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Mathematical context. Interval computation is a collaboration between human program-
mer and machine infrastructure which, correctly done, produces mathematically proven numerical
results about continuous problems|for instance, rigorous bounds on the global minimum of a
function or the solution of a dierential equation. It is part of the discipline of constructive real
analysis". In the long term, the results of such computations may become suciently trusted to be
accepted as contributing to legal decisions. The machine infrastructure acts as a body of theorems
on which the correctness of an interval algorithm relies, so it must be made as reliable as is prac-
tical. In its logical chain are many links|hardware, underlying
oating-point system, etc.|over
which this standard has no control. The standard aims to strengthen one specic link, by dening
interval objects and operations that are theoretically well-founded and practical to implement.
There are several mathematical bases of interval computation, which are not wholly compatible
with each other. Thus it was necessary to make a choice for the purposes of this standard. The
working group generally agreed that a high priority should be given to making the mathematical
basis easy to grasp, easy to teach, and easy to interpret in the context of real-world applications.
We believe the one we have chosen is the best match for those aims. In this theory
 Intervals are sets.
 They are subsets of the real numbers R.
 An interval operation is dened algebraically, in contrast to topologically. The interval
version of an elementary function such as sin x is essentially the natural extension to sets
of the corresponding pointwise function on real numbers.
This contrasts on the one hand with Kaucher or modal interval theory, where an interval is a
formal object, an ordered pair of real numbers x; x, of which those having x  x are interpretable
as intervals in the set sense; and on the other hand with containment set (cset) theory, where
intervals are subsets of the extended reals R, and operations are dened topologically, in terms
of limits. In this standard, the set IR of intervals comprises precisely the closed and connected
(in the topological sense) subsets of R. This includes the empty set, as well as intervals that are
unbounded on one or both sides.
1.2. Specication Levels. The 754-2008 standard describes itself as layered into four Specica-
tion Levels. To manage complexity, P1788 uses a corresponding structure: level 1, of mathematical
interval theory; level 2, the nite set of interval datums in terms of which nite-precision interval
computation is dened; level 3, of representations of intervals by
oating-point numbers; level 4,
of bit strings and memory.
There is another important player: the programming language. We acknowledge the experience
of the 754-2008 working group, who recognized a serious defect of the 754-1985 standard, namely
that it specied individual operations but not how they should be used in expressions. Over the
years, compilers made clever transformations so that it became impossible to know the precisions
used and the roundings performed while evaluating an expression, or whether the compiler had
even optimized away" (1:0 + x) ??? 1:0 to become simply x.
This is also a problem for intervals. Thus the standard makes requirements and recommenda-
tions on language implementations, thereby dening the notion of a standard-conforming imple-
mentation of intervals within a language.
The language does not constitute a fth level in some linear sequence; from the user's viewpoint
it sits above datum level 2, alongside theory level 1, as a practical means to implement interval
algorithms by manipulating level 2 entities (though most languages have in
uence on levels 3 and
4 also).
1.3. The Fundamental Theorem. Ramon Moore's Fundamental Theorem of Interval Arith-
metic (FTIA) is central to interval computation. Roughly, it says that if f is an explicit expression
dening a real function f(x1; : : : ; xn), then evaluating f in interval mode" over any interval inputs
(x1; : : : ; xn) is guaranteed to give an enclosure of the range of f over those inputs. A version of
the FTIA holds in all variants of interval theory, but with varying hypotheses and conclusions.