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precision. The new polynomial has degree 4 instead of 10, and a maximum
error of 2**-30.04 ulps instead of 2**-33.15. This doesn't affect the
final error significantly; the maximum error was and is about 0.5015
ulps on i386 -O1, and the number of cases with an error of > 0.5 ulps
is increased from 13851 to 14407.
Note that the error is only this close to 0.5 ulps due to excessive
extra precision caused by compiler bugs on i386. The extra precision
could be obtained intentionally, and is useful for keeping the error
of the hyperbolic float functions below 1 ulp, since these functions
are implemented using expm1f. My recent change for scaling by 2**k
had the unintentional side effect of retaining extra precision for
longer, so callers of expm1f see errors of more like 0.0015 ulps than
0.5015 ulps, and for the hyperbolic functions this reduces the maximum
error from nearly about 2 ulps to about 0.75 ulps.
This is about 10% faster on i386 (A64). expm1* is still very slow,
but now the float version is actually significantly faster. The
algorithm is very sophisticated but not very good except on machines
with fast division.
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exponent bits of the reduced result, construct 2**k (hopefully in
parallel with the construction of the reduced result) and multiply by
it. This tends to be much faster if the construction of 2**k is
actually in parallel, and might be faster even with no parallelism
since adjustment of the exponent requires a read-modify-wrtite at an
unfortunate time for pipelines.
In some cases involving exp2* on amd64 (A64), this change saves about
40 cycles or 30%. I think it is inherently only about 12 cycles faster
in these cases and the rest of the speedup is from partly-accidentally
avoiding compiler pessimizations (the construction of 2**k is now
manually scheduled for good results, and -O2 doesn't always mess this
up). In most cases on amd64 (A64) and i386 (A64) the speedup is about
20 cycles. The worst case that I found is expf on ia64 where this
change is a pessimization of about 10 cycles or 5%. The manual
scheduling for plain exp[f] is harder and not as tuned.
Details specific to expm1*:
- the saving is closer to 12 cycles than to 40 for expm1* on i386 (A64).
For some reason it is much larger for negative args.
- also convert to __FBSDID().
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- float ynf(int n, float x) /* wrapper ynf */
+float
+ynf(int n, float x) /* wrapper ynf */
This is because the __STDC__ stuff was indented.
Reviewed by: md5
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Reviewed by: md5
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This will make a number of things easier in the future, as well as (finally!)
avoiding the Id-smashing problem which has plagued developers for so long.
Boy, I'm glad we're not using sup anymore. This update would have been
insane otherwise.
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-- Begin comments from J.T. Conklin:
The most significant improvement is the addition of "float" versions
of the math functions that take float arguments, return floats, and do
all operations in floating point. This doesn't help (performance)
much on the i386, but they are still nice to have.
The float versions were orginally done by Cygnus' Ian Taylor when
fdlibm was integrated into the libm we support for embedded systems.
I gave Ian a copy of my libm as a starting point since I had already
fixed a lot of bugs & problems in Sun's original code. After he was
done, I cleaned it up a bit and integrated the changes back into my
libm.
-- End comments
Reviewed by: jkh
Submitted by: jtc
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