Computable Performance Metrics
Module 1: Floating Point Precision
MACHAR – Test for IEEE 754
Final
Module Author
Kris Stewart
Computer Science Department
San Diego State University
stewart@sdsu.edu
http://www.stewart.cs.sdsu.edu/
W.M. Keck Foundation
This file: ComputablePerformanceMetrics1.doc
Overview:
Module 0: Floating Point Precision covered the task to find a “computable” way to detect
the actual manner in which a computer performed arithmetic, developed by W.J. Cody
[1] in 1988. In that module, a minimal test was examined to compute a machine’s unit
round-off, the value eps so that 1.0 + eps = 1.0. Considering the extent to which the
FORTRAN subroutine MACHAR can compute the properties of a processor, some may
wish to examine the full range of values computed. This can be used to test if the user’s
computer platform actually follows the IEEE 754 Floating Point Standard [2].
Introduction to problem:
There are several properties that characterize the behavior of a floating processing unit
(FPU) in performing arithmetic. Module 0 focused on the machine round-off and for
most situations this is adequate as a background for Module 2: Computing Error and
Work Estimators. Given the effort the numerical analysis community put into debating
and then specifying the IEEE-754 Floating-point standard, the acceptance by most
computer floating point chip manufacturers of IEEE-754 and the current reconsideration
of the 21 year old standard [3], an interesting class exploration of Cody’s computational
experiment to determine the processor’s behavior on the system students actually use can
be valuable. The FORTRAN code for MACHAR.F is available online from NETLIB [4]
and can be used to determine the performance of the compute system you may use in
your own courses or research.
Statement of the problem:
What are the floating point properties of the machine rohan.sdsu.edu which is used for
classes at San Diego State University? NOTE: The portable routine MACHAR.F can be
used to extend to other platforms with a FORTRAN compiler.
Background Information:
The reader should review the information presented in Module 0: Floating Point
Precision and its references.
Explanation of the module used to solve the problem:
We will compile and run the MACHAR.F program from W.F. Cody.
Conceptual questions to examine student’s understanding:
Does your own computer satisfy the IEEE 754 Floating-point standard?
Does the Studio 11 FORTRAN 95 compiler, version 8.2, show that rohan.sdsu.edu
satisfies IEEE 754 floating-point arithmetic?
Problems and projects:
Download the routine MACHAR.F from NETLIB [4] and write your own driver program
to call the subroutine and print out its results.
Solutions:
We provide a zip file containing MACHAR.F from NETLIB and a simple driver
program, MACHAR_DRIVER.F. This routine was successfully compiled using
f90 machar_driver.f machar.f -o machar
machar > machar_out.txt
on the system rohan.sdsu.edu and resulted in the following output:
using machar_driver.f with call to machar.f
machar.f actually "computes" the machine constants
MACHAR CONSTANTS
IBETA = 2 radix
IT = 24 number of base ibeta digits in fraction
IRND = 5 0 if fl pt add chops; 1 if rounds
NGRD = 0 number guard digits for multiply
MACHEP = -23 largest negative integer 1+ibeta**macheps != 1. [!= means not equal]
NEGEP = -24 1-ibeta**negeps != 1.
IEXP = 8 number of bits in exponent
MINEXP = -126 largest negative integer ibeta^minexp for valid fl pt number
MAXEXP = 128 largest positive integer exponent for fl pt number
EPS = 1.1920929E-07 fl pt unit round-off of machine
Note: This is the same
value as our simple loop in Module 0
EPSNEG = 5.9604645E-08
XMIN = 1.1754944E-38 smallest nonvanishing fl pt power of radix
XMAX = 3.4028235E+38 largest finite fl pt number
===========================================================
Suggestions to instructors for using the module:
The author was in the position of writing mathematical software for the Zilog Z80 chip,
one of the first floating point chips that was produced for computer hobbyists in the years
before the IBM PC arrived. She built her own computer from a kit distributed by IMSAI
in 1978 as the compute platform for her Masters Thesis at SDSU. The codes developed
for her thesis were translations into BASIC of the software in Computer Methods for
Mathematical Computations [5] , a useful library for numerical analysis published in
1977. This was a demonstration that small systems could be used for numerical analysis
on the 16 bit architecture of the Z80 Floating Point Unit. The codes in BASIC are
archived in the Guide to Available Mathematical Software (GAMS) at NIST [6]. With
her Masters, she began to work with Fred Krogh at the Jet Propulsion Laboratory, who
had very strong feelings on the process of writing quality mathematical software that
would run effectively on a wide range of computing platforms [7].
For anyone who wrote scientific software in the days before the standardization of the
IEEE 754, the care that was needed to write code to be portable over the wide variety of
computing platforms was important. Before the 1980’s, it might take a large investment
of program designer time to move a code from, for example, an octal-based UNIVAC
machine and have it run effectively on a hexadecimal-based IBM system. Luckily this is
now history that supports the importance of standards and user-community involvement
with the computer-producing marketplace. The IEEE 754-1985 standard was in place
well before most current college students were born. One could say that the continued
increase in performance predicted by Dr. Moore in 1965 is assisted by the tremendous
reduction in cost by commoditization of the components of computers now. For those
who wish to examine this exercise using the C programming language, a version of
MACHAR.C is also available from NETLIB [8]. Additional background on the
Anatomy of a Floating Point Number, with diagrams such as
and more are available, online, from WIKIPEDIA [9]. The author feels the class
discussion can be completed in one 50 minute lecture, but it is an individual decision for
each instructor how to weight the usefulness of this information. As John Seely Brown,
former Chief Scientist at XEROX PARC, says our students are “Growing Up Digital”
[10]. Our students studying computational science were born IEEE floating-point digital!
References:
[1] W.J. Cody, Jr., Algorithm 665: {MACHAR}: A Subroutine to Dynamically
Determine Machine Parameters". ACM Trans. on Mathematical Software (14), 1988.
[2] IEEE 754 Floating-point Standard FAQ (Frequently Asked Questions) is available
online from http://grouper.ieee.org/groups/754/faq.html
[3] IEEE has a group discussing updating the 21 year old 754 standard. Available online
from http://grouper.ieee.org/groups/754/index.html
[4] NETLIB is the numerical analysis community collection of mathematical software,
papers and databases available online from http://www.netlib.org/
MACHAR.F is part of the Basic Linear Algebra Subroutines (BLAS) in
FORTRAN. Available online from http://www.netlib.org/blas/machar.f
[5] G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computer Methods for Mathematical
Computations, Prentice-Hall, Englewood Cliffs, NY, 1977.
[6] http://gams.nist.gov/serve.cgi/Package/SCRUNCH/
K. Stewart, Master’s Project at SDSU, 1979. Code archived at NIST through
collaboration with David Kahaner. SCRUNCH = Small Crunch – A Numerical Analysis
Library in Basic for the Intel 8080 with Z80 floating point processor computer. (Note,
this was prior to the IBM PC based on the 8086 chipset).
[7] F. Krogh, http://mathalacarte.com/, now retired from the Jet Propulsion Lab and still
very active in the development of high-quality mathematical software, especially for
optimization.
[8] NETLIB distributes a main program and C version also. Available online from
http://www.netlib.org/blas/machar.c
[9] IEEE floating-point standard from Wikipedia gives an excellent article on the
“Anatomy of a floating-point number”. Available on line from
http://en.wikipedia.org/wiki/IEEE_floating-point_standard
[10] John Seely Brown, “Growing up Digital”, Change, March/April 2000, pp. 10-20.
Also available online from Dr. Brown’s home page (under Publications):
http://www.johnseelybrown.com/Growing_up_digital.pdf