Remove the slow paths from pow. Like several other double precision math
functions, pow is exactly rounded. This is not required from math functions
and causes major overheads as it requires multiple fallbacks using higher
precision arithmetic if a result is close to 0.5ULP. Ridiculous slowdowns
of up to 100000x have been reported when the highest precision path triggers.
All GLIBC math tests pass on AArch64 and x64 (with ULP of pow set to 1).
The worst case error is ~0.506ULP. A simple test over a few hundred million
values shows pow is 10% faster on average. This fixes BZ #13932.
[BZ #13932]
* sysdeps/ieee754/dbl-64/uexp.h (err_1): Remove.
* benchtests/pow-inputs: Update comment for slow path cases.
* manual/probes.texi (slowpow_p10): Delete removed probe.
(slowpow_p10): Likewise.
* math/Makefile: Remove halfulp.c and slowpow.c.
* sysdeps/aarch64/libm-test-ulps: Set ULP of pow to 1.
* sysdeps/generic/math_private.h (__exp1): Remove error argument.
(__halfulp): Remove.
(__slowpow): Remove.
* sysdeps/i386/fpu/halfulp.c: Delete file.
* sysdeps/i386/fpu/slowpow.c: Likewise.
* sysdeps/ia64/fpu/halfulp.c: Likewise.
* sysdeps/ia64/fpu/slowpow.c: Likewise.
* sysdeps/ieee754/dbl-64/e_exp.c (__exp1): Remove error argument,
improve comments and add error analysis.
* sysdeps/ieee754/dbl-64/e_pow.c (__ieee754_pow): Add error analysis.
(power1): Remove function:
(log1): Remove error argument, add error analysis.
(my_log2): Remove function.
* sysdeps/ieee754/dbl-64/halfulp.c: Delete file.
* sysdeps/ieee754/dbl-64/slowpow.c: Likewise.
* sysdeps/m68k/m680x0/fpu/halfulp.c: Likewise.
* sysdeps/m68k/m680x0/fpu/slowpow.c: Likewise.
* sysdeps/powerpc/power4/fpu/Makefile: Remove CPPFLAGS-slowpow.c.
* sysdeps/x86_64/fpu/libm-test-ulps: Set ULP of pow to 1.
* sysdeps/x86_64/fpu/multiarch/Makefile: Remove slowpow-fma.c,
slowpow-fma4.c, halfulp-fma.c, halfulp-fma4.c.
* sysdeps/x86_64/fpu/multiarch/e_pow-fma.c (__slowpow): Remove define.
* sysdeps/x86_64/fpu/multiarch/e_pow-fma4.c (__slowpow): Likewise.
* sysdeps/x86_64/fpu/multiarch/halfulp-fma.c: Delete file.
* sysdeps/x86_64/fpu/multiarch/halfulp-fma4.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/slowpow-fma.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/slowpow-fma4.c: Likewise.
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 395 5173 144
min 399 54 15 13
mean 5317 200 1349 23
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.
* sysdeps/ieee754/dbl-64/e_exp.c: Include <math-svid-compat.h> and
<errno.h>. Include "eexp.tbl".
(half): New constant.
(one): Likewise.
(__ieee754_exp): Rewrite.
(__slowexp): Remove prototype.
* sysdeps/ieee754/dbl-64/eexp.tbl: New file.
* sysdeps/ieee754/dbl-64/slowexp.c: Remove file.
* sysdeps/i386/fpu/slowexp.c: Likewise.
* sysdeps/ia64/fpu/slowexp.c: Likewise.
* sysdeps/m68k/m680x0/fpu/slowexp.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/slowexp-avx.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/slowexp-fma.c: Likewise.
* sysdeps/x86_64/fpu/multiarch/slowexp-fma4.c: Likewise.
* sysdeps/generic/math_private.h (__slowexp): Remove prototype.
* sysdeps/ieee754/dbl-64/e_pow.c: Remove mention of slowexp.c in
comment.
* sysdeps/powerpc/power4/fpu/Makefile [$(subdir) = math]
(CPPFLAGS-slowexp.c): Remove variable.
* sysdeps/x86_64/fpu/multiarch/Makefile (libm-sysdep_routines):
Remove slowexp-fma, slowexp-fma4 and slowexp-avx.
(CFLAGS-slowexp-fma.c): Remove variable.
(CFLAGS-slowexp-fma4.c): Likewise.
(CFLAGS-slowexp-avx.c): Likewise.
* sysdeps/x86_64/fpu/multiarch/e_exp-avx.c (__slowexp): Do not
define as macro.
* sysdeps/x86_64/fpu/multiarch/e_exp-fma.c (__slowexp): Likewise.
* sysdeps/x86_64/fpu/multiarch/e_exp-fma4.c (__slowexp): Likewise.
* math/Makefile (type-double-routines): Remove slowexp.
* manual/probes.texi (slowexp_p6): Remove.
(slowexp_p32): Likewise.
The mantissa of mp_no is intended to take only integral values. This
is a relatively good choice for powerpc due to its 4 fpus, but not for
other architectures, which suffer due to this choice. This change
makes the default mantissa a long integer and allows powerpc to
override it. Additionally, some operations have been optimized for
integer manipulation, resulting in a significant improvement in
performance.
