017323fd9002ep-1,
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-0x1.67fffffffffb0p-1, 0x1.faeab0ae9386cp-2,
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-0x1.6ffffffffffb2p-1, 0x1.f30ec837503d7p-2,
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-0x1.77fffffffff7fp-1, 0x1.eb5210d627133p-2,
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-0x1.7ffffffffffe8p-1, 0x1.e3b40ebefcd95p-2,
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-0x1.87fffffffffc8p-1, 0x1.dc3448110dae2p-2,
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-0x1.8fffffffffb30p-1, 0x1.d4d244cf4ef06p-2,
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-0x1.97fffffffffefp-1, 0x1.cd8d8ed8ee395p-2,
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-0x1.9ffffffffffa7p-1, 0x1.c665b1e1f1e5cp-2,
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-0x1.a7fffffffffdcp-1, 0x1.bf5a3b6bf18d6p-2,
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-0x1.affffffffff95p-1, 0x1.b86ababeef93bp-2,
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-0x1.b7fffffffffcbp-1, 0x1.b196c0e24d256p-2,
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-0x1.bffffffffff32p-1, 0x1.aadde095dadf7p-2,
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-0x1.c7fffffffff6ap-1, 0x1.a43fae4b047c9p-2,
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-0x1.cffffffffffb6p-1, 0x1.9dbbc01e182a4p-2,
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-0x1.d7fffffffffcap-1, 0x1.9751adcfa81ecp-2,
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-0x1.dffffffffffcdp-1, 0x1.910110be0699ep-2,
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-0x1.e7ffffffffffbp-1, 0x1.8ac983dedbc69p-2,
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-0x1.effffffffff88p-1, 0x1.84aaa3b8d51a9p-2,
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-0x1.f7fffffffffbbp-1, 0x1.7ea40e5d6d92ep-2,
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-0x1.fffffffffffdbp-1, 0x1.78b56362cef53p-2,
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-0x1.03fffffffff00p+0, 0x1.72de43ddcb1f2p-2,
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-0x1.07ffffffffe6fp+0, 0x1.6d1e525bed085p-2,
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-0x1.0bfffffffffd6p+0, 0x1.677532dda1c57p-2};
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static const double
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half = 0.5,
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one = 1.0,
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/* t2-t5 terms used for polynomial computation. */
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t2 = 0x1.5555555555555p-3, /* 1.6666666666666665741e-1 */
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t3 = 0x1.5555555555555p-5, /* 4.1666666666666664354e-2 */
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t4 = 0x1.1111111111111p-7, /* 8.3333333333333332177e-3 */
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t5 = 0x1.6c16c16c16c17p-10; /* 1.3888888888888889419e-3 */
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