Floating-point arithmetic

Definition

Fix integers $\beta \ge 2$ (the base), $p \ge 1$ (the precision), and $e_{\min} < e_{\max}$. A floating-point number of this system is $0$ or a real number of the form $$x = \pm\, (d_0 . d_1 d_2 \ldots d_{p-1})_\beta \times \beta^{e} = \pm \left( \sum_{i=0}^{p-1} d_i \beta^{-i} \right) \beta^{e}.$$ Here the digits $d_i$ are integers with $0 \le d_i \le \beta - 1$, and the exponent $e$ is an integer with $e_{\min} \le e \le e_{\max}$. The number is normalized if $d_0 \ne 0$. Let $F$ be the set of these numbers. Let $\mathrm{fl}$ be a rounding to $F$ of the real numbers between $\min F$ and $\max F$, usually rounding to nearest. Floating-point arithmetic replaces each operation $\circ \in \set{+, -, \times, /}$ on $x, y \in F$, whenever $x \circ y$ lies in this range, by $$x \circledast y = \mathrm{fl}(x \circ y),$$ that is, the exact result rounded to $F$.