@@ -40,6 +40,7 @@
#include <math_private.h>
#include <fenv.h>
#include <float.h>
+#include "eexp.tbl"
#ifndef SECTION
# define SECTION
@@ -50,8 +51,10 @@ SECTION
__ieee754_exp (double x)
{
double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
+ double z;
mynumber junk1, junk2, binexp = {{0, 0}};
int4 i, j, m, n, ex;
+ int4 k;
double retval;
{
@@ -61,7 +64,42 @@ __ieee754_exp (double x)
m = junk1.i[HIGH_HALF];
n = m & hugeint;
- if (n > smallint && n < bigint)
+ if (n < 0x3ff0a2b2) /* |x| < 1.03972053527832 */
+ {
+ if (n < 0x3f862e42) /* |x| < 3/2 ln 2 */
+ {
+ if (n < 0x3ed00000) /* |x| < 1/64 ln 2 */
+ {
+ if (n < 0x3e300000) /* |x| < 2^18 */
+ {
+ retval = one + junk1.x;
+ goto ret;
+ }
+ retval = one + junk1.x * (one + half * junk1.x);
+ goto ret;
+ }
+ t = junk1.x * junk1.x;
+ retval = junk1.x + (t * (half + junk1.x * t2) +
+ (t * t) * (t3 + junk1.x * t4 + t * t5));
+ retval = one + retval;
+ goto ret;
+ }
+
+ /* Find the multiple of 2^-6 nearest x. */
+ k = n >> 20;
+ j = (0x00100000 | (n & 0x000fffff)) >> (0x40c - k);
+ j = (j - 1) & ~1;
+ if (m < 0)
+ j += 134;
+ z = junk1.x - TBL2[j];
+ t = z * z;
+ retval = z + (t * (half + (z * t2))
+ + (t * t) * (t3 + z * t4 + t * t5));
+ retval = TBL2[j + 1] + TBL2[j + 1] * retval;
+ goto ret;
+ }
+
+ if (n < bigint) /* && |x| >= 1.03972053527832 */
{
y = x * log2e.x + three51.x;
bexp = y - three51.x; /* multiply the result by 2**bexp */
@@ -94,12 +132,6 @@ __ieee754_exp (double x)
goto ret;
}
- if (n <= smallint)
- {
- retval = 1.0;
- goto ret;
- }
-
if (n >= badint)
{
if (n > infint)
new file mode 100644
@@ -0,0 +1,172 @@
+/* EXP function tables - for use in computing double precision exponential
+ Copyright (C) 2018 Free Software Foundation, Inc.
+ This file is part of the GNU C Library.
+
+ The GNU C Library is free software; you can redistribute it and/or
+ modify it under the terms of the GNU Lesser General Public
+ License as published by the Free Software Foundation; either
+ version 2.1 of the License, or (at your option) any later version.
+
+ The GNU C Library is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ Lesser General Public License for more details.
+
+ You should have received a copy of the GNU Lesser General Public
+ License along with the GNU C Library; if not, see
+ <http://www.gnu.org/licenses/>. */
+
+/* For i = 0, ..., 66,
+ TBL2[2*i] is a double precision number near (i+1)*2^-6, and
+ TBL2[2*i+1] = exp(TBL2[2*i]) to within a relative error less
+ than 2^-60.
+
+ For i = 67, ..., 133,
+ TBL2[2*i] is a double precision number near -(i+1)*2^-6, and
+ TBL2[2*i+1] = exp(TBL2[2*i]) to within a relative error less
+ than 2^-60. */
+
+static const double TBL2[268] = {
+ 0x1.ffffffffffc82p-7, 0x1.04080ab55de32p+0,
+ 0x1.fffffffffffdbp-6, 0x1.08205601127ecp+0,
+ 0x1.80000000000a0p-5, 0x1.0c49236829e91p+0,
+ 0x1.fffffffffff79p-5, 0x1.1082b577d34e9p+0,
+ 0x1.3fffffffffffcp-4, 0x1.14cd4fc989cd6p+0,
+ 0x1.8000000000060p-4, 0x1.192937074e0d4p+0,
+ 0x1.c000000000061p-4, 0x1.1d96b0eff0e80p+0,
+ 0x1.fffffffffffd6p-4, 0x1.2216045b6f5cap+0,
+ 0x1.1ffffffffff58p-3, 0x1.26a7793f6014cp+0,
+ 0x1.3ffffffffff75p-3, 0x1.2b4b58b372c65p+0,
+ 0x1.5ffffffffff00p-3, 0x1.3001ecf601ad1p+0,
+ 0x1.8000000000020p-3, 0x1.34cb8170b583ap+0,
+ 0x1.9ffffffffa629p-3, 0x1.39a862bd3b344p+0,
+ 0x1.c00000000000fp-3, 0x1.3e98deaa11dcep+0,
+ 0x1.e00000000007fp-3, 0x1.439d443f5f16dp+0,
+ 0x1.0000000000072p-2, 0x1.48b5e3c3e81abp+0,
+ 0x1.0fffffffffecap-2, 0x1.4de30ec211dfbp+0,
+ 0x1.1ffffffffff8fp-2, 0x1.5325180cfacd2p+0,
+ 0x1.300000000003bp-2, 0x1.587c53c5a7b04p+0,
+ 0x1.4000000000034p-2, 0x1.5de9176046007p+0,
+ 0x1.4ffffffffff89p-2, 0x1.636bb9a98322fp+0,
+ 0x1.5ffffffffffe7p-2, 0x1.690492cbf942ap+0,
+ 0x1.6ffffffffff78p-2, 0x1.6eb3fc55b1e45p+0,
+ 0x1.7ffffffffff65p-2, 0x1.747a513dbef32p+0,
+ 0x1.8ffffffffffd5p-2, 0x1.7a57ede9ea22ep+0,
+ 0x1.9ffffffffff6ep-2, 0x1.804d30347b50fp+0,
+ 0x1.affffffffffc3p-2, 0x1.865a7772164aep+0,
+ 0x1.c000000000053p-2, 0x1.8c802477b0030p+0,
+ 0x1.d00000000004dp-2, 0x1.92be99a09bf1ep+0,
+ 0x1.e000000000096p-2, 0x1.99163ad4b1e08p+0,
+ 0x1.efffffffffefap-2, 0x1.9f876d8e8c4fcp+0,
+ 0x1.fffffffffffd0p-2, 0x1.a61298e1e0688p+0,
+ 0x1.0800000000002p-1, 0x1.acb82581eee56p+0,
+ 0x1.100000000001fp-1, 0x1.b3787dc80f979p+0,
+ 0x1.17ffffffffff8p-1, 0x1.ba540dba56e4fp+0,
+ 0x1.1fffffffffffap-1, 0x1.c14b431256441p+0,
+ 0x1.27fffffffffc4p-1, 0x1.c85e8d43f7c9bp+0,
+ 0x1.2fffffffffffdp-1, 0x1.cf8e5d84758a6p+0,
+ 0x1.380000000001fp-1, 0x1.d6db26d16cd84p+0,
+ 0x1.3ffffffffffd8p-1, 0x1.de455df80e39bp+0,
+ 0x1.4800000000052p-1, 0x1.e5cd799c6a59cp+0,
+ 0x1.4ffffffffffc8p-1, 0x1.ed73f240dc10cp+0,
+ 0x1.5800000000013p-1, 0x1.f539424d90f71p+0,
+ 0x1.5ffffffffffbcp-1, 0x1.fd1de6182f885p+0,
+ 0x1.680000000002dp-1, 0x1.02912df5ce741p+1,
+ 0x1.7000000000040p-1, 0x1.06a39207f0a2ap+1,
+ 0x1.780000000004fp-1, 0x1.0ac660691652ap+1,
+ 0x1.7ffffffffff6fp-1, 0x1.0ef9db467dcabp+1,
+ 0x1.87fffffffffe5p-1, 0x1.133e45d82e943p+1,
+ 0x1.9000000000035p-1, 0x1.1793e4652cc6dp+1,
+ 0x1.97fffffffffb3p-1, 0x1.1bfafc47bda48p+1,
+ 0x1.a000000000000p-1, 0x1.2073d3f1bd518p+1,
+ 0x1.a80000000004ap-1, 0x1.24feb2f105ce2p+1,
+ 0x1.affffffffffedp-1, 0x1.299be1f3e7f11p+1,
+ 0x1.b7ffffffffffbp-1, 0x1.2e4baacdb6611p+1,
+ 0x1.c00000000001dp-1, 0x1.330e587b62b39p+1,
+ 0x1.c800000000079p-1, 0x1.37e437282d538p+1,
+ 0x1.cffffffffff51p-1, 0x1.3ccd943268248p+1,
+ 0x1.d7fffffffff74p-1, 0x1.41cabe304cadcp+1,
+ 0x1.e000000000011p-1, 0x1.46dc04f4e5343p+1,
+ 0x1.e80000000001ep-1, 0x1.4c01b9950a124p+1,
+ 0x1.effffffffff9ep-1, 0x1.513c2e6c73196p+1,
+ 0x1.f7fffffffffedp-1, 0x1.568bb722dd586p+1,
+ 0x1.0000000000034p+0, 0x1.5bf0a8b1457b0p+1,
+ 0x1.03fffffffffe2p+0, 0x1.616b5967376dfp+1,
+ 0x1.07fffffffff4bp+0, 0x1.66fc20f0337a9p+1,
+ 0x1.0bffffffffffdp+0, 0x1.6ca35859290f5p+1,
+ -0x1.fffffffffffe4p-7, 0x1.f80feabfeefa5p-1,
+ -0x1.ffffffffffb0bp-6, 0x1.f03f56a88b5fep-1,
+ -0x1.7ffffffffffa7p-5, 0x1.e88dc6afecfc5p-1,
+ -0x1.ffffffffffea8p-5, 0x1.e0fabfbc702b8p-1,
+ -0x1.3ffffffffffb3p-4, 0x1.d985c89d041acp-1,
+ -0x1.7ffffffffffe3p-4, 0x1.d22e6a0197c06p-1,
+ -0x1.bffffffffff9ap-4, 0x1.caf42e73a4c89p-1,
+ -0x1.fffffffffff98p-4, 0x1.c3d6a24ed822dp-1,
+ -0x1.1ffffffffffe9p-3, 0x1.bcd553b9d7b67p-1,
+ -0x1.3ffffffffffe0p-3, 0x1.b5efd29f24c2dp-1,
+ -0x1.5fffffffff553p-3, 0x1.af25b0a61a9f4p-1,
+ -0x1.7ffffffffff8bp-3, 0x1.a876812c08794p-1,
+ -0x1.9fffffffffe51p-3, 0x1.a1e1d93d68828p-1,
+ -0x1.bffffffffff6ep-3, 0x1.9b674f8f2f3f5p-1,
+ -0x1.dffffffffff7fp-3, 0x1.95067c7837a0cp-1,
+ -0x1.fffffffffff7ap-3, 0x1.8ebef9eac8225p-1,
+ -0x1.0fffffffffffep-2, 0x1.8890636e31f55p-1,
+ -0x1.1ffffffffff41p-2, 0x1.827a56188975ep-1,
+ -0x1.2ffffffffffbap-2, 0x1.7c7c708877656p-1,
+ -0x1.3fffffffffff8p-2, 0x1.769652df22f81p-1,
+ -0x1.4ffffffffff90p-2, 0x1.70c79eba33c2fp-1,
+ -0x1.5ffffffffffdbp-2, 0x1.6b0ff72deb8aap-1,
+ -0x1.6ffffffffff9ap-2, 0x1.656f00bf5798ep-1,
+ -0x1.7ffffffffff9fp-2, 0x1.5fe4615e98eb0p-1,
+ -0x1.8ffffffffffeep-2, 0x1.5a6fc061433cep-1,
+ -0x1.9fffffffffc4ap-2, 0x1.5510c67cd26cdp-1,
+ -0x1.affffffffff30p-2, 0x1.4fc71dc13566bp-1,
+ -0x1.bfffffffffff0p-2, 0x1.4a9271936fd0ep-1,
+ -0x1.cfffffffffff3p-2, 0x1.45726ea84fb8cp-1,
+ -0x1.dfffffffffff3p-2, 0x1.4066c2ff3912bp-1,
+ -0x1.effffffffff80p-2, 0x1.3b6f1ddd05ab9p-1,
+ -0x1.fffffffffffdfp-2, 0x1.368b2fc6f9614p-1,
+ -0x1.0800000000000p-1, 0x1.31baaa7dca843p-1,
+ -0x1.0ffffffffffa4p-1, 0x1.2cfd40f8bdce4p-1,
+ -0x1.17fffffffff0ap-1, 0x1.2852a760d5ce7p-1,
+ -0x1.2000000000000p-1, 0x1.23ba930c1568bp-1,
+ -0x1.27fffffffffbbp-1, 0x1.1f34ba78d568dp-1,
+ -0x1.2fffffffffe32p-1, 0x1.1ac0d5492c1dbp-1,
+ -0x1.37ffffffff042p-1, 0x1.165e9c3e67ef2p-1,
+ -0x1.3ffffffffff77p-1, 0x1.120dc93499431p-1,
+ -0x1.47fffffffff6bp-1, 0x1.0dce171e34ecep-1,
+ -0x1.4fffffffffff1p-1, 0x1.099f41ffbe588p-1,
+ -0x1.57ffffffffe02p-1, 0x1.058106eb8a7aep-1,
+ -0x1.5ffffffffffe5p-1, 0x1.017323fd9002ep-1,
+ -0x1.67fffffffffb0p-1, 0x1.faeab0ae9386cp-2,
+ -0x1.6ffffffffffb2p-1, 0x1.f30ec837503d7p-2,
+ -0x1.77fffffffff7fp-1, 0x1.eb5210d627133p-2,
+ -0x1.7ffffffffffe8p-1, 0x1.e3b40ebefcd95p-2,
+ -0x1.87fffffffffc8p-1, 0x1.dc3448110dae2p-2,
+ -0x1.8fffffffffb30p-1, 0x1.d4d244cf4ef06p-2,
+ -0x1.97fffffffffefp-1, 0x1.cd8d8ed8ee395p-2,
+ -0x1.9ffffffffffa7p-1, 0x1.c665b1e1f1e5cp-2,
+ -0x1.a7fffffffffdcp-1, 0x1.bf5a3b6bf18d6p-2,
+ -0x1.affffffffff95p-1, 0x1.b86ababeef93bp-2,
+ -0x1.b7fffffffffcbp-1, 0x1.b196c0e24d256p-2,
+ -0x1.bffffffffff32p-1, 0x1.aadde095dadf7p-2,
+ -0x1.c7fffffffff6ap-1, 0x1.a43fae4b047c9p-2,
+ -0x1.cffffffffffb6p-1, 0x1.9dbbc01e182a4p-2,
+ -0x1.d7fffffffffcap-1, 0x1.9751adcfa81ecp-2,
+ -0x1.dffffffffffcdp-1, 0x1.910110be0699ep-2,
+ -0x1.e7ffffffffffbp-1, 0x1.8ac983dedbc69p-2,
+ -0x1.effffffffff88p-1, 0x1.84aaa3b8d51a9p-2,
+ -0x1.f7fffffffffbbp-1, 0x1.7ea40e5d6d92ep-2,
+ -0x1.fffffffffffdbp-1, 0x1.78b56362cef53p-2,
+ -0x1.03fffffffff00p+0, 0x1.72de43ddcb1f2p-2,
+ -0x1.07ffffffffe6fp+0, 0x1.6d1e525bed085p-2,
+ -0x1.0bfffffffffd6p+0, 0x1.677532dda1c57p-2};
+
+static const double
+ half = 0.5,
+ one = 1.0,
+/* t2-t5 terms used for polynomial computation. */
+ t2 = 0x1.5555555555555p-3, /* 1.6666666666666665741e-1 */
+ t3 = 0x1.5555555555555p-5, /* 4.1666666666666664354e-2 */
+ t4 = 0x1.1111111111111p-7, /* 8.3333333333333332177e-3 */
+ t5 = 0x1.6c16c16c16c17p-10; /* 1.3888888888888889419e-3 */
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. New in this version: Copyright year for eexp.tbl changed from 2017 to 2018. Spurious white space diff e_exp.c removed. ChangeLog: 2018-03-20 Patrick McGehearty <patrick.mcgehearty@oracle.com> * sysdeps/ieee754/dbl-64/e_exp.c: faster __ieee754_exp() * sysdeps/ieee754/dbl-64/eexp.tbl: New file for e_exp.c --- sysdeps/ieee754/dbl-64/e_exp.c | 46 +++++++++-- sysdeps/ieee754/dbl-64/eexp.tbl | 172 +++++++++++++++++++++++++++++++++++++++ 2 files changed, 211 insertions(+), 7 deletions(-) create mode 100644 sysdeps/ieee754/dbl-64/eexp.tbl