Copyright 2000, 2001, 2002, 2004 Free Software Foundation, Inc.

This file is part of the GNU MP Library.

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it under the terms of the GNU Lesser General Public License as published by
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along with the GNU MP Library.  If not, see http://www.gnu.org/licenses/.





               GMP SPEED MEASURING AND PARAMETER TUNING


The programs in this directory are for knowledgeable users who want to
measure GMP routines on their machine, and perhaps tweak some settings or
identify things that can be improved.

The programs here are tools, not ready to run solutions.  Nothing is built
in a normal "make all", but various Makefile targets described below exist.

Relatively few systems and CPUs have been tested, so be sure to verify that
results are sensible before relying on them.




MISCELLANEOUS NOTES

--enable-assert

    Don't configure with --enable-assert, since the extra code added by
    assertion checking may influence measurements.

Direct mapped caches

    Some effort has been made to accommodate CPUs with direct mapped caches,
    by putting data blocks more or less contiguously on the stack.  But this
    will depend on TMP_ALLOC using alloca, and even then it may or may not
    be enough.

FreeBSD 4.2 i486 getrusage

    This getrusage seems to be a bit doubtful, it looks like it's
    microsecond accurate, but sometimes ru_utime remains unchanged after a
    time of many microseconds has elapsed.  It'd be good to detect this in
    the time.c initializations, but for now the suggestion is to pretend it
    doesn't exist.

        ./configure ac_cv_func_getrusage=no

NetBSD 1.4.1 m68k macintosh time base

    On this system it's been found getrusage often goes backwards, making it
    unusable (time.c getrusage_backwards_p detects this).  gettimeofday
    sometimes doesn't update atomically when it crosses a 1 second boundary.
    Not sure what to do about this.  Expect possible intermittent failures.

SCO OpenUNIX 8 /etc/hw

    /etc/hw takes about a second to return the cpu frequency, which suggests
    perhaps it's measuring each time it runs.  If this is annoying when
    running the speed program repeatedly then set a GMP_CPU_FREQUENCY
    environment variable (see TIME BASE section below).

Low resolution timebase

    Parameter tuning can be very time consuming if the only timebase
    available is a 10 millisecond clock tick, to the point of being
    unusable.  This is currently the case on VAX and ARM systems.




PARAMETER TUNING

The "tuneup" program runs some tests designed to find the best settings for
various thresholds, like MUL_KARATSUBA_THRESHOLD.  Its output can be put
into gmp-mparam.h.  The program is built and run with

        make tune

If the thresholds indicated are grossly different from the values in the
selected gmp-mparam.h then there may be a performance boost in applicable
size ranges by changing gmp-mparam.h accordingly.

Be sure to do a full reconfigure and rebuild to get any newly set thresholds
to take effect.  A partial rebuild is enough sometimes, but a fresh
configure and make is certain to be correct.

If a CPU has specific tuned parameters coming from a gmp-mparam.h in one of
the mpn subdirectories then the values from "make tune" should be similar.
But check that the configured CPU is right and there are no machine specific
effects causing a difference.

It's hoped the compiler and options used won't have too much effect on
thresholds, since for most CPUs they ultimately come down to comparisons
between assembler subroutines.  Missing out on the longlong.h macros by not
using gcc will probably have an effect.

Some thresholds produced by the tune program are merely single values chosen
from what's a range of sizes where two algorithms are pretty much the same
speed.  When this happens the program is likely to give somewhat different
values on successive runs.  This is noticeable on the toom3 thresholds for
instance.




SPEED PROGRAM

The "speed" program can be used for measuring and comparing various
routines, and producing tables of data or gnuplot graphs.  Compile it with

	make speed

(Or on DOS systems "make speed.exe".)

Here are some examples of how to use it.  Check the code for all the
options.

Draw a graph of mpn_mul_n, stepping through sizes by 10 or a factor of 1.05
(whichever is greater).

        ./speed -s 10-5000 -t 10 -f 1.05 -P foo mpn_mul_n
	gnuplot foo.gnuplot

Compare mpn_add_n and an mpn_lshift by 1, showing times in cycles and
showing under mpn_lshift the difference between it and mpn_add_n.

	./speed -s 1-40 -c -d mpn_add_n mpn_lshift.1

Using option -c for times in cycles is interesting but normally only
necessary when looking carefully at assembler subroutines.  You might think
it would always give an integer value, but this doesn't happen in practice,
probably due to overheads in the time measurements.

In the free-form output the "#" symbol against a measurement means the
corresponding routine is fastest at that size.  This is a convenient visual
cue when comparing different routines.  The graph data files <name>.data
don't get this since it would upset gnuplot or other data viewers.




TIME BASE

The time measuring method is determined in time.c, based on what the
configured host has available.  A cycle counter is preferred, possibly
supplemented by another method if the counter has a limited range.  A
microsecond accurate getrusage() or gettimeofday() will work quite well too.

The cycle counters (except possibly on alpha) and gettimeofday() will depend
on the machine being otherwise idle, or rather on other jobs not stealing
CPU time from the measuring program.  Short routines (those that complete
within a timeslice) should work even on a busy machine.

Some trouble is taken by speed_measure() in common.c to avoid ill effects
from sporadic interrupts, or other intermittent things (like cron waking up
every minute).  But generally an idle machine will be necessary to be
certain of consistent results.

The CPU frequency is needed to convert between cycles and seconds, or for
when a cycle counter is supplemented by getrusage() etc.  The speed program
will convert as necessary according to the output format requested.  The
tune program will work with either cycles or seconds.

freq.c knows how to get the frequency on some systems, or can measure a
cycle counter against gettimeofday() or getrusage(), but when that fails, or
needs to be overridden, an environment variable GMP_CPU_FREQUENCY can be
used (in Hertz).  For example in "bash" on a 650 MHz machine,

	export GMP_CPU_FREQUENCY=650e6

A high precision time base makes it possible to get accurate measurements in
a shorter time.




EXAMPLE COMPARISONS - VARIOUS

Here are some ideas for things that can be done with the speed program.

There's always going to be a certain amount of overhead in the time
measurements, due to reading the time base, and in the loop that runs a
routine enough times to get a reading of the desired precision.  Noop
functions taking various arguments are available to measure this.  The
"overhead" printed by the speed program each time in its intro is the "noop"
routine, but note that this is just for information, it isn't deducted from
the times printed or anything.

	./speed -s 1 noop noop_wxs noop_wxys

To see how many cycles per limb a routine is taking, look at the time
increase when the size increments, using option -D.  This avoids fixed
overheads in the measuring.  Also, remember many of the assembler routines
have unrolled loops, so it might be necessary to compare times at, say, 16,
32, 48, 64 etc to see what the unrolled part is taking, as opposed to any
finishing off.

        ./speed -s 16-64 -t 16 -C -D mpn_add_n

The -C option on its own gives cycles per limb, but is really only useful at
big sizes where fixed overheads are small compared to the code doing the
real work.  Remember of course memory caching and/or page swapping will
affect results at large sizes.

        ./speed -s 500000 -C mpn_add_n

Once a calculation stops fitting in the CPU data cache, it's going to start
taking longer.  Exactly where this happens depends on the cache priming in
the measuring routines, and on what sort of "least recently used" the
hardware does.  Here's an example for a CPU with a 16kbyte L1 data cache and
32-bit limb, showing a suddenly steeper curve for mpn_add_n at about 2000
limbs.

        ./speed -s 1-4000 -t 5 -f 1.02 -P foo mpn_add_n
	gnuplot foo.gnuplot

When a routine has an unrolled loop for, say, multiples of 8 limbs and then
an ordinary loop for the remainder, it can happen that it's actually faster
to do an operation on, say, 8 limbs than it is on 7 limbs.  The following
draws a graph of mpn_sub_n, to see whether times smoothly increase with
size.

        ./speed -s 1-100 -c -P foo mpn_sub_n
	gnuplot foo.gnuplot

If mpn_lshift and mpn_rshift have special case code for shifts by 1, it
ought to be faster (or at least not slower) than shifting by, say, 2 bits.

        ./speed -s 1-200 -c mpn_rshift.1 mpn_rshift.2

An mpn_lshift by 1 can be done by mpn_add_n adding a number to itself, and
if the lshift isn't faster there's an obvious improvement that's possible.

        ./speed -s 1-200 -c mpn_lshift.1 mpn_add_n_self

On some CPUs (AMD K6 for example) an "in-place" mpn_add_n where the
destination is one of the sources is faster than a separate destination.
Here's an example to see this.  ".1" selects dst==src1 for mpn_add_n (and
mpn_sub_n), for other values see speed.h SPEED_ROUTINE_MPN_BINARY_N_CALL.

        ./speed -s 1-200 -c mpn_add_n mpn_add_n.1

The gmp manual points out that divisions by powers of two should be done
using a right shift because it'll be significantly faster than an actual
division.  The following shows by what factor mpn_rshift is faster than
mpn_divrem_1, using division by 32 as an example.

        ./speed -s 10-20 -r mpn_rshift.5 mpn_divrem_1.32




EXAMPLE COMPARISONS - MULTIPLICATION

mul_basecase takes a ".<r>" parameter which is the first (larger) size
parameter.  For example to show speeds for 20x1 up to 20x15 in cycles,

        ./speed -s 1-15 -c mpn_mul_basecase.20

mul_basecase with no parameter does an NxN multiply, so for example to show
speeds in cycles for 1x1, 2x2, 3x3, etc, up to 20x20, in cycles,

        ./speed -s 1-20 -c mpn_mul_basecase

sqr_basecase is implemented by a "triangular" method on most CPUs, making it
up to twice as fast as mul_basecase.  In practice loop overheads and the
products on the diagonal mean it falls short of this.  Here's an example
running the two and showing by what factor an NxN mul_basecase is slower
than an NxN sqr_basecase.  (Some versions of sqr_basecase only allow sizes
below SQR_KARATSUBA_THRESHOLD, so if it crashes at that point don't worry.)

        ./speed -s 1-20 -r mpn_sqr_basecase mpn_mul_basecase

The technique described above with -CD for showing the time difference in
cycles per limb between two size operations can be done on an NxN
mul_basecase using -E to change the basis for the size increment to N*N.
For instance a 20x20 operation is taken to be doing 400 limbs, and a 16x16
doing 256 limbs.  The following therefore shows the per crossproduct speed
of mul_basecase and sqr_basecase at around 20x20 limbs.

        ./speed -s 16-20 -t 4 -CDE mpn_mul_basecase mpn_sqr_basecase

Of course sqr_basecase isn't really doing NxN crossproducts, but it can be
interesting to compare it to mul_basecase as if it was.  For sqr_basecase
the -F option can be used to base the deltas on N*(N+1)/2 operations, which
is the triangular products sqr_basecase does.  For example,

        ./speed -s 16-20 -t 4 -CDF mpn_sqr_basecase

Both -E and -F are preliminary and might change.  A consistent approach to
using them when claiming certain per crossproduct or per triangularproduct
speeds hasn't really been established, but the increment between speeds in
the range karatsuba will call seems sensible, that being k to k/2.  For
instance, if the karatsuba threshold was 20 for the multiply and 30 for the
square,

        ./speed -s 10-20 -t 10 -CDE mpn_mul_basecase
        ./speed -s 15-30 -t 15 -CDF mpn_sqr_basecase



EXAMPLE COMPARISONS - MALLOC

The gmp manual recommends application programs avoid excessive initializing
and clearing of mpz_t variables (and mpq_t and mpf_t too).  Every new
variable will at a minimum go through an init, a realloc for its first
store, and finally a clear.  Quite how long that takes depends on the C
library.  The following compares an mpz_init/realloc/clear to a 10 limb
mpz_add.  Don't be surprised if the mallocing is quite slow.

        ./speed -s 10 -c mpz_init_realloc_clear mpz_add

On some systems malloc and free are much slower when dynamic linked.  The
speed-dynamic program can be used to see this.  For example the following
measures malloc/free, first static then dynamic.

        ./speed -s 10 -c malloc_free
        ./speed-dynamic -s 10 -c malloc_free

Of course a real world program has big problems if it's doing so many
mallocs and frees that it gets slowed down by a dynamic linked malloc.





EXAMPLE COMPARISONS - STRING CONVERSIONS

mpn_get_str does a binary to string conversion.  The base is specified with
a ".<r>" parameter, or decimal by default.  Power of 2 bases are much faster
than general bases.  The following compares decimal and hex for instance.

        ./speed -s 1-20 -c mpn_get_str mpn_get_str.16

Smaller bases need more divisions to split a given size number, and so are
slower.  The following compares base 3 and base 9.  On small operands 9 will
be nearly twice as fast, though at bigger sizes this reduces since in the
current implementation both divide repeatedly by 3^20 (or 3^40 for 64 bit
limbs) and those divisions come to dominate.

        ./speed -s 1-20 -cr mpn_get_str.3 mpn_get_str.9

mpn_set_str does a string to binary conversion.  The base is specified with
a ".<r>" parameter, or decimal by default.  Power of 2 bases are faster than
general bases on large conversions.

	./speed -s 1-512 -f 2 -c mpn_set_str.8 mpn_set_str.10

mpn_set_str also has some special case code for decimal which is a bit
faster than the general case, basically by giving the compiler a chance to
optimize some multiplications by 10.

	./speed -s 20-40 -c mpn_set_str.9 mpn_set_str.10 mpn_set_str.11




EXAMPLE COMPARISONS - GCDs

mpn_gcd_1 has a threshold for when to reduce using an initial x%y when both
x and y are single limbs.  This isn't tuned currently, but a value can be
established by a measurement like

	./speed -s 10-32 mpn_gcd_1.10

This runs src[0] from 10 to 32 bits, and y fixed at 10 bits.  If the div
threshold is high, say 31 so it's effectively disabled then a 32x10 bit gcd
is done by nibbling away at the 32-bit operands bit-by-bit.  When the
threshold is small, say 1 bit, then an initial x%y is done to reduce it to a
10x10 bit operation.

The threshold in mpn/generic/gcd_1.c or the various assembler
implementations can be tweaked up or down until there's no more speedups on
interesting combinations of sizes.  Note that this affects only a 1x1 limb
operation and so isn't very important.  (An Nx1 limb operation always does
an initial modular reduction, using mpn_mod_1 or mpn_modexact_1_odd.)




SPEED PROGRAM EXTENSIONS

Potentially lots of things could be made available in the program, but it's
been left at only the things that have actually been wanted and are likely
to be reasonably useful in the future.

Extensions should be fairly easy to make though.  speed-ext.c is an example,
in a style that should suit one-off tests, or new code fragments under
development.

many.pl is a script for generating a new speed program supplemented with
alternate versions of the standard routines.  It can be used for measuring
experimental code, or for comparing different implementations that exist
within a CPU family.




THRESHOLD EXAMINING

The speed program can be used to examine the speeds of different algorithms
to check the tune program has done the right thing.  For example to examine
the karatsuba multiply threshold,

	./speed -s 5-40 mpn_mul_basecase mpn_kara_mul_n

When examining the toom3 threshold, remember it depends on the karatsuba
threshold, so the right karatsuba threshold needs to be compiled into the
library first.  The tune program uses specially recompiled versions of
mpn/mul_n.c etc for this reason, but the speed program simply uses the
normal libgmp.la.

Note further that the various routines may recurse into themselves on sizes
far enough above applicable thresholds.  For example, mpn_kara_mul_n will
recurse into itself on sizes greater than twice the compiled-in
MUL_KARATSUBA_THRESHOLD.

When doing the above comparison between mul_basecase and kara_mul_n what's
probably of interest is mul_basecase versus a kara_mul_n that does one level
of Karatsuba then calls to mul_basecase, but this only happens on sizes less
than twice the compiled MUL_KARATSUBA_THRESHOLD.  A larger value for that
setting can be compiled-in to avoid the problem if necessary.  The same
applies to toom3 and DC, though in a trickier fashion.

There are some upper limits on some of the thresholds, arising from arrays
dimensioned according to a threshold (mpn_mul_n), or asm code with certain
sized displacements (some x86 versions of sqr_basecase).  So putting huge
values for the thresholds, even just for testing, may fail.




FUTURE

Make a program to check the time base is working properly, for small and
large measurements.  Make it able to test each available method, including
perhaps the apparent resolution of each.

Make a general mechanism for specifying operand overlap, and a syntax like
maybe "mpn_add_n.dst=src2" to select it.  Some measuring routines do this
sort of thing with the "r" parameter currently.



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