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Subsection: A Primer On Interval Arithmetic

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\subsection{A primer on Interval Arithmetic}
\label{prec_section}
In this section, we present briefly Interval Arithmetic and focus on its interaction with floating point approximation of reals. For more details on
Interval Arithmetic, the interested reader can consult a more extensive reference, such as~\cite{InterBook}.\medskip

\subsubsection{Bird's eye view on Interval Arithmetic.}
Interval arithmetic is a representation of reals by intervals that contain them. For instance, one can specify that a value $x$ is given with an error $\epsilon$ by considering the interval
$[x-\epsilon, x+\epsilon]$, or even manipulating the transcendent constant $\pi$ as the interval $[3.14, 3.15]$. Interval arithmetic is crucial the context of numerical
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The \emph{diameter} of the interval
$\inter{x}$ is defined as $\inter{x}^+ - \inter^{x}^-$, and its \emph{center} is the real $\frac{1}{2}(\inter{x}^+ + \inter^{x}^-)$.

We can now define abstractly an arithmetic on intervals:

\begin{definition} Let $\bowtie$ be a binary operation --- resp.\ f be a function ---, then the result $\inter{x} \bowtie \inter{y}$ of the operation between the intervals $\inter{x}$ and ${\inter{y}}$ --- resp $f(\inter{x})$, result of the application of $f$ --- is the smallest interval, in the sense of inclusion, containing \[ \{x\bowtie y | (x,y)\in \inter{x}\times\inter{y}\} \qquad \textrm{--- resp.~} \{f(x) | x\in \inter{x}\} \textrm{~---}.
\] \end{definition}

In the context of actual computations, requiring the equality in the above definition is in most case illusory, since reals cannot be represented exactly.
Yet, only the inclusion of the resulting interval in the last defined set
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\subsection{Representations of reals}

\subsubsection{Integral representation}
\label{sec:intgral_representation}
As evoked, real numbers can only be computationally handled at finite precision.
A naive yet convenient way to do so is to use integers to represent the bounds of intervals (specifically, integers made from the most-significant bits of the binary expansion of the desired reals). Formally we have:

\begin{definition}[Integral representation of reals]
\label{def:intgral_representation}
Let $x\in\RR$ be an arbitrary real number and $n>0$ a non-negative integer. Define its \emph{integral representation at accuracy\footnote{% We use here the denomination of "accuracy" instead of "precision" to avoid confusions with the floating point precision as defined in paragraph~\ref{sec:fp_representation}.}~% $n$} as the symetric interval centered on $\lfloor2^{n}x\rceil$ of diameter $1$: \[ \inter{x}_n = \left[\lfloor2^{n}x\rceil-\frac{1}{2}, \lfloor2^{n}x\rceil+\frac{1}{2} \right] \]
\end{definition}

\begin{remark} The interval $\inter{x}_n$ only needs to be represented as its center,
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