Conference rusure::math

426.0. "Floating Point Accuracy" by TOOLS::STAN () Tue Jan 14 1986 15:17

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Subject: Technical Report of Possible Interest
Posted: 8 Jan 86 21:45:54 GMT
Organization: The Clouds Project, School of ICS, Georgia Tech
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Now available:
 
"A Report on the Accuracy of Some Floating Point Math Functions
 on Selected Computers," Technical Report GIT-ICS-85/06, 
 by Eugene Spafford and John Flaspohler, Georgia Tech
 
Abstract
--------
The Unix operating system and the C programming language have gained a
large following in recent years, especially in research and academic
settings.  C and Unix-like environments are available on a wide variety
of machines, from personal computers to mainframe computers; however,
few, if any, of these implementations provide accurate floating point
libraries, although users tend to believe they do.  This paper presents
the results of running a set of accuracy tests on more than a dozen
different computer systems under various versions of Unix and Unix-like
environments.
 
We believe the results presented in our report should be of major
concern to anyone using floating point library routines in their work
or research, including users of "packaged" software like statistics
systems and spreadsheets.  The 14 systems we tested for our report
included Vaxen running 4.2 BSD, a Pyramid 90x, an AT&T 3B20S, an AT&T
7300, a Sun 2, a Ridge, a Cyber and a Masscomp.  To summarize, we found
*no* system with an adequate set of functions.  More than a few of the
systems were 100% inaccurate in some of the tests.
 
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A floating point number x on a computer can be represented as follows:
      __                     ___
      | d1     d2          dt  |   e
x= +  |  --- + --- + ... + --- |* b
   -  |   1      2           t |
      |  b      b           b  |
      --                     ---    

b = base of the computer system
t = number of digits
e = exponent and L<= e <= U

0<=di<=b-1 and dt is always suspect
                                     1-t           1-t
Machine precision can be defined as b    (ie. 1 + b   = 1)
If a number falls below this value, it is zero.

To look at a few interesting machines:           1-t
machine		b	t	L	U	b   
-----------------------------------------------------------------------
CRAY-1		2	48	-16384	8191	7.1ee-15
IBM 360/370	16	6	-64	63	9.5ee-7	single precision
IBM 360/370	16	14	-64	63	2.2ee-16 double precision
ILLIAC IV	2	48	-16384	16383	7.1ee-15
B5500		8	13	-51	77	1.5ee-11
SETUN		3	18	?	?	7.7e-9
MANICA II	65536	2 11/16	-7	7	7.3e-9

>                                     1-t           1-t
>Machine precision can be defined as b    (ie. 1 + b   = 1)
>If a number falls below this value, it is zero.
>
This is not correct; b^(1-t) is typically 1e-15 while 1e-30 is representable
and not zero. You have confused roundoff (or truncation) with underflow.

re .2
What I was trying to show was machine precision, not machine accuracy.
There is a difference.