Real intervals, complex rectangles and ball enclosures

A real interval is a closed set $[a,b]\subset \mathbb{R}$ with finite dyadic endpoints $a\le b$. Its width is $b-a$, midpoint is $(a+b)\mathrm{/}2$, and its midpoint-radius form is $m+[-r,r]$ with $m=(a+b)\mathrm{/}2$ and $r=(b-a)\mathrm{/}2$. The empty set is the status empty; the whole real line is entire. overflow, domain-error, split-domain, and inconclusive are computation statuses, not finite intervals.

A complex rectangle is $X+iY=\{x+iy:x\in X,y\in Y\}$ for real intervals $X,Y$. Complex addition and multiplication use $(x+iy)+(u+iv)=(x+u)+i(y+v)$ and $(x+iy)(u+iv)=(xu-yv)+i(xv+yu)$. For $z=x+iy$, its complex modulus is $\mathrm{\mid }z\mathrm{\mid }=\sqrt{x^2+y^2}$, and for $z\mathrm{\ne }0$ its reciprocal is $1\mathrm{/}z=(x-iy)\mathrm{/}(x^2+y^2)$. A complex ball $B(c,r)$ is $\{z\in \mathbb{C}:\mathrm{\mid }z-c\mathrm{\mid }\le r\}$ with dyadic complex center $c$ and dyadic $r\ge 0$. For any two represented sets $S,T$, the inclusion order is $S⊑T$ iff $S\subseteq T$.

The width of a complex rectangle with coordinate widths $w_X,w_Y$ is its Euclidean diameter $\sqrt{w_X^2+w_Y^2}$; the width of a ball $B(c,r)$ is its diameter $2r$. Width comparisons against a dyadic tolerance can be decided by exact rational comparisons after squaring the nonnegative quantities.

The range of a real or complex function $f$ on a set $S$ is $f(S)=\{f(x):x\in S\cap \mathrm{dom}(f)\}$. A finite expression DAG is a finite directed acyclic graph with input nodes, constant nodes, and operation nodes; constants are exact dyadics or are supplied with certified enclosing intervals. Sharing a node means the exact same intermediate value is reused.