Date: Wed, 29 May 2013 09:24:41 -0700 From: Steve Kargl <sgk@troutmask.apl.washington.edu> To: Bruce Evans <brde@optusnet.com.au> Cc: freebsd-numerics@freebsd.org Subject: Re: Patches for s_expl.c Message-ID: <20130529162441.GA58773@troutmask.apl.washington.edu> In-Reply-To: <20130529062437.V4648@besplex.bde.org> References: <20130528172242.GA51485@troutmask.apl.washington.edu> <20130529062437.V4648@besplex.bde.org>
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On Wed, May 29, 2013 at 07:39:04AM +1000, Bruce Evans wrote:
> On Tue, 28 May 2013, Steve Kargl wrote:
>
> > Here are two patches for ld80/s_expl.c and ld128/s_expl.c.
> > Instead of committing the one large patch that I have spent
> > hours testing, I have split it into two. One patch fixes/updates
> > expl(). The other patch is the implementation of expm1l().
> >
> > My commit messages will be:
> >
> > Patch 1:
> >
> > ld80/s_expl.c:
> >
> > * Use the LOG2_INTERVALS macro instead of hardcoding 7.
>
> The use of LOG2_INTERVALS isn't merged into the ld128 version. Patch 2
> merges its use for expm1l() only.
Hopefully, fixed.
> > * Use LD80C to set overflow and underflow thresholds, and then use
> > #defines to access the .e component to reduce diffs with ld128 version.
> > * Rename polynomial coefficients P# to A#, which is used in Tang.
>
> Almost all the declarations polynomial coefficients are still formatted
> in a nonstandard way, but differently than in previous development
> versions. I keep sending you patches for this.
Hopefully, fixed. All fancy whitespace has been removed including
in comments with hex values.
> > * Compute expm1l(x) for IEEE 754 128-bit format.
>
> There is a fairly large bug in this, from only merging half of the
> most recent micro-optimization in the development version of the ld80
> version. This might only be an efficiency bug, but I haven't tested
> the ld128 version with either the full merge or the half merge.
>
> The ld128 version still has excessive optimizations for |x| near 0.
> It uses a slightly different high-degree polynomial on each side of
> 0. The ld80 version uses the same poly on each side. Most of the
> style bugs in the 4 exp[!2]l functions are in the coeffs for the
> polys on each side. I haven't tried so hard to get you to fix them
> since I want to remove them.
Hopefully, fixed to the extent that opened ld80/s_expl.c in one
nedit window and ld128/s_expl.c in another. I copied everything
from ld80 to ld128 except of course literal constants and
polynomials that must be different.
> >
> > These are based on:
> >
> > PTP Tang, "Table-driven implementation of the Expm1 function
> > in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
> > 211-222 (1992).
> >
> > These commit logs may be too terse for some, but quite frankly after
> > 2 or 3 years of submitting and resubmitting diffs, I've forgotten
> > why some changes have or have not been made.
> >
> > expm1l() resides in s_expl.c because she shares the same table,
> > polynomial coefficients, and some numerical constants with expl().
>
> There are some minor style regressions relative to previous development
> versions outside of poly coeffs. Patches later.
I'm sure you're going to hate the new patch at the end.
> > Index: ld80/s_expl.c
> > ===================================================================
> > --- ld80/s_expl.c (revision 251062)
> > +++ ld80/s_expl.c (working copy)
> > ...
> > @@ -78,11 +82,11 @@
> > * |exp(x) - p(x)| < 2**-77.2
> > * (0.002708 is ln2/(2*INTERVALS) rounded up a little).
> > */
> > -P2 = 0.5,
> > -P3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
> > -P4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
> > -P5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
> > -P6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
> > +A2 = 0.5,
> > +A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
> > +A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
> > +A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
> > +A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
>
> Example of a formatting regression. The extra space that was before the
> values is for a possible minus sign. This space is still there for the
> hex values. The extra space before the equals sign is used for fancy
> formatting to line up the values when the variable names reach A10. Since
> thee variable names only reach A6, this is not needed.
All coefficient are now formatted with the form:
A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
ie., 1 space before and 1 space after =. The space in the comments
for the implicit + sign has been removed.
> > Index: ld128/s_expl.c
> > ===================================================================
> > --- ld128/s_expl.c (revision 251062)
> > +++ ld128/s_expl.c (working copy)
> > ...
> > @@ -38,34 +40,56 @@
> > #include "math_private.h"
> >
> > #define INTERVALS 128
> > +#define LOG2_INTERVALS 7
>
> Not used.
Hopefully, fixed.
> > n2 = (unsigned)n % INTERVALS;
> > k = (n - n2) / INTERVALS;
> > r1 = x - fn * L1;
> > - r2 = -fn * L2;
> > + r2 = fn * -L2;
> > + r = r1 + r2;
>
> 1 micro-optimization (that uses LOG2_INTERVALS) not merrged here.
>
Hopefully, fixed.
> > ...
> > + if (k > LDBL_MANT_DIG - 1)
> > + t = s[n2].lo - twomk + t * (q + r1) + s[n2].hi;
> > + else
> > + t = s[n2].lo + t * (q + r1) + (s[n2].hi - twomk);
>
> The last statement isn't accurate enough for k = 0 and k = -1, so
> handling of those cases were moved earlier so that this statement
> could be optimized to what it is now. The ld128 version is missing
> this.
ld80 code merged into ld128.
> > --- ld128/s_expl.c 2013-05-28 09:36:11.000000000 -0700
> > +++ ld128/s_expl.c.all 2013-05-28 09:34:52.000000000 -0700
> > ...
> > + if (k == 0) {
> > + t = s[n2].lo * (r1 + 1) + t * q + s[n2].hi * r1 +
> > + (s[n2].hi - 1);
> > + RETURNI(t);
> > + }
> > +
> > + if (k == -1) {
> > + t = s[n2].lo * (r1 + 1) + t * q + s[n2].hi * r1 +
> > + (s[n2].hi - 2);
> > + RETURNI(t / 2);
> > + }
> > +
> > +
>
> Same as for ld808, except for 2 style bugs instead of 1 (1 more extra
> blank line).
Hopefully, fixed.
> > + if (k > LDBL_MANT_DIG - 1)
> > + t = s[n2].lo - twomk + t * (q + r1) + s[n2].hi;
> > + else if (k < 1)
> > + t = s[n2].lo * (r1 + 1) + t * q + s[n2].hi * r1 +
> > + (s[n2].hi - twomk);
> > + else
> > + t = s[n2].lo * (q + r1 + 1) + s[n2].hi * (q + r1) +
> > + (s[n2].hi - twomk);
>
> Not the same as for ld128. Still has the old slower code, so it probably
> still works, but even more slowly than before except for k == 0 and k == -1,
> since there are extra branches to filter out those values.
ld80 and ld128 now use identical code.
>
> Some patches relative to my version now instead of later:
>
> @ --- z22/s_expl.c Wed May 29 04:48:10 2013
> @ +++ ./s_expl.c Wed May 29 06:16:29 2013
> @ @@ -30,5 +30,5 @@
> @ __FBSDID("$FreeBSD: src/lib/msun/ld80/s_expl.c,v 1.10 2012/10/13 19:53:11 kargl Exp $");
> @
> @ -/*-
> @ +/**
> @ * Compute the exponential of x for Intel 80-bit format. This is based on:
> @ *
>
> This ugliness is now required by style(9) :-(. You only made this change in
> some places places.
Hopefully, fixed.
> @ @@ -83,9 +83,9 @@
> @ * (0.002708 is ln2/(2*INTERVALS) rounded up a little).
> @ */
> @ -A2 = 0.5,
> @ -A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
> @ -A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
> @ -A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
> @ -A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
> @ +A2 = 0.5,
> @ +A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
> @ +A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
> @ +A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
> @ +A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
> @
> @ /*
>
> Fix regressions relative to a previous development version.
I made this conform to style(9).
> @ @@ -267,11 +275,12 @@
> @ r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */
> @ #if defined(HAVE_EFFICIENT_IRINTL)
> @ - n = irintl(fn);
> @ + n = irintl(fn);
> @ #elif defined(HAVE_EFFICIENT_IRINT)
> @ - n = irint(fn);
> @ + n = irint(fn);
> @ #else
> @ - n = (int)fn;
> @ + n = (int)fn;
>
> Fix more regressions.
Hopefully, fixed.
> @ #endif
> @ n2 = (unsigned)n % INTERVALS;
> @ + /* Depend on the sign bit being propagated: */
> @ k = n >> LOG2_INTERVALS;
> @ r1 = x - fn * L1;
>
> I think a comment is needed. This micro-optimization was merged from
> s_exp2*.c, where it is commented on more prominently for the long
> double versions only.
Ignored adding a comment.
>
> The coeffs have lots of style bugs, though not as many as for ld128.
>
Hopefully, fixed.
> @ @@ -389,4 +409,9 @@
> @ x4 = x2 * x2;
> @ q = x4 * (x2 * (x4 *
> @ + /*
> @ + * XXX the number of terms is no longer good for
> @ + * pairwise grouping of all except B3, and the
> @ + * grouping is no longer from highest down.
> @ + */
> @ (x2 * B12 + (x * B11 + B10)) +
> @ (x2 * (x * B9 + B8) + (x * B7 + B6))) +
I left this as-is with whitespace and did not add the comment.
This should be the only place where there is a substantial
deviation from style(9).
> @ @@ -407,9 +432,9 @@
> @ fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
> @ #if defined(HAVE_EFFICIENT_IRINTL)
> @ - n = irintl(fn);
> @ + n = irintl(fn);
> @ #elif defined(HAVE_EFFICIENT_IRINT)
> @ - n = irint(fn);
> @ + n = irint(fn);
> @ #else
> @ - n = (int)fn;
> @ + n = (int)fn;
> @ #endif
Hopefully, fixed.
> @ n2 = (unsigned)n % INTERVALS;
> @ @@ -434,22 +459,21 @@
> @
> @ if (k == 0) {
> @ - t = s[n2].lo * (r1 + 1) + t * q + s[n2].hi * r1 +
> @ - (s[n2].hi - 1);
> @ + t = SUM2P(s[n2].hi - 1, s[n2].lo * (r1 + 1) + t * q +
> @ + s[n2].hi * r1);
> @ RETURNI(t);
> @ }
> @ -
>
> Style bug (extra blank line between related statements).
Hopefully, fixed.
>
> @ if (k == -1) {
> @ - t = s[n2].lo * (r1 + 1) + t * q + s[n2].hi * r1 +
> @ - (s[n2].hi - 2);
> @ + t = SUM2P(s[n2].hi - 2, s[n2].lo * (r1 + 1) + t * q +
> @ + s[n2].hi * r1);
> @ RETURNI(t / 2);
> @ }
> @
>
> This blank line is correct since the statements are unrelated -- the
> evaluation method changes significantly. For k = 0 and k = -1, the
> evaluation is the same but we repeat it all to avoid using a variable
> for (k - 1) for the 2 values of k.
>
> @ if (k < -7) {
> @ - t = s[n2].lo + t * (q + r1) + s[n2].hi;
> @ + t = SUM2P(s[n2].hi, s[n2].lo + t * (q + r1));
> @ RETURNI(t * twopk - 1);
> @ }
> @
> @ if (k > 2 * LDBL_MANT_DIG - 1) {
> @ - t = s[n2].lo + t * (q + r1) + s[n2].hi;
> @ + t = SUM2P(s[n2].hi, s[n2].lo + t * (q + r1));
> @ if (k == LDBL_MAX_EXP)
> @ RETURNI(t * 2 * 0x1p16383L - 1);
>
> Ignore all the other changes in this hunk.
After making the changes, current unscientific testing gives
(best viewed in a 95 column window):
expl
Timing:
1M 2M 10M 100M
i386 [-11355.0:11356.0] 0.088302 0.867567 8.64871
amd64 [-11355.0:11356.0] 0.062994 0.631960 6.30295
sparc64 [-11355.0:11356.0] 39.5309 79.1927
Accuracy:
M Max ULP x at Max ULP
i386 [-11355.0:11356.0] 1 0.50465 -3.5510383760383760e+03 -0x1.bbe13a6062b8cdd4p+11
i386 [-11355.0:11356.0] 10 0.50556 -9.6479456830945683e+03 -0x1.2d7f90c24c5c686p+13
i386 [-11355.0:11356.0] 100 0.50654 -7.9982712426427124e+03 -0x1.f3e45702867bb01p+12
amd64 [-11355.0:11356.0] 1 0.50465 -3.5510383760383760e+03 -0x1.bbe13a6062b8cdd4p+11
amd64 [-11355.0:11356.0] 10 0.50556 -9.6479456830945683e+03 -0x1.2d7f90c24c5c686p+13
amd64 [-11355.0:11356.0] 100 0.50654 -7.9982712426427124e+03 -0x1.f3e45702867bb01p+12
sparc64 [-11355.0:11356.0] 1 0.50619 1.79779355979355979355979355979355983e+03
sparc64 {-11355.0:11356.0] 2 0.50541 1.11496704618352309176154588077294027e+04
expm1l
Timing:
1M 10M 100M
i386 [-64.0000:-0.1659] 0.435783 4.342621 43.41397
i386 [ -0.1659: 0.1659] 0.082880 0.829142 8.28948
i386 [ 0.1659:11356.0] 0.110590 1.096098 10.96253
amd64 [-64.0000:-0.1659] 0.066751 0.648734 6.46649
amd64 [ -0.1659: 0.1659] 0.061531 0.614824 6.14377
amd64 [ 0.1659:11356.0] 0.071677 0.716927 7.16819
sparc64 [-113.000:-0.1659] 37.84224
sparc64 [ -0.1659: 0.1659] 66.28533
sparc64 [ 0.1659:11356.0] 41.20714
Accuracy:
M Max ULP x at Max ULP
i386 [-64.0000:-0.1659] 1 0.50824 -1.7579429539429599e-01 -0x1.6806d6ec55bd2cp-3
i386 [ -0.1659: 0.1659] 1 0.50807 1.5765476175476175e-01 0x1.42e07fee5cecaa04p-3
i386 [ 0.1659:11356.0] 1 0.50533 4.6558240641420642e+03 0x1.22fd2f5de1bf8cb2p+12
i386 [-64.0000:-0.1659] 10 0.51163 -1.8666523480652408e-01 -0x1.7e4a57b65a7cp-3
i386 [ -0.1659: 0.1659] 10 0.51031 -1.6139564864956486e-01 -0x1.4a89cd45552be4a8p-3
i386 [ 0.1659:11356.0] 10 0.50597 7.2029609713952472e+03 0x1.c22f60238aafa618p+12
i386 [-64.0000:-0.1659] 100 0.51520 -1.8119337383093434e-01 -0x1.731582f6d89b72p-3
i386 [ -0.1659: 0.1659] 100 0.51161 1.6120475455904754e-01 0x1.4a25b7e6539760ecp-3
i386 [ 0.1659:11356.0] 100 0.50645 1.5581592136564341e+03 0x1.858a308e79dd8494p+10
amd64 [-64.0000:-0.1659] 1 0.50502 -1.8115636515636515e-01 -0x1.73021bbe7877ccp-3
amd64 [ -0.1659: 0.1659] 1 0.50807 1.5765476175476175e-01 0x1.42e07fee5cecaa04p-3
amd64 [ 0.1659:11356.0] 1 0.50522 5.3732636683514684e+03 0x1.4fd437fc4e28bfb6p+12
amd64 [-64.0000:-0.1659] 10 0.51363 -1.7086629347662934e-01 -0x1.5def25b3c452dap-3
amd64 [ -0.1659: 0.1659] 10 0.51031 -1.6139564864956486e-01 -0x1.4a89cd45552be4a8p-3
amd64 [ 0.1659:11356.0] 10 0.50595 2.2495034322503431e-01 0x1.ccb2c3fb0104dbe4p-3
amd64 [-64.0000:-0.1659] 100 0.51376 -2.7335577165055771e-01 -0x1.17ea934da5e086p-2
amd64 [ -0.1659: 0.1659] 100 0.51161 1.6120475455904754e-01 0x1.4a25b7e6539760ecp-3
amd64 [ 0.1659:11356.0] 100 0.50662 3.9436528827225188e+02 0x1.8a5d83883eef2676p+8
sparc64 [-113.000:-0.1659] 1 0.50339 -4.89331501511501510727132103685011835e+00
sparc64 [ -0.1659:0.1659] 1 0.50837 -1.28120218820218813976976441251060453e-01
sparc64 [ 0.1659:11356.] 1 0.50514 6.45515777662077662077313264157127259e+03
Testing on flame is excrudiating slow especially because rdivacky
is building clang. Yes, the following is one massive patch.
--
Steve
Index: ld80/s_expl.c
===================================================================
--- ld80/s_expl.c (revision 251067)
+++ ld80/s_expl.c (working copy)
@@ -29,7 +29,7 @@
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
-/*-
+/**
* Compute the exponential of x for Intel 80-bit format. This is based on:
*
* PTP Tang, "Table-driven implementation of the exponential function
@@ -50,6 +50,7 @@
#include "math_private.h"
#define INTERVALS 128
+#define LOG2_INTERVALS 7
#define BIAS (LDBL_MAX_EXP - 1)
static const long double
@@ -60,9 +61,12 @@
static const union IEEEl2bits
/* log(2**16384 - 0.5) rounded towards zero: */
-o_threshold = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L),
+/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
+o_thresholdu = LD80C(0xb17217f7d1cf79ab, 13, 11356.5234062941439488L),
+#define o_threshold (o_thresholdu.e)
/* log(2**(-16381-64-1)) rounded towards zero: */
-u_threshold = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
+u_thresholdu = LD80C(0xb21dfe7f09e2baa9, 13, -11399.4985314888605581L);
+#define u_threshold (u_thresholdu.e)
static const double
/*
@@ -70,19 +74,19 @@
* have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest
* bits zero so that multiplication of it by n is exact.
*/
-INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */
-L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */
+INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */
+L1 = 5.4152123484527692e-3, /* 0x162e42ff000000.0p-60 */
L2 = -3.2819649005320973e-13, /* -0x1718432a1b0e26.0p-94 */
/*
* Domain [-0.002708, 0.002708], range ~[-5.7136e-24, 5.7110e-24]:
* |exp(x) - p(x)| < 2**-77.2
* (0.002708 is ln2/(2*INTERVALS) rounded up a little).
*/
-P2 = 0.5,
-P3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
-P4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
-P5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
-P6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
+A2 = 0.5,
+A3 = 1.6666666666666119e-1, /* 0x15555555555490.0p-55 */
+A4 = 4.1666666666665887e-2, /* 0x155555555554e5.0p-57 */
+A5 = 8.3333354987869413e-3, /* 0x1111115b789919.0p-59 */
+A6 = 1.3888891738560272e-3; /* 0x16c16c651633ae.0p-62 */
/*
* 2^(i/INTERVALS) for i in [0,INTERVALS] is represented by two values where
@@ -96,8 +100,7 @@
static const struct {
double hi;
double lo;
-/* XXX should rename 's'. */
-} s[INTERVALS] = {
+} tbl[INTERVALS] = {
0x1p+0, 0x0p+0,
0x1.0163da9fb3335p+0, 0x1.b61299ab8cdb7p-54,
0x1.02c9a3e778060p+0, 0x1.dcdef95949ef4p-53,
@@ -232,7 +235,8 @@
expl(long double x)
{
union IEEEl2bits u, v;
- long double fn, q, r, r1, r2, t, t23, t45, twopk, twopkp10000, z;
+ long double fn, q, r, r1, r2, t, twopk, twopkp10000;
+ long double z;
int k, n, n2;
uint16_t hx, ix;
@@ -242,40 +246,38 @@
ix = hx & 0x7fff;
if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
if (ix == BIAS + LDBL_MAX_EXP) {
- if (hx & 0x8000 && u.xbits.man == 1ULL << 63)
- return (0.0L); /* x is -Inf */
- return (x + x); /* x is +Inf, NaN or unsupported */
+ if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
+ return (-1 / x);
+ return (x + x); /* x is +Inf, +NaN or unsupported */
}
- if (x > o_threshold.e)
+ if (x > o_threshold)
return (huge * huge);
- if (x < u_threshold.e)
+ if (x < u_threshold)
return (tiny * tiny);
- } else if (ix < BIAS - 66) { /* |x| < 0x1p-66 */
- /* includes pseudo-denormals */
- if (huge + x > 1.0L) /* trigger inexact iff x != 0 */
- return (1.0L + x);
+ } else if (ix < BIAS - 65) { /* |x| < 0x1p-65 (includes pseudos) */
+ return (1 + x); /* 1 with inexact iff x != 0 */
}
ENTERI();
- /* Reduce x to (k*ln2 + midpoint[n2] + r1 + r2). */
+ /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
/* Use a specialized rint() to get fn. Assume round-to-nearest. */
fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */
#if defined(HAVE_EFFICIENT_IRINTL)
- n = irintl(fn);
+ n = irintl(fn);
#elif defined(HAVE_EFFICIENT_IRINT)
- n = irint(fn);
+ n = irint(fn);
#else
- n = (int)fn;
+ n = (int)fn;
#endif
n2 = (unsigned)n % INTERVALS;
- k = (n - n2) / INTERVALS;
+ k = n >> LOG2_INTERVALS;
r1 = x - fn * L1;
- r2 = -fn * L2;
+ r2 = fn * -L2;
/* Prepare scale factors. */
- v.xbits.man = 1ULL << 63;
+ v.e = 1;
if (k >= LDBL_MIN_EXP) {
v.xbits.expsign = BIAS + k;
twopk = v.e;
@@ -284,21 +286,181 @@
twopkp10000 = v.e;
}
- /* Evaluate expl(midpoint[n2] + r1 + r2) = s[n2] * expl(r1 + r2). */
- /* Here q = q(r), not q(r1), since r1 is lopped like L1. */
- t45 = r * P5 + P4;
+ /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */
z = r * r;
- t23 = r * P3 + P2;
- q = r2 + z * t23 + z * z * t45 + z * z * z * P6;
- t = (long double)s[n2].lo + s[n2].hi;
- t = s[n2].lo + t * (q + r1) + s[n2].hi;
+ q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
+ t = (long double)tbl[n2].lo + tbl[n2].hi;
+ t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
/* Scale by 2**k. */
if (k >= LDBL_MIN_EXP) {
if (k == LDBL_MAX_EXP)
- RETURNI(t * 2.0L * 0x1p16383L);
+ RETURNI(t * 2 * 0x1p16383L);
RETURNI(t * twopk);
} else {
RETURNI(t * twopkp10000 * twom10000);
}
}
+
+/**
+ * Compute expm1l(x) for Intel 80-bit format. This is based on:
+ *
+ * PTP Tang, "Table-driven implementation of the Expm1 function
+ * in IEEE floating-point arithmetic," ACM Trans. Math. Soft., 18,
+ * 211-222 (1992).
+ */
+
+/*
+ * Our T1 and T2 are chosen to be approximately the points where method
+ * A and method B have the same accuracy. Tang's T1 and T2 are the
+ * points where method A's accuracy changes by a full bit. For Tang,
+ * this drop in accuracy makes method A immediately less accurate than
+ * method B, but our larger INTERVALS makes method A 2 bits more
+ * accurate so it remains the most accurate method significantly
+ * closer to the origin despite losing the full bit in our extended
+ * range for it.
+ */
+static const double
+T1 = -0.1659, /* ~-30.625/128 * log(2) */
+T2 = 0.1659; /* ~30.625/128 * log(2) */
+
+/*
+ * Domain [-0.1659, 0.1659], range ~[-1.2027e-22, 3.4417e-22]:
+ * |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-71.2
+ */
+static const union IEEEl2bits
+B3 = LD80C(0xaaaaaaaaaaaaaaab, -3, 1.66666666666666666671e-01L),
+B4 = LD80C(0xaaaaaaaaaaaaaaac, -5, 4.16666666666666666712e-02L);
+
+static const double
+B5 = 8.3333333333333245e-03, /* 0x1.111111111110cp-7 */
+B6 = 1.3888888888888861e-03, /* 0x1.6c16c16c16c0ap-10 */
+B7 = 1.9841269841532042e-04, /* 0x1.a01a01a0319f9p-13 */
+B8 = 2.4801587302069236e-05, /* 0x1.a01a01a03cbbcp-16 */
+B9 = 2.7557316558468562e-06, /* 0x1.71de37fd33d67p-19 */
+B10 = 2.7557315829785151e-07, /* 0x1.27e4f91418144p-22 */
+B11 = 2.5063168199779829e-08, /* 0x1.ae94fabdc6b27p-26 */
+B12 = 2.0887164654459567e-09; /* 0x1.1f122d6413fe1p-29 */
+
+long double
+expm1l(long double x)
+{
+ union IEEEl2bits u, v;
+ long double fn, hx2_hi, hx2_lo, q, r, r1, r2, t, twomk, twopk, x_hi;
+ long double x_lo, x2, z;
+ long double x4;
+ int k, n, n2;
+ uint16_t hx, ix;
+
+ /* Filter out exceptional cases. */
+ u.e = x;
+ hx = u.xbits.expsign;
+ ix = hx & 0x7fff;
+ if (ix >= BIAS + 6) { /* |x| >= 64 or x is NaN */
+ if (ix == BIAS + LDBL_MAX_EXP) {
+ if (hx & 0x8000) /* x is -Inf, -NaN or unsupported */
+ return (-1 / x - 1);
+ return (x + x); /* x is +Inf, +NaN or unsupported */
+ }
+ if (x > o_threshold)
+ return (huge * huge);
+ /*
+ * expm1l() never underflows, but it must avoid
+ * unrepresentable large negative exponents. We used a
+ * much smaller threshold for large |x| above than in
+ * expl() so as to handle not so large negative exponents
+ * in the same way as large ones here.
+ */
+ if (hx & 0x8000) /* x <= -64 */
+ return (tiny - 1); /* good for x < -65ln2 - eps */
+ }
+
+ ENTERI();
+
+ if (T1 < x && x < T2) {
+ if (ix < BIAS - 64) { /* |x| < 0x1p-64 (includes pseudos) */
+ /* x (rounded) with inexact if x != 0: */
+ RETURNI(x == 0 ? x :
+ (0x1p100 * x + fabsl(x)) * 0x1p-100);
+ }
+
+ x2 = x * x;
+ x4 = x2 * x2;
+
+ q = x4 * (x2 * (x4 *
+ (x2 * B12 + (x * B11 + B10)) +
+ (x2 * (x * B9 + B8) + (x * B7 + B6))) +
+ (x * B5 + B4.e)) + x2 * x * B3.e;
+
+ x_hi = (float)x;
+ x_lo = x - x_hi;
+ hx2_hi = x_hi * x_hi / 2;
+ hx2_lo = x_lo * (x + x_hi) / 2;
+ if (ix >= BIAS - 7)
+ RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi));
+ else
+ RETURNI(hx2_lo + q + hx2_hi + x);
+ }
+
+ /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
+ /* Use a specialized rint() to get fn. Assume round-to-nearest. */
+ fn = x * INV_L + 0x1.8p63 - 0x1.8p63;
+#if defined(HAVE_EFFICIENT_IRINTL)
+ n = irintl(fn);
+#elif defined(HAVE_EFFICIENT_IRINT)
+ n = irint(fn);
+#else
+ n = (int)fn;
+#endif
+ n2 = (unsigned)n % INTERVALS;
+ k = n >> LOG2_INTERVALS;
+ r1 = x - fn * L1;
+ r2 = fn * -L2;
+ r = r1 + r2;
+
+ /* Prepare scale factor. */
+ v.e = 1;
+ v.xbits.expsign = BIAS + k;
+ twopk = v.e;
+
+ /*
+ * Evaluate lower terms of
+ * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
+ */
+ z = r * r;
+ q = r2 + z * (A2 + r * A3) + z * z * (A4 + r * A5) + z * z * z * A6;
+
+ t = (long double)tbl[n2].lo + tbl[n2].hi;
+
+ if (k == 0) {
+ t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 +
+ (tbl[n2].hi - 1);
+ RETURNI(t);
+ }
+
+ if (k == -1) {
+ t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 +
+ (tbl[n2].hi - 2);
+ RETURNI(t / 2);
+ }
+
+ if (k < -7) {
+ t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
+ RETURNI(t * twopk - 1);
+ }
+
+ if (k > 2 * LDBL_MANT_DIG - 1) {
+ t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
+ if (k == LDBL_MAX_EXP)
+ RETURNI(t * 2 * 0x1p16383L - 1);
+ RETURNI(t * twopk - 1);
+ }
+
+ v.xbits.expsign = BIAS - k;
+ twomk = v.e;
+ if (k > LDBL_MANT_DIG - 1)
+ t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi;
+ else
+ t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk);
+ RETURNI(t * twopk);
+}
Index: ld128/s_expl.c
===================================================================
--- ld128/s_expl.c (revision 251067)
+++ ld128/s_expl.c (working copy)
@@ -1,5 +1,5 @@
/*-
- * Copyright (c) 2012 Steven G. Kargl
+ * Copyright (c) 2009-2012 Steven G. Kargl
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
@@ -22,6 +22,8 @@
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+ *
+ * Optimized by Bruce D. Evans.
*/
#include <sys/cdefs.h>
@@ -38,35 +40,56 @@
#include "math_private.h"
#define INTERVALS 128
+#define LOG2_INTERVALS 7
#define BIAS (LDBL_MAX_EXP - 1)
+static const long double
+huge = 0x1p10000L,
+twom10000 = 0x1p-10000L;
+/* XXX Prevent gcc from erroneously constant folding this: */
static volatile const long double tiny = 0x1p-10000L;
static const long double
-INV_L = 1.84664965233787316142070359168242182e+02L,
-L1 = 5.41521234812457272982212595914567508e-03L,
-L2 = -1.02536706388947310094527932552595546e-29L,
-huge = 0x1p10000L,
-o_threshold = 11356.523406294143949491931077970763428L,
-twom10000 = 0x1p-10000L,
+/* log(2**16384 - 0.5) rounded towards zero: */
+/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
+o_threshold = 11356.523406294143949491931077970763428L,
+/* log(2**(-16381-64-1)) rounded towards zero: */
u_threshold = -11433.462743336297878837243843452621503L;
+static const double
+/*
+ * ln2/INTERVALS = L1+L2 (hi+lo decomposition for multiplication). L1 must
+ * have at least 22 (= log2(|LDBL_MIN_EXP-extras|) + log2(INTERVALS)) lowest
+ * bits zero so that multiplication of it by n is exact.
+ */
+INV_L = 1.8466496523378731e+2, /* 0x171547652b82fe.0p-45 */
+L2 = -1.0253670638894731e-29; /* -0x1.9ff0342542fc3p-97 */
static const long double
-P2 = 5.00000000000000000000000000000000000e-1L,
-P3 = 1.66666666666666666666666666666666972e-1L,
-P4 = 4.16666666666666666666666666653708268e-2L,
-P5 = 8.33333333333333333333333315069867254e-3L,
-P6 = 1.38888888888888888888996596213795377e-3L,
-P7 = 1.98412698412698412718821436278644414e-4L,
-P8 = 2.48015873015869681884882576649543128e-5L,
-P9 = 2.75573192240103867817876199544468806e-6L,
-P10 = 2.75573236172670046201884000197885520e-7L,
-P11 = 2.50517544183909126492878226167697856e-8L;
+/* 0x1.62e42fefa39ef35793c768000000p-8 */
+L1 = 5.41521234812457272982212595914567508e-03L;
+static const long double
+/*
+ * Domain [-0.002708, 0.002708], range ~[-2.4011e-38, 2.4244e-38]:
+ * |exp(x) - p(x)| < 2**-124.9
+ * (0.002708 is ln2/(2*INTERVALS) rounded up a little).
+ */
+A2 = 0.5,
+A3 = 1.66666666666666666666666666651085500e-01L,
+A4 = 4.16666666666666666666666666425885320e-02L,
+A5 = 8.33333333333333333334522877160175842e-03L,
+A6 = 1.38888888888888888889971139751596836e-03L;
+
+static const double
+A7 = 1.9841269841269471e-04,
+A8 = 2.4801587301585284e-05,
+A9 = 2.7557324277411234e-06,
+A10 = 2.7557333722375072e-07;
+
static const struct {
long double hi;
long double lo;
-} s[INTERVALS] = {
+} tbl[INTERVALS] = {
0x1p0L, 0x0p0L,
0x1.0163da9fb33356d84a66aep0L, 0x3.36dcdfa4003ec04c360be2404078p-92L,
0x1.02c9a3e778060ee6f7cacap0L, 0x4.f7a29bde93d70a2cabc5cb89ba10p-92L,
@@ -201,9 +224,10 @@
expl(long double x)
{
union IEEEl2bits u, v;
- long double fn, r, r1, r2, q, t, twopk, twopkp10000;
+ long double q, r, r1, t, twopk, twopkp10000;
+ double dr, fn, r2;
int k, n, n2;
- uint32_t hx, ix;
+ uint16_t hx, ix;
/* Filter out exceptional cases. */
u.e = x;
@@ -211,31 +235,36 @@
ix = hx & 0x7fff;
if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
if (ix == BIAS + LDBL_MAX_EXP) {
- if (hx & 0x8000 && u.xbits.manh == 0 &&
- u.xbits.manl == 0)
- return (0.0L); /* x is -Inf */
- return (x + x); /* x is +Inf or NaN */
+ if (hx & 0x8000) /* x is -Inf or -NaN */
+ return (-1 / x);
+ return (x + x); /* x is +Inf or +NaN */
}
if (x > o_threshold)
return (huge * huge);
if (x < u_threshold)
return (tiny * tiny);
- } else if (ix < BIAS - 115) { /* |x| < 0x1p-115 */
- if (huge + x > 1.0L) /* trigger inexact iff x != 0 */
- return (1.0L + x);
+ } else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */
+ return (1 + x); /* 1 with inexact iff x != 0 */
}
- /* Reduce x to (k*ln2 + midpoint[n2] + r1 + r2). */
- fn = x * INV_L + 0x1.8p112 - 0x1.8p112;
- n = (int)fn;
+ ENTERI();
+
+ /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
+ /* Use a specialized rint() to get fn. Assume round-to-nearest. */
+ fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52;
+ r = x - fn * L1 - fn * L2; /* r = r1 + r2 done independently. */
+#if defined(HAVE_EFFICIENT_IRINT)
+ n = irint(fn);
+#else
+ n = (int)fn;
+#endif
n2 = (unsigned)n % INTERVALS;
- k = (n - n2) / INTERVALS;
+ k = n >> LOG2_INTERVALS;
r1 = x - fn * L1;
- r2 = -fn * L2;
+ r2 = fn * -L2;
/* Prepare scale factors. */
- v.xbits.manh = 0;
- v.xbits.manl = 0;
+ v.e = 1;
if (k >= LDBL_MIN_EXP) {
v.xbits.expsign = BIAS + k;
twopk = v.e;
@@ -244,18 +273,223 @@
twopkp10000 = v.e;
}
- r = r1 + r2;
- q = r * r * (P2 + r * (P3 + r * (P4 + r * (P5 + r * (P6 + r * (P7 +
- r * (P8 + r * (P9 + r * (P10 + r * P11)))))))));
- t = s[n2].lo + s[n2].hi;
- t = s[n2].hi + (s[n2].lo + t * (r2 + q + r1));
+ /* Evaluate expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2). */
+ dr = r;
+ q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 +
+ dr * (A7 + dr * (A8 + dr * (A9 + dr * A10))))))));
+ t = tbl[n2].lo + tbl[n2].hi;
+ t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
/* Scale by 2**k. */
if (k >= LDBL_MIN_EXP) {
if (k == LDBL_MAX_EXP)
- return (t * 2.0L * 0x1p16383L);
- return (t * twopk);
+ RETURNI(t * 2 * 0x1p16383L);
+ RETURNI(t * twopk);
} else {
- return (t * twopkp10000 * twom10000);
+ RETURNI(t * twopkp10000 * twom10000);
}
}
+
+/*
+ * Our T1 and T2 are chosen to be approximately the points where method
+ * A and method B have the same accuracy. Tang's T1 and T2 are the
+ * points where method A's accuracy changes by a full bit. For Tang,
+ * this drop in accuracy makes method A immediately less accurate than
+ * method B, but our larger INTERVALS makes method A 2 bits more
+ * accurate so it remains the most accurate method significantly
+ * closer to the origin despite losing the full bit in our extended
+ * range for it.
+ */
+static const double
+T1 = -0.1659, /* ~-30.625/128 * log(2) */
+T2 = 0.1659; /* ~30.625/128 * log(2) */
+
+/*
+ * Split the interval [T1:T2] into two intervals [T1:T3] and [T3:T2].
+ * Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear
+ * in both subintervals, so set T3 = 2**-5, which places the condition
+ * into the [T1:T3] interval.
+ */
+static const double
+T3 = 0.03125;
+
+/*
+ * XXX Estimated range is for absolute error.
+ * Domain [-0.1659, 0.03125], range ~[-1.8933e-38, 1.8943e-38]:
+ * |(exp(x)-1-x-x**2/2)/x**3 - p(x)| < 2**-125.3
+ */
+static const long double
+C3 = 1.66666666666666666666666666666666667e-01L,
+C4 = 4.16666666666666666666666666666666645e-02L,
+C5 = 8.33333333333333333333333333333371638e-03L,
+C6 = 1.38888888888888888888888888891188658e-03L,
+C7 = 1.98412698412698412698412697235950394e-04L,
+C8 = 2.48015873015873015873015112487849040e-05L,
+C9 = 2.75573192239858906525606685484412005e-06L,
+C10 = 2.75573192239858906612966093057020362e-07L,
+C11 = 2.50521083854417203619031960151253944e-08L,
+C12 = 2.08767569878679576457272282566520649e-09L,
+C13 = 1.60590438367252471783548748824255707e-10L;
+
+static const double
+C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae3p-37 */
+C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */
+C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */
+C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */
+C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */
+
+/*
+ * XXX Estimated range is for absolute error.
+ * Domain [0.03125, 0.1659], range ~[-2.7597e-38, 2.7602e-38]:
+ * |(exp(x)-1-x-x**2/2)/x**3 - p(x)| < 2**-124.8
+ */
+static const long double
+D3 = 1.66666666666666666666666666666682245e-01L,
+D4 = 4.16666666666666666666666666634228324e-02L,
+D5 = 8.33333333333333333333333364022244481e-03L,
+D6 = 1.38888888888888888888887138722762072e-03L,
+D7 = 1.98412698412698412699085805424661471e-04L,
+D8 = 2.48015873015873015687993712101479612e-05L,
+D9 = 2.75573192239858944101036288338208042e-06L,
+D10 = 2.75573192239853161148064676533754048e-07L,
+D11 = 2.50521083855084570046480450935267433e-08L,
+D12 = 2.08767569819738524488686318024854942e-09L,
+D13 = 1.60590442297008495301927448122499313e-10L;
+
+static const double
+D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */
+D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */
+D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */
+D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */
+
+long double
+expm1l(long double x)
+{
+ union IEEEl2bits u, v;
+ long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi;
+ long double x_lo, x2;
+ double dr, dx, fn, r2;
+ int k, n, n2;
+ uint16_t hx, ix;
+
+ /* Filter out exceptional cases. */
+ u.e = x;
+ hx = u.xbits.expsign;
+ ix = hx & 0x7fff;
+ if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */
+ if (ix == BIAS + LDBL_MAX_EXP) {
+ if (hx & 0x8000) /* x is -Inf or -NaN */
+ return (-1 / x - 1);
+ return (x + x); /* x is +Inf or +NaN */
+ }
+ if (x > o_threshold)
+ return (huge * huge);
+ /*
+ * expm1l() never underflows, but it must avoid
+ * unrepresentable large negative exponents. We used a
+ * much smaller threshold for large |x| above than in
+ * expl() so as to handle not so large negative exponents
+ * in the same way as large ones here.
+ */
+ if (hx & 0x8000) /* x <= -128 */
+ return (tiny - 1); /* good for x < -114ln2 - eps */
+ }
+
+ ENTERI();
+
+ if (T1 < x && x < T2) {
+ if (ix < BIAS - 113) { /* |x| < 0x1p-113 */
+ /* x (rounded) with inexact if x != 0: */
+ RETURNI(x == 0 ? x :
+ (0x1p200 * x + fabsl(x)) * 0x1p-200);
+ }
+
+ x2 = x * x;
+ dx = x;
+
+ if (x < T3) {
+ q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 +
+ x * (C7 + x * (C8 + x * (C9 + x * (C10 +
+ x * (C11 + x * (C12 + x * (C13 +
+ dx * (C14 + dx * (C15 + dx * (C16 +
+ dx * (C17 + dx * C18))))))))))))));
+ } else {
+ q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 +
+ x * (D7 + x * (D8 + x * (D9 + x * (D10 +
+ x * (D11 + x * (D12 + x * (D13 +
+ dx * (D14 + dx * (D15 + dx * (D16 +
+ dx * D17)))))))))))));
+ }
+
+ x_hi = (float)x;
+ x_lo = x - x_hi;
+ hx2_hi = x_hi * x_hi / 2;
+ hx2_lo = x_lo * (x + x_hi) / 2;
+ if (ix >= BIAS - 7)
+ RETURNI(hx2_lo + x_lo + q + (hx2_hi + x_hi));
+ else
+ RETURNI(hx2_lo + q + hx2_hi + x);
+ }
+
+ /* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
+ /* Use a specialized rint() to get fn. Assume round-to-nearest. */
+ fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52;
+#if defined(HAVE_EFFICIENT_IRINT)
+ n = irint(fn);
+#else
+ n = (int)fn;
+#endif
+ n2 = (unsigned)n % INTERVALS;
+ k = n >> LOG2_INTERVALS;
+ r1 = x - fn * L1;
+ r2 = fn * -L2;
+ r = r1 + r2;
+
+ /* Prepare scale factor. */
+ v.e = 1;
+ v.xbits.expsign = BIAS + k;
+ twopk = v.e;
+
+ /*
+ * Evaluate lower terms of
+ * expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
+ */
+ dr = r;
+ q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 +
+ dr * (A7 + dr * (A8 + dr * (A9 + dr * A10))))))));
+
+ t = tbl[n2].lo + tbl[n2].hi;
+
+ if (k == 0) {
+ t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 +
+ (tbl[n2].hi - 1);
+ RETURNI(t);
+ }
+
+ if (k == -1) {
+ t = tbl[n2].lo * (r1 + 1) + t * q + tbl[n2].hi * r1 +
+ (tbl[n2].hi - 2);
+ RETURNI(t / 2);
+ }
+
+ if (k < -7) {
+ t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
+ RETURNI(t * twopk - 1);
+ }
+
+ if (k > 2 * LDBL_MANT_DIG - 1) {
+ t = tbl[n2].lo + t * (q + r1) + tbl[n2].hi;
+ if (k == LDBL_MAX_EXP)
+ RETURNI(t * 2 * 0x1p16383L - 1);
+ RETURNI(t * twopk - 1);
+ }
+
+ v.xbits.expsign = BIAS - k;
+ twomk = v.e;
+
+ if (k > LDBL_MANT_DIG - 1)
+ t = tbl[n2].lo - twomk + t * (q + r1) + tbl[n2].hi;
+ else
+ t = tbl[n2].lo + t * (q + r1) + (tbl[n2].hi - twomk);
+ RETURNI(t * twopk);
+}
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