Reliable Computing 3: 219–227, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Computing on Sequences of Embedded
DAVID BERTHELOT and MARC DAUMAS
Laboratoire de l’Informatique du Parall
elisme, ENS Lyon, Lyon, France,
(Received: 29 November 1996; accepted: 28 February 1997)
Abstract. What will prevent most users to turn from standard arithmetic to interval arithmetic is the
commonbeliefthatonanyreal-life program, interval arithmetic will return most pessimistic bounds
to their problem. Although some adequate set of reﬁnements usually yields satisfactory results with
interval arithmetic, an interval-novice would most certainly not spend the time to look into the details
and learn about his code mathematic peculiarities.
We present in this work the prototype of a software library that deals almost transparently with
intervals. Directly from its model of execution, the library is able to automatically decide to reﬁne a
part or the totality of an evaluation as it is needed. We also introduce a new technique based on an
extended number representation that even improves the performances of the library and reduces the
interval computed after a numerical evaluation.
A real number is usually represented in computers as a ﬂoating point approximation
with a signiﬁcant ﬁeld and an exponent ﬁeld both bounded in length . Some
existing software libraries use adapted algebraic extensions on the ﬁeld of rational
numbers to compute exactly the result only with the error-free integer-arithmetic
With today’s computers, it is crucial to be able to return each result to the
accuracy demanded by the user. Interval arithmetic allows extensive control of the
error at each step of a program. Yet, many algorithms are not well suited to straight-
forward evaluation with an interval unit. In some cases, the mathematical equations
can be modiﬁed in order to create a strictly contracting application over a compact
set. Thereafter, the result of the problem is the necessary ﬁxed point of the function.
In all other cases, the user is supposed to test his algorithm with a given starting
precision for example the ﬂoating point double format; if the accuracy of the result
is not sufﬁcient, the program should be run again with an extended precision.
With the development of industry-level libraries for interval computation , the
software groups have directly built some automatic precision control procedures
in their softwares. We shall present in this work a library for computation on
sequences of Cauchy’s embedded interval. For a ﬁnite precision usage, the library
is able, by construction, to adapt its precision to yield the correct answer directly to