path: root/bdoc.txt

http://www.byte.com/bmark/bmark.htm
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BYTEmark

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This is release 2 of BYTE Magazine's BYTEmark benchmark program (previously
known as BYTE's Native Mode Benchmarks). This document covers the Native
Mode (a.k.a. Algorithm Level) tests; benchmarks designed to expose the
capabilities of a system's CPU, FPU, and memory system. Another group of
benchmarks within the BYTEmark suite includes the Application Simulation
Benchmarks. They are detailed in a separate document. [NOTE: The
documentation for the Application simulation benchmarks should appear before
the end of March, 95. -- RG].

The Tests

The Native Mode portion of the BYTEmark consists of a number of well-known
algorithms; some BYTE has used before in earlier versions of the benchmark,
others are new. The complete suite consists of 10 tests:

Numeric sort - Sorts an array of 32-bit integers.

String sort - Sorts an array of strings of arbitrary length.

Bitfield - Executes a variety of bit manipulation functions.

Emulated floating-point - A small software floating-point package.

Fourier coefficients - A numerical analysis routine for calculating series
approximations of waveforms.

Assignment algorithm - A well-known task allocation algorithm.

Huffman compression - A well-known text and graphics compression algorithm.

IDEA encryption - A relatively new block cipher algorithm.

Neural Net - A small but functional back-propagation network simulator.

LU Decomposition - A robust algorithm for solving linear equations.

A more complete description of each test can be found in later sections of
this document.

BYTE built the BYTEmark with the multiplatform world foremost in mind. There
were, of course, other considerations that we kept high on the list:

Real-world algorithms. The algorithms should actually do something. Previous
benchmarks often moved gobs of bytes from one point to another, added or
subtracted piles and piles of numbers, or (in some cases) actually executed
NOP instructions. We should not belittle those tests of yesterday, they had
their place. However, we think it better that tests be based on activities
that are more complex in nature.

Easy to port. All the benchmarks are written in "vanilla" ANSI C. This
provides us with the best chance of moving them quickly and accurately to
new processors and operating systems as they appear. It also simplifies
maintenance.

This means that as new 64-bit (and, perhaps, 128-bit) processors appear, the
benchmarks can test them as soon as a compiler is available.

Comprehensive. The algorithms were derived from a variety of sources. Some
are routines that BYTE had been using for some time. Others are routines
derived from well-known texts in the computer science world. Furthermore,
the algorithms differ in structure. Some simply "walk" sequentially through
one-dimensional arrays. Others build and manipulate two-dimensional arrays.
Finally, some benchmarks are "integer" tests, while others exercise the
floating-point coprocessor (if one is available).

Scalable. We wanted these benchmarks to be useful across as wide a variety
of systems as possible. We also wanted to give them a lifetime beyond the
next wave of new processors.

To that end, we incorporated "dynamic workload adjustment." A complete
description of this appears in a later section. In a nutshell, this allows
the tests to "expand or contract" depending on the capabilities of the
system under test, all the while providing consistent results so that fair
and accurate comparisons are possible.

Honesty In Advertising

We'd be lying if we said that the BYTEmark was all the benchmarking that
anyone would ever need to run on a system. It would be equally inaccurate to
suggest that the tests are completely free of inadequacies. There are many
things the tests do not do, there are shortcomings, and there are problems.

BYTE will continue to improve the BYTEmark. The source code is freely
available, and we encourage vendors and users to examine the routines and
provide us with their feedback. In this way, we assure fairness,
comprehensiveness, and accuracy.

Still, as we mentioned, there are some shortcomings. Here are those we
consider the most significant. Keep them in mind as you examine the results
of the benchmarks now and in the future.

At the mercy of C compilers. Being written in ANSI C, the benchmark program
is highly portable. This is a reflection of the "world we live in." If this
were a one-processor world, we might stand a chance at hand-crafting a
benchmark in assembly language. (At one time, that's exactly what BYTE did.)
Not today, no way.

The upshot is that the benchmarks must be compiled. For broadest coverage,
we selected ANSI C. And when they're compiled, the resulting executable's
performance can be highly dependent on the capabilities of the C compiler.
Today's benchmark results can be blown out of the water tomorrow if someone
new enters the scene with an optimizing strategy that outperforms existing
competition.

This concern is not easily waved off. It will require you to keep careful
track of compiler version and optimization switches. As BYTE builds its
database of benchmark results, version number and switch setting will become
an integral part of that data. This will be true for published information
as well, so that you can make comparisons fairly and accurately. BYTE will
control the distribution of test results so that all relevant compiler
information is attached to the data.

As a faint justification -- for those who think this situation results in
"polluted" tests -- we should point out that we are in the same boat as all
the other developers (at least, all those using C compilers -- and that's
quite a sizeable group). If the only C compilers for a given system happen
to be poor ones, everyone suffers. It's a fact that a given platform's
ultimate potential depends as much on the development software available as
on the technical achievements of the hardware design.

It's just CPU and FPU. It's very tempting to try to capture the performance
of a machine in a single number. That has never been possible -- though it's
been tried a lot -- and the gap between that ideal and reality will forever
widen.

These benchmarks are meant to expose the theoretical upper limit of the CPU,
FPU, and memory architecture of a system. They cannot measure video, disk,
or network throughput (those are the domains of a different set of
benchmarks). You should, therefore, use the results of these tests as part,
not all, of any evaluation of a system.

Single threaded. Currently, each benchmark test uses only a single execution
thread. It's unlikely that you'll find any modern operating system that does
not have some multitasking component. How a system "scales" as more tasks
are run simultaneously is an effect that the current benchmarks cannot
explore.

BYTE is working on a future version of the tests that will solve this
problem.

The tests are synthetic. This quite reasonable argument is based on the fact
that people don't run benchmarks for a living, they run applications.
Consequently, the only true measure of a system is how well it performs
whatever applications you will be running. This, in fact, is the philosophy
behind the BAPCo benchmarks.

This is not a point with which we would disagree. BYTE regularly makes use
of a variety of application benchmarks. None of this suggests, however, that
the BYTEmark benchmarks serve no purpose.

BYTEmark's results should be used as predictors. They can be moved to a new
platform long before native applications will be ported. The BYTEmark
benchmarks will therefore provide an early look at the potential of the
machine. Additionally, the BYTEmark permits you to "home in" on an aspect of
the overall architecture. How well does the system perform when executing
floating-point computations? Does its memory architecture help or hinder the
management of memory buffers that may fall on arbitrary address boundaries?
How does the cache work with a program whose memory access favors moving
randomly through memory as opposed to moving sequentially through memory?

The answers to these questions can give you a good idea of how well a system
would support a particular class of applications. Only a synthetic benchmark
can give the narrow view necessary to find the answers.

Dynamic Workloads

Our long history of benchmarking has taught us one thing above all others:
Tomorrow's system will go faster than today's by an amount exceeding your
wildest guess -- and then some. Dealing with this can become an unending
race.

It goes like this: You design a benchmark algorithm, you specify its
parameters (how big the array is, how many loops, etc.), you run it on
today's latest super-microcomputer, collect your data, and go home. A new
machine arrives the next day, you run your benchmark, and discover that the
test executes so quickly that the resolution of the clock routine you're
using can't keep up with it (i.e., the test is over and done before the
system clock even has a chance to tick).

If you modify your routine, the figures you collected yesterday are no good.
If you create a better clock routine by sneaking down into the system
hardware, you can kiss portability goodbye.

The BYTEmark benchmarks solve this problem by a process we'll refer to as
"dynamic workload adjustment." In principle, it simply means that if the
test runs so fast that the system clock can't time it, the benchmark
increases the test workload -- and keeps increasing it -- until enough time
is consumed to gather reliable test results.

Here's an example.

The BYTEmark benchmarks perform timing using a "stopwatch" paradigm. The
routine StartStopwatch() begins timing; StopStopwatch() ends timing and
reports the elapsed time in clock ticks. Now, "clock ticks" is a value that
varies from system to system. We'll presume that our test system provides
1000 clock ticks per second. (We'll also presume that the system actually
updates its clock 1000 times per second. Surprisingly, some systems don't do
that. One we know of will tell you that the clock provides 100 ticks per
second, but updates the clock in 5- or 6-tick increments. The resolution is
no better than somewhere around 1/18th of a second.) Here, when we say
"system" we mean not only the computer system, but the environment provided
by the C compiler. Interestingly, different C compilers for the same system
will report different clock ticks per second.

Built into the benchmarks is a global variable called GLOBALMINTICKS. This
variable is the minimum number of clock ticks that the benchmark will allow
StopStopwatch() to report.

Suppose you run the Numeric Sort benchmark. The benchmark program will
construct an array filled with random numbers, call StartStopwatch(), sort
the array, and call StopStopwatch(). If the time reported in StopStopwatch()
is less than GLOBALMINTICKS, then the benchmark will build two arrays, and
try again. If sorting two arrays took less time than GLOBALMINTICKS, the
process repeats with more arrays.

This goes on until the benchmark makes enough work so that an interval
between StartStopwatch() and StopStopwatch() exceeds GLOBALMINTICKS. Once
that happens, the test is actually run, and scores are calculated.

Notice that the benchmark didn't make bigger arrays, it made more arrays.
That's because the time taken by the sort test does not increase linearly as
the array grows, it increases by a factor of N*log(N) (where N is the size
of the array).

This principle is applied to all the benchmark tests. A machine with a less
accurate clock may be forced to sort more arrays at a time, but the results
are given in arrays per second. In this way fast machines, slow machines,
machines with accurate clocks, machines with less accurate clocks, can all
be tested with the same code.

Confidence Intervals

Another built-in feature of the BYTEmark is a set of statistical-analysis
routines. Running benchmarks is one thing; the question arises as to how
many times should a test be run until you know you have a good sampling.
Also, can you determine whether the test is stable (i.e., do results vary
widely from one execution of the benchmark to the next)?

The BYTEmark keeps score as follows: Each test (a test being a numeric
sort, a string sort, etc.) is run five times. These five scores are
averaged, the standard deviation is determined, and a 95% confidence
half-interval for the mean is calculated (using the student t
distribution). This tells us that the true average lies -- with a 95%
probability -- within plus or minus the confidence half-interval of
the calculated average. If this half-interval is within 5% of the
calculated average, the benchmarking stops. Otherwise, a new test is
run and the calculations are repeated with all of the runs done so
far, including the new one. The benchmark proceeds this way up to a
total of 30 runs. If the length of the half-interval is still bigger
than 5% of the calculated average then a warning issued that the
results might not be statistically certain before the average is
displayed.

** Fixed a statistical bug here. Uwe F. Mayer

The upshot is that, for each benchmark test, the true average is -- with a
95% level of confidence -- within 5% of the average reported. Here, the
"true average" is the average we would get were we able to run the tests
over and over again an infinite number of times.

This specification ensures that the calculation of results is controlled;
that someone running the tests in California will use the same technique for
determining benchmark results as someone running the tests in New York.

In case there is uneven system load due to other processes while this
benchmark suite executes, it might take longer to run the benchmark suite
as compared to a run an unloaded system. This is because the benchmark does
some statistical analysis to make sure that the reported results are
statistically significant (as explained above), and a high variation in
individual runs requires more runs to achieve the required statistical
confidence.

*** added last the paragraph, Uwe F. Mayer

Interpreting Results

Of course, running the benchmarks can present you with a boatload of data.
It can get mystifying, and some of the more esoteric statistical information
is valuable only to a limited audience. The big question is: What does it
all mean?

First, we should point out that the BYTEmark reports both "raw" and indexed
scores for each test. The raw score for a particular test amounts to the
"iterations per second" of that test. For example, the numeric sort test
reports as its raw score the number of arrays it was able to sort per
second.

The indexed score is the raw score of the system under test divided by the
raw score obtained on the baseline machine. As of this release, the
baseline machine is a DELL 90 Mhz Pentium XPS/90 with 16 MB of RAM and 256K
of external processor cache. (The compiler used was the Watcom C/C++ 10.0
compiler; optimizations set to "fastest possible code", 4-byte structure
alignment, Pentium code generation with Pentium register-based calling. The
operating system was MSDOS.) The indexed score serves to "normalize" the
raw scores, reducing their dynamic range and making them easier to
grasp. Simply put, if your machine has an index score of 2.0 on the numeric
sort test, it performed that test twice as fast as this 90 Mhz Pentium.

If you run all the tests (as you'll see, it is possible to perform "custom
runs", which execute only a subset of the tests) the BYTEmark will also
produce two overall index figures: Integer index and Floating-point index.
The Integer index is the geometric mean of those tests that involve only
integer processing -- numeric sort, string sort, bitfield, emulated
floating-point, assignment, Huffman, and IDEA -- while the Floating-point
index is the geometric mean of those tests that require the floating-point
coprocessor -- Fourier, neural net, and LU decomposition. You can use these
scores to get a general feel for the performance of the machine under test
as compared to the baseline 90 Mhz Pentium.

The Linux/Unix port has a second baseline machine, it is an AMD K6/233 with
32 MB RAM and 512 KB L2-cache running Linux 2.0.32 and using GNU gcc
version 2.7.2.3 and libc-5.4.38.  The integer index was split as suggested
by Andrew D. Balsa <andrewbalsa@usa.net>, and reflects the realization that
memory management is important in CPU design. The original tests have been
left alone, however, the geometric mean of the tests NUMERIC SORT, FP
EMULATION, IDEA, and HUFFMAN now constitutes the integer-arithmetic focused
benchmark index, while the geometric mean of the tests STRING SORT,
BITFIELD, and ASSIGNMENT makes up the new memory index. The floating point
index has been left alone, it is still the geometric mean of FOURIER,
NEURAL NET, and LU DECOMPOSITION.

*** added the section on Linux, Uwe F. Mayer

What follows is a list of the benchmarks and associated brief remarks that
describe what the tests do: What they exercise; what a "good" result or a
"bad" result means. Keep in mind that, in this expanding universe of faster
processors, bigger caches, more elaborate memory architectures, "good" and
"bad" are indeed relative terms. A good score on today's hot new processor
will be a bad score on tomorrow's hot new processor.

These remarks are based on empirical data and profiling that we have done to
date. (NOTE: The profiling is limited to Intel and Motorola 68K on this
release. As more data is gathered, we will be refining this section.
3/14/95--RG)

Benchmark                            Description

Numeric sort                         Generic integer performance.  Should
                                     exercise non-sequential performance
                                     of cache (or memory if cache is less
                                     than 8K).  Moves 32-bit longs at a
                                     time, so 16-bit processors will be
                                     at a disadvantage.



String sort                          Tests memory-move performance.
                                     Should exercise non-sequential
                                     performance of cache, with added
                                     burden that moves are byte-wide and
                                     can occur on odd address boundaries.
                                      May tax the performance of
                                     cell-based processors that must
                                     perform additional shift operations
                                     to deal with bytes.



Bitfield                             Exercises "bit twiddling"
                                     performance.  Travels through memory
                                     in a somewhat sequential fashion;
                                     different from sorts in that data is
                                     merely altered in place.  If
                                     properly compiled, takes into
                                     account 64-bit processors, which
                                     should see a boost.



Emulated F.P.                        Past experience has shown this test
                                     to be a good measurement of overall
                                     performance.



Fourier                              Good measure of transcendental and
                                     trigonometric performance of FPU.
                                     Little array activity, so this test
                                     should not be dependent of cache or
                                     memory architecture.



Assignment                           The test moves through large integer
                                     arrays in both row-wise and
                                     column-wise fashion.  Cache/memory
                                     with good sequential performance
                                     should see a boost (memory is
                                     altered in place -- no moving as in
                                     a sort operation).   Processing is
                                     done in 32-bit chunks -- no
                                     advantage given to 64-bit
                                     processors.



Huffman                              A combination of byte operations,
                                     bit twiddling, and overall integer
                                     manipulation.  Should be a good
                                     general measurement.



IDEA                                 Moves through data sequentially in
                                     16-bit chunks.  Should provide a
                                     good indication of raw speed.



Neural Net                           Small-array floating-point test
                                     heavily dependent on the exponential
                                     function; less dependent on overall
                                     FPU performance.  Small arrays, so
                                     cache/memory architecture should not
                                     come into play.



LU decomposition.                           A floating-point test that moves
                                     through arrays in both row-wise and
                                     column-wise fashion.  Exercises only
                                     fundamental math operations (+, -,
                                     *, /).

The Command File

Purpose

The BYTEmark program allows you to override many of its default parameters
using a command file. The command file also lets you request statistical
information, as well as specify an output file to hold the test results for
later use.

You identify the command file using a command-line argument. E.G.,

C:NBENCH -cCOMFILE.DAT

tells the benchmark program to read from COMFILE.DAT in the current
directory.

The content of the command file is simply a series of parameter names and
values, each on a single line. The parameters control internal variables
that are either global in nature (i.e., they effect all tests in the
program) or are specific to a given benchmark test.

The parameters are listed in a reference guide that follows, arranged in the
following groups:

Global Parameters

Numeric Sort

String Sort

Bitfield

Emulated floating-point

Fourier coefficients

Assignment algorithm

IDEA encryption

Huffman compression

Neural net

LU decomposition

As mentioned above, those items listed under "Global Parameters" affect all
tests; the rest deal with specific benchmarks. There is no required ordering
to parameters as they appear in the command file. You can specify them in
any sequence you wish.

You should be judicious in your use of a command file. Some parameters will
override the "dynamic workload" adjustment that each test performs. Doing
this completely bypasses the benchmark code that is designed to produce an
accurate reading from your system clock. Other parameters will alter default
settings, yielding test results that cannot be compared with published
benchmark results.

A Sample Command File

Suppose you built a command file that contained the following:

ALLSTATS=T

CUSTOMRUN=T

OUTFILE=D:\DATA.DAT

DONUMSORT=T

DOLU=T

Here's what this file tells the benchmark program:

ALLSTATS=T means that you've requested a "dump" of all the statistics the
test gathers. This includes not only the standard deviations of tests run,
it also produces test-specific information such as the number of arrays
built, the array size, etc.

CUSTOMRUN=T tells the system that this is a custom run. Only tests
explicitly specified will be executed.

OUTFILE=D:\DATA.DAT will write the output of the benchmark to the file
DATA.DAT on the root of the D: drive. (If DATA.DAT already exists, output
will be appended to the file.)

DONUMSORT=T tells the system to run the numeric sort benchmark. (This was
necessary on account of the CUSTOMRUN=T line, above.)

DOLU=T tells the system to run the LU decomposition benchmark.

Command File Parameters Reference

(NOTE: Altering some global parameters can invalidate results for comparison
purposes. Those parameters are indicated in the following section by a bold
asterisk (*). If you alter any parameters so indicated, you may NOT publish
the resulting data as BYTEmark scores.)

Global Parameters

GLOBALMINTICKS=<n>

This overrides the default global_min_ticks value (defined in NBENCH1.H).
The global_min_ticks value is defined as the minimum number of clock ticks
per iteration of a particular benchmark. For example, if global_min_ticks is
set to 100 and the numeric sort benchmark is run; each iteration MUST take
at least 100 ticks, or the system will expand the work-per-iteration.

MINSECONDS=<n>

Sets the minimum number of seconds any particular test will run. This has
the effect of controlling the number of repetitions done. Default: 5.

ALLSTATS=<T|F>

Set this flag to T for a "dump" of all statistics. The information displayed
varies from test to test. Default: F.

OUTFILE=<path>

Specifies that output should go to the specified output file. Any test
results and statistical data displayed on-screen will also be written to the
file. If the file does not exist, it will be created; otherwise, new output
will be appended to an existing file. This allows you to "capture" several
runs into a single file for later review.

Note: the path should not appear in quotes. For example, something like the
following would work: OUTFILE=C:\BENCH\DUMP.DAT

CUSTOMRUN=<T|F>

Set this flag to T for a custom run. A "custom run" means that the program
will run only the benchmark tests that you explicitly specify. So, use this
flag to run a subset of the tests. Default: F.

Numeric Sort

DONUMSORT=<T|F>

Indicates whether to do the numeric sort. Default is T, unless this is a
custom run (CUSTOMRUN=T), in which case default is F.

NUMNUMARRAYS=<n>

Indicates the number of numeric arrays the system will build. Setting this
value will override the program's "dynamic workload" adjustment for this
test.*

NUMARRAYSIZE=<n>

Indicates the number of elements in each numeric array. Default is 8001
entries. (NOTE: Altering this value will invalidate the test for comparison
purposes. The performance of the numeric sort test is not related to the
array size as a linear function; i.e., an array twice as big will not take
twice as long. The relationship involves a logarithmic function.)*

NUMMINSECONDS=<n>

Overrides MINSECONDS for the numeric sort test.

String Sort

DOSTRINGSORT=<T|F>

Indicates whether to do the string sort. Default is T, unless this is a
custom run (CUSTOMRUN=T), in which case the default is F.

STRARRAYSIZE=<n>

Sets the size of the string array. Default is 8111. (NOTE: Altering this
value will invalidate the test for comparison purposes. The performance of
the string sort test is not related to the array size as a linear function;
i.e., an array twice as big will not take twice as long. The relationship
involves a logarithmic function.)*

NUMSTRARRAYS=<n>

Sets the number of string arrays that will be created to run the test.
Setting this value will override the program's "dynamic workload" adjustment
for this test.*

STRMINSECONDS=<n>

Overrides MINSECONDS for the string sort test.

Bitfield

DOBITFIELD=<T|F>

Indicates whether to do the bitfield test. Default is T, unless this is a
custom run (CUSTOMRUN=T), in which case the default is F.

NUMBITOPS=<n>

Sets the number of bitfield operations that will be performed. Setting this
value will override the program's "dynamic workload" adjustment for this
test.*

BITFIELDSIZE=<n>

Sets the number of 32-bit elements in the bitfield arrays. The default value
is dependent on the size of a long as defined by the current compiler. For a
typical compiler that defines a long to be 32 bits, the default is 32768.
(NOTE: Altering this parameter will invalidate test results for comparison
purposes.)*

BITMINSECONDS=<n>

Overrides MINSECONDS for the bitfield test.

Emulated floating-point

DOEMF=<T|F>

Indicates whether to do the emulated floating-point test. Default is T,
unless this is a custom run (CUSTOMRUN=T), in which case the default is F.

EMFARRAYSIZE=<n>

Sets the size (number of elements) of the emulated floating-point benchmark.
Default is 3000. The test builds three arrays, each of equal size. This
parameter sets the number of elements for EACH array. (NOTE: Altering this
parameter will invalidate test results for comparison purposes.)*

EMFLOOPS=<n>

Sets the number of loops per iteration of the floating-point test. Setting
this value will override the program's "dynamic workload" adjustment for
this test.*

EMFMINSECONDS=<n>

Overrides MINSECONDS for the emulated floating-point test.

Fourier coefficients

DOFOUR=<T|F>

Indicates whether to do the Fourier test. Default is T, unless this is a
custom run (CUSTOMRUN=T), in which case the default is F.

FOURASIZE=<n>

Sets the size of the array for the Fourier test. This sets the number of
coefficients the test will derive. NOTE: Specifying this value will override
the system's "dynamic workload" adjustment for this test, and may make the
results invalid for comparison purposes.*

FOURMINSECONDS=<n>

Overrides MINSECONDS for the Fourier test.

Assignment Algorithm

DOASSIGN=<T|F>

Indicates whether to do the assignment algorithm test. Default is T, unless
this is a custom run (CUSTOMRUN=T), in which case the default is F.

ASSIGNARRAYS=<n>

Indicates the number of arrays that will be built for the test. Specifying
this value will override the system's "dynamic workload" adjustment for this
test. (NOTE: The size of the arrays in the assignment algorithm is fixed at
101 x 101. Altering the array size requires adjusting global constants and
recompiling; to do so, however, would invalidate test results.)*

ASSIGNMINSECONDS=<n>

Overrides MINSECONDS for the assignment algorithm test.

IDEA encryption

DOIDEA=<T|F>

Indicates whether to do the IDEA encryption test. Default is T, unless this
is a custom run (CUSTOMRUN=T), in which case the default is F.

IDEAARRAYSIZE=<n>

Sets the size of the plain-text character array that will be encrypted by the
test. Default is 4000. The benchmark actually builds 3 arrays: 1st
plain-text, encrypted version, and 2nd plain-text. The 2nd plain-text array is
the destination for the decryption process [part of the test]. All arrays
are set to the same size. (NOTE: Specifying this value will invalidate test
results for comparison purposes.)*

IDEALOOPS=<n>

Indicates the number of loops in the IDEA test. Specifying this value will
override the system's "dynamic workload" adjustment for this test.*

IDEAMINSECONDS=<n>

Overrides MINSECONDS for the IDEA test.

Huffman compression

DOHUFF=<T|F>

Indicates whether to do the Huffman test. Default is T, unless this is a
custom run (CUSTOMRUN=T), in which case the default is F.

HUFFARRAYSIZE=<n>

Sets the size of the string buffer that will be compressed using the Huffman
test. The default is 5000. (NOTE: Altering this value will invalidate test
results for comparison purposes.)*

HUFFLOOPS=<n>

Sets the number of loops in the Huffman test. Specifying this value will
override the system's "dynamic workload" adjustment for this test.*

HUFFMINSECONDS=<n>

Overrides MINSECONDS for the Huffman test.

Neural net

DONNET=<T|F>

Indicates whether to do the Neural Net test. Default is T, unless this is a
custom run (CUSTOMRUN=T), in which case the default is F.

NNETLOOPS=<n>

Sets the number of loops in the Neural Net test. NOTE: Altering this value
overrides the benchmark's "dynamic workload" adjustment algorithm, and may
invalidate the results for comparison purposes.*

NNETMINSECONDS=<n>

Overrides MINSECONDS for the Neural Net test.

LU decomposition

DOLU=<T|F>

Indicates whether to do the LU decomposition test. Default is T, unless this
is a custom run (CUSTOMRUN=T), in which case the default is F.

LUNUMARRAYS=<n>

Sets the number of arrays in each iteration of the LU decomposition test.
Specifying this value will override the system's "dynamic workload"
adjustment for this test.*

LUMINSECONDS=<n>

Overrides MINSECONDS for the LU decomposition test.

Numeric Sort

Description

This benchmark is designed to explore how well the system sorts a numeric
array. In this case, a numeric array is a one-dimensional collection of
signed, 32-bit integers. The actual sorting is performed by a heapsort
algorithm (see the text box following for a description of the heapsort
algorithm).

It's probably unnecessary to point out (but we'll do it anyway) that sorting
is a fundamental operation in computer application software. You'll likely
find sorting routines nestled deep inside a variety of applications;
everything from database systems to operating-systems kernels.

The numeric sort benchmark reports the number of arrays it was able to sort
per second. The array size is set by a global constant (it can be overridden
by the command file -- see below).

Analysis

Optimized 486 code: Profiling of the numeric sort benchmark using Watcom's
profiler (Watcom C/C++ 10.0) indicates that the algorithm spends most of its
time in the numsift() function (specifically, about 90% of the benchmark's
time takes place in numsift()). Within numsift(), two if statements dominate
time spent:

if(array[k]<array[k+1L]) and if(array[i]<array[k])

Both statements involve indexes into arrays, so it's likely the processor is
spending a lot of time resolving the array references. (Though both
statements involve "less-than" comparisons, we doubt that much time is
consumed in performing the signed compare operation.) Though the first
statement involves array elements that are adjacent to one another, the
second does not. In fact, the second statement will probably involve
elements that are far apart from one another during early passes through the
sifting process. We expect that systems whose caching system pre-fetches
contiguous elements (often in "burst" line fills) will not have any great
advantage of systems without pre-fetch mechanisms.

Similar results were found when we profiled the numeric sort algorithm under
the Borland C/C++ compiler.

680x0 Code (Macintosh CodeWarrior): CodeWarrior's profiler is function
based; consequently, it does not allow for line-by-line analysis as does the
Watcom compiler's profiler.

However, the CodeWarrior profiler does give us enough information to note
that NumSift() only accounts for about 28% of the time consumed by the
benchmark. The outer routine, NumHeapSort() accounts for around 71% of the
time taken. It will require additional analysis to determine why the two
compilers -- Watcom and CodeWarrior divide the workload so differently. (It
may have something to do with compiler architecture, or the act of profiling
the code may produce results that are significantly different than how the
program runs under normal conditions, though that would lead one to wonder
what use profilers would be.)

Porting Considerations

The numeric sort routine should represent a trivial porting exercise. It is
not an overly large benchmark in terms of source code. Additionally, the
only external routines it calls on are for allocating and releasing memory,
and managing the stopwatch.

The numeric sort benchmark depends on the following global definitions (note
that these may be overridden by the command file):

NUMNUMARRAYS -- Sets the upper limit on the number of arrays that the
benchmark will attempt to build. The numeric sort benchmark creates work for
itself by requiring the system to sort more and more arrays...not bigger and
bigger arrays. (The latter case would skew results, because the sorting time
for heapsort is N log2 N - e.g., doubling the array size does not double the
sort time.) This constant sets the upper limit to the number of arrays the
system will build before it signals an error. The default value is 100, and
may be changed if your system exceeds this limit.

NUMARRAYSIZE - Determines the size of each array built. It has been set to
8111L and should not be tampered with. The command file entry
NUMARRAYSIZE=<n> can be used to change this value, but results produced by
doing this will make your results incompatible with other runs of the
benchmark (since results will be skewed -- see preceding paragraph).

To test for a correct execution of the numeric sort benchmark, #define the
DEBUG symbol. This will enable code that verifies that arrays are properly
sorted. You should run the benchmark program using a command file that has
only the numeric sort test enabled. If there is an error, the program will
display "SORT ERROR" (If this happens, it's possible that tons of "SORT
ERROR" messages will be emitted, so it's best not to redirect output to a
file), otherwise it will print "Numeric sort: OK" (also quite a few times).

References

Gonnet, G.H. 1984, Handbook of Algorithms and Data Structures (Reading, MA:
Addison-Wesley).

Knuth, Donald E. 1968, Fundamental Algorithms, vol 1 of The Art of Computer
Programming (Reading, MA: Addison-Wesley).

Press, William H., Flannery, Brian P., Teukolsky, Saul A., and Vetterling,
William T. 1989, Numerical Recipes in Pascal (Cambridge: Cambridge
University Press).

Heapsort

The heapsort algorithm is well-covered in a number of the popular
computer-science textbooks. In fact, it gets a pat on the back in Numerical
Recipes (Press et. al.), where the authors write:

Heapsort is our favorite sorting routine. It can be recommended
wholeheartedly for a variety of sorting applications. It is a true
"in-place" sort, requiring no auxiliary storage.

Heapsort works by building the array into a kind of a queue called a heap.
You can imagine this heap as being a form of in-memory binary tree. The
topmost (root) element of the tree is the element that -- were the array
sorted -- would be the largest element in the array. Sorting takes place by
first constructing the heap, then pulling the root off the tree, promoting
the next largest element to the root, pulling it off, and so on. (The
promotion process is known as "sifting up.")

Heapsort executes in N log2 N time even in its worst case. Unlike some other
sorting algorithms, it does not benefit from a partially sorted array
(though Gonnet does refer to a variation of heapsort, called "smoothsort,"
which does -- see references).

String Sort

Description

This benchmark is designed to gauge how well the system moves bytes around.
By that we mean, how well the system can copy a string of bytes from one
location to another; source and destination being aligned to arbitrary
addresses. (This is unlike the numeric sort array, which moves bytes
longword-at-a-time.) The strings themselves are built so as to be of random
length, ranging from no fewer than 4 bytes and no greater than 80 bytes. The
mixture of random lengths means that processors will be forced to deal with
strings that begin and end on arbitrary address boundaries.

The string sort benchmark uses the heapsort algorithm; this is the same
algorithm as is used in the numeric sort benchmark (see the sidebar on the
heapsort for a detailed description of the algorithm).

Manipulation of the strings is actually handled by two arrays. One array
holds the strings themselves; the other is a pointers array. Each member of
the pointers array carries an offset that points into the string array, so
that the ith pointer carries the offset to the ith string. This allows the
benchmark to rapidly locate the position of the ith string. (The sorting
algorithm requires exchanges of items that might be "distant" from one
another in the array. It's critical that the routine be able to rapidly find
a string based on its indexed position in the array.)

The string sort benchmark reports the number of string arrays it was able to
sort per second. The size of the array is set by a global constant.

Analysis

Optimized 486 code (Watcom C/C++ 10.0): Profiling of the string sort
benchmark indicates that it spends most of its time in the C library routine
memmove(). Within that routine, most of the execution is consumed by a pair
of instructions: rep movsw and rep movsd. These are repeated string move --
word width and repeated string move -- doubleword width, respectively.

This is precisely where we want to see the time spent. It's interesting to
note that the memmove() of the particular compiler/profiler tested (Watcom
C/C++ 10.0) was "smart" enough to do most of the moving on word or
doubleword boundaries. The string sort benchmark specifically sets arbitrary
boundaries, so we'd expect to see lots of byte-wide moves. The "smart"
memmove() is able to move bytes only when it has to, and does the remainder
of the work via words and doublewords (which can move more bits at a time).

680x0 Code (Macintosh CodeWarrior): Because CodeWarrior's profiler is
function based, it is impossible to get an idea of how much time the test
spends in library routines such as memmove(). Fortunately, as an artifact of
the early version of the benchmark, the string sort algorithm makes use of
the MoveMemory() routine in the sysspec.c file (system specific routines).
This call, on anything other than a 16-bit DOS system, calls memmove()
directly. Hence, we can get a good approximation of how much time is spent
moving bytes.

The answer is that nearly 78% of the benchmark's time is consumed by
MoveMemory(), the rest being taken up by the other routines (the
str_is_less() routine, which performs string comparisons, takes about 7% of
the time). As above, we can guess that most of the benchmark's time is
dependent on the performance of the library's memmove() routine.

Porting Considerations

As with the numeric sort routine, the string sort benchmark should be simple
to port. Simpler, in fact. The string sort benchmark routine is not
dependent on any typedef that may change from machine to machine (unless a
char type is not 8 bits).

The string sort benchmark depends on the following global definitions:

NUMSTRARRAYS - Sets the upper limit on the number of arrays that the
benchmark will attempt to build. The string sort benchmark creates work for
itself by requiring the system to sort more and more arrays, not bigger and
bigger arrays. (See section on Numeric Sort for an explanation.) This
constant sets the upper limit to the number of arrays the system will build
before it signals an error. The default value is 100, and may be changed if
your system exceeds this limit.

STRARRAYSIZE - Sets the default size of the string arrays built. We say
"arrays" because, as with the numeric sort benchmark, the system adds work
not by expanding the size of the array, but by adding more arrays. This
value is set to 8111, and should not be modified, since results would not be
comparable with other runs of the same benchmark on other machines.

To test for a correct execution of the string sort benchmark, #define
the DEBUG symbol. This will enable code that verifies the arrays are
properly sorted. Set up a command file that runs only the string sort,
and execute the benchmark program. If the routine is operating
properly, the benchmark will print "String sort: OK", this message is
printed quite often. Otherwise, the program will display "SORT ERROR"
for each pair of strings it finds out of order (which can be really
often).

References

See the references for the Numeric Sort benchmark.

Bitfield Operations

Description

The purpose of this benchmark is to explore how efficiently the system
executes operations that deal with "twiddling bits." The test is set up to
simulate a "bit map"; a data structure used to keep track of storage usage.
(Don't confuse this meaning of "bitmap" with its use in describing a
graphics data structure.)

Systems often use bit maps to keep an inventory of memory blocks or (more
frequently) disk blocks. In the case of a bit map that manages disk usage,
an operating system will set aside a buffer in memory so that each bit in
that buffer corresponds to a block on the disk drive. A 0 bit means that the
corresponding block is free; a 1 bit means the block is in use. Whenever a
file requests a new block of disk storage, the operating system searches the
bit map for the first 0 bit, sets the bit (to indicate that the block is now
spoken for), and returns the number of the corresponding disk block to the
requesting file.

These types of operations are precisely what this test simulates. A block of
memory is set allocated for the bit map. Another block of memory is
allocated, and set up to hold a series of "bit map commands". Each bitmap
command tells the simulation to do 1 of 3 things:

1) Clear a series of consecutive bits,

2) Set a series of consecutive bits, or

3) Complement (1->0 and 0->1) a series of consecutive bits.

The bit map command block is loaded with a set of random bit map commands
(each command covers an random number of bits), and simulation routine steps
sequentially through the command block, grabbing a command and executing it.

The bitfield benchmark reports the number of bits it was able to operate on
per second. The size of the bit map is constant; the bitfield operations
array is adjusted based on the capabilities of the processor. (See the
section describing the auto-adjust feature of the benchmarks.)

Analysis

Optimized 486 code: Using the Watcom C/C++ 10.0 profiler, the Bitfield
benchmark appears to spend all of its time in two routines: ToggleBitRun()
(74% of the time) and DoBitFieldIteration() (24% of the time). We say
"appears" because this is misleading, as we will explain.

First, it is important to recall that the test performs one of three
operations for each run of bits (see above). The routine ToggleBitRun()
handles two of those three operations: setting a run of bits and clearing a
run of bits. An if() statement inside ToggleBitRun() decides which of the
two operations is performed. (Speed freaks will quite rightly point out that
this slows the entire algorithm. ToggleBitRun() is called by a switch()
statement which has already decided whether bits should be set or cleared;
it's a waste of time to have ToggleBitRun() have to make that decision yet
again.)

DoBitFieldIteration() is the "outer" routine that calls ToggleBitRun().
DoBitFieldIteration() also calls FlipBitRun(). This latter routine is the
one that performs the third bitfield operation: complementing a run of bits.
FlipBitRun() gets no "air time" at all (while DoBitFieldIteration() gets 24
% of the time) simply because the compiler's optimizer recognizes that
FlipBitRun() is only called by DoBitFieldIteration(), and is called only
once. Consequently, the optimizer moves FlipBitRun() "inline", i.e., into
DoBitFieldIteration(). This removes an unnecessary call/return cycle (and is
probably part of the reason why the FlipBitRun() code gets 24% of the
algorithm's time, instead of something closer to 30% of its time.)

Within the routines, those lines of code that actually do the shifting, the
and operations, and the or operations, consume time evenly. This should make
for a good test of a processor's "bit twiddling" capabilities.

680x0 Code (Macintosh CodeWarrior): The CodeWarrior profiler is function
based. Consequently, it is impossible to produce a profile of machine
instruction execution time. We can, however, get a good picture of how the
algorithm divides its time among the various functions.

Unlike the 486 compiler, the CodeWarrior compiler did not appear to collapse
the FlipBitRun() routine into the outer DoBitFieldIteration() routine. (We
don't know this for certain, of course. It's possible that the compiler
would have done this had we not been profiling.)

In any case, the time spent in the two "core" routines of the bitfield test
are shown below:

FlipBitRun() - 18031.2 microsecs (called 509 times)

ToggleBitRun() - 50770.6 microsecs (called 1031 times)

In terms of total time, FlipBitRun() takes about 35% of the time (it gets
about 33% of the calls). Remember, ToggleBitRun() is a single routine that
is called both to set and clear bits. Hence, ToggleBitRun() is called twice
as often as FlipBitRun().

We can conclude that time spent setting bits to 1, setting bits to 0, and
changing the state of bits, is about equal; the load is balanced close to
what we'd expect it to be, based on the structure of the algorithm.

Porting Considerations

The bitfield operations benchmark is dependent on the size of the long
datatype. On most systems, this is 32 bits. However, on some of the newer
RISC chips, a long can be 64 bits long. If your system does use 64-bit
longs, you'll need to #define the symbol LONG64.

If you are unsure of the size of a long in your system (some C compiler
manuals make it difficult to discover), simply place an ALLSTATS=T line in
the command file and run the benchmarks. This will cause the benchmark
program to display (among other things) the size of the data types int,
short, and long in bytes.

BITFARRAYSIZE - Sets the number of longs in the bit map array. This number
is fixed, and should not be altered. The bitfield test adjusts itself by
adding more bitfield commands (see above), not by creating a larger bit map.

Currently, there is no code added to test for correct execution. If you are
concerned that your port was incorrect, you'll need to step through your
favorite debugger and verify execution against the original source code.

** I added a resetting of the random number generator, and a resetting
** of the bitfield to each loop. Those operations are outside of the
** timed loop, and should add to make the benchmark more consistent.
** There also is now debugging information available. If you define
** DEBUG then the program will write a file named "debugbit.dat",
** which is the contents of the bitfield after the calibration loop of
** 30 operations. You can compare this file with the file
** "debugbit.good" that comes with the distribution.
** Uwe F. Mayer <mayer@tux.edu>

References

None.

Emulated Floating-point

Description

The emulated floating-point benchmark includes routines that are similar to
those that would be executed whenever a system performs floating-point
operations in the absence of a coprocessor. In general, this amounts to a
mixture of integer instructions, including shift operations, integer
addition and subtraction, and bit testing (among others).

The benchmark itself is remarkably simple. The test builds three
1-dimensional arrays and loads the first two up with random floating-point
numbers. The arrays are then partitioned into 4 equal-sized groups, and the
test proceeds by performing addition, subtraction, multiplication, and
division -- one operation on each group. (For example, for the addition
group, an element from the first array is added to the second array and the
result is placed in the third array.)

Of course, most of the work takes place inside the routines that perform the
addition, subtraction, multiplication, and division. These routines operate
on a special data type (referred to as an InternalFPF number) that -- though
not strictly IEEE compliant -- carries all the necessary data fields to
support an IEEE-compatible floating-point system. Specifically, an
InternalFPF number is built up of the following fields:

Type (indicates a NORMAL, SUBNORMAL, etc.)

Mantissa sign

Unbiased, signed 16-bit exponent

4-word (16 bits) mantissa.

The emulated floating-point test reports its results in number of loops per
second (where a "loop" is one pass through the arrays as described above).

Finally, we are aware that this test could be on its way to becoming an
anachronism. A growing number of systems are appearing that have
coprocessors built into the main CPU. It's possible that floating-point
emulation will one day be a thing of the past.

Analysis

Optimized 486 code (Watcom C/C++ 10.0): The algorithm's time is distributed
across a number of routines. The distribution is:

ShiftMantLeft1() - 60% of the time

ShiftMantRight1() - 17% of the time

DivideInternalFPF() - 14% of the time

MultiplyInternalFPF() - 5% of the time.

The first two routines are similar to one another; both shift bits about in
a floating-point number's mantissa. It's reasonable that ShiftMantLeft1()
should take a larger share of the system's time; it is called as part of the
normalization process that concludes every emulated addition, subtraction,
mutiplication, and division.

680x0 Code (Macintosh CodeWarrior): CodeWarrior's profiler is
function-based; consequently, it isn't possible to get timing at the machine
instruction level. However, the output to CodeWarrior's profiler has
provided insight into the breakdown of time spent in various functions that
forces us to rethink our 486 code analysis.

Analyzing what goes on inside the emulated floating-point tests is a tough
one to call because some of the routines that are part of the test are
called by the function that builds the arrays. Consequently, a quick look at
the profiler's output can be misleading; it's not obvious how much time a
particular routine is spending in the test and how much time that same
routine is spending setting up the test (an operation that does not get
timed).

Specifically, the routine that loads up the arrays with test data calls
LongToInternalFPF() and DivideInternalFPF(). LongToInternalFPF() makes one
call to normalize() if the number is not a true zero. In turn, normalize()
makes an indeterminate number of calls to ShiftMantLeft1(), depending on the
structure of the mantissa being normalized.

What's worse, DivideInternalFPF() makes all sorts of calls to all kinds of
important low-level routines such as Sub16Bits() and ShiftMantLeft1().
Untangling the wiring of which routine is being called as part of the test,
and which is being called as part of the setup could probably be done with
the computer equivalent of detective work and spelunking, but in the
interest of time we'll opt for approximation.

Here's a breakdown of some of the important routines and their times:

AddSubInternalFPF() - 1003.9 microsecs (called 9024 times)

MultiplyInternalFPF() - 20143 microsecs (called 5610 times)

DivideInternalFPF() - 18820.9 microsecs (called 3366 times).

The 3366 calls to DivideInternalFPF() are timed calls, not setup calls --
the profiler at least gives outputs of separate calls made to the same
routine, so we can determine which call is being made by the benchmark, and
which is being made by the setup routine. It turns out that the setup
routine calls DivideInternalFPF() 30,000 times.

Notice that though addition/subtraction are called most often,
multiplication next, then finally division; the time spent in each is the
reverse. Division takes the most time, then multiplication, finally
addition/subtraction. (There's probably some universal truth lurking here
somewhere, but we haven't found it yet.)

Other routines, and their breakdown:

Add16Bits() - 115.3 microsecs

ShiftMantRight1() - 574.2 microsecs

Sub16Bits() - 1762 microsecs

StickySiftRightMant - 40.4 microsecs

ShiftMantLeft1() - 17486.1 microsecs

The times for the last three routines are suspect, since they are called by
DivideInternalFPF(), and a large portion of their time could be part of the
setup process. This is what leads us to question the results obtained in the
486 analysis, since it, too, is unable to determine precisely who is calling
whom.

Porting Considerations

Earlier versions of this benchmark were extremely sensitive to porting;
particularly to the "endianism" of the target system. We have tried to
eliminate many of these problems. The test is nonetheless more "sensitive"
to porting than most others.

Pay close attention to the following defines and typedefs. They can be found
in the files EMFLOAT.H, NMGLOBAL.H, and NBENCH1.H:

u8 - Stands for unsigned, 8-bit. Usually defined to be unsigned char.

u16 - Stands for unsigned, 16-bit. Usually defined to be unsigned short.

u32 - Stands for unsigned, 32-bit. Usually defined to be unsigned long.

INTERNAL_FPF_PRECISION - Indicates the number of elements in the mantissa of
an InternalFPF number. Should be set to 4.

The exponent field of an InternalFPF number is of type short. It should be
set to whatever minimal data type can hold a signed, 16-bit number.

Other global definitions you will want to be aware of:

CPUEMFLOATLOOPMAX - Sets the maximum number of loops the benchmark will
attempt before flagging an error. Each execution of a loop in the emulated
floating-point test is "non-destructive," since the test takes factors from
two arrays, operates on the factors, and places the result in a third array.
Consequently, the test makes more work for itself by increasing the number
of times it passes through the arrays (# of loops). If the system exceeds
the limit set by CPUEMFLOATLOOPMAX, it will signal an error.

This value may be altered to suit your system; it will not effect the
benchmark results (unless you reduce it so much the system can never
generate enough loops to produce a good test run).

EMFARRAYSIZE - Sets the size of the arrays to be used in the test. This
value is the number of entries (InternalFPF numbers) per array. Currently,
the number is fixed at 3000, and should not be altered.

Currently, there is no means of testing correct execution of the benchmark
other than via debugger. There are routines available to decode the internal
floating point format and print out the numbers, but no formal correctness
test has been constructed. (This should be available soon. -- 3/14/95 RG)

** It now prints out the operations of 8 of the entries used in the
** test. Assuming you leave EMFARRAYSIZE at 3000, your results should
** look like the ones below. The number in front of the colon is the
** index of the entry.
**  
**  	 2: (-1.1160E   0) + (-4.5159E   0) = -5.6320E   0
**  	 6: (-4.4507E  -1) - (-8.2050E  -1) = +3.7543E  -1
**  	10: (+1.2465E   0) * (+7.4667E  -1) = +9.3075E  -1
**  	14: (-1.2781E   0) / (-1.7367E   0) = +7.3596E  -1
**    2986: (-7.0390E   0) * (-2.0752E   0) = +1.4607E   1
**    2990: (+8.3753E  -1) / (+2.3876E   1) = +3.5078E  -2
**    2994: (-1.1393E   0) + (-1.6080E   1) = -1.7219E   1
**    2998: (+7.2450E   0) - (-8.2654E  -1) = +8.0716E   0
**
** Uwe F. Mayer <mayer@tux.edu>

References

Microprocessor Programming for Computer Hobbyists, Neill Graham, Tab Books,
Blue Ridge Summit, PA, 1977.

Apple Numerica Manual, Second edition, Apple Computer, Addison-Wesley
Publishing Co., Reading, MA, 1988.

Fourier Series

Description

This is a floating-point benchmark designed primarily to exercise the
trigonometric and transcendental functions of the system. It calculates the
first n Fourier coefficients of the function (x+1)x on the interval 0,2. In
this case, the function (x+1)x is being treated as a cyclic waveform with a
period of 2.

The Fourier coefficients, when applied as factors to a properly constructed
series of sine and cosine functions, allow you to approximate the original
waveform. (In fact, if you can calculate all the Fourier coefficients --
there'll be an infinite number -- you can reconstruct the waveform exactly).
You have to calculate the coefficients via integration, and the algorithm
does this using a simple trapezoidal rule for its numeric integration
function.

The upshot of all this is that it provides an exercise for the
floating-point routines that calculate sine, cosine, and raising a number to
a power. There are also some floating-point multiplications, divisions,
additions, and subtractions mixed in.

The benchmark reports its results as the number of coefficients calculated
per second.

As an additional note, we should point out that the performance of this
benchmark is heavily dependent on how well-built the compiler's math library
is. We have seen at least two cases where recompilation with new (and
improved!) math libraries have resulted in two-fold and five-fold
performance improvements. (Apparently, when a compiler gets moved to a new
platform, the trigonometric and transcendental functions in the math
libraries are among the last routines to be "hand optimized" for the new
platform.) About all we can say about this is that whenever you run this
test, verify that you have the latest and greatest math libraries.

Analysis

Optimized 486 code: The benchmark partitions its time almost evenly among
the modules pow387, exp386, and trig387; giving between 25% and 28% of its
time to each. This is based on profiling with the Watcom compiler running
under Windows NT. These modules hold the routines that handle raising a
number to a power and performing trigonometric (sine and cosine)
calculations. For example, within trig387, time was nearly equally divided
between the routine that calculates sine and the routine that calculates
cosine.

The remaining time (between 17% and 18%) was spent in the balance of the
test. We noticed that most of that time occurred in the routine
thefunction(). This is at the heart of the numerical integration routine the
benchmark uses.

Consequently, this benchmark should be a good test of the exponential and
trigonometric capabilities of a processor. (Note that we recognize that the
performance also depends on how well the compiler's math library is built.)

680x0 Code (Macintosh CodeWarrior): The CodeWarrior profiler is function
based, therefore it is impossible to get performance results for individual
machine instructions. The CodeWarrior compiler is also unable to tell us how
much time is spent within a given library routine; we can't see how much
time gets spent executing the sin(), cos(), or pow() functions (which,
unfortunately, was the whole idea behind the benchmark).

About all we can glean from the results is that thefunction() takes about
74% of the time in the test (this is where the heavy math calculations take
place) while trapezoidintegrate() accounts for about 26% of the time on its
own.

Porting Considerations

Necessarily, this benchmark is at the mercy of the efficiency of the
floating-point support provided by whatever compiler you are using. It is
recommended that, if you are doing the port yourself, you contact the
designers of the compiler, and discuss with them what optimization switches
should be set to produce the fastest code. (This sounds simple; usually it's
not. Some systems let you decide between speed and true IEEE compliance.)

As far as global definitions go, this benchmark is happily free of them. All
the math is done using double data types. We have noticed that, on some Unix
systems, you must be careful to include the correct math libraries.
Typically, you'll discover this at link time.

To test for correct execution of the benchmark: It's unlikely you'll need to
do this, since the algorithm is so cut-and-dried. Furthermore, there are no
explicit provisions made to verify the correctness. You can, however, either
dip into your favorite debugger, or alter the code to print out the contents
of the abase (which holds the A[i] terms) and bbase (which holds the B[i]
terms) arrays as they are being filled (see routine DoFPUTransIteration).
** This is exactly what I have done, it now prints out A[i] and B[i] data.
** Uwe F. Mayer <mayer@tux.edu>
Run the benchmark with a command file set to execute only the Fourier test,
and examine the contents of the arrays. The first 100 are listed below.

A[i]=
   2.84 1.05 0.274 0.0824 0.0102 -0.024 -0.0426 -0.0536 -0.0605 -0.065
-0.0679 -0.0698 -0.0709 -0.0715 -0.0717 -0.0715 -0.0711 -0.0704
-0.0696 -0.0685 -0.0674 -0.0661 -0.0647 -0.0632 -0.0615 -0.0598 -0.058
-0.0561 -0.0542 -0.0521 -0.0501 -0.0479 -0.0457 -0.0434 -0.0411
-0.0387 -0.0363 -0.0338 -0.0313 -0.0288 -0.0262 -0.0236 -0.0209
-0.0183 -0.0156 -0.0129 -0.0102 -0.00744 -0.0047 -0.00196 0.000794
0.00355 0.0063 0.00905 0.0118 0.0145 0.0172 0.0199 0.0226 0.0253
0.0279 0.0305 0.0331 0.0357 0.0382 0.0407 0.0431 0.0455 0.0479 0.0502
0.0525 0.0547 0.0569 0.059 0.061 0.063 0.0649 0.0668 0.0686 0.0703
0.072 0.0736 0.0751 0.0765 0.0779 0.0792 0.0804 0.0816 0.0826 0.0836
0.0845 0.0853 0.0861 0.0867 0.0873 0.0877 0.0881 0.0884 0.0887 0.0888

B[i]= 
(undefined) -1.88 -1.16 -0.806 -0.61 -0.487 -0.402 -0.34 -0.293 -0.255
-0.224 -0.199 -0.177 -0.158 -0.141 -0.126 -0.113 -0.101 -0.0901
-0.0802 -0.071 -0.0625 -0.0546 -0.0473 -0.0404 -0.034 -0.0279 -0.0222
-0.0168 -0.0117 -0.00693 -0.00238 0.00193 0.00601 0.00988 0.0135 0.017
0.0203 0.0234 0.0263 0.0291 0.0317 0.0341 0.0364 0.0385 0.0405 0.0424
0.0441 0.0457 0.0471 0.0484 0.0496 0.0507 0.0516 0.0525 0.0532 0.0538
0.0543 0.0546 0.0549 0.055 0.0551 0.055 0.0549 0.0546 0.0543 0.0538
0.0533 0.0527 0.052 0.0512 0.0503 0.0493 0.0483 0.0472 0.046 0.0447
0.0434 0.042 0.0405 0.039 0.0374 0.0358 0.0341 0.0323 0.0305 0.0287
0.0268 0.0249 0.023 0.021 0.019 0.0169 0.0149 0.0128 0.0107 0.00857
0.00644 0.0043 0.00215

Note that there is no B[0] coefficient. If the above numbers are in the
arrays shown, you can feel pretty confident that the benchmark it working
properly.

References

Engineering and Scientific Computations in Pascal, Lawrence P. Huelsman,
Harper & Row, New York, 1986.

Assignment Algorithm

Description

This test is built on an algorithm with direct application to the business
world. The assignment algorithm solves the following problem: Say you have X
machines and Y jobs. Any of the machines can do any of the jobs; however, the
machines are sufficiently different so that the cost of doing a particular
job can vary depending what machine does it. Furthermore, the jobs are
sufficiently different that the cost varies depending on which job a given
machine does. You therefore construct a matrix; machines are the rows, jobs
are the columns, and the [i,j] element of the array is the cost of doing the
jth job on the ith machine. How can you assign the jobs so that the cost of
completing them all is minimal? (This also assumes that one machine does one
job.)

Did you get that?

The assignment algorithm benchmark is largely a test of how well the
processor handles problems built around array manipulation. It is not a
floating-point test; the "cost matrix" built by the algorithm is simply a 2D
array of long integers. This benchmark considers an iteration to be a run of
the assignment algorithm on a 101 x 101 - element matrix. It reports its
results in iterations per second.

Analysis

Optimized 486 code (Watcom C/C++ 10.0): There are numerous loops within the
assignment algorithm. The development system we were using (Watcom C/C++
10.0) appears to have a fine time unrolling many of them. Consequently, it
is difficult to pin down the execution impact of single lines (as in, for
example, the numeric sort benchmark).

On the level of functions, the benchmark spends around 70% of its time in
the routine first_assignments(). This is where a) lone zeros in rows and
columns are found and selected, and b) a choice is made between duplicate
zeros. Around 23% of the time is spent in the second_assignments() routine
where (if first_assignments() fails) the matrix is partitioned into smaller
submatrices.

Overall, we did a tally of instruction mix execution. The approximate
breakdowns are:

move - 38%

conditional jump - 12%

unconditional jump - 11%

comparison - 14%

math/logical/shift - 24%

Many of the move instructions that appeared to consume the most amounts of
time were referencing items on the local stack frame. This required an
indirect reference through EBP, plus a constant offset to resolve the
address.

This should be a good exercise of a cache, since operations in the
first_assignments() routine require both row-wise and column-wise movement
through the array. Note that the routine could be made more "severe" by
chancing the assignedtableau[][] array to an array of unsigned char --
forcing fetches on byte boundaries.

680x0 Code (CodeWarrior): The CodeWarrior profiler is function-based.
Consequently, it's not possible to determine what's going on at the machine
instruction level. We can, however, get a good idea of how much time the
algorithm spends in each routine. The important routines are broken down as
follows:

calc_minimum_costs() - approximately 0.3% of the time

(250 microsecs)

first_assignments() - approximately 79% of the time

(96284.6 microsecs)

second_assignments() - approximately 19% of the time

(22758 microsecs)

These times are approximate; some time is spent in the Assignment() routine
itself.

These figures are reasonably close to those of the 486, at least in terms of
the mixture of time spent in a particular routine. Hence, this should still
be a good test of system cache (as described in the preceding section),
given the behavior of the first_assignments() routine.

Porting Considerations

The assignment algorithm test is purely an integer benchmark, and requires
no special data types that might be affected by ports to different
architectures. There are only two global constants that affect the
algorithm:

ASSIGNROWS and ASSIGNCOLS - These set the size of the assignment array. Both
are defined to be 101 (so, the array that is benchmarked is a 101 x 101
-element array of longs). These values should not be altered.

To test for correct execution of the benchmark: #define the symbol DEBUG,
recompile, set up a command file that executes only the assignment
algorithm, and run the benchmark. (You may want to pipe the output through a
paging filter, like the more program.) The act of defining DEBUG will enable
a section of code that displays the assigned columns on a per-row basis. If
the benchmark is working properly, the numbers to be displayed
should be:

R000: 056 R001: 066 R002: 052 R003: 065 R004: 043 R005: 023 R006: 016
R007: 077 R008: 095 R009: 004 R010: 064 R011: 076 R012: 078 R013: 091
R014: 013 R015: 029 R016: 044 R017: 014 R018: 041 R019: 042 R020: 020
R021: 071 R022: 024 R023: 017 R024: 055 R025: 040 R026: 070 R027: 025
R028: 031 R029: 019 R030: 073 R031: 002 R032: 047 R033: 009 R034: 035
R035: 045 R036: 005 R037: 063 R038: 081 R039: 039 R040: 087 R041: 008
R042: 053 R043: 093 R044: 049 R045: 092 R046: 061 R047: 046 R048: 026
R049: 034 R050: 088 R051: 000 R052: 028 R053: 018 R054: 072 R055: 021
R056: 037 R057: 082 R058: 006 R059: 058 R060: 096 R061: 068 R062: 069
R063: 054 R064: 057 R065: 086 R066: 097 R067: 084 R068: 099 R069: 051
R070: 098 R071: 003 R072: 074 R073: 062 R074: 080 R075: 033 R076: 011
R077: 094 R078: 012 R079: 050 R080: 010 R081: 038 R082: 089 R083: 059
R084: 022 R085: 079 R086: 015 R087: 007 R088: 075 R089: 083 R090: 060
R091: 048 R092: 032 R093: 067 R094: 001 R095: 030 R096: 027 R097: 085
R098: 090 R099: 036 R100: 100

These are the column choices for each row made by the algorithm. If
you see these numbers displayed, the algorithm is working correctly.

*** The original debugging information was incorrect, as it not only
*** display the chosen columns, but also displayed eliminated columns.
*** Changed to show all 101 entries. Uwe F. Mayer <mayer@tux.edu>

References

Quantitative Decision Making for Business, Gordon, Pressman, and Cohn,
Prentice-Hall, Englewood Cliffs, NJ, 1990.

Quantitative Decision Making, Guiseppi A. Forgionne, Wadsworth Publishing
Co., California, 1986.

Huffman Compression

Description

This is a compression algorithm that -- while helpful for some time as a
text compression technique -- has since fallen out of fashion on account of
the superior performance by algorithms such as LZW compression. It is,
however, still used in some graphics file formats in one form or another.

The benchmark consists of three parts:

Building a "Huffman Tree" (explained below),

Compression, and

Decompression.

A "Huffman Tree" is a special data structure that guides the compression and
decompression processes. If you were to diagram one, it would look like a
large binary tree (i.e., two branches per each node). Describing its
function in detail is beyond the scope of this paper (see the references for
more information). We should, however, point out that the tree is built from
the "bottom up"; and the procedure for constructing it requires that the
algorithm scan the uncompressed buffer, building a frequency table for all
the characters appearing in the buffer. (This version of the Huffman
algorithm compresses byte-at-a-time, though there's no reason why the same
principle could not be applied to tokens larger than one byte.)

Once the tree is built, text compression is relatively straightforward. The
algorithm fetches a character from the uncompressed buffer, navigates the
tree based on the character's value, and produces a bit stream that is
concatenated to the compressed buffer. Decompression is the reverse of that
process. (We recognize that we are simplifying the algorithm. Again, we
recommend you check the references.)

The Huffman Compression benchmark considers an iteration to be the three
operations described above, performed on an uncompressed text buffer of 5000
bytes. It reports its results in iterations per second.

Analysis

Optimized 486 code (Watcom C/C++ 10.0): The Huffman compression algorithm --
tree building, compression, and decompression -- is written as a single,
large routine: DoHuffIteration(). All the benchmark's time is spent within
that routine.

Components of DoHuffIteration() that consume the most time are those that
perform the compression and decompression .

The code for performing the compression spends most of its time (accounting
for about 13%) constructing the bit string for a character that is being
compressed. It does this by seeking up the tree from a leaf, emitting 1's
and 0's in the process, until it reaches the root. The stream of 1's and 0's
are loaded into a character array; the algorithm then walks "backward"
through the array, setting (or clearing) bits in the compression buffer as
it goes.

Similarly, the decompression portion takes about 12% of the time as the
algorithm pulls bits out of the compressed buffer -- using them to navigate
the Huffman tree -- and reconstructs the original text.

680x0 Code (Macintosh CodeWarrior): CodeWarrior's profiler is function
based. Consequently, it's impossible to get performance scores for
individual machine instructions. Furthermore, as mentioned above, the
Huffman compression algorithm is written as a monolithic routine. This makes
the results from the CodeWarrior profiler all the more sparse.

We can at least point out that the lowmost routines (GetCompBit() and
SetCompBit()) that read and write individual bits, though called nearly 13
million times each, account for only 0.7% and 0.3% of the total time,
respectively.

Porting Considerations

The Huffman algorithm relies on no special data types. It should port
readily. Global constants of interest include:

EXCLUDED - This is a large, positive value. Currently it is set to 32000,
and should be left alone. Basically, this is a token that the system uses to
indicate an excluded character (one that does not appear in the plain-text).
It is set to a ridiculously high value that will never appear in the
pointers of the tree during normal construction.

MAXHUFFLOOPS - This is another one of those "governor" constants. The
Huffman benchmark creates more work for itself by doing multiple
compression/decompression loops. This constant sets the maximum number of
loops it will attempt per iteration before it gives up. Currently, it is set
to 50000. Though it is unlikely you'll ever need to modify this value, you
can increase it if your machine is too fast for the adjustment algorithm. Do
not reduce the number.

HUFFARRAYSIZE - This value sets the size of the plain-text array to be
compressed. You can override this value with the command file to see how
well your machine performs for larger or smaller arrays. The subsequent
results, however, are invalid for comparison with other systems.

To test for correct execution of the benchmark: #define the symbol DEBUG,
recompile, build a command file that executes only the Huffman compression
algorithm, and run the benchmark. Defining DEBUG will enable a section of
code that verifies the decompression as it takes place (i.e., the routine
compares -- character at a time -- the uncompressed data with the original
plain-text). If there's an error, the program will repeatedly display: "Error
at textoffset xxx". 

** If everything is correct it will emit quite a few "Huffman: OK" messages.
**
** I added a resetting of the random number generator, outside of the
** timed loop, and a resetting of the Huffman tree, inside of the
** timed loop. That should help to make the benchmark more consistent.
** The program did originally only reset half of the tree, which lead
** to runtime errors on some systems. The effect on the benchmark
** should be negligible, and in fact comes out as being of the order
** of less than 1% on my test system.
** Uwe F. Mayer <mayer@tux.edu>

References

Data Compression: Methods and Theory, James A. Storer, Computer Science
Press, Rockville, MD, 1988.

An Introduction to Text Processing, Peter D. Smith, MIT Press, Cambridge,
MA, 1990.

IDEA Encryption

Description

This is another benchmark based on a "higher-level" algorithm; "higher
-level" in the sense that it is more complex than a sort or a search
operation.

Security -- and, therefore, cryptography -- are becoming increasingly
important issues in the computer realm. It's likely that more and more
machines will be running routines like the IDEA encryption algorithm. (IDEA
is an acronym for the International Data Encryption Algorithm.)

A good description of the algorithm (and, in fact, the reference we used to
create the source code for the test) can be found in Bruce Schneier's
exhaustive exploration of encryption, "Applied Cryptography" (see
references). To quote Mr. Schneier: "In my opinion, it [IDEA] is the best
and most secure block algorithm available to the public at this time."

IDEA is a symmetrical, block cipher algorithm. Symmetrical means that the
same routine used to encrypt the data also decrypts the data. A block cipher
works on the plain-text (the message to be encrypted) in fixed, discrete
chunks. In the case of IDEA, the algorithm encrypts and decrypts 64 bits at
a time.

As pointed out in Schneier's book, there are three operations that the IDEA
uses to do its work:

XOR (exclusive-or)

Addition modulo 216 (ignoring overflow)

Multiplication modulo 216+1 (ignoring overflow).

IDEA requires a key of 128 bits. However, keys and blocks are further
subdivided into 16-bit chunks, so that any given operation within the IDEA
encryption is performed on 16-bit quantities. (This is one of the many
advantages of the algorithm, it is efficient even on 16-bit processors.)

The IDEA benchmark considers an "iteration" to be an encryption and
decryption of a buffer of 4000 bytes. The test actually builds 3 buffers:
The first to hold the original plain-text, the second to hold the encrypted
text, and the third to hold the decrypted text (the contents of which should
match that of the first buffer). It reports its results in iterations per
second.

Analysis

Optimized 486 code: The algorithm actually spends most of its time (nearly
75%) within the mul() routine, which performs the multiplication modulo
216+1. This is a super-simple routine, consisting primarily of if
statements, shifts, and additions.

The remaining time (around 24%) is spent in the balance of the cipher_idea()
routine. (Note that cipher_idea() calls the mul() routine frequently; so,
the 24% is comprised of the other lines of cipher_idea()). cipher_idea() is
littered with simple pointer-fetch-and-increment operations, some addition,
and some exclusive-or operations.

Note that IDEA's exercise of system capabilities probably doesn't extend
beyond testing simple integer math operations. Since the buffer size is set
to 4000 bytes, the test will run entirely in processor cache on most
systems. Even the cache won't get a heavy "internal" workout, since the
algorithm proceeds sequentially through each buffer from lower to higher
addresses.

680x0 code (Macintosh CodeWarrior): CodeWarrior's profiler is function
based; consequently, it is impossible to determine execution profiles for
individual machine instructions. We can, however, get an idea of how much
time is spent in each routine.

As with Huffman compression, the IDEA algorithm is written monolithically --
a single, large routine does most of the work. However, a special
multiplication routine, mul(), is frequently called within each
encryption/decryption iteration (see above).

In this instance, the results for the 68K system diverges widely from those
of the 486 system. The CodeWarrior profiler shows the mul() routine as
taking only 4% of the total time in the benchmark, even though it is called
over 20 million times. The outer routine is called 600,000 times, and
accounts for about 96% of the whole program's entire time.

Porting Considerations

Since IDEA does its work in 16-bit units, it is particularly important that
u16 be defined to whatever datatype provides an unsigned 16-bit integer on
the test platform. Usually, unsigned short works for this. (You can verify
the size of a short by running the benchmarks with a command file that
includes ALLSTATS=T as one of the commands. This will cause the benchmark
program to display a message that tells the size of the int, short, and long
data-types in bytes.)

Also, the mul() routine in IDEA requires the u32 datatype to define an
unsigned 32-bit integer. In most cases, unsigned long works.

To test for correct execution of the benchmark: #define the symbol DEBUG,
recompile, build a command file that executes only the IDEA algorithm, and
run the benchmark. Defining DEBUG will enable a section of code that
compares the original plain-text with the output of the test. (Remember, the
benchmark performs both encryption and decryption.) If the algorithm has
failed, the output will not match the input, and you'll see "IDEA Error"
messages all over your display.

References

Applied Cryptography: Protocols, Algorithms, and Source Code in C, Bruce
Schneier, John Wiley & Sons, Inc., New York, 1994.

Neural Net

Description

The Neural Net simulation benchmark is based on a simple back-propagation
neural network presented by Maureen Caudill as part of a BYTE article that
appeared in the October, 1991 issue (see "Expert Networks" in that issue).
The network involved is a simple 3-layer (input neurodes, middle-layer
neurodes, and output neurodes) network that accepts a number of 5 x 7 input
patterns and produce a single 8-bit output pattern.

The test involves sending the network an input pattern that is the 5 x 7
"image" of a character (1's and 0's -- 1's representing lit pixels, 0's
representing unlit pixels), and teaching it the 8-bit ASCII code for the
character.

A thorough description of how the back propagation algorithm works is beyond
the scope of this paper. We recommend you search through the references
given at the end of this paper, particularly Ms. Caudill's article, for
detailed discussion. In brief, the benchmark is primarily an exercise in
floating-point operations, with some frequent use of the exp() function. It
also performs a great deal of array references, though the arrays in use are
well under 300 elements each (and less than 100 in most cases).

The Neural Net benchmark considers an iteration to be a single learning
cycle. (A "learning cycle" is defined as the time it takes the network to be
able to associate all input patterns to the correct output patterns within a
specified tolerance.) It reports its results in iterations per second.

Analysis

Optimized 486 code: The forward pass of the network (i.e., calculating
outputs from inputs) utilize a sigmoid function. This function has, at its
heart, a call to the exp() library routine. A small but non-negligible
amount of time is spent in that function (a little over 5% for the 486
system we tested).

The learning portion of the network benchmark depends on the derivative of
the sigmoid function, which turns out to require only multiplications and
subtractions. Consequently, each learning pass exercises only simple
floating-point operations.

If we divide the time spent in the test into two parts -- forward pass and
backward pass (the latter being the learning pass) -- then the test appears
to spend the greatest part of its time in the learning phase. In fact, most
time is spent in the adjust_mid_wts() routine. This is the part of the
routine that alters the weights on the middle layer neurodes. (It accounts
for over 40% of the benchmark's time.)

680x0 Code (Macintosh CodeWarrior): Though CodeWarrior's profiler is
function based, the neural net benchmark is highly modular. We can therefore
get a good breakdown of routine usage:

worst_pass_error() - 304 microsecs (called 4680 times)

adjust_mid_wts() - 83277 microsecs (called 46800 times)

adjust_out_wts() - 17394 microsecs (called 46800 times)

do_mid_error() - 11512 microsecs (called 46800 times)

do_out_error() - 3002 microsecs (called 46800 times)

do_mid_forward() - 49559 microsecs (called 46800 times)

do_out_forward() - 20634 microsecs (called 46800 times)

Again, most time was spent in adjust_mid_wts() (as on the 486), accounting
for almost twice as much time as do_mid_forward().

Porting Consideration

The Neural Net benchmark is not dependent on any special data types. There
are a number of global variables and arrays that should not be altered in
any way. Most importantly, the #defines found in NBENCH1.H under the Neural
Net section should not be changed. These control not only the number of
neurodes in each layer; they also include constants that govern the learning
processes.

Other globals to be aware of:

MAXNNETLOOPS - This constant simply sets the upper limit on the number of
training loops the test will permit per iteration. The Neural Net benchmark
adjusts its workload by re-teaching itself over and over (each time it
begins a new training session, the network is "cleared" -- loaded with
random values). It is unlikely you will ever need to modify this constant.

inpath - This string pointer is set to the path from which the neural net's
input data is read. It is currently hardwired to "NNET.DAT". You shouldn't
have to change this name, unless your file system requires directory
information as part of the path.

Note that the Neural Net benchmark is the only test that requires an
external data file. The contents of the file are listed in an attachment to
this paper. You should use the attachment to reconstruct the file should it
become lost or corrupted. Any changes to the file will invalidate the test
results.

To test for correct execution of the benchmark: #define the symbol DEBUG,
recompile, build a command file that executes only the Neural Net test, and
run the benchmark. Defining DEBUG will enable a section of code that
displays how many passes through the learning process were required for the
net to learn. It should learn in 780 passes.

References

"Expert Networks," Maureen Caudill, BYTE Magazine, October, 1991.

Simulating Neural Networks, Norbert Hoffmann, Verlag Vieweg, Wiesbaden,
1994.

Signal and Image Processing with Neural Networks, Timothy Masters, John
Wiley and Sons, New York, 1994.

Introduction to Neural Networks, Jeannette Stanley, California Scientific
Software, CA, 1989.

LU Decomposition

Description

LU Decomposition is an algorithm that can be used as the heart of a program
for solving linear equations. Suppose you have a matrix A. LU Decomposition
determines the matrices L and U such that

L . U = A

where L is a lower triangular matrix and U is an upper triangular matrix. (A
lower triangular matrix has nonzero elements only on the main diagonal and
below. An upper triangular matrix has nonzero elements only on the main
diagonal and above.)

Without going into the mathematical details too deeply, having the L and U
matrices makes the solution of linear equations (i.e., equations of the form
A . x = b) quite easy. It turns out that you can also use LU decomposition
to determine matrix inverses and determinants.

The algorithm used in the benchmarks was derived from Numerical Recipes in
Pascal (there is a C version of the book, which we did not have on hand), a
book we heartily recommend to anyone serious about mathematical and
scientific computing. The authors are approving of LU decomposition as a
means of solving linear equations, pointing out that their version (which
makes use of what we would have to call "Crout's method with partial
implicit pivoting") is a factor of 3 better than one of their Gauss-Jordan
routines, a factor of 1.5 better than another. They go on to demonstrate the
use of LU decomposition for iterative improvement of linear equation
solutions.

The benchmark begins by creating a "solvable" linear system. This is easily
done by loading up the column vector b with random integers, then
initializing A with an identity matrix. The equations are then "scrambled"
by either multiplying a row by a constant, or adding one row to another. The
scrambled matrices are handed to the LU algorithm.

The LU Decomposition benchmark considers a single iteration to be the
solution of one set of equations (the size of A is fixed at 101 x 101
elements). It reports its results in iterations per second.

Analysis

Optimized 486 code (Watcom C/C++ 10.0): The entire algorithm consists of two
parts: the LU decomposition itself, and the back substitution algorithm that
builds the solution vector. The majority of the algorithm's time takes place
within the former; the algorithm that builds the L and U matrices (this
takes place in routine ludcmp()).

Within ludcmp(), there are two extremely tight for loops forming the heart
of Crout's algorithm that consume the majority of the time. The loops are
"tight" in that they each consist of only one line of code; in both cases,
the line of code is a "multiply and accumulate" operation (actually, it's
sort of a multiply and de-accumulate, since the result of the multiplication
is subtracted, not added).

In both cases, the items multiplied are elements from the A array; and one
factor's row index is varying more rapidly, while another factor's column
index is varying more rapidly.

Note that this is a good overall test of floating-point operations within
matrices. Most of the math is floating-point; primarily additions,
subtractions, and multiplications (only a few divisions).

680x0 Code (Macintosh CodeWarrior): CodeWarrior's profiler is function
based. It is therefore impossible to determine execution profiles at the
machine-code level. The profiler does, however, allow us to determine how
much time the benchmark spends in each routine. This breakdown is as
follows:

lusolve() - 3.4 microsecs (about 0% of the time)

lubksb() 1198 microsec (about 2% of the time)

ludcmp() - 63171 microsec (about 91% of the time)

The above percentages are for the whole program. Consequently, as a portion
of actual benchmark time, the amount attributed to each will be slightly
larger (though the proportions will remain the same).

Since ludcmp() performs the actual LU decomposition, this is exactly where
we'd want the benchmark to spend its time. The lubksb() routine calls
ludcmp(), using the resulting matrix to "back-solve" the linear equation.

Porting Considerations

The LU Decomposition routine requires no special data types, and is immune
to byte ordering. It does make use of a typedef (LUdblptr) that includes an
embedded union; this allows the benchmark to "coerce" a pointer to double
into a pointer to a 2D array of double. This arrangement has not caused
problems with the compilers we have tested to date.

Other constants and globals to be aware of:

LUARRAYROWS and LUARRAYCOLS - These constants set the size of the
coefficient matrix, A. They cannot be altered by command file. In fact, you
shouldn't alter them at all, or your results will be invalid. Currently,
they are both set to 101.

MAXLUARRAYS - This is another "governor" constant. The algorithm performs
dynamic workload adjustment by building more and more arrays to solve per
timing round. This sets the maximum upper limit of arrays that it will
build. Currently, it is set to 1000, which should be more than enough for
the reasonable future (1000 arrays of 101 x 101 floating-point doubles would
require somewhere around 80 megabytes of RAM -- and that's not counting the
column vectors).

To test for correct execution of the benchmark: Currently, there is no
simple technique for doing this. You can, however, either use your favorite
debugger (or embed a printf() statement) at the conclusion of the lubksb()
routine. When this routine concludes, the array b will hold the solution
vector. These items will be stored as floating-point doubles, and the first
14 are (with rounding):

46 20 23 22 85 86 97 95 8 89 75 67 6 86

If you find these numbers as the first 14 in the array b[], then you're
virtually guaranteed that the algorithm is working correctly.

*** The above is not correct, as the initial matrix is not the identity,
*** but a matrix with random nonzero entries on the diagonal (they have
*** altered the algorithm since they wrote the documentation).
*** I changed the output of the debugging routine, it now prints first
*** what the array b should hold (as righthand side divided by diagonal
*** entry), and then it prints what the array b does hold after the
*** decomposition has been done to compute the solution of the system. If
*** you get the same, then fine.
*** And, by the way, my original right hand sides are
***  46  23  85  97   8  75   6  81  88  76   6  84  31  53   2 ...
*** and the diagonal entries are
*** 520 922 186 495  89 267 786 571 175 600 738 321 897 541 859 ...
*** You notice that one has every other number of the original sequence.
*** This is due to BYTE's change of the algorithm, as they now also use the
*** random number generator to generate the diagonal elements.
*** Here is the complete set of data:
*** 46/520=0.09  23/922=0.02  85/186=0.46   97/495=0.20  8/89=0.09
*** 75/267=0.28  6/786=0.01   81/571=0.14   88/175=0.50  76/600=0.13
*** 6/738=0.01   84/321=0.26  31/897=0.03   53/541=0.10  2/859=0.00
*** 86/92=0.93   51/121=0.42  29/248=0.12   51/789=0.06  84/6=14.00
*** 21/180=0.12  33/48=0.69   2/899=0.00    12/820=0.01  69/372=0.19
*** 59/809=0.07  74/18=4.11   40/788=0.05   39/56=0.70   86/91=0.95
*** 33/878=0.04  82/165=0.50  42/561=0.07   8/274=0.03   84/694=0.12
*** 32/352=0.09  25/969=0.03  59/816=0.07   33/112=0.29  5/125=0.04
*** 89/740=0.12  7/223=0.03   54/994=0.05   33/80=0.41   55/676=0.08
*** 6/524=0.01   36/544=0.07  21/160=0.13   58/596=0.10  15/717=0.02
*** 84/311=0.27  98/530=0.18  46/713=0.06   41/233=0.18  73/640=0.11
*** 40/343=0.12  72/586=0.12  100/965=0.10  59/764=0.08  37/866=0.04
*** 27/682=0.04  3/652=0.00   41/352=0.12   87/786=0.11  45/79=0.57
*** 83/761=0.11  41/817=0.05  46/209=0.22   78/930=0.08  85/210=0.40
*** 80/756=0.11  18/931=0.02  30/669=0.04   47/127=0.37  85/891=0.10
*** 66/364=0.18  83/955=0.09  58/637=0.09   58/778=0.07  82/288=0.28
*** 42/540=0.08  76/290=0.26  59/36=1.64    29/463=0.06  63/476=0.13
*** 6/340=0.02   73/341=0.21  59/737=0.08   81/492=0.16  98/443=0.22
*** 58/32=1.81   53/562=0.09  54/263=0.21   46/367=0.13  58/390=0.15
*** 96/845=0.11  30/746=0.04  2/687=0.00    28/849=0.03  84/180=0.47
*** 85/382=0.22
*** Uwe F. Mayer <mayer@tux.edu>

References

Numerical Recipes in Pascal: The Art of Scientific Computing, Press,
Flannery, Teukolsky, Vetterling, Cambridge University Press, New York, 1989.