@@ -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[]);
deleted file mode 100644
@@ -1 +0,0 @@
-/* Not needed. */
deleted file mode 100644
@@ -1 +0,0 @@
-/* Not needed. */
@@ -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 +33,197 @@
/* */
/***************************************************************************/
+/* IBM exp(x) replaced by following exp(x) in 2018. 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, Jan 2018 to use j/64 instead of j/32
+ and to round with FE_TONEAREST if another rounding mode is set on entry.
+ Method (large arguments):
+ 1. Argument Reduction: given the input x, find r and integer k
+ and j such that
+ x = (k+j/64)*(ln2) + r, |r| <= (1/128)*ln2
+
+ 2. exp(x) = 2^k * (2^(j/64) + 2^(j/64)*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/64) 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_64 * 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 & 0x3f) << 1;
+ m = k >> 6;
+ z = (xx.x - k * ln2_64hi) - k * ln2_64lo;
+
+ /* 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 +1,255 @@
+/* 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/>. */
+
+
+/*
+ TBL[2*j] is 2**(j/64), rounded to nearest.
+ TBL[2*j+1] is 2**(j/64) - 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[128] = {
+ 0x1.0000000000000p+0, 0x0.0000000000000p+0,
+ 0x1.02c9a3e778061p+0, -0x1.19083535b085dp-56,
+ 0x1.059b0d3158574p+0, 0x1.d73e2a475b465p-55,
+ 0x1.0874518759bc8p+0, 0x1.186be4bb284ffp-57,
+ 0x1.0b5586cf9890fp+0, 0x1.8a62e4adc610bp-54,
+ 0x1.0e3ec32d3d1a2p+0, 0x1.03a1727c57b52p-59,
+ 0x1.11301d0125b51p+0, -0x1.6c51039449b3ap-54,
+ 0x1.1429aaea92de0p+0, -0x1.32fbf9af1369ep-54,
+ 0x1.172b83c7d517bp+0, -0x1.19041b9d78a76p-55,
+ 0x1.1a35beb6fcb75p+0, 0x1.e5b4c7b4968e4p-55,
+ 0x1.1d4873168b9aap+0, 0x1.e016e00a2643cp-54,
+ 0x1.2063b88628cd6p+0, 0x1.dc775814a8495p-55,
+ 0x1.2387a6e756238p+0, 0x1.9b07eb6c70573p-54,
+ 0x1.26b4565e27cddp+0, 0x1.2bd339940e9d9p-55,
+ 0x1.29e9df51fdee1p+0, 0x1.612e8afad1255p-55,
+ 0x1.2d285a6e4030bp+0, 0x1.0024754db41d5p-54,
+ 0x1.306fe0a31b715p+0, 0x1.6f46ad23182e4p-55,
+ 0x1.33c08b26416ffp+0, 0x1.32721843659a6p-54,
+ 0x1.371a7373aa9cbp+0, -0x1.63aeabf42eae2p-54,
+ 0x1.3a7db34e59ff7p+0, -0x1.5e436d661f5e3p-56,
+ 0x1.3dea64c123422p+0, 0x1.ada0911f09ebcp-55,
+ 0x1.4160a21f72e2ap+0, -0x1.ef3691c309278p-58,
+ 0x1.44e086061892dp+0, 0x1.89b7a04ef80d0p-59,
+ 0x1.486a2b5c13cd0p+0, 0x1.3c1a3b69062f0p-56,
+ 0x1.4bfdad5362a27p+0, 0x1.d4397afec42e2p-56,
+ 0x1.4f9b2769d2ca7p+0, -0x1.4b309d25957e3p-54,
+ 0x1.5342b569d4f82p+0, -0x1.07abe1db13cadp-55,
+ 0x1.56f4736b527dap+0, 0x1.9bb2c011d93adp-54,
+ 0x1.5ab07dd485429p+0, 0x1.6324c054647adp-54,
+ 0x1.5e76f15ad2148p+0, 0x1.ba6f93080e65ep-54,
+ 0x1.6247eb03a5585p+0, -0x1.383c17e40b497p-54,
+ 0x1.6623882552225p+0, -0x1.bb60987591c34p-54,
+ 0x1.6a09e667f3bcdp+0, -0x1.bdd3413b26456p-54,
+ 0x1.6dfb23c651a2fp+0, -0x1.bbe3a683c88abp-57,
+ 0x1.71f75e8ec5f74p+0, -0x1.16e4786887a99p-55,
+ 0x1.75feb564267c9p+0, -0x1.0245957316dd3p-54,
+ 0x1.7a11473eb0187p+0, -0x1.41577ee04992fp-55,
+ 0x1.7e2f336cf4e62p+0, 0x1.05d02ba15797ep-56,
+ 0x1.82589994cce13p+0, -0x1.d4c1dd41532d8p-54,
+ 0x1.868d99b4492edp+0, -0x1.fc6f89bd4f6bap-54,
+ 0x1.8ace5422aa0dbp+0, 0x1.6e9f156864b27p-54,
+ 0x1.8f1ae99157736p+0, 0x1.5cc13a2e3976cp-55,
+ 0x1.93737b0cdc5e5p+0, -0x1.75fc781b57ebcp-57,
+ 0x1.97d829fde4e50p+0, -0x1.d185b7c1b85d1p-54,
+ 0x1.9c49182a3f090p+0, 0x1.c7c46b071f2bep-56,
+ 0x1.a0c667b5de565p+0, -0x1.359495d1cd533p-54,
+ 0x1.a5503b23e255dp+0, -0x1.d2f6edb8d41e1p-54,
+ 0x1.a9e6b5579fdbfp+0, 0x1.0fac90ef7fd31p-54,
+ 0x1.ae89f995ad3adp+0, 0x1.7a1cd345dcc81p-54,
+ 0x1.b33a2b84f15fbp+0, -0x1.2805e3084d708p-57,
+ 0x1.b7f76f2fb5e47p+0, -0x1.5584f7e54ac3bp-56,
+ 0x1.bcc1e904bc1d2p+0, 0x1.23dd07a2d9e84p-55,
+ 0x1.c199bdd85529cp+0, 0x1.11065895048ddp-55,
+ 0x1.c67f12e57d14bp+0, 0x1.2884dff483cadp-54,
+ 0x1.cb720dcef9069p+0, 0x1.503cbd1e949dbp-56,
+ 0x1.d072d4a07897cp+0, -0x1.cbc3743797a9cp-54,
+ 0x1.d5818dcfba487p+0, 0x1.2ed02d75b3707p-55,
+ 0x1.da9e603db3285p+0, 0x1.c2300696db532p-54,
+ 0x1.dfc97337b9b5fp+0, -0x1.1a5cd4f184b5cp-54,
+ 0x1.e502ee78b3ff6p+0, 0x1.39e8980a9cc8fp-55,
+ 0x1.ea4afa2a490dap+0, -0x1.e9c23179c2893p-54,
+ 0x1.efa1bee615a27p+0, 0x1.dc7f486a4b6b0p-54,
+ 0x1.f50765b6e4540p+0, 0x1.9d3e12dd8a18bp-54,
+ 0x1.fa7c1819e90d8p+0, 0x1.74853f3a5931ep-55};
+
+/* 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_64 = 64/ln2 - used to scale x to primary range. */
+ invln2_64 = 0x1.71547652b82fep+6,
+/* ln2_64hi = high 32 bits of log(2.)/64. */
+ ln2_64hi = 0x1.62e42fee00000p-7,
+/* ln2_64lo = remainder bits for log(2.)/64 - ln2_64hi. */
+ ln2_64lo = 0x1.a39ef35793c76p-39,
+/* t2-t5 terms used for polynomial computation. */
+ t2 = 0x1.5555555555555p-3, /* 1.6666666666526086527e-1 */
+ t3 = 0x1.5555555555555p-5, /* 4.1666666666226079285e-2 */
+ t4 = 0x1.1111111111111p-7, /* 8.3333679843421958056e-3 */
+ t5 = 0x1.6c16c16c16c17p-10, /* 1.3888949086377719040e-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) 2001-2018 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
-}
deleted file mode 100644
@@ -1 +0,0 @@
-/* Not needed. */
@@ -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
@@ -1508,7 +1508,9 @@ ildouble: 5
ldouble: 5
Function: "exp":
+double: 1
float: 1
+idouble: 1
ifloat: 1
ildouble: 1
ldouble: 1
@@ -1902,7 +1902,9 @@ ildouble: 5
ldouble: 5
Function: "exp":
+double: 1
float128: 1
+idouble: 1
ifloat128: 1
ildouble: 1
ldouble: 1
@@ -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
@@ -53,7 +52,7 @@ CFLAGS-s_sincosf-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
@@ -75,14 +74,13 @@ 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
@@ -93,7 +91,6 @@ 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>
New with this version: Adds updates sparc and x86_64 libm-test-ulps files (1 ulp for various exp tests). Rewrite of full comment to reflect current state of patch. Summary of patch rationale 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 are 2x on Sparc s7 and 5x on x86_64. The former code included a slow path to assure no 1 ulp errors that could be 50-200 times slower than the normal path. Informal testing suggests perhaps 1 in 200 values might invoke the slow path. Using the glibc_perf tests: sparc (nsec) x86 (nsec) old new old new max 18180 936 4863 275 min 399 96 15 15 mean 5499 419 1336 24 Glibc correctness tests for exp() and expf() were run. Within the test suite 1 input value was found to cause a 1 ulp difference when "FE_TONEAREST" rounding mode is set. No differences in exp() were seen for the tested values for the other rounding modes. When tested over a range of 10 million input values, the new code gets a 1 ulp error approximately 1.6 times per 1000 values. That rate was similar for all four rounding modes. The patch uses a 64 entry scaling table. 32, 128, and 256 entry tables were also examined with the following error rates: Table 1 ulp/ entries 1000 32 2.9 64 1.6 128 1.0 256 0.6 Each table entry takes 16 bytes meaning a 256 entry table requires 4K bytes. That large of a table was thought to possibly have an impact in overall performance by displacing other data in an exp() heavy application. Further optimization is possible in the handling of rounding modes. Using get_rounding_mode and libc_fesetround() instead of SET_RESTORE_ROUND provides a measurable gain for Sparc. Unfortunately, on x86, one works with sse fp unit rounding mode while the other works on x87 fp unit rounding mode. Adding libc_fegetround, libc_fegetroundf and libc_fegetroundl to to match libc_fesetround() should not be too large a task but outside the scope of this patch. --- manual/probes.texi | 14 - math/Makefile | 2 +- sysdeps/generic/math_private.h | 1 - sysdeps/i386/fpu/slowexp.c | 1 - sysdeps/ia64/fpu/slowexp.c | 1 - sysdeps/ieee754/dbl-64/e_exp.c | 341 +++++++++++++-------------- sysdeps/ieee754/dbl-64/e_pow.c | 2 +- sysdeps/ieee754/dbl-64/eexp.tbl | 255 ++++++++++++++++++++ sysdeps/ieee754/dbl-64/slowexp.c | 86 ------- sysdeps/m68k/m680x0/fpu/slowexp.c | 1 - sysdeps/powerpc/power4/fpu/Makefile | 1 - sysdeps/sparc/fpu/libm-test-ulps | 2 + sysdeps/x86_64/fpu/libm-test-ulps | 2 + 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 - 20 files changed, 430 insertions(+), 318 deletions(-) delete mode 100644 sysdeps/i386/fpu/slowexp.c delete mode 100644 sysdeps/ia64/fpu/slowexp.c create mode 100644 sysdeps/ieee754/dbl-64/eexp.tbl delete mode 100644 sysdeps/ieee754/dbl-64/slowexp.c delete mode 100644 sysdeps/m68k/m680x0/fpu/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