@@ -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 +33,239 @@
/* */
/***************************************************************************/
+/* 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.
+ Method (large arguments):
+ 1. Argument Reduction: given the input x, find r and integer k
+ and j such that
+ x = (k+j/32)*(ln2) + r, |r| <= (1/64)*ln2
+
+ 2. exp(x) = 2^k * (2^(j/32) + 2^(j/32)*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/32) 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;
+ int fe_val;
+ 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))
- {
- retval = res * binexp.x;
- goto ret;
+ 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 */
+ /* raise inexact if x != 0 */
+ if (hx < 0x3e300000)
+ {
+ retval = one + xx.x;
+ return (retval);
+ }
+ retval = one + xx.x * (one + half * xx.x);
+ return (retval);
+ }
+ /*
+ Use FE_TONEAREST rounding mode for computing yy.y
+ Avoid set/reset of rounding mode if already in FE_TONEAREST mode
+ */
+ fe_val = get_rounding_mode ();
+ if (fe_val == FE_TONEAREST) {
+ t = xx.x * xx.x;
+ yy.y = xx.x + (t * (half + xx.x * t2.x) +
+ (t * t) * (t3.x + xx.x * t4.x + t * t5.x));
+ retval = one + yy.y;
+ } else {
+ libc_fesetround (FE_TONEAREST);
+ t = xx.x * xx.x;
+ yy.y = xx.x + (t * (half + xx.x * t2.x) +
+ (t * t) * (t3.x + xx.x * t4.x + t * t5.x));
+ retval = one + yy.y;
+ libc_fesetround (fe_val);
}
- else
- {
- retval = __slowexp (x);
- goto ret;
- } /*if error is over bound */
- }
+ return (retval);
+ }
- if (n <= smallint)
- {
- retval = 1.0;
- goto ret;
+ /* 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;
+ /*
+ Use FE_TONEAREST rounding mode for computing yy.y
+ Avoid set/reset of rounding mode if already in FE_TONEAREST mode
+ */
+ fe_val = get_rounding_mode ();
+ if (fe_val == FE_TONEAREST) {
+ z = xx.x - TBL2.x[j];
+ t = z * z;
+ yy.y = z + (t * (half + (z * t2.x)) +
+ (t * t) * (t3.x + z * t4.x + t * t5.x));
+ retval = TBL2.x[j + 1] + TBL2.x[j + 1] * yy.y;
+ } else {
+ libc_fesetround (FE_TONEAREST);
+ z = xx.x - TBL2.x[j];
+ t = z * z;
+ yy.y = z + (t * (half + (z * t2.x)) +
+ (t * t) * (t3.x + z * t4.x + t * t5.x));
+ retval = TBL2.x[j + 1] + TBL2.x[j + 1] * yy.y;
+ libc_fesetround (fe_val);
}
+ 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.x)
+ { /* set overflow error condition */
+ retval = hhuge * hhuge;
+ return retval;
+ }
+ if (-xx.x > threshold2.x)
+ { /* set underflow error condition */
+ double force_underflow = tiny * tiny;
+ math_force_eval (force_underflow);
+ retval = zero;
+ 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;
- }
+ /*
+ Use FE_TONEAREST rounding mode for computing yy.y
+ Avoid set/reset of rounding mode if already in FE_TONEAREST mode
+ */
+ fe_val = get_rounding_mode ();
+ if (fe_val == FE_TONEAREST) {
+ t = invln2_32.x * 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 & 0x1f) << 1;
+ m = k >> 5;
+ z = (xx.x - k * ln2_32hi.x) - k * ln2_32lo.x;
+
+ /* z is now in primary range */
+ t = z * z;
+ yy.y = z + (t * (half + z * t2.x) +
+ (t * t) * (t3.x + z * t4.x + t * t5.x));
+ yy.y = TBL.x[j] + (TBL.x[j + 1] + TBL.x[j] * yy.y);
+ } else {
+ libc_fesetround (FE_TONEAREST);
+ t = invln2_32.x * xx.x;
+ if (ix < 0)
+ t -= half;
+ else
+ t += half;
+ k = (int) t;
+ j = (k & 0x1f) << 1;
+ m = k >> 5;
+ z = (xx.x - k * ln2_32hi.x) - k * ln2_32lo.x;
+
+ /* z is now in primary range */
+ t = z * z;
+ yy.y = z + (t * (half + z * t2.x) +
+ (t * t) * (t3.x + z * t4.x + t * t5.x));
+ yy.y = TBL.x[j] + (TBL.x[j + 1] + TBL.x[j] * yy.y);
+ libc_fesetround (fe_val);
}
-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.x * 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 +1,440 @@
+/* EXP function tables - for use in ocmputing double precisoin 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/>. */
+
+#ifdef BIG_ENDI
+static const union {
+ int i[128];
+ double x[64];
+} TBL = { .i = {
+ 0x3FF00000, 0x00000000, 0x00000000, 0x00000000,
+ 0x3FF059B0, 0xD3158574, 0x3C8D73E2, 0xA475B465,
+ 0x3FF0B558, 0x6CF9890F, 0x3C98A62E, 0x4ADC610A,
+ 0x3FF11301, 0xD0125B51, 0xBC96C510, 0x39449B3A,
+ 0x3FF172B8, 0x3C7D517B, 0xBC819041, 0xB9D78A76,
+ 0x3FF1D487, 0x3168B9AA, 0x3C9E016E, 0x00A2643C,
+ 0x3FF2387A, 0x6E756238, 0x3C99B07E, 0xB6C70573,
+ 0x3FF29E9D, 0xF51FDEE1, 0x3C8612E8, 0xAFAD1255,
+ 0x3FF306FE, 0x0A31B715, 0x3C86F46A, 0xD23182E4,
+ 0x3FF371A7, 0x373AA9CB, 0xBC963AEA, 0xBF42EAE2,
+ 0x3FF3DEA6, 0x4C123422, 0x3C8ADA09, 0x11F09EBC,
+ 0x3FF44E08, 0x6061892D, 0x3C489B7A, 0x04EF80D0,
+ 0x3FF4BFDA, 0xD5362A27, 0x3C7D4397, 0xAFEC42E2,
+ 0x3FF5342B, 0x569D4F82, 0xBC807ABE, 0x1DB13CAC,
+ 0x3FF5AB07, 0xDD485429, 0x3C96324C, 0x054647AD,
+ 0x3FF6247E, 0xB03A5585, 0xBC9383C1, 0x7E40B497,
+ 0x3FF6A09E, 0x667F3BCD, 0xBC9BDD34, 0x13B26456,
+ 0x3FF71F75, 0xE8EC5F74, 0xBC816E47, 0x86887A99,
+ 0x3FF7A114, 0x73EB0187, 0xBC841577, 0xEE04992F,
+ 0x3FF82589, 0x994CCE13, 0xBC9D4C1D, 0xD41532D8,
+ 0x3FF8ACE5, 0x422AA0DB, 0x3C96E9F1, 0x56864B27,
+ 0x3FF93737, 0xB0CDC5E5, 0xBC675FC7, 0x81B57EBC,
+ 0x3FF9C491, 0x82A3F090, 0x3C7C7C46, 0xB071F2BE,
+ 0x3FFA5503, 0xB23E255D, 0xBC9D2F6E, 0xDB8D41E1,
+ 0x3FFAE89F, 0x995AD3AD, 0x3C97A1CD, 0x345DCC81,
+ 0x3FFB7F76, 0xF2FB5E47, 0xBC75584F, 0x7E54AC3B,
+ 0x3FFC199B, 0xDD85529C, 0x3C811065, 0x895048DD,
+ 0x3FFCB720, 0xDCEF9069, 0x3C7503CB, 0xD1E949DB,
+ 0x3FFD5818, 0xDCFBA487, 0x3C82ED02, 0xD75B3706,
+ 0x3FFDFC97, 0x337B9B5F, 0xBC91A5CD, 0x4F184B5C,
+ 0x3FFEA4AF, 0xA2A490DA, 0xBC9E9C23, 0x179C2893,
+ 0x3FFF5076, 0x5B6E4540, 0x3C99D3E1, 0x2DD8A18B } };
+
+/*
+ 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 union {
+ int i[536];
+ double x[268];
+} TBL2 = { .i = {
+ 0x3F8FFFFF, 0xFFFFFC82, 0x3FF04080, 0xAB55DE32,
+ 0x3F9FFFFF, 0xFFFFFFDB, 0x3FF08205, 0x601127EC,
+ 0x3FA80000, 0x000000A0, 0x3FF0C492, 0x36829E91,
+ 0x3FAFFFFF, 0xFFFFFF79, 0x3FF1082B, 0x577D34E9,
+ 0x3FB3FFFF, 0xFFFFFFFC, 0x3FF14CD4, 0xFC989CD6,
+ 0x3FB80000, 0x00000060, 0x3FF19293, 0x7074E0D4,
+ 0x3FBC0000, 0x00000061, 0x3FF1D96B, 0x0EFF0E80,
+ 0x3FBFFFFF, 0xFFFFFFD6, 0x3FF22160, 0x45B6F5CA,
+ 0x3FC1FFFF, 0xFFFFFF58, 0x3FF26A77, 0x93F6014C,
+ 0x3FC3FFFF, 0xFFFFFF75, 0x3FF2B4B5, 0x8B372C65,
+ 0x3FC5FFFF, 0xFFFFFF00, 0x3FF3001E, 0xCF601AD1,
+ 0x3FC80000, 0x00000020, 0x3FF34CB8, 0x170B583A,
+ 0x3FC9FFFF, 0xFFFFA629, 0x3FF39A86, 0x2BD3B344,
+ 0x3FCC0000, 0x0000000F, 0x3FF3E98D, 0xEAA11DCE,
+ 0x3FCE0000, 0x0000007F, 0x3FF439D4, 0x43F5F16D,
+ 0x3FD00000, 0x00000072, 0x3FF48B5E, 0x3C3E81AB,
+ 0x3FD0FFFF, 0xFFFFFECA, 0x3FF4DE30, 0xEC211DFB,
+ 0x3FD1FFFF, 0xFFFFFF8F, 0x3FF53251, 0x80CFACD2,
+ 0x3FD30000, 0x0000003B, 0x3FF587C5, 0x3C5A7B04,
+ 0x3FD40000, 0x00000034, 0x3FF5DE91, 0x76046007,
+ 0x3FD4FFFF, 0xFFFFFF89, 0x3FF636BB, 0x9A98322F,
+ 0x3FD5FFFF, 0xFFFFFFE7, 0x3FF69049, 0x2CBF942A,
+ 0x3FD6FFFF, 0xFFFFFF78, 0x3FF6EB3F, 0xC55B1E45,
+ 0x3FD7FFFF, 0xFFFFFF65, 0x3FF747A5, 0x13DBEF32,
+ 0x3FD8FFFF, 0xFFFFFFD5, 0x3FF7A57E, 0xDE9EA22E,
+ 0x3FD9FFFF, 0xFFFFFF6E, 0x3FF804D3, 0x0347B50F,
+ 0x3FDAFFFF, 0xFFFFFFC3, 0x3FF865A7, 0x772164AE,
+ 0x3FDC0000, 0x00000053, 0x3FF8C802, 0x477B0030,
+ 0x3FDD0000, 0x0000004D, 0x3FF92BE9, 0x9A09BF1E,
+ 0x3FDE0000, 0x00000096, 0x3FF99163, 0xAD4B1E08,
+ 0x3FDEFFFF, 0xFFFFFEFA, 0x3FF9F876, 0xD8E8C4FC,
+ 0x3FDFFFFF, 0xFFFFFFD0, 0x3FFA6129, 0x8E1E0688,
+ 0x3FE08000, 0x00000002, 0x3FFACB82, 0x581EEE56,
+ 0x3FE10000, 0x0000001F, 0x3FFB3787, 0xDC80F979,
+ 0x3FE17FFF, 0xFFFFFFF8, 0x3FFBA540, 0xDBA56E4F,
+ 0x3FE1FFFF, 0xFFFFFFFA, 0x3FFC14B4, 0x31256441,
+ 0x3FE27FFF, 0xFFFFFFC4, 0x3FFC85E8, 0xD43F7C9B,
+ 0x3FE2FFFF, 0xFFFFFFFD, 0x3FFCF8E5, 0xD84758A6,
+ 0x3FE38000, 0x0000001F, 0x3FFD6DB2, 0x6D16CD84,
+ 0x3FE3FFFF, 0xFFFFFFD8, 0x3FFDE455, 0xDF80E39B,
+ 0x3FE48000, 0x00000052, 0x3FFE5CD7, 0x99C6A59C,
+ 0x3FE4FFFF, 0xFFFFFFC8, 0x3FFED73F, 0x240DC10C,
+ 0x3FE58000, 0x00000013, 0x3FFF5394, 0x24D90F71,
+ 0x3FE5FFFF, 0xFFFFFFBC, 0x3FFFD1DE, 0x6182F885,
+ 0x3FE68000, 0x0000002D, 0x40002912, 0xDF5CE741,
+ 0x3FE70000, 0x00000040, 0x40006A39, 0x207F0A2A,
+ 0x3FE78000, 0x0000004F, 0x4000AC66, 0x0691652A,
+ 0x3FE7FFFF, 0xFFFFFF6F, 0x4000EF9D, 0xB467DCAB,
+ 0x3FE87FFF, 0xFFFFFFE5, 0x400133E4, 0x5D82E943,
+ 0x3FE90000, 0x00000035, 0x4001793E, 0x4652CC6D,
+ 0x3FE97FFF, 0xFFFFFFB3, 0x4001BFAF, 0xC47BDA48,
+ 0x3FEA0000, 0x00000000, 0x4002073D, 0x3F1BD518,
+ 0x3FEA8000, 0x0000004A, 0x40024FEB, 0x2F105CE2,
+ 0x3FEAFFFF, 0xFFFFFFED, 0x400299BE, 0x1F3E7F11,
+ 0x3FEB7FFF, 0xFFFFFFFB, 0x4002E4BA, 0xACDB6611,
+ 0x3FEC0000, 0x0000001D, 0x400330E5, 0x87B62B39,
+ 0x3FEC8000, 0x00000079, 0x40037E43, 0x7282D538,
+ 0x3FECFFFF, 0xFFFFFF51, 0x4003CCD9, 0x43268248,
+ 0x3FED7FFF, 0xFFFFFF74, 0x40041CAB, 0xE304CADC,
+ 0x3FEE0000, 0x00000011, 0x40046DC0, 0x4F4E5343,
+ 0x3FEE8000, 0x0000001E, 0x4004C01B, 0x9950A124,
+ 0x3FEEFFFF, 0xFFFFFF9E, 0x400513C2, 0xE6C73196,
+ 0x3FEF7FFF, 0xFFFFFFED, 0x400568BB, 0x722DD586,
+ 0x3FF00000, 0x00000034, 0x4005BF0A, 0x8B1457B0,
+ 0x3FF03FFF, 0xFFFFFFE2, 0x400616B5, 0x967376DF,
+ 0x3FF07FFF, 0xFFFFFF4B, 0x40066FC2, 0x0F0337A9,
+ 0x3FF0BFFF, 0xFFFFFFFD, 0x4006CA35, 0x859290F5,
+ 0xBF8FFFFF, 0xFFFFFFE4, 0x3FEF80FE, 0xABFEEFA5,
+ 0xBF9FFFFF, 0xFFFFFB0B, 0x3FEF03F5, 0x6A88B5FE,
+ 0xBFA7FFFF, 0xFFFFFFA7, 0x3FEE88DC, 0x6AFECFC5,
+ 0xBFAFFFFF, 0xFFFFFEA8, 0x3FEE0FAB, 0xFBC702B8,
+ 0xBFB3FFFF, 0xFFFFFFB3, 0x3FED985C, 0x89D041AC,
+ 0xBFB7FFFF, 0xFFFFFFE3, 0x3FED22E6, 0xA0197C06,
+ 0xBFBBFFFF, 0xFFFFFF9A, 0x3FECAF42, 0xE73A4C89,
+ 0xBFBFFFFF, 0xFFFFFF98, 0x3FEC3D6A, 0x24ED822D,
+ 0xBFC1FFFF, 0xFFFFFFE9, 0x3FEBCD55, 0x3B9D7B67,
+ 0xBFC3FFFF, 0xFFFFFFE0, 0x3FEB5EFD, 0x29F24C2D,
+ 0xBFC5FFFF, 0xFFFFF553, 0x3FEAF25B, 0x0A61A9F4,
+ 0xBFC7FFFF, 0xFFFFFF8B, 0x3FEA8768, 0x12C08794,
+ 0xBFC9FFFF, 0xFFFFFE51, 0x3FEA1E1D, 0x93D68828,
+ 0xBFCBFFFF, 0xFFFFFF6E, 0x3FE9B674, 0xF8F2F3F5,
+ 0xBFCDFFFF, 0xFFFFFF7F, 0x3FE95067, 0xC7837A0C,
+ 0xBFCFFFFF, 0xFFFFFF7A, 0x3FE8EBEF, 0x9EAC8225,
+ 0xBFD0FFFF, 0xFFFFFFFE, 0x3FE88906, 0x36E31F55,
+ 0xBFD1FFFF, 0xFFFFFF41, 0x3FE827A5, 0x6188975E,
+ 0xBFD2FFFF, 0xFFFFFFBA, 0x3FE7C7C7, 0x08877656,
+ 0xBFD3FFFF, 0xFFFFFFF8, 0x3FE76965, 0x2DF22F81,
+ 0xBFD4FFFF, 0xFFFFFF90, 0x3FE70C79, 0xEBA33C2F,
+ 0xBFD5FFFF, 0xFFFFFFDB, 0x3FE6B0FF, 0x72DEB8AA,
+ 0xBFD6FFFF, 0xFFFFFF9A, 0x3FE656F0, 0x0BF5798E,
+ 0xBFD7FFFF, 0xFFFFFF9F, 0x3FE5FE46, 0x15E98EB0,
+ 0xBFD8FFFF, 0xFFFFFFEE, 0x3FE5A6FC, 0x061433CE,
+ 0xBFD9FFFF, 0xFFFFFC4A, 0x3FE5510C, 0x67CD26CD,
+ 0xBFDAFFFF, 0xFFFFFF30, 0x3FE4FC71, 0xDC13566B,
+ 0xBFDBFFFF, 0xFFFFFFF0, 0x3FE4A927, 0x1936FD0E,
+ 0xBFDCFFFF, 0xFFFFFFF3, 0x3FE45726, 0xEA84FB8C,
+ 0xBFDDFFFF, 0xFFFFFFF3, 0x3FE4066C, 0x2FF3912B,
+ 0xBFDEFFFF, 0xFFFFFF80, 0x3FE3B6F1, 0xDDD05AB9,
+ 0xBFDFFFFF, 0xFFFFFFDF, 0x3FE368B2, 0xFC6F9614,
+ 0xBFE08000, 0x00000000, 0x3FE31BAA, 0xA7DCA843,
+ 0xBFE0FFFF, 0xFFFFFFA4, 0x3FE2CFD4, 0x0F8BDCE4,
+ 0xBFE17FFF, 0xFFFFFF0A, 0x3FE2852A, 0x760D5CE7,
+ 0xBFE20000, 0x00000000, 0x3FE23BA9, 0x30C1568B,
+ 0xBFE27FFF, 0xFFFFFFBB, 0x3FE1F34B, 0xA78D568D,
+ 0xBFE2FFFF, 0xFFFFFE32, 0x3FE1AC0D, 0x5492C1DB,
+ 0xBFE37FFF, 0xFFFFF042, 0x3FE165E9, 0xC3E67EF2,
+ 0xBFE3FFFF, 0xFFFFFF77, 0x3FE120DC, 0x93499431,
+ 0xBFE47FFF, 0xFFFFFF6B, 0x3FE0DCE1, 0x71E34ECE,
+ 0xBFE4FFFF, 0xFFFFFFF1, 0x3FE099F4, 0x1FFBE588,
+ 0xBFE57FFF, 0xFFFFFE02, 0x3FE05810, 0x6EB8A7AE,
+ 0xBFE5FFFF, 0xFFFFFFE5, 0x3FE01732, 0x3FD9002E,
+ 0xBFE67FFF, 0xFFFFFFB0, 0x3FDFAEAB, 0x0AE9386C,
+ 0xBFE6FFFF, 0xFFFFFFB2, 0x3FDF30EC, 0x837503D7,
+ 0xBFE77FFF, 0xFFFFFF7F, 0x3FDEB521, 0x0D627133,
+ 0xBFE7FFFF, 0xFFFFFFE8, 0x3FDE3B40, 0xEBEFCD95,
+ 0xBFE87FFF, 0xFFFFFFC8, 0x3FDDC344, 0x8110DAE2,
+ 0xBFE8FFFF, 0xFFFFFB30, 0x3FDD4D24, 0x4CF4EF06,
+ 0xBFE97FFF, 0xFFFFFFEF, 0x3FDCD8D8, 0xED8EE395,
+ 0xBFE9FFFF, 0xFFFFFFA7, 0x3FDC665B, 0x1E1F1E5C,
+ 0xBFEA7FFF, 0xFFFFFFDC, 0x3FDBF5A3, 0xB6BF18D6,
+ 0xBFEAFFFF, 0xFFFFFF95, 0x3FDB86AB, 0xABEEF93B,
+ 0xBFEB7FFF, 0xFFFFFFCB, 0x3FDB196C, 0x0E24D256,
+ 0xBFEBFFFF, 0xFFFFFF32, 0x3FDAADDE, 0x095DADF7,
+ 0xBFEC7FFF, 0xFFFFFF6A, 0x3FDA43FA, 0xE4B047C9,
+ 0xBFECFFFF, 0xFFFFFFB6, 0x3FD9DBBC, 0x01E182A4,
+ 0xBFED7FFF, 0xFFFFFFCA, 0x3FD9751A, 0xDCFA81EC,
+ 0xBFEDFFFF, 0xFFFFFFCD, 0x3FD91011, 0x0BE0699E,
+ 0xBFEE7FFF, 0xFFFFFFFB, 0x3FD8AC98, 0x3DEDBC69,
+ 0xBFEEFFFF, 0xFFFFFF88, 0x3FD84AAA, 0x3B8D51A9,
+ 0xBFEF7FFF, 0xFFFFFFBB, 0x3FD7EA40, 0xE5D6D92E,
+ 0xBFEFFFFF, 0xFFFFFFDB, 0x3FD78B56, 0x362CEF53,
+ 0xBFF03FFF, 0xFFFFFF00, 0x3FD72DE4, 0x3DDCB1F2,
+ 0xBFF07FFF, 0xFFFFFE6F, 0x3FD6D1E5, 0x25BED085,
+ 0xBFF0BFFF, 0xFFFFFFD6, 0x3FD67753, 0x2DDA1C57 } };
+
+#else
+#ifdef LITTLE_ENDI
+
+static const union {
+ int i[128];
+ double x[64];
+} TBL = { .i = {
+ 0x00000000, 0x3FF00000, 0x00000000, 0x00000000,
+ 0xD3158574, 0x3FF059B0, 0xA475B465, 0x3C8D73E2,
+ 0x6CF9890F, 0x3FF0B558, 0x4ADC610A, 0x3C98A62E,
+ 0xD0125B51, 0x3FF11301, 0x39449B3A, 0xBC96C510,
+ 0x3C7D517B, 0x3FF172B8, 0xB9D78A76, 0xBC819041,
+ 0x3168B9AA, 0x3FF1D487, 0x00A2643C, 0x3C9E016E,
+ 0x6E756238, 0x3FF2387A, 0xB6C70573, 0x3C99B07E,
+ 0xF51FDEE1, 0x3FF29E9D, 0xAFAD1255, 0x3C8612E8,
+ 0x0A31B715, 0x3FF306FE, 0xD23182E4, 0x3C86F46A,
+ 0x373AA9CB, 0x3FF371A7, 0xBF42EAE2, 0xBC963AEA,
+ 0x4C123422, 0x3FF3DEA6, 0x11F09EBC, 0x3C8ADA09,
+ 0x6061892D, 0x3FF44E08, 0x04EF80D0, 0x3C489B7A,
+ 0xD5362A27, 0x3FF4BFDA, 0xAFEC42E2, 0x3C7D4397,
+ 0x569D4F82, 0x3FF5342B, 0x1DB13CAC, 0xBC807ABE,
+ 0xDD485429, 0x3FF5AB07, 0x054647AD, 0x3C96324C,
+ 0xB03A5585, 0x3FF6247E, 0x7E40B497, 0xBC9383C1,
+ 0x667F3BCD, 0x3FF6A09E, 0x13B26456, 0xBC9BDD34,
+ 0xE8EC5F74, 0x3FF71F75, 0x86887A99, 0xBC816E47,
+ 0x73EB0187, 0x3FF7A114, 0xEE04992F, 0xBC841577,
+ 0x994CCE13, 0x3FF82589, 0xD41532D8, 0xBC9D4C1D,
+ 0x422AA0DB, 0x3FF8ACE5, 0x56864B27, 0x3C96E9F1,
+ 0xB0CDC5E5, 0x3FF93737, 0x81B57EBC, 0xBC675FC7,
+ 0x82A3F090, 0x3FF9C491, 0xB071F2BE, 0x3C7C7C46,
+ 0xB23E255D, 0x3FFA5503, 0xDB8D41E1, 0xBC9D2F6E,
+ 0x995AD3AD, 0x3FFAE89F, 0x345DCC81, 0x3C97A1CD,
+ 0xF2FB5E47, 0x3FFB7F76, 0x7E54AC3B, 0xBC75584F,
+ 0xDD85529C, 0x3FFC199B, 0x895048DD, 0x3C811065,
+ 0xDCEF9069, 0x3FFCB720, 0xD1E949DB, 0x3C7503CB,
+ 0xDCFBA487, 0x3FFD5818, 0xD75B3706, 0x3C82ED02,
+ 0x337B9B5F, 0x3FFDFC97, 0x4F184B5C, 0xBC91A5CD,
+ 0xA2A490DA, 0x3FFEA4AF, 0x179C2893, 0xBC9E9C23,
+ 0x5B6E4540, 0x3FFF5076, 0x2DD8A18B, 0x3C99D3E1 } };
+/*
+ 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 union {
+ int i[536];
+ double x[268];
+} TBL2 = { .i = {
+ 0xFFFFFC82, 0x3F8FFFFF, 0xAB55DE32, 0x3FF04080,
+ 0xFFFFFFDB, 0x3F9FFFFF, 0x601127EC, 0x3FF08205,
+ 0x000000A0, 0x3FA80000, 0x36829E91, 0x3FF0C492,
+ 0xFFFFFF79, 0x3FAFFFFF, 0x577D34E9, 0x3FF1082B,
+ 0xFFFFFFFC, 0x3FB3FFFF, 0xFC989CD6, 0x3FF14CD4,
+ 0x00000060, 0x3FB80000, 0x7074E0D4, 0x3FF19293,
+ 0x00000061, 0x3FBC0000, 0x0EFF0E80, 0x3FF1D96B,
+ 0xFFFFFFD6, 0x3FBFFFFF, 0x45B6F5CA, 0x3FF22160,
+ 0xFFFFFF58, 0x3FC1FFFF, 0x93F6014C, 0x3FF26A77,
+ 0xFFFFFF75, 0x3FC3FFFF, 0x8B372C65, 0x3FF2B4B5,
+ 0xFFFFFF00, 0x3FC5FFFF, 0xCF601AD1, 0x3FF3001E,
+ 0x00000020, 0x3FC80000, 0x170B583A, 0x3FF34CB8,
+ 0xFFFFA629, 0x3FC9FFFF, 0x2BD3B344, 0x3FF39A86,
+ 0x0000000F, 0x3FCC0000, 0xEAA11DCE, 0x3FF3E98D,
+ 0x0000007F, 0x3FCE0000, 0x43F5F16D, 0x3FF439D4,
+ 0x00000072, 0x3FD00000, 0x3C3E81AB, 0x3FF48B5E,
+ 0xFFFFFECA, 0x3FD0FFFF, 0xEC211DFB, 0x3FF4DE30,
+ 0xFFFFFF8F, 0x3FD1FFFF, 0x80CFACD2, 0x3FF53251,
+ 0x0000003B, 0x3FD30000, 0x3C5A7B04, 0x3FF587C5,
+ 0x00000034, 0x3FD40000, 0x76046007, 0x3FF5DE91,
+ 0xFFFFFF89, 0x3FD4FFFF, 0x9A98322F, 0x3FF636BB,
+ 0xFFFFFFE7, 0x3FD5FFFF, 0x2CBF942A, 0x3FF69049,
+ 0xFFFFFF78, 0x3FD6FFFF, 0xC55B1E45, 0x3FF6EB3F,
+ 0xFFFFFF65, 0x3FD7FFFF, 0x13DBEF32, 0x3FF747A5,
+ 0xFFFFFFD5, 0x3FD8FFFF, 0xDE9EA22E, 0x3FF7A57E,
+ 0xFFFFFF6E, 0x3FD9FFFF, 0x0347B50F, 0x3FF804D3,
+ 0xFFFFFFC3, 0x3FDAFFFF, 0x772164AE, 0x3FF865A7,
+ 0x00000053, 0x3FDC0000, 0x477B0030, 0x3FF8C802,
+ 0x0000004D, 0x3FDD0000, 0x9A09BF1E, 0x3FF92BE9,
+ 0x00000096, 0x3FDE0000, 0xAD4B1E08, 0x3FF99163,
+ 0xFFFFFEFA, 0x3FDEFFFF, 0xD8E8C4FC, 0x3FF9F876,
+ 0xFFFFFFD0, 0x3FDFFFFF, 0x8E1E0688, 0x3FFA6129,
+ 0x00000002, 0x3FE08000, 0x581EEE56, 0x3FFACB82,
+ 0x0000001F, 0x3FE10000, 0xDC80F979, 0x3FFB3787,
+ 0xFFFFFFF8, 0x3FE17FFF, 0xDBA56E4F, 0x3FFBA540,
+ 0xFFFFFFFA, 0x3FE1FFFF, 0x31256441, 0x3FFC14B4,
+ 0xFFFFFFC4, 0x3FE27FFF, 0xD43F7C9B, 0x3FFC85E8,
+ 0xFFFFFFFD, 0x3FE2FFFF, 0xD84758A6, 0x3FFCF8E5,
+ 0x0000001F, 0x3FE38000, 0x6D16CD84, 0x3FFD6DB2,
+ 0xFFFFFFD8, 0x3FE3FFFF, 0xDF80E39B, 0x3FFDE455,
+ 0x00000052, 0x3FE48000, 0x99C6A59C, 0x3FFE5CD7,
+ 0xFFFFFFC8, 0x3FE4FFFF, 0x240DC10C, 0x3FFED73F,
+ 0x00000013, 0x3FE58000, 0x24D90F71, 0x3FFF5394,
+ 0xFFFFFFBC, 0x3FE5FFFF, 0x6182F885, 0x3FFFD1DE,
+ 0x0000002D, 0x3FE68000, 0xDF5CE741, 0x40002912,
+ 0x00000040, 0x3FE70000, 0x207F0A2A, 0x40006A39,
+ 0x0000004F, 0x3FE78000, 0x0691652A, 0x4000AC66,
+ 0xFFFFFF6F, 0x3FE7FFFF, 0xB467DCAB, 0x4000EF9D,
+ 0xFFFFFFE5, 0x3FE87FFF, 0x5D82E943, 0x400133E4,
+ 0x00000035, 0x3FE90000, 0x4652CC6D, 0x4001793E,
+ 0xFFFFFFB3, 0x3FE97FFF, 0xC47BDA48, 0x4001BFAF,
+ 0x00000000, 0x3FEA0000, 0x3F1BD518, 0x4002073D,
+ 0x0000004A, 0x3FEA8000, 0x2F105CE2, 0x40024FEB,
+ 0xFFFFFFED, 0x3FEAFFFF, 0x1F3E7F11, 0x400299BE,
+ 0xFFFFFFFB, 0x3FEB7FFF, 0xACDB6611, 0x4002E4BA,
+ 0x0000001D, 0x3FEC0000, 0x87B62B39, 0x400330E5,
+ 0x00000079, 0x3FEC8000, 0x7282D538, 0x40037E43,
+ 0xFFFFFF51, 0x3FECFFFF, 0x43268248, 0x4003CCD9,
+ 0xFFFFFF74, 0x3FED7FFF, 0xE304CADC, 0x40041CAB,
+ 0x00000011, 0x3FEE0000, 0x4F4E5343, 0x40046DC0,
+ 0x0000001E, 0x3FEE8000, 0x9950A124, 0x4004C01B,
+ 0xFFFFFF9E, 0x3FEEFFFF, 0xE6C73196, 0x400513C2,
+ 0xFFFFFFED, 0x3FEF7FFF, 0x722DD586, 0x400568BB,
+ 0x00000034, 0x3FF00000, 0x8B1457B0, 0x4005BF0A,
+ 0xFFFFFFE2, 0x3FF03FFF, 0x967376DF, 0x400616B5,
+ 0xFFFFFF4B, 0x3FF07FFF, 0x0F0337A9, 0x40066FC2,
+ 0xFFFFFFFD, 0x3FF0BFFF, 0x859290F5, 0x4006CA35,
+ 0xFFFFFFE4, 0xBF8FFFFF, 0xABFEEFA5, 0x3FEF80FE,
+ 0xFFFFFB0B, 0xBF9FFFFF, 0x6A88B5FE, 0x3FEF03F5,
+ 0xFFFFFFA7, 0xBFA7FFFF, 0x6AFECFC5, 0x3FEE88DC,
+ 0xFFFFFEA8, 0xBFAFFFFF, 0xFBC702B8, 0x3FEE0FAB,
+ 0xFFFFFFB3, 0xBFB3FFFF, 0x89D041AC, 0x3FED985C,
+ 0xFFFFFFE3, 0xBFB7FFFF, 0xA0197C06, 0x3FED22E6,
+ 0xFFFFFF9A, 0xBFBBFFFF, 0xE73A4C89, 0x3FECAF42,
+ 0xFFFFFF98, 0xBFBFFFFF, 0x24ED822D, 0x3FEC3D6A,
+ 0xFFFFFFE9, 0xBFC1FFFF, 0x3B9D7B67, 0x3FEBCD55,
+ 0xFFFFFFE0, 0xBFC3FFFF, 0x29F24C2D, 0x3FEB5EFD,
+ 0xFFFFF553, 0xBFC5FFFF, 0x0A61A9F4, 0x3FEAF25B,
+ 0xFFFFFF8B, 0xBFC7FFFF, 0x12C08794, 0x3FEA8768,
+ 0xFFFFFE51, 0xBFC9FFFF, 0x93D68828, 0x3FEA1E1D,
+ 0xFFFFFF6E, 0xBFCBFFFF, 0xF8F2F3F5, 0x3FE9B674,
+ 0xFFFFFF7F, 0xBFCDFFFF, 0xC7837A0C, 0x3FE95067,
+ 0xFFFFFF7A, 0xBFCFFFFF, 0x9EAC8225, 0x3FE8EBEF,
+ 0xFFFFFFFE, 0xBFD0FFFF, 0x36E31F55, 0x3FE88906,
+ 0xFFFFFF41, 0xBFD1FFFF, 0x6188975E, 0x3FE827A5,
+ 0xFFFFFFBA, 0xBFD2FFFF, 0x08877656, 0x3FE7C7C7,
+ 0xFFFFFFF8, 0xBFD3FFFF, 0x2DF22F81, 0x3FE76965,
+ 0xFFFFFF90, 0xBFD4FFFF, 0xEBA33C2F, 0x3FE70C79,
+ 0xFFFFFFDB, 0xBFD5FFFF, 0x72DEB8AA, 0x3FE6B0FF,
+ 0xFFFFFF9A, 0xBFD6FFFF, 0x0BF5798E, 0x3FE656F0,
+ 0xFFFFFF9F, 0xBFD7FFFF, 0x15E98EB0, 0x3FE5FE46,
+ 0xFFFFFFEE, 0xBFD8FFFF, 0x061433CE, 0x3FE5A6FC,
+ 0xFFFFFC4A, 0xBFD9FFFF, 0x67CD26CD, 0x3FE5510C,
+ 0xFFFFFF30, 0xBFDAFFFF, 0xDC13566B, 0x3FE4FC71,
+ 0xFFFFFFF0, 0xBFDBFFFF, 0x1936FD0E, 0x3FE4A927,
+ 0xFFFFFFF3, 0xBFDCFFFF, 0xEA84FB8C, 0x3FE45726,
+ 0xFFFFFFF3, 0xBFDDFFFF, 0x2FF3912B, 0x3FE4066C,
+ 0xFFFFFF80, 0xBFDEFFFF, 0xDDD05AB9, 0x3FE3B6F1,
+ 0xFFFFFFDF, 0xBFDFFFFF, 0xFC6F9614, 0x3FE368B2,
+ 0x00000000, 0xBFE08000, 0xA7DCA843, 0x3FE31BAA,
+ 0xFFFFFFA4, 0xBFE0FFFF, 0x0F8BDCE4, 0x3FE2CFD4,
+ 0xFFFFFF0A, 0xBFE17FFF, 0x760D5CE7, 0x3FE2852A,
+ 0x00000000, 0xBFE20000, 0x30C1568B, 0x3FE23BA9,
+ 0xFFFFFFBB, 0xBFE27FFF, 0xA78D568D, 0x3FE1F34B,
+ 0xFFFFFE32, 0xBFE2FFFF, 0x5492C1DB, 0x3FE1AC0D,
+ 0xFFFFF042, 0xBFE37FFF, 0xC3E67EF2, 0x3FE165E9,
+ 0xFFFFFF77, 0xBFE3FFFF, 0x93499431, 0x3FE120DC,
+ 0xFFFFFF6B, 0xBFE47FFF, 0x71E34ECE, 0x3FE0DCE1,
+ 0xFFFFFFF1, 0xBFE4FFFF, 0x1FFBE588, 0x3FE099F4,
+ 0xFFFFFE02, 0xBFE57FFF, 0x6EB8A7AE, 0x3FE05810,
+ 0xFFFFFFE5, 0xBFE5FFFF, 0x3FD9002E, 0x3FE01732,
+ 0xFFFFFFB0, 0xBFE67FFF, 0x0AE9386C, 0x3FDFAEAB,
+ 0xFFFFFFB2, 0xBFE6FFFF, 0x837503D7, 0x3FDF30EC,
+ 0xFFFFFF7F, 0xBFE77FFF, 0x0D627133, 0x3FDEB521,
+ 0xFFFFFFE8, 0xBFE7FFFF, 0xEBEFCD95, 0x3FDE3B40,
+ 0xFFFFFFC8, 0xBFE87FFF, 0x8110DAE2, 0x3FDDC344,
+ 0xFFFFFB30, 0xBFE8FFFF, 0x4CF4EF06, 0x3FDD4D24,
+ 0xFFFFFFEF, 0xBFE97FFF, 0xED8EE395, 0x3FDCD8D8,
+ 0xFFFFFFA7, 0xBFE9FFFF, 0x1E1F1E5C, 0x3FDC665B,
+ 0xFFFFFFDC, 0xBFEA7FFF, 0xB6BF18D6, 0x3FDBF5A3,
+ 0xFFFFFF95, 0xBFEAFFFF, 0xABEEF93B, 0x3FDB86AB,
+ 0xFFFFFFCB, 0xBFEB7FFF, 0x0E24D256, 0x3FDB196C,
+ 0xFFFFFF32, 0xBFEBFFFF, 0x095DADF7, 0x3FDAADDE,
+ 0xFFFFFF6A, 0xBFEC7FFF, 0xE4B047C9, 0x3FDA43FA,
+ 0xFFFFFFB6, 0xBFECFFFF, 0x01E182A4, 0x3FD9DBBC,
+ 0xFFFFFFCA, 0xBFED7FFF, 0xDCFA81EC, 0x3FD9751A,
+ 0xFFFFFFCD, 0xBFEDFFFF, 0x0BE0699E, 0x3FD91011,
+ 0xFFFFFFFB, 0xBFEE7FFF, 0x3DEDBC69, 0x3FD8AC98,
+ 0xFFFFFF88, 0xBFEEFFFF, 0x3B8D51A9, 0x3FD84AAA,
+ 0xFFFFFFBB, 0xBFEF7FFF, 0xE5D6D92E, 0x3FD7EA40,
+ 0xFFFFFFDB, 0xBFEFFFFF, 0x362CEF53, 0x3FD78B56,
+ 0xFFFFFF00, 0xBFF03FFF, 0x3DDCB1F2, 0x3FD72DE4,
+ 0xFFFFFE6F, 0xBFF07FFF, 0x25BED085, 0x3FD6D1E5,
+ 0xFFFFFFD6, 0xBFF0BFFF, 0x2DDA1C57, 0x3FD67753 } };
+
+#endif
+#endif
+
+
+#ifdef BIG_ENDI
+
+static const mynumber
+/* Following three values used to scale x to primary range */
+ invln2_32 = {{0x40471547, 0x652b82fe}}, /* 4.61662413084468283841e+01 */
+ ln2_32hi = {{0x3f962e42, 0xfee00000}}, /* 2.16608493865351192653e-02 */
+ ln2_32lo = {{0x3d9a39ef, 0x35793c76}}, /* 5.96317165397058656257e-12 */
+/* t2-t5 terms used for polynomial computation */
+ t2 = {{0x3fc55555, 0x55548f7c}}, /* 1.6666666666526086527e-1 */
+ t3 = {{0x3fa55555, 0x55545d4e}}, /* 4.1666666666226079285e-2 */
+ t4 = {{0x3f811115, 0xb7aa905e}}, /* 8.3333679843421958056e-3 */
+ t5 = {{0x3f56c172, 0x8d739765}}, /* 1.3888949086377719040e-3 */
+/* maximum value for x to not overflow */
+ threshold1 = {{0x40862E42, 0xFEFA39EF}}, /* 7.09782712893383973096e+02 */
+/* maximum value for -x to not underflow */
+ threshold2 = {{0x40874910, 0xD52D3051}}, /* 7.45133219101941108420e+02 */
+/* scaling factor used when result near zero*/
+ twom54 = {{0x3c900000, 0x00000000}}; /* 5.55111512312578270212e-17 */
+
+#else
+#ifdef LITTLE_ENDI
+
+static const mynumber
+/* Following three values used to scale x to primary range */
+ invln2_32 = {{0x652b82fe, 0x40471547}}, /* 4.61662413084468283841e+01 */
+ ln2_32hi = {{0xfee00000, 0x3f962e42}}, /* 2.16608493865351192653e-02 */
+ ln2_32lo = {{0x35793c76, 0x3d9a39ef}}, /* 5.96317165397058656257e-12 */
+/* t2-t5 terms used for polynomial computation */
+ t2 = {{0x55548f7c, 0x3fc55555}}, /* 1.6666666666526086527e-1 */
+ t3 = {{0x55545d4e, 0x3fa55555}}, /* 4.1666666666226079285e-2 */
+ t4 = {{0xb7aa905e, 0x3f811115}}, /* 8.3333679843421958056e-3 */
+ t5 = {{0x8d739765, 0x3f56c172}}, /* 1.3888949086377719040e-3 */
+/* maximum value for x to not overflow */
+ threshold1 = {{0xFEFA39EF, 0x40862E42}}, /* 7.09782712893383973096e+02 */
+/* maximum value for -x to not underflow */
+ threshold2 = {{0xD52D3051, 0x40874910}}, /* 7.45133219101941108420e+02 */
+/* scaling factor used when result near zero*/
+ twom54 = {{0x00000000, 0x3c900000}}; /* 5.55111512312578270212e-17 */
+
+#endif
+#endif
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
@@ -42,7 +41,7 @@ libm-sysdep_routines += e_expf-fma
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
@@ -64,14 +63,13 @@ 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
@@ -82,7 +80,6 @@ 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 2 of proposed patch. Revised copyright notice and formatting issues. Removed slowexp.c and related references. Replaced tables of double float 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 381 5173 766 min 399 54 15 13 mean 5317 199 1349 24 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 | 379 +++++++++++++----------- sysdeps/ieee754/dbl-64/e_pow.c | 2 +- sysdeps/ieee754/dbl-64/eexp.tbl | 440 +++++++++++++++++++++++++++ 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, 651 insertions(+), 313 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