Outward-rounded interval, rectangle and ball operations

For real intervals $X=[a,b]$ and $Y=[c,d]$, compute the displayed exact endpoint formulas and round the lower endpoint down and the upper endpoint up in the format of Dyadic floating-point format and directed rounding. If either exact endpoint overflows, return overflow with the outward extended endpoints; never substitute a finite endpoint for an overflow sentinel. Addition is $[a+c,b+d]$, subtraction is $[a-d,b-c]$, and multiplication is $[\min (ac,ad,bc,bd),\max (ac,ad,bc,bd)]$. Division is defined only when $0\mathrm{\notin }Y$ and is $X[1\mathrm{/}d,1\mathrm{/}c]$, with reciprocal endpoints computed as exact rationals before outward rounding. Squaring is $[0,\max (a^2,b^2)]$ when $0\in X$ and $[\min (a^2,b^2),\max (a^2,b^2)]$ otherwise. Square root is defined only for $X\subseteq [0,\mathrm{\infty })$ and is enclosed by applying Exact dyadic bisection for square-root endpoints to the exact dyadic endpoints with request $n=p$, where $p$ is the selected format precision, and rounding outward for storage.

If $0\in Y$ for division, return split-domain with the two subdomains $Y\cap (-\mathrm{\infty },0)$ and $Y\cap (0,\mathrm{\infty })$, or entire when the caller does not request subdivision. No finite quotient interval is returned in this case. An invalid square-root domain returns domain-error or an explicitly restricted nonnegative subdomain.

Complex-rectangle addition, subtraction and multiplication use $(x+iy)(u+iv)=(xu-yv)+i(xv+yu)$, applying the real interval operations to each component. For a rectangle $X+iY$ excluding $0$, define its reciprocal by $(X-iY)\mathrm{/}(X^2+Y^2)$, computing the square hulls, their sum, and the component quotient endpoint formulas as exact rationals before rounding the final reciprocal components outward. The exact denominator interval is strictly positive because at least one coordinate interval excludes zero; do not round it before division, since underflow could introduce zero. Propagate any final endpoint overflow. Division multiplies by this reciprocal.

For a ball $B(c,r)$, let $X_B=[\mathrm{\Re }c-r,\mathrm{\Re }c+r]$ and $Y_B=[\mathrm{\Im }c-r,\mathrm{\Im }c+r]$ be its coordinate projection intervals. For a real interval $J$, let $M(J)=\max (\mathrm{\mid }\inf J\mathrm{\mid },\mathrm{\mid }\sup J\mathrm{\mid })$ and let $d(J)=0$ if $0\in J$, otherwise $d(J)=\min (\mathrm{\mid }\inf J\mathrm{\mid },\mathrm{\mid }\sup J\mathrm{\mid })$. Put $M_B=M(X_B)+M(Y_B)$ and $m_B=\max (d(X_B),d(Y_B))$. Then $\mathrm{\mid }z\mathrm{\mid }\le M_B$ for every $z\in B(c,r)$ and $\mathrm{\mid }z\mathrm{\mid }\ge m_B$ for every such $z$. Define ball addition and subtraction as $B(0,\mathrm{ru}(M_{B_1}+M_{B_2}))$, multiplication as $B(0,\mathrm{ru}(M_{B_1}{M}_{B_2}))$, and division as $B(0,\mathrm{ru}(M_{B_1}\mathrm{/}{m}_{B_2}))$ only when $m_{B_2}>0$. These origin-centered output balls are conservative but inclusion-isotone. If $m_{B_2}=0$, return split-domain, entire, or inconclusive; if a radius overlows, return overflow.

Dyadic endpoints are rational, so every real arithmetic endpoint and every complex-division component before square root is rational. Reciprocals are formed only for nonzero rational endpoints. In a complex rectangle excluding zero, at least one coordinate interval omits zero; its exact squared interval has a strictly positive lower endpoint, so the unrounded $X^2+Y^2$ does also. The prescribed exact rational component quotients therefore have nonzero denominators even when storage rounding would underflow that lower bound. For ball division, $m_{B_2}>0$ gives $\mathrm{\mid }y\mathrm{\mid }\ge m_{B_2}>0$ for every $y\in B_2$. Every endpoint overflow is reported explicitly and cannot produce a finite-result claim.