Reliable Computing 7: 41–47, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
Roundoff-Free Number Fields for Interval
Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968,
(Received: 15 March 1994; accepted: 10 January 1997)
Abstract. Roundoff errors are inevitable if the exact result of some operation is not representable in
a computer, and has, therefore, to be approximated. To avoid roundoff errors, it is hence necessary to
choose a set of computer-representable numbers in such a way that the results of all basic operations
will be still in this set. In this paper, we prove that if we include arithmetic operations and computing
the interval range into this operations list, then the set F of numbers will be roundoff-free iff F is a
real closed ﬁeld; therefore, the smallest such set is the set of all real algebraic numbers (i.e., solutions
of polynomial equations with rational coefﬁcients).
One the sources of computational errors is roundoff. The necessity of a roundoff
is caused by the fact that in the computer, not all real numbers can be represented,
and therefore, if the result of an arithmetic operation cannot be represented, we
must approximate this result by a representable number. A natural question is: what
numbers must we represent to avoid roundoff?
In traditional computers, real numbers are represented by binary fractions of a ﬁxed
ﬁxed point numbers are of the type
(m, n non-negative integers) for ﬁxed
n and m ≤ 2
ﬂoating point numbers are of the type
with m ≤ 2
and p in some limited
In this representation, the sum of the representable ﬂoating point numbers is
not always computer representable: e.g., 2 + 2
is not, because its representation
This work was partially supported by NSF Grants No. CDA-9015006 and EEC-9322370, and
by NASA Grants No. NAG 9-757 and NCCS-209. The author is thankful to the anonymous referees
for their valuable suggestions that improved the readability of the paper.