Exception-free arithmetic on the extended reals
Abstract
Interval arithmetic is a way to produce guaranteed enclosures of the
results of numerical calculations. Suppose $f(x)$ is a real
expression in real variables $x= (x_1, \ldots, x_n)$, built up from
the 4 basic arithmetic operations and other 'standard functions'. Let
$X_1, \ldots, X_n$ be (compact) real intervals. The process of {\em
interval evaluation} of $f(X_1, ..., X_n)$ replaces each real
operation by the corresponding interval operation wherever it occurs
in $f$, e.g. $A \times B$ is the smallest interval containing $\{a
\times b \mid a \in A, b \in B\}$.
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As is well known, it yields a guaranteed enclosure for the true range
$\{f(x_1, \ldots, x_n) \mid x_1 \in X_1, \ldots, x_n \in X_n\}$,
provided no exceptions such as division by (an interval containing)
zero occur during evaluation.
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Interval arithmetic takes set inputs and produces set outputs. Noting
this, we show there is a consistent way to extend arithmetic to $R^* =
R \cup \{-\infty, +\infty\}$, such that interval evaluation continues
to give enclosures, and there are {\em no exceptions}. The basic
ideas are: the usual set-theory meaning of evaluating a relation at a
set; and taking topological closure of the graph of a function in a
suitable $(R^{*})^n$. It gives rigorous meaning to intuitively
sensible statements like $1/0 = \{-\infty, +\infty\}$, $0/0 = R^*$
(but $(x/x)_{|x=0} = 1$), $\sin(+\infty) = [-1,1]$, etc.
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A practical consequence is that an exception-free floating-point
interval arithmetic system is possible. Such a system is implemented
at hardware level in the new Sun Fortran compiler, currently on
beta-release.