arg to __kernel_rem_pio2() gives 53-bit (double) precision, not single
precision and/or the array dimension like I thought. prec == 2 is
used in e_rem_pio2.c for double precision although it is documented
to be for 64-bit (extended) precision, and I just reduced it by 1
thinking that this would give the value suitable for 24-bit (float)
precision. Reducing it 1 more to the documented value for float
precision doesn't actually work (it gives errors of ~0.75 ulps in the
reduced arg, but errors of much less than 0.5 ulps are needed; the bug
seems to be in kernel_rem_pio2.c). Keep using a value 1 larger than
the documented value but supply an array large enough hold the extra
unused result from this.
The bug can also be fixed quickly by increasing init_jk[0] in
k_rem_pio2.c from 2 to 3. This gives behaviour identical to using
prec == 1 except it doesn't create the extra result. It isn't clear
how the precision bug affects higher precisions. 113-bit (quad) is
the largest precision, so there is no way to use a large precision
to fix it.
they can be #included in other .c files to give inline functions, and
use them to inline the functions in most callers (not in e_lgammaf_r.c).
__kernel_tanf() is too large and complicated for gcc to inline very well.
An athlons, this gives a speed increase under favourable pipeline
conditions of about 10% overall (larger for AXP, smaller for A64).
E.g., on AXP, sinf() on uniformly distributed args in [-2Pi, 2Pi]
now takes 30-56 cycles; it used to take 45-61 cycles; hardware fsin
takes 65-129.
On athlons, this gives a speedup of 10-20% for tanf() on uniformly
distributed args in [-2Pi, 2Pi]. (It only directly applies for 43%
of the args and gives a 16-20% speedup for these (more for AXP than
A64) and this gives an overall speedup of 10-12% which is all that it
should; however, it gives an overall speedup of 17-20% with gcc-3.3
on AXP-A64 by mysteriously effected cases where it isn't executed.)
I originally intended to use double precision for all internals of
float trig functions and will probably still do this, but benchmarking
showed that converting to double precision and back is a pessimization
in cases where a simple float precision calculation works, so it may
be optimal to switch precisions only when using extra precision is
much simpler.
__ieee754_rem_pio2f() to its 3 callers and manually inline them.
On Athlons, with favourable compiler flags and optimizations and
favourable pipeline conditions, this gives a speedup of 30-40 cycles
for cosf(), sinf() and tanf() on the range pi/4 < |x| <= 9pi/4, so
thes functions are now signifcantly faster than the hardware trig
functions in many cases. E.g., in a benchmark with uniformly distributed
x in [-2pi, 2pi], A64 hardware fcos took 72-129 cycles and cosf() took
37-55 cycles. Out-of-order execution is needed to get both of these
times. The optimizations in this commit apparently work more by
removing 1 serialization point than by reducing latency.
s_cosf.c and s_sinf.c:
Use a non-bogus magic constant for the threshold of pi/4. It was 2 ulps
smaller than pi/4 rounded down, but its value is not critical so it should
be the result of natural rounding.
s_cosf.c and s_tanf.c:
Use a literal 0.0 instead of an unnecessary variable initialized to
[(float)]0.0. Let the function prototype convert to 0.0F.
Improved wording in some comments.
Attempted to improve indentation of comments.
number of branches.
Use a non-bogus magic constant for the threshold of pi/4. It was 2 ulps
smaller than pi/4 rounded down, but its value is not critical so it should
be the result of natural rounding. Use "<=" comparisons with rounded-
down thresholds for all small multiples of pi/4.
Cleaned up previous commit:
- use static const variables instead of expressions for multiples of pi/2
to ensure that they are evaluated at compile time. gcc currently
evaluates them at compile time but C99 compilers are not required
to do so. We want compile time evaluation for optimization and don't
care about side effects.
- use M_PI_2 instead of a magic constant for pi/2. We need magic constants
related to pi/2 elsewhere but not here since we just want pi/2 rounded
to double and even prefer it to be rounded in the default rounding mode.
We can depend on the cmpiler being C99ish enough to round M_PI_2 correctly
just as much as we depended on it handling hex constants correctly. This
also fixes a harmless rounding error in the hex constant.
- keep using expressions n*<value for pi/2> in the initializers for the
static const variables. 2*M_PI_2 and 4*M_PI_2 are obviously rounded in
the same way as the corresponding infinite precision expressions for
multiples of pi/2, and 3*M_PI_2 happens to be rounded like this, so we
don't need magic constants for the multiples.
- fixed and/or updated some comments.
define also, but res_config.h was not included into libc/net/name6.c.
So getipnodebyaddr() ignored the multiple PTRs.
PR: kern/88241
Submitted by: Dan Lukes <dan__at__obluda.cz>
MFC after: 3 days
The threshold for not being tiny was too small. Use the usual 2**-12
threshold. This change is not just an optimization, since the general
code that we fell into has accuracy problems even for tiny x. Avoiding
it fixes 2*1366 args with errors of more than 1 ulp, with a maximum
error of 1.167 ulps.
The magic number 22 is log(DBL_EPSILON)/2 plus slop. This is bogus
for float precision. Use 9 (~log(FLT_EPSILON)/2 plus less slop than
for double precision). The code for handling the interval
[2**-28, 9_was_22] has accuracy problems even for [9, 22], so this
change happens to fix errors of more than 1 ulp in about 2*17000
cases. It leaves such errors in about 2*1074000 cases, with a max
error of 1.242 ulps.
The threshold for switching from returning exp(x)/2 to returning
exp(x/2)^2/2 was a little smaller than necessary. As for coshf(),
This was not quite harmless since the exp(x/2)^2/2 case is inaccurate,
and fixing it avoids accuracy problems in 2*6 cases, leaving problems
in 2*19997 cases.
Fixed naming errors in pseudo-code in comments.
The threshold for not being tiny was confusing and too small. Use the
usual 2**-12 threshold and simplify the algorithm slightly so that
this threshold works (now use the threshold for sinhf() instead of one
for 1+expm1()). This is just a small optimization.
The magic number 22 is log(DBL_EPSILON)/2 plus slop. This is bogus
for float precision. Use 9 (~log(FLT_EPSILON)/2 plus less slop than
for double precision).
The threshold for switching from returning exp(x)/2 to returning
exp(x/2)^2/2 was a little smaller than necessary. This was not quite
harmless since the exp(x/2)^2/2 case is inaccurate. Fixing it happens
to avoid accuracy problems for 2*6 of the 2*151 args that were handled
by the exp(x)/2 case. This leaves accuracy problems for about 2*19997
args near the overflow threshold (~89); the maximum error there is
2.5029 ulps.
There are also accuracy probles for args in +-[0.5*ln2, 9] -- 2*188885
args with errors of more than 1 ulp, with a maximum error of 1.384 ulps.
Fixed a syntax error and naming errors in pseudo-code in comments.
specialized for float precision. The new polynomial has degree 8
instead of 14, and a maximum error of 2**-34.34 (absolute) instead of
2**-30.66. This doesn't affect the final error significantly; the
maximum error was and is about 0.8879 ulps on amd64 -01.
The fdlibm expf() is not used on i386's (the "optimized" asm version
is used), but probably should be since it was already significantly
faster than the asm version on athlons. The asm version has the
advantage of being more accurate, so keep using it for now.
polynomial for __kernel_tanf(). The old one was the double-precision
polynomial with coefficients truncated to float. Truncation is not
a good way to convert minimax polynomials to lower precision. Optimize
for efficiency and use the lowest-degree polynomial that gives a
relative error of less than 1 ulp. It has degree 13 instead of 27,
and happens to be 2.5 times more accurate (in infinite precision) than
the old polynomial (the maximum error is 0.017 ulps instead of 0.041
ulps).
Unlike for cosf and sinf, the old accuracy was close to being inadequate
-- the polynomial for double precision has a max error of 0.014 ulps
and nearly this small an error is needed. The new accuracy is also a
bit small, but exhaustive checking shows that even the old accuracy
was enough. The increased accuracy reduces the maximum relative error
in the final result on amd64 -O1 from 0.9588 ulps to 0.9044 ulps.
special case pi/4 <= |x| < 3*pi/4. This gives a tiny optimization (it
saves 2 subtractions, which are scheduled well so they take a whole 1
cycle extra on an AthlonXP), and simplifies the code so that the
following optimization is not so ugly.
Optimize for the range 3*pi/4 < |x| < 9*Pi/2 in the same way. On
Athlon{XP,64} systems, this gives a 25-40% optimization (depending a
lot on CFLAGS) for the cosf() and sinf() consumers on this range.
Relative to i387 hardware fcos and fsin, it makes the software versions
faster in most cases instead of slower in most cases. The relative
optimization is smaller for tanf() the inefficient part is elsewhere.
The 53-bit approximation to pi/2 is good enough for pi/4 <= |x| <
3*pi/4 because after losing up to 24 bits to subtraction, we still
have 29 bits of precision and only need 25 bits. Even with only 5
extra bits, it is possible to get perfectly rounded results starting
with the reduced x, since if x is nearly a multiple of pi/2 then x is
not near a half-way case and if x is not nearly a multiple of pi/2
then we don't lose many bits. With our intentionally imperfect rounding
we get the same results for cosf(), sinf() and tanf() as without this
optimization.
standard in C99 and POSIX.1-2001+. They are also not deprecated, since
apart from being standard they can handle special args slightly better
than the ilogb() functions.
Move their documentation to ilogb.3. Try to use consistent and improved
wording for both sets of functions. All of ieee854, C99 and POSIX
have better wording and more details for special args.
Add history for the logb() functions and ilogbl(). Fix history for
ilogb().
