1 Introduction A real number x is usually approximated in a digital computer by an element fl(x) of a finite set F of &quot;floating-point &quot; numbers. We regard the elements of F as exactly representable real numbers, and take fl(x) as the floating-point number closest to x. The definition of &quot;closest&quot;, rules for breaking ties, and the possibility of truncating instead of rounding are discussed later. We restrict our attention to binary computers in which floating-point numbers are represented in a word (or multiple word) of fixed length w bits, using some convenient (possibly redundant) code. Usually F is a set of numbers of the for
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