                            BYTE Benchmarks

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. 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.

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.

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 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 a 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 comprocessor -- 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.

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 sequentally 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 decomp.                           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 onscreen 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 plaintext character array that will be encrypted
by the test. Default is 4000. The benchmark actually builds 3 arrays:
1st plaintext, encrypted version, and 2nd plaintext. The 2nd plaintext
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.)

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 complete with no error messages.
Otherwise, the program will display "Sort Error" for each pair of
strings it finds out of order.

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, seting 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.

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 floatingpoint 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)

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 coefficents, 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 intergration, 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). Run the benchmark with a command file set
to execute only the Fourier test, and examine the contents of the
arrays. The first 4 elements of each array should be:

A[i] B[i]

2.837770756 n/a

1.045784473 -1.879103261

.2741002242 -1.158835123

.0824148217 -.8057591902

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 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
first 25 numbers to be displayed should be:

37 58 95 99 100 66 9 52 4 65 43 23 16 19 62 13 77 10 11 95 4 64 2 76 78

These are the column choices for each row made by the algorithm. (For
example, row 0 selects column 37, row 1 selects column 58, etc.) Odds
are extremely good that, if you see these numbers displayed, the
algorithm is working correctly.

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 plaintext). 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 plaintext 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 plaintext). If there's an error,
the program will repeatedly display: "Error at textoffset xxx".

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 cypher algorithm. Symmetrical means that
the same routine used to encrypt the data also decrypts the data. A
block cipher works on the plaintext (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 plaintext, 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/decription 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 datatypes 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 plaintext 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 filesystem 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.

References

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