@@ -258,20 +258,6 @@ Unless explicitly mentioned otherwise, a precision of 1 implies 24 bits of
precision in the mantissa of the multiple precision number. Hence, a precision
level of 32 implies 768 bits of precision in the mantissa.
-@deftp Probe slowexp_p6 (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{exp} function is called with an
-input that results in multiple precision computation with precision
-6. Argument @var{$arg1} is the input value and @var{$arg2} is the
-computed output.
-@end deftp
-
-@deftp Probe slowexp_p32 (double @var{$arg1}, double @var{$arg2})
-This probe is triggered when the @code{exp} function is called with an
-input that results in multiple precision computation with precision
-32. Argument @var{$arg1} is the input value and @var{$arg2} is the
-computed output.
-@end deftp
-
@deftp Probe slowpow_p10 (double @var{$arg1}, double @var{$arg2}, double @var{$arg3}, double @var{$arg4})
This probe is triggered when the @code{pow} function is called with
inputs that result in multiple precision computation with precision
@@ -114,7 +114,7 @@ type-ldouble-yes := ldouble
# double support
type-double-suffix :=
type-double-routines := branred doasin dosincos halfulp mpa mpatan2 \
- mpatan mpexp mplog mpsqrt mptan sincos32 slowexp \
+ mpatan mpexp mplog mpsqrt mptan sincos32 \
slowpow sincostab k_rem_pio2
# float support
@@ -262,7 +262,6 @@ extern double __sin32 (double __x, double __res, double __res1);
extern double __cos32 (double __x, double __res, double __res1);
extern double __mpsin (double __x, double __dx, bool __range_reduce);
extern double __mpcos (double __x, double __dx, bool __range_reduce);
-extern double __slowexp (double __x);
extern double __slowpow (double __x, double __y, double __z);
extern void __docos (double __x, double __dx, double __v[]);
@@ -1,3 +1,4 @@
+/* EXP function - Compute double precision exponential */
/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
@@ -23,7 +24,7 @@
/* exp1 */
/* */
/* FILES NEEDED:dla.h endian.h mpa.h mydefs.h uexp.h */
-/* mpa.c mpexp.x slowexp.c */
+/* mpa.c mpexp.x */
/* */
/* An ultimate exp routine. Given an IEEE double machine number x */
/* it computes the correctly rounded (to nearest) value of e^x */
@@ -32,207 +33,196 @@
/* */
/***************************************************************************/
+/* IBM exp(x) replaced by following exp(x) in 2017. IBM exp1(x,xx) remains. */
+/* exp(x)
+ Hybrid algorithm of Peter Tang's Table driven method (for large
+ arguments) and an accurate table (for small arguments).
+ Written by K.C. Ng, November 1988.
+ Revised by Patrick McGehearty, Dec 2017 to use j/256 instead of j/32
+ Method (large arguments):
+ 1. Argument Reduction: given the input x, find r and integer k
+ and j such that
+ x = (k+j/256)*(ln2) + r, |r| <= (1/512)*ln2
+
+ 2. exp(x) = 2^k * (2^(j/256) + 2^(j/256)*expm1(r))
+ a. expm1(r) is approximated by a polynomial:
+ expm1(r) ~ r + t1*r^2 + t2*r^3 + ... + t5*r^6
+ Here t1 = 1/2 exactly.
+ b. 2^(j/256) is represented to twice double precision
+ as TBL[2j]+TBL[2j+1].
+
+ Note: If divide were fast enough, we could use another approximation
+ in 2.a:
+ expm1(r) ~ (2r)/(2-R), R = r - r^2*(t1 + t2*r^2)
+ (for the same t1 and t2 as above)
+
+ Special cases:
+ exp(INF) is INF, exp(NaN) is NaN;
+ exp(-INF)= 0;
+ for finite argument, only exp(0)=1 is exact.
+
+ Accuracy:
+ According to an error analysis, the error is always less than
+ an ulp (unit in the last place). The largest errors observed
+ are less than 0.55 ulp for normal results and less than 0.75 ulp
+ for subnormal results.
+
+ Misc. info.
+ For IEEE double
+ if x > 7.09782712893383973096e+02 then exp(x) overflow
+ if x < -7.45133219101941108420e+02 then exp(x) underflow. */
+
#include <math.h>
+#include <math-svid-compat.h>
+#include <math_private.h>
+#include <errno.h>
#include "endian.h"
#include "uexp.h"
+#include "uexp.tbl"
#include "mydefs.h"
#include "MathLib.h"
-#include "uexp.tbl"
-#include <math_private.h>
#include <fenv.h>
#include <float.h>
-#ifndef SECTION
-# define SECTION
-#endif
+extern double __ieee754_exp (double);
+
+#include "eexp.tbl"
+
+static const double
+ half = 0.5,
+ one = 1.0;
-double __slowexp (double);
-/* An ultimate exp routine. Given an IEEE double machine number x it computes
- the correctly rounded (to nearest) value of e^x. */
double
-SECTION
-__ieee754_exp (double x)
+__ieee754_exp (double x_arg)
{
- double bexp, t, eps, del, base, y, al, bet, res, rem, cor;
- mynumber junk1, junk2, binexp = {{0, 0}};
- int4 i, j, m, n, ex;
+ double z, t;
double retval;
-
+ int hx, ix, k, j, m;
+ union
{
- SET_RESTORE_ROUND (FE_TONEAREST);
-
- junk1.x = x;
- m = junk1.i[HIGH_HALF];
- n = m & hugeint;
-
- if (n > smallint && n < bigint)
- {
- y = x * log2e.x + three51.x;
- bexp = y - three51.x; /* multiply the result by 2**bexp */
-
- junk1.x = y;
-
- eps = bexp * ln_two2.x; /* x = bexp*ln(2) + t - eps */
- t = x - bexp * ln_two1.x;
-
- y = t + three33.x;
- base = y - three33.x; /* t rounded to a multiple of 2**-18 */
- junk2.x = y;
- del = (t - base) - eps; /* x = bexp*ln(2) + base + del */
- eps = del + del * del * (p3.x * del + p2.x);
-
- binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 1023) << 20;
-
- i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
- j = (junk2.i[LOW_HALF] & 511) << 1;
-
- al = coar.x[i] * fine.x[j];
- bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
- + coar.x[i + 1] * fine.x[j + 1]);
-
- rem = (bet + bet * eps) + al * eps;
- res = al + rem;
- cor = (al - res) + rem;
- if (res == (res + cor * err_0))
+ int i_part[2];
+ double x;
+ } xx;
+ union
+ {
+ int y_part[2];
+ double y;
+ } yy;
+ xx.x = x_arg;
+
+ ix = xx.i_part[HIGH_HALF];
+ hx = ix & ~0x80000000;
+
+ if (hx < 0x3ff0a2b2)
+ { /* |x| < 3/2 ln 2 */
+ if (hx < 0x3f862e42)
+ { /* |x| < 1/64 ln 2 */
+ if (hx < 0x3ed00000)
+ { /* |x| < 2^-18 */
+ if (hx < 0x3e300000)
+ {
+ {
+ SET_RESTORE_ROUND (FE_TONEAREST);
+ retval = one + xx.x;
+ }
+ return retval;
+ }
+ {
+ SET_RESTORE_ROUND (FE_TONEAREST);
+ retval = one + xx.x * (one + half * xx.x);
+ }
+ return retval;
+ }
{
- retval = res * binexp.x;
- goto ret;
+ SET_RESTORE_ROUND (FE_TONEAREST);
+ t = xx.x * xx.x;
+ yy.y = xx.x + (t * (half + xx.x * t2)
+ + (t * t) * (t3 + xx.x * t4 + t * t5));
+ retval = one + yy.y;
}
- else
- {
- retval = __slowexp (x);
- goto ret;
- } /*if error is over bound */
- }
+ return retval;
+ }
- if (n <= smallint)
+ /* Find the multiple of 2^-6 nearest x. */
+ k = hx >> 20;
+ j = (0x00100000 | (hx & 0x000fffff)) >> (0x40c - k);
+ j = (j - 1) & ~1;
+ if (ix < 0)
+ j += 134;
{
- retval = 1.0;
- goto ret;
+ SET_RESTORE_ROUND (FE_TONEAREST);
+ z = xx.x - TBL2[j];
+ t = z * z;
+ yy.y = z + (t * (half + (z * t2))
+ + (t * t) * (t3 + z * t4 + t * t5));
+ retval = TBL2[j + 1] + TBL2[j + 1] * yy.y;
}
+ return retval;
+ }
- if (n >= badint)
- {
- if (n > infint)
- {
- retval = x + x;
- goto ret;
- } /* x is NaN */
- if (n < infint)
- {
- if (x > 0)
- goto ret_huge;
- else
- goto ret_tiny;
- }
- /* x is finite, cause either overflow or underflow */
- if (junk1.i[LOW_HALF] != 0)
- {
- retval = x + x;
- goto ret;
- } /* x is NaN */
- retval = (x > 0) ? inf.x : zero; /* |x| = inf; return either inf or 0 */
- goto ret;
- }
+ if (hx >= 0x40862e42)
+ { /* x is large, infinite, or nan. */
+ if (hx >= 0x7ff00000)
+ {
+ if (ix == 0xfff00000 && xx.i_part[LOW_HALF] == 0)
+ return zero; /* exp(-inf) = 0. */
+ return (xx.x * xx.x); /* exp(nan/inf) is nan or inf. */
+ }
+ if (xx.x > threshold1)
+ { /* Set overflow error condition. */
+ retval = hhuge * hhuge;
+ return retval;
+ }
+ if (-xx.x > threshold2)
+ { /* Set underflow error condition. */
+ double force_underflow = tiny * tiny;
+ math_force_eval (force_underflow);
+ retval = force_underflow;
+ return retval;
+ }
+ }
- y = x * log2e.x + three51.x;
- bexp = y - three51.x;
- junk1.x = y;
- eps = bexp * ln_two2.x;
- t = x - bexp * ln_two1.x;
- y = t + three33.x;
- base = y - three33.x;
- junk2.x = y;
- del = (t - base) - eps;
- eps = del + del * del * (p3.x * del + p2.x);
- i = ((junk2.i[LOW_HALF] >> 8) & 0xfffffffe) + 356;
- j = (junk2.i[LOW_HALF] & 511) << 1;
- al = coar.x[i] * fine.x[j];
- bet = ((coar.x[i] * fine.x[j + 1] + coar.x[i + 1] * fine.x[j])
- + coar.x[i + 1] * fine.x[j + 1]);
- rem = (bet + bet * eps) + al * eps;
- res = al + rem;
- cor = (al - res) + rem;
- if (m >> 31)
- {
- ex = junk1.i[LOW_HALF];
- if (res < 1.0)
- {
- res += res;
- cor += cor;
- ex -= 1;
- }
- if (ex >= -1022)
- {
- binexp.i[HIGH_HALF] = (1023 + ex) << 20;
- if (res == (res + cor * err_0))
- {
- retval = res * binexp.x;
- goto ret;
- }
- else
- {
- retval = __slowexp (x);
- goto check_uflow_ret;
- } /*if error is over bound */
- }
- ex = -(1022 + ex);
- binexp.i[HIGH_HALF] = (1023 - ex) << 20;
- res *= binexp.x;
- cor *= binexp.x;
- eps = 1.0000000001 + err_0 * binexp.x;
- t = 1.0 + res;
- y = ((1.0 - t) + res) + cor;
- res = t + y;
- cor = (t - res) + y;
- if (res == (res + eps * cor))
- {
- binexp.i[HIGH_HALF] = 0x00100000;
- retval = (res - 1.0) * binexp.x;
- goto check_uflow_ret;
- }
- else
- {
- retval = __slowexp (x);
- goto check_uflow_ret;
- } /* if error is over bound */
- check_uflow_ret:
- if (retval < DBL_MIN)
- {
- double force_underflow = tiny * tiny;
- math_force_eval (force_underflow);
- }
- if (retval == 0)
- goto ret_tiny;
- goto ret;
- }
+ {
+ SET_RESTORE_ROUND (FE_TONEAREST);
+ t = invln2_256 * xx.x;
+ if (ix < 0)
+ t -= half;
else
- {
- binexp.i[HIGH_HALF] = (junk1.i[LOW_HALF] + 767) << 20;
- if (res == (res + cor * err_0))
- retval = res * binexp.x * t256.x;
- else
- retval = __slowexp (x);
- if (isinf (retval))
- goto ret_huge;
- else
- goto ret;
- }
+ t += half;
+ k = (int) t;
+ j = (k & 0xff) << 1;
+ m = k >> 8;
+ z = (xx.x - k * ln2_256hi) - k * ln2_256lo;
+
+ /* z is now in primary range. */
+ t = z * z;
+ yy.y = z + (t * (half + z * t2) + (t * t) * (t3 + z * t4 + t * t5));
+ yy.y = TBL[j] + (TBL[j + 1] + TBL[j] * yy.y);
}
-ret:
- return retval;
-
- ret_huge:
- return hhuge * hhuge;
- ret_tiny:
- return tiny * tiny;
+ if (m < -1021)
+ {
+ yy.y_part[HIGH_HALF] += (m + 54) << 20;
+ retval = twom54 * yy.y;
+ if (retval < DBL_MIN)
+ {
+ double force_underflow = tiny * tiny;
+ math_force_eval (force_underflow);
+ }
+ return retval;
+ }
+ yy.y_part[HIGH_HALF] += m << 20;
+ return yy.y;
}
#ifndef __ieee754_exp
strong_alias (__ieee754_exp, __exp_finite)
#endif
+#ifndef SECTION
+# define SECTION
+#endif
+
/* Compute e^(x+xx). The routine also receives bound of error of previous
calculation. If after computing exp the error exceeds the allowed bounds,
the routine returns a non-positive number. Otherwise it returns the
@@ -25,7 +25,7 @@
/* log1 */
/* checkint */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h */
-/* halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c */
+/* halfulp.c mpexp.c mplog.c slowpow.c mpa.c */
/* uexp.c upow.c */
/* root.tbl uexp.tbl upow.tbl */
/* An ultimate power routine. Given two IEEE double machine numbers y,x */
new file mode 100644
@@ -0,0 +1,447 @@
+/* EXP function tables - for use in computing double precision exponential
+ Copyright (C) 2017 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/>. */
+
+
+/*
+ TBL[2*j] is 2**(j/256), rounded to nearest.
+ TBL[2*j+1] is 2**(j/256) - TBL[2*j], rounded to nearest.
+ These values are used to approximate exp(x) using the formula
+ given in the comments for e_exp.c. */
+
+static const double TBL[512] = {
+ 0x1.0000000000000p+0, 0x0.0000000000000p+0,
+ 0x1.00b1afa5abcbfp+0, -0x1.4f6b2a7609f71p-55,
+ 0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54,
+ 0x1.02168143b0281p+0, -0x1.2bf310fc54eb6p-55,
+ 0x1.02c9a3e778061p+0, -0x1.19083535b085dp-56,
+ 0x1.037d42e11bbccp+0, 0x1.56811eeade11ap-57,
+ 0x1.04315e86e7f85p+0, -0x1.0a31c1977c96ep-54,
+ 0x1.04e5f72f654b1p+0, 0x1.4c3793aa0d08cp-55,
+ 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55,
+ 0x1.0650a0e3c1f89p+0, -0x1.5cb7b5799c397p-54,
+ 0x1.0706b29ddf6dep+0, -0x1.c91dfe2b13c27p-55,
+ 0x1.07bd42b72a836p+0, 0x1.3233454458700p-55,
+ 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57,
+ 0x1.092bdf66607e0p+0, -0x1.68063800a3fd1p-54,
+ 0x1.09e3ecac6f383p+0, 0x1.1487818316136p-54,
+ 0x1.0a9c79b1f3919p+0, 0x1.5d16c873d1d38p-55,
+ 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54,
+ 0x1.0c0f145e46c85p+0, 0x1.4f98906d21cefp-54,
+ 0x1.0cc922b7247f7p+0, 0x1.01edc16e24f71p-54,
+ 0x1.0d83b23395decp+0, -0x1.bc14de43f316ap-54,
+ 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b52p-59,
+ 0x1.0efa55fdfa9c5p+0, -0x1.49db9bc54021bp-54,
+ 0x1.0fb66affed31bp+0, -0x1.b9bedc44ebd7bp-57,
+ 0x1.1073028d7233ep+0, 0x1.d46eb1692fdd5p-55,
+ 0x1.11301d0125b51p+0, -0x1.6c51039449b3ap-54,
+ 0x1.11edbab5e2ab6p+0, -0x1.ca454f703fb72p-54,
+ 0x1.12abdc06c31ccp+0, -0x1.1b514b36ca5c7p-58,
+ 0x1.136a814f204abp+0, -0x1.7108fba48dcf0p-57,
+ 0x1.1429aaea92de0p+0, -0x1.32fbf9af1369ep-54,
+ 0x1.14e95934f312ep+0, -0x1.b91e839bf44abp-55,
+ 0x1.15a98c8a58e51p+0, 0x1.2406ab9eeab0ap-55,
+ 0x1.166a45471c3c2p+0, 0x1.8f23b82ea1a32p-58,
+ 0x1.172b83c7d517bp+0, -0x1.19041b9d78a76p-55,
+ 0x1.17ed48695bbc0p+0, 0x1.09e3fe2ac5a64p-56,
+ 0x1.18af9388c8deap+0, -0x1.11023d1970f6cp-54,
+ 0x1.1972658375d2fp+0, 0x1.4aadd85f17e08p-54,
+ 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55,
+ 0x1.1af99f8138a1cp+0, 0x1.7bf85a4b69280p-54,
+ 0x1.1bbe084045cd4p+0, -0x1.95386352ef607p-54,
+ 0x1.1c82f95281c6bp+0, 0x1.009778010f8c9p-54,
+ 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54,
+ 0x1.1e0e75eb44027p+0, -0x1.6fdd8088cb6dep-54,
+ 0x1.1ed5022fcd91dp+0, -0x1.1df98027bb78cp-54,
+ 0x1.1f9c18438ce4dp+0, -0x1.bf524a097af5cp-54,
+ 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55,
+ 0x1.212be3578a819p+0, 0x1.3592d2cfcaac9p-54,
+ 0x1.21f49917ddc96p+0, 0x1.2a97e9494a5eep-55,
+ 0x1.22bdda27912d1p+0, 0x1.d34fb5577d69fp-55,
+ 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54,
+ 0x1.2451ffb82140ap+0, 0x1.acfcc911ca996p-55,
+ 0x1.251ce4fb2a63fp+0, 0x1.ac155bef4f4a4p-55,
+ 0x1.25e85711ece75p+0, 0x1.3e1a24ac31b2cp-54,
+ 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55,
+ 0x1.2780e341ddf29p+0, 0x1.e067c05f9e76cp-54,
+ 0x1.284dfe1f56381p+0, -0x1.a4c3a8c3f0d7ep-54,
+ 0x1.291ba7591bb70p+0, -0x1.2cc7228401cbdp-55,
+ 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55,
+ 0x1.2ab8a66d10f13p+0, -0x1.95743191690a7p-54,
+ 0x1.2b87fd0dad990p+0, -0x1.10adcd6381aa4p-59,
+ 0x1.2c57e39771b2fp+0, -0x1.50145a6eb5124p-54,
+ 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54,
+ 0x1.2df961f641589p+0, 0x1.d16cffbbce198p-54,
+ 0x1.2ecafa93e2f56p+0, 0x1.1ca0f45d52383p-56,
+ 0x1.2f9d24abd886bp+0, -0x1.53c55532bda93p-57,
+ 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55,
+ 0x1.31432edeeb2fdp+0, 0x1.959a3f3f3fcd1p-55,
+ 0x1.32170fc4cd831p+0, 0x1.a9ce78e18047cp-55,
+ 0x1.32eb83ba8ea32p+0, -0x1.c45e83cb4f318p-54,
+ 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54,
+ 0x1.3496266e3fa2dp+0, -0x1.35a75930881a4p-55,
+ 0x1.356c55f929ff1p+0, -0x1.b5cee5c4e4628p-55,
+ 0x1.36431a2de883bp+0, -0x1.c3144a06cb85ep-55,
+ 0x1.371a7373aa9cbp+0, -0x1.63aeabf42eae2p-54,
+ 0x1.37f26231e754ap+0, -0x1.9f5ca9eceb23cp-54,
+ 0x1.38cae6d05d866p+0, -0x1.e958d3c9904bdp-54,
+ 0x1.39a401b7140efp+0, -0x1.9a9a5fc8e2934p-54,
+ 0x1.3a7db34e59ff7p+0, -0x1.5e436d661f5e3p-56,
+ 0x1.3b57fbfec6cf4p+0, 0x1.54c66e26fff18p-54,
+ 0x1.3c32dc313a8e5p+0, -0x1.efff8375d29c3p-54,
+ 0x1.3d0e544ede173p+0, 0x1.fe8d08c284c71p-56,
+ 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55,
+ 0x1.3ec70df1c5175p+0, -0x1.af6637b8c9bcap-55,
+ 0x1.3fa4504ac801cp+0, -0x1.7d023f956f9f3p-54,
+ 0x1.40822c367a024p+0, 0x1.bddf8b6f4d048p-55,
+ 0x1.4160a21f72e2ap+0, -0x1.ef3691c309278p-58,
+ 0x1.423fb2709468ap+0, -0x1.8462dc0b314ddp-54,
+ 0x1.431f5d950a897p+0, -0x1.1c7dde35f7999p-55,
+ 0x1.43ffa3f84b9d4p+0, 0x1.880be9704c003p-55,
+ 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59,
+ 0x1.45c2042a7d232p+0, -0x1.8641982fb1f8ep-57,
+ 0x1.46a41ed1d0057p+0, 0x1.c944bd1648a76p-54,
+ 0x1.4786d668b3237p+0, -0x1.c20f0ed445733p-54,
+ 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56,
+ 0x1.494e1e192aed2p+0, -0x1.3b2895e499ea0p-55,
+ 0x1.4a32af0d7d3dep+0, 0x1.9cb62f3d1be56p-54,
+ 0x1.4b17dea6db7d7p+0, -0x1.125b87f2897f0p-55,
+ 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56,
+ 0x1.4ce41b817c114p+0, 0x1.05e29690abd5dp-54,
+ 0x1.4dcb299fddd0dp+0, 0x1.8ecdbbc6a7833p-54,
+ 0x1.4eb2d81d8abffp+0, -0x1.5257d2e5d7a52p-54,
+ 0x1.4f9b2769d2ca7p+0, -0x1.4b309d25957e3p-54,
+ 0x1.508417f4531eep+0, 0x1.a249b49b7465fp-56,
+ 0x1.516daa2cf6642p+0, -0x1.f768569bd93efp-55,
+ 0x1.5257de83f4eefp+0, -0x1.c998d43efef71p-56,
+ 0x1.5342b569d4f82p+0, -0x1.07abe1db13cadp-55,
+ 0x1.542e2f4f6ad27p+0, 0x1.7926d192d5f7ep-55,
+ 0x1.551a4ca5d920fp+0, -0x1.d689cefede59bp-55,
+ 0x1.56070dde910d2p+0, -0x1.0fb6e168eebf0p-54,
+ 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54,
+ 0x1.57e27dbe2c4cfp+0, -0x1.0b98c8a57b9c4p-54,
+ 0x1.58d12d497c7fdp+0, 0x1.295e15b9a1de8p-55,
+ 0x1.59c0827ff07ccp+0, -0x1.7e2cee467e60fp-54,
+ 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54,
+ 0x1.5ba11fba87a03p+0, -0x1.b77a14c233e1ap-54,
+ 0x1.5c9268a5946b7p+0, 0x1.c4b1b816986a2p-60,
+ 0x1.5d84590998b93p+0, -0x1.cd6a7a8b45643p-54,
+ 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54,
+ 0x1.5f6a320dceb71p+0, -0x1.9eadde3cdcf92p-55,
+ 0x1.605e1b976dc09p+0, -0x1.3e2429b56de47p-54,
+ 0x1.6152ae6cdf6f4p+0, 0x1.e4b3e4ab84c27p-54,
+ 0x1.6247eb03a5585p+0, -0x1.383c17e40b497p-54,
+ 0x1.633dd1d1929fdp+0, 0x1.84710beb964e5p-54,
+ 0x1.6434634ccc320p+0, -0x1.c483c759d8933p-55,
+ 0x1.652b9febc8fb7p+0, -0x1.ae3d5c9a73e09p-54,
+ 0x1.6623882552225p+0, -0x1.bb60987591c34p-54,
+ 0x1.671c1c70833f6p+0, -0x1.e8732586c6134p-55,
+ 0x1.68155d44ca973p+0, 0x1.038ae44f73e65p-57,
+ 0x1.690f4b19e9538p+0, 0x1.804bd9aeb445dp-55,
+ 0x1.6a09e667f3bcdp+0, -0x1.bdd3413b26456p-54,
+ 0x1.6b052fa75173ep+0, 0x1.a38f52c9a9d0ep-56,
+ 0x1.6c012750bdabfp+0, -0x1.2895667ff0b0dp-56,
+ 0x1.6cfdcddd47645p+0, 0x1.c7aa9b6f17309p-54,
+ 0x1.6dfb23c651a2fp+0, -0x1.bbe3a683c88abp-57,
+ 0x1.6ef9298593ae5p+0, -0x1.0b9749e1ac8b2p-54,
+ 0x1.6ff7df9519484p+0, -0x1.83c0f25860ef6p-55,
+ 0x1.70f7466f42e87p+0, 0x1.9d644d45aa65fp-58,
+ 0x1.71f75e8ec5f74p+0, -0x1.16e4786887a99p-55,
+ 0x1.72f8286ead08ap+0, -0x1.20aa02cd62c72p-54,
+ 0x1.73f9a48a58174p+0, -0x1.0a8d96c65d53cp-54,
+ 0x1.74fbd35d7cbfdp+0, 0x1.047fd618a6e1cp-54,
+ 0x1.75feb564267c9p+0, -0x1.0245957316dd3p-54,
+ 0x1.77024b1ab6e09p+0, 0x1.b7877169147f8p-54,
+ 0x1.780694fde5d3fp+0, 0x1.866b80a02162dp-54,
+ 0x1.790b938ac1cf6p+0, 0x1.349a862aadd3ep-54,
+ 0x1.7a11473eb0187p+0, -0x1.41577ee04992fp-55,
+ 0x1.7b17b0976cfdbp+0, -0x1.bebb58468dc88p-54,
+ 0x1.7c1ed0130c132p+0, 0x1.f124cd1164dd6p-54,
+ 0x1.7d26a62ff86f0p+0, 0x1.1bddbfb72b8b4p-54,
+ 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56,
+ 0x1.7f3878491c491p+0, -0x1.07f11cf9311aep-55,
+ 0x1.80427543e1a12p+0, -0x1.27c86626d972bp-54,
+ 0x1.814d2add106d9p+0, 0x1.464370d151d4dp-54,
+ 0x1.82589994cce13p+0, -0x1.d4c1dd41532d8p-54,
+ 0x1.8364c1eb941f7p+0, 0x1.99b9a31df2bd5p-54,
+ 0x1.8471a4623c7adp+0, -0x1.8d684a341cdfbp-55,
+ 0x1.857f4179f5b21p+0, -0x1.ba748f8b216d0p-58,
+ 0x1.868d99b4492edp+0, -0x1.fc6f89bd4f6bap-54,
+ 0x1.879cad931a436p+0, 0x1.5d2d7d2db47bdp-55,
+ 0x1.88ac7d98a6699p+0, 0x1.994c2f37cb53ap-54,
+ 0x1.89bd0a478580fp+0, 0x1.d53954475202ap-54,
+ 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54,
+ 0x1.8be05bad61778p+0, 0x1.ecb5efc43446ep-54,
+ 0x1.8cf3216b5448cp+0, -0x1.0d55e32e9e3aap-56,
+ 0x1.8e06a5e0866d9p+0, -0x1.7114a6fc9b2e6p-54,
+ 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55,
+ 0x1.902fed0282c8ap+0, 0x1.592ca85fe3fd2p-54,
+ 0x1.9145b0b91ffc6p+0, -0x1.dd6792e582524p-54,
+ 0x1.925c353aa2fe2p+0, -0x1.3455fa639db7fp-55,
+ 0x1.93737b0cdc5e5p+0, -0x1.75fc781b57ebcp-57,
+ 0x1.948b82b5f98e5p+0, -0x1.dc3d6797d2d99p-55,
+ 0x1.95a44cbc8520fp+0, -0x1.64b7c96a5f039p-56,
+ 0x1.96bdd9a7670b3p+0, -0x1.ba5967f19c896p-58,
+ 0x1.97d829fde4e50p+0, -0x1.d185b7c1b85d1p-54,
+ 0x1.98f33e47a22a2p+0, 0x1.cabdaa24c78ecp-56,
+ 0x1.9a0f170ca07bap+0, -0x1.173bd91cee632p-54,
+ 0x1.9b2bb4d53fe0dp+0, -0x1.dd84e4df6d518p-54,
+ 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56,
+ 0x1.9d674194bb8d5p+0, -0x1.516bea3dd8233p-54,
+ 0x1.9e86319e32323p+0, 0x1.824ca78e64c6ep-56,
+ 0x1.9fa5e8d07f29ep+0, -0x1.4a9ceaaf1facep-55,
+ 0x1.a0c667b5de565p+0, -0x1.359495d1cd533p-54,
+ 0x1.a1e7aed8eb8bbp+0, 0x1.c6618ee8be70ep-54,
+ 0x1.a309bec4a2d33p+0, 0x1.6305c7ddc36abp-54,
+ 0x1.a42c980460ad8p+0, -0x1.aa780589fb120p-54,
+ 0x1.a5503b23e255dp+0, -0x1.d2f6edb8d41e1p-54,
+ 0x1.a674a8af46052p+0, 0x1.50f5630670366p-57,
+ 0x1.a799e1330b358p+0, 0x1.bcb7ecac563c7p-54,
+ 0x1.a8bfe53c12e59p+0, -0x1.4f867b2ba15a9p-54,
+ 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54,
+ 0x1.ab0e521356ebap+0, 0x1.89c31dae94545p-55,
+ 0x1.ac36bbfd3f37ap+0, -0x1.f9234cae76cd0p-55,
+ 0x1.ad5ff3a3c2774p+0, 0x1.7ef3bb6b1b8e5p-54,
+ 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54,
+ 0x1.afb4ce622f2ffp+0, -0x1.4b2fc0f315ecdp-54,
+ 0x1.b0e07298db666p+0, -0x1.bdef54c80e425p-54,
+ 0x1.b20ce6c9a8952p+0, 0x1.4dd024a0756ccp-54,
+ 0x1.b33a2b84f15fbp+0, -0x1.2805e3084d708p-57,
+ 0x1.b468415b749b1p+0, -0x1.f763de9df7c90p-56,
+ 0x1.b59728de5593ap+0, -0x1.c71dfbbba6de3p-54,
+ 0x1.b6c6e29f1c52ap+0, 0x1.2a8f352883f6ep-54,
+ 0x1.b7f76f2fb5e47p+0, -0x1.5584f7e54ac3bp-56,
+ 0x1.b928cf22749e4p+0, -0x1.b721654cb65c6p-54,
+ 0x1.ba5b030a1064ap+0, -0x1.efcd30e54292ep-54,
+ 0x1.bb8e0b79a6f1fp+0, -0x1.f52d1c9696205p-60,
+ 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55,
+ 0x1.bdf69c3f3a207p+0, -0x1.c262360ea5b52p-60,
+ 0x1.bf2c25bd71e09p+0, -0x1.efdca3f6b9c73p-54,
+ 0x1.c06286141b33dp+0, -0x1.d8a5aa1fbca34p-55,
+ 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55,
+ 0x1.c2d1cd9fa652cp+0, -0x1.6e51617c8a5d7p-54,
+ 0x1.c40ab5fffd07ap+0, 0x1.b4537e083c60ap-54,
+ 0x1.c544778fafb22p+0, 0x1.12f072493b5afp-54,
+ 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54,
+ 0x1.c7ba88988c933p+0, -0x1.e76bbbe255559p-55,
+ 0x1.c8f6d9406e7b5p+0, 0x1.1acbc48805c44p-56,
+ 0x1.ca3405751c4dbp+0, -0x1.7f2bed10d08f5p-55,
+ 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56,
+ 0x1.ccb0f2e6d1675p+0, -0x1.d220f86009092p-56,
+ 0x1.cdf0b555dc3fap+0, -0x1.dd83b53829d72p-55,
+ 0x1.cf3155b5bab74p+0, -0x1.a08e9b86dff57p-54,
+ 0x1.d072d4a07897cp+0, -0x1.cbc3743797a9cp-54,
+ 0x1.d1b532b08c968p+0, 0x1.55636219a36eep-54,
+ 0x1.d2f87080d89f2p+0, -0x1.d487b719d8578p-54,
+ 0x1.d43c8eacaa1d6p+0, 0x1.3db53bf5a1614p-54,
+ 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55,
+ 0x1.d6c76e862e6d3p+0, 0x1.fe87a4a8165a0p-58,
+ 0x1.d80e316c98398p+0, -0x1.11ec18beddfe8p-54,
+ 0x1.d955d71ff6075p+0, 0x1.a052dbb9af6bep-54,
+ 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54,
+ 0x1.dbe7cd63a8315p+0, -0x1.b76f1926b8be4p-54,
+ 0x1.dd321f301b460p+0, 0x1.2da5778f018c3p-54,
+ 0x1.de7d5641c0658p+0, -0x1.ca5528e79ba8fp-54,
+ 0x1.dfc97337b9b5fp+0, -0x1.1a5cd4f184b5cp-54,
+ 0x1.e11676b197d17p+0, -0x1.2b529bd5c7f44p-56,
+ 0x1.e264614f5a129p+0, -0x1.7b627817a1496p-54,
+ 0x1.e3b333b16ee12p+0, -0x1.9f4a431fdc68bp-54,
+ 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55,
+ 0x1.e653924676d76p+0, -0x1.63ff87522b735p-55,
+ 0x1.e7a51fbc74c83p+0, 0x1.2d522ca0c8de2p-54,
+ 0x1.e8f7977cdb740p+0, -0x1.1089480b054b1p-54,
+ 0x1.ea4afa2a490dap+0, -0x1.e9c23179c2893p-54,
+ 0x1.eb9f4867cca6ep+0, 0x1.4832f2293e4f2p-54,
+ 0x1.ecf482d8e67f1p+0, -0x1.c93f3b411ad8cp-54,
+ 0x1.ee4aaa2188510p+0, 0x1.1c68da487568dp-54,
+ 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54,
+ 0x1.f0f9c1cb6412ap+0, -0x1.3220065181d45p-54,
+ 0x1.f252b376bba97p+0, 0x1.3a1a5bf0d8e43p-54,
+ 0x1.f3ac948dd7274p+0, -0x1.95a5a3ed837dep-56,
+ 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54,
+ 0x1.f6632798844f8p+0, 0x1.fa37b3539343ep-54,
+ 0x1.f7bfdad9cbe14p+0, -0x1.dbb12d006350ap-54,
+ 0x1.f91d802243c89p+0, -0x1.12ea8a779f689p-57,
+ 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55,
+ 0x1.fbdba3692d514p+0, -0x1.9677315098eb6p-56,
+ 0x1.fd3c22b8f71f1p+0, 0x1.2eb74966579e7p-57,
+ 0x1.fe9d96b2a23d9p+0, 0x1.4a6037442fde3p-56};
+
+/* 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
+/* invln2_256 = 256/ln2 - used to scale x to primary range. */
+ invln2_256 = 0x1.71547652b82fep+8,
+/* ln2_256hi = high 32 bits of log(2.)/256. */
+ ln2_256hi = 0x1.62e42fee00000p-9,
+/* ln2_256lo = remainder bits for log(2.)/256 - ln2_256hi. */
+ ln2_256lo = 0x1.a39ef35793c76p-41,
+/* 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, /* 1.388888888888889419e-3 */
+/* Maximum value for x to not overflow. */
+ threshold1 = 0x1.62e42fefa39efp+9, /* 7.09782712893383973096e+02 */
+/* Maximum value for -x to not underflow to zero in FE_TONEAREST mode. */
+ threshold2 = 0x1.74910d52d3051p+9, /* 7.45133219101941108420e+02 */
+/* Scaling factor used when result near zero. */
+ twom54 = 0x1.0000000000000p-54; /* 5.55111512312578270212e-17 */
deleted file mode 100644
@@ -1,86 +0,0 @@
-/*
- * IBM Accurate Mathematical Library
- * written by International Business Machines Corp.
- * Copyright (C) 2001-2017 Free Software Foundation, Inc.
- *
- * This program 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.
- *
- * This program 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 this program; if not, see <http://www.gnu.org/licenses/>.
- */
-/**************************************************************************/
-/* MODULE_NAME:slowexp.c */
-/* */
-/* FUNCTION:slowexp */
-/* */
-/* FILES NEEDED:mpa.h */
-/* mpa.c mpexp.c */
-/* */
-/*Converting from double precision to Multi-precision and calculating */
-/* e^x */
-/**************************************************************************/
-#include <math_private.h>
-
-#include <stap-probe.h>
-
-#ifndef USE_LONG_DOUBLE_FOR_MP
-# include "mpa.h"
-void __mpexp (mp_no *x, mp_no *y, int p);
-#endif
-
-#ifndef SECTION
-# define SECTION
-#endif
-
-/*Converting from double precision to Multi-precision and calculating e^x */
-double
-SECTION
-__slowexp (double x)
-{
-#ifndef USE_LONG_DOUBLE_FOR_MP
- double w, z, res, eps = 3.0e-26;
- int p;
- mp_no mpx, mpy, mpz, mpw, mpeps, mpcor;
-
- /* Use the multiple precision __MPEXP function to compute the exponential
- First at 144 bits and if it is not accurate enough, at 768 bits. */
- p = 6;
- __dbl_mp (x, &mpx, p);
- __mpexp (&mpx, &mpy, p);
- __dbl_mp (eps, &mpeps, p);
- __mul (&mpeps, &mpy, &mpcor, p);
- __add (&mpy, &mpcor, &mpw, p);
- __sub (&mpy, &mpcor, &mpz, p);
- __mp_dbl (&mpw, &w, p);
- __mp_dbl (&mpz, &z, p);
- if (w == z)
- {
- /* Track how often we get to the slow exp code plus
- its input/output values. */
- LIBC_PROBE (slowexp_p6, 2, &x, &w);
- return w;
- }
- else
- {
- p = 32;
- __dbl_mp (x, &mpx, p);
- __mpexp (&mpx, &mpy, p);
- __mp_dbl (&mpy, &res, p);
-
- /* Track how often we get to the uber-slow exp code plus
- its input/output values. */
- LIBC_PROBE (slowexp_p32, 2, &x, &res);
- return res;
- }
-#else
- return (double) __ieee754_expl((long double)x);
-#endif
-}
@@ -3,5 +3,4 @@
ifeq ($(subdir),math)
CFLAGS-mpa.c += --param max-unroll-times=4 -funroll-loops -fpeel-loops
CPPFLAGS-slowpow.c += -DUSE_LONG_DOUBLE_FOR_MP=1
-CPPFLAGS-slowexp.c += -DUSE_LONG_DOUBLE_FOR_MP=1
endif
@@ -10,7 +10,7 @@ libm-sysdep_routines += s_ceil-sse4_1 s_ceilf-sse4_1 s_floor-sse4_1 \
libm-sysdep_routines += e_exp-fma e_log-fma e_pow-fma s_atan-fma \
e_asin-fma e_atan2-fma s_sin-fma s_tan-fma \
- mplog-fma mpa-fma slowexp-fma slowpow-fma \
+ mplog-fma mpa-fma slowpow-fma \
sincos32-fma doasin-fma dosincos-fma \
halfulp-fma mpexp-fma \
mpatan2-fma mpatan-fma mpsqrt-fma mptan-fma
@@ -32,7 +32,6 @@ CFLAGS-mpsqrt-fma.c = -mfma -mavx2
CFLAGS-mptan-fma.c = -mfma -mavx2
CFLAGS-s_atan-fma.c = -mfma -mavx2
CFLAGS-sincos32-fma.c = -mfma -mavx2
-CFLAGS-slowexp-fma.c = -mfma -mavx2
CFLAGS-slowpow-fma.c = -mfma -mavx2
CFLAGS-s_sin-fma.c = -mfma -mavx2
CFLAGS-s_tan-fma.c = -mfma -mavx2
@@ -52,7 +51,7 @@ CFLAGS-s_cosf-fma.c = -mfma -mavx2
libm-sysdep_routines += e_exp-fma4 e_log-fma4 e_pow-fma4 s_atan-fma4 \
e_asin-fma4 e_atan2-fma4 s_sin-fma4 s_tan-fma4 \
- mplog-fma4 mpa-fma4 slowexp-fma4 slowpow-fma4 \
+ mplog-fma4 mpa-fma4 slowpow-fma4 \
sincos32-fma4 doasin-fma4 dosincos-fma4 \
halfulp-fma4 mpexp-fma4 \
mpatan2-fma4 mpatan-fma4 mpsqrt-fma4 mptan-fma4
@@ -74,14 +73,13 @@ CFLAGS-mpsqrt-fma4.c = -mfma4
CFLAGS-mptan-fma4.c = -mfma4
CFLAGS-s_atan-fma4.c = -mfma4
CFLAGS-sincos32-fma4.c = -mfma4
-CFLAGS-slowexp-fma4.c = -mfma4
CFLAGS-slowpow-fma4.c = -mfma4
CFLAGS-s_sin-fma4.c = -mfma4
CFLAGS-s_tan-fma4.c = -mfma4
libm-sysdep_routines += e_exp-avx e_log-avx s_atan-avx \
e_atan2-avx s_sin-avx s_tan-avx \
- mplog-avx mpa-avx slowexp-avx \
+ mplog-avx mpa-avx \
mpexp-avx
CFLAGS-e_atan2-avx.c = -msse2avx -DSSE2AVX
@@ -92,7 +90,6 @@ CFLAGS-mpexp-avx.c = -msse2avx -DSSE2AVX
CFLAGS-mplog-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_atan-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_sin-avx.c = -msse2avx -DSSE2AVX
-CFLAGS-slowexp-avx.c = -msse2avx -DSSE2AVX
CFLAGS-s_tan-avx.c = -msse2avx -DSSE2AVX
endif
@@ -1,6 +1,5 @@
#define __ieee754_exp __ieee754_exp_avx
#define __exp1 __exp1_avx
-#define __slowexp __slowexp_avx
#define SECTION __attribute__ ((section (".text.avx")))
#include <sysdeps/ieee754/dbl-64/e_exp.c>
@@ -1,6 +1,5 @@
#define __ieee754_exp __ieee754_exp_fma
#define __exp1 __exp1_fma
-#define __slowexp __slowexp_fma
#define SECTION __attribute__ ((section (".text.fma")))
#include <sysdeps/ieee754/dbl-64/e_exp.c>
@@ -1,6 +1,5 @@
#define __ieee754_exp __ieee754_exp_fma4
#define __exp1 __exp1_fma4
-#define __slowexp __slowexp_fma4
#define SECTION __attribute__ ((section (".text.fma4")))
#include <sysdeps/ieee754/dbl-64/e_exp.c>
deleted file mode 100644
@@ -1,9 +0,0 @@
-#define __slowexp __slowexp_avx
-#define __add __add_avx
-#define __dbl_mp __dbl_mp_avx
-#define __mpexp __mpexp_avx
-#define __mul __mul_avx
-#define __sub __sub_avx
-#define SECTION __attribute__ ((section (".text.avx")))
-
-#include <sysdeps/ieee754/dbl-64/slowexp.c>
deleted file mode 100644
@@ -1,9 +0,0 @@
-#define __slowexp __slowexp_fma
-#define __add __add_fma
-#define __dbl_mp __dbl_mp_fma
-#define __mpexp __mpexp_fma
-#define __mul __mul_fma
-#define __sub __sub_fma
-#define SECTION __attribute__ ((section (".text.fma")))
-
-#include <sysdeps/ieee754/dbl-64/slowexp.c>
deleted file mode 100644
@@ -1,9 +0,0 @@
-#define __slowexp __slowexp_fma4
-#define __add __add_fma4
-#define __dbl_mp __dbl_mp_fma4
-#define __mpexp __mpexp_fma4
-#define __mul __mul_fma4
-#define __sub __sub_fma4
-#define SECTION __attribute__ ((section (".text.fma4")))
-
-#include <sysdeps/ieee754/dbl-64/slowexp.c>
Version 9 of proposed patch. Replaced get_rounding_mode and libc_fesetround() with SET_RESTORE_ROUND to avoid Intel rounding mode issue which showed as test failures in tgamma_upward. Adds noticable overhead for platforms that incur significant cost when rounding mode is already FE_TONEAREST. Added SET_RESTORE_ROUND to two more cases which resolved 1 ulp rounding errors for cexp(). Expanded the scaling table from 64 entries to 128 entries, renaming invln2_64 to invln2_256 as well as ln2_64hi and ln2_64lo to ln2_256hi and ln2_256lo. That reduces the 1 ulp error rate per 1000 values from 1.6 to 0.6. Even with all changes, performance gain is still quite dramatic. Adding libc_fegetround, libc_fegetroundf and libc_fegetroundl to the math_private and fenv_private.h macros would allow using libc_fegetround and libc_fesetround as a future performance enhancement. Version 8 of proposed patch. Renamed ln2_32hi2 and ln2_32lo2 to be ln2_64hi and ln2_64lo. Revised comments to more accurately describe these constants. Revised constants t2, t3, t4, t5 to better match values of 1/n factorial. Change eliminated 1 ulp error in 942 tests out 40 million values tested. Version 7 of proposed patch. Fixed formatting issue in sysdeps/ieee754/dbl-64/e_exp.c Version 6 of proposed patch. Fixed error in patch revision. Cleaned up formatting of return () and location of '+' for line breaks. Fixed comments in eexp.tbl. Adjusted 3 values in eexp.tbl to be correctly rounded in ulp as computed by quad precision. Modified e_exp.c and eexp.tbl to use table of 64 intervals instead of 32 intervals for computing exp(x). That change reduced the differences from the prior ieee754 exp(x) to 16 in 10,000 from 29 in 10,000. Also reduced the make check differences for exp to 1 from 3. No observed change in performance for using the larger table on either x86 or Sparc. Version 5 of proposed patch. Cleaned up formatting of comments and braces. Returned to single patch for submission. Version 4 of proposed patch. New comments revised to use GNU standard comment formating. Limited comment added in eexp.tbl for TBL[]. The original src used for porting to Linux did not have a comment about TBL[]. The new comment is limited to the current worker's level of understanding. The (-xx.x > threshold2) case is changed to return force_underflow. For FE_TONEAREST, tiny*tiny will always be zero but for FE_UPWARD, it will be the smallest representable value. That change caused no change in the math test results for Sparc or x86. Version 3 changes All hex constants in version 2 replaced with C99 double hex constants, allowing Big Endian and Little Endian versions to be merged. Only e_exp.c and eexp.tbl changed from version 2. Minor changes in performance results due to system noise. No other changes from version 2. Version 2 of proposed patch. Revised copyright notice and formatting issues. Removed slowexp.c and related references. Replaced tables of double constants with hex constants, taking special attention to correctly handle little endian and big endian versions. Using hex initialization also required changing variables to be declared as unions. Tables moved from e_exp.c to sysdeps/ieee754/dbl-64/eexp.tbl. Replaced __fegetround(), __fesetround() with get_rounding_mode and libc_fesetround(). Removed use of "small". "inexact mode" now ignored. Retested and rebenchmarked on sparc and x86 with the above changes. 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. Typical performance gains is typically around 5x when measured on Sparc s7 for common values between exp(1) and exp(40). Using the glibc perf tests on sparc, sparc (nsec) x86 (nsec) old new old new max 17629 395 5173 144 min 399 54 15 13 mean 5317 200 1349 23 The extreme max times for the old (ieee754) exp are due to the multiprecision computation in the old algorithm when the true value is very near 0.5 ulp away from an value representable in double precision. The new algorithm does not take special measures for those cases. The current glibc exp perf tests overrepresent those values. Informal testing suggests approximately one in 200 cases might invoke the high cost computation. The performance advantage of the new algorithm for other values is still large but not as large as indicated by the chart above. Glibc correctness tests for exp() and expf() were run. Within the test suite 3 input values were found to cause 1 bit differences (ulp) when "FE_TONEAREST" rounding mode is set. No differences in exp() were seen for the tested values for the other rounding modes. Typical example: exp(-0x1.760cd2p+0) (-1.46113312244415283203125) new code: 2.31973271630014299393707e-01 0x1.db14cd799387ap-3 old code: 2.31973271630014271638132e-01 0x1.db14cd7993879p-3 exp = 2.31973271630014285508337 (high precision) Old delta: off by 0.49 ulp New delta: off by 0.51 ulp In addition, because ieee754_exp() is used by other routines, cexp() showed test results with very small imaginary input values where the imaginary portion of the result was off by 3 ulp when in upward rounding mode, but not in the other rounding modes. For x86, tgamma showed a few values where the ulp increased to 6 (max ulp for tgamma is 5). Sparc tgamma did not show these failures. I presume the tgamma differences are due to compiler optimization differences within the gamma function.The gamma function is known to be difficult to compute accurately. --- manual/probes.texi | 14 - math/Makefile | 2 +- sysdeps/generic/math_private.h | 1 - sysdeps/ieee754/dbl-64/e_exp.c | 340 ++++++++++----------- sysdeps/ieee754/dbl-64/e_pow.c | 2 +- sysdeps/ieee754/dbl-64/eexp.tbl | 447 +++++++++++++++++++++++++++ sysdeps/ieee754/dbl-64/slowexp.c | 86 ----- sysdeps/powerpc/power4/fpu/Makefile | 1 - sysdeps/x86_64/fpu/multiarch/Makefile | 9 +- sysdeps/x86_64/fpu/multiarch/e_exp-avx.c | 1 - sysdeps/x86_64/fpu/multiarch/e_exp-fma.c | 1 - sysdeps/x86_64/fpu/multiarch/e_exp-fma4.c | 1 - sysdeps/x86_64/fpu/multiarch/slowexp-avx.c | 9 - sysdeps/x86_64/fpu/multiarch/slowexp-fma.c | 9 - sysdeps/x86_64/fpu/multiarch/slowexp-fma4.c | 9 - 15 files changed, 617 insertions(+), 315 deletions(-) create mode 100644 sysdeps/ieee754/dbl-64/eexp.tbl delete mode 100644 sysdeps/ieee754/dbl-64/slowexp.c delete mode 100644 sysdeps/x86_64/fpu/multiarch/slowexp-avx.c delete mode 100644 sysdeps/x86_64/fpu/multiarch/slowexp-fma.c delete mode 100644 sysdeps/x86_64/fpu/multiarch/slowexp-fma4.c