Reliable Computing 8: 245–248, 2002.
2002 Kluwer Academic Publishers. Printed in the Netherlands.
Rump’s Example Revisited
EUGENE LOH and G. WILLIAM WALSTER
Sun Microsystems, Inc., 901 San Antonio Road, UMPK16–304, Palo Alto, CA 94303–4900, USA
(Received: 26 April 2001; accepted: 23 December 2001)
Abstract. Rump published an example (Algorithms for Veriﬁed Inclusions: Theory and Practice, in:
Moore, R. E. (ed.), Reliability in Computing: The Role of Interval Methods in Scientiﬁc Computing,
Academic Press, Boston, 1988, pp. 109–126) in which numerical evaluation of an expression gave a
misleading result, even though use of increasing arithmetic precision suggested reliable computation.
This oft-cited example happens not to be reproducible on many modern computers. The expression
is rewritten so that Rump’s result is reproducible using IEEE 754 arithmetic.
Rump’s example  is to compute the expression
− 2) + 5
with a = 77617 and b = 33096. On an IBM S/370 main frame he computed
in (1.1) using single, double, and extended-precision arithmetic, to produce the
This suggests a reliable result of approximately 1.172603 or even
1.1726039400532. In fact, however, the correct result (within one unit of the
last digit) is
Even the sign was wrong.
This example is often cited (for example, see , ) in the interval literature
to illustrate the fact that increasing precision does not always expose numerical
instability, whereas interval arithmetic must. Other similar examples are not known
to the authors. As noted in ,
the striking results reported by Rump are not
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The authors learned of this paper as the present paper was going to press. Neither the content
nor the conclusions that follow have changed as a consequence.