Certified special-function enclosure result

A certified enclosure for a real value is a nonempty closed interval $B=[a,b]$ with finite dyadic endpoints; for a complex value it is a nonempty closed rectangle $B=[a_1,b_1]+i[a_2,b_2]$ with finite dyadic endpoints. Its radius is half its length in the real case and half its Euclidean diagonal in the complex case; $d(B,0)$ is the Euclidean distance from $B$ to $0$. An evaluator item declares whether its input is an exact dyadic point or a dyadic interval/rectangle enclosure; a successful range result contains the exact function image of every admissible point in an input enclosure. Constants not exactly dyadic must be supplied with certified enclosing intervals or rectangles.

The interval backend uses the endpoint operations in Outward-rounded interval, rectangle and ball operations; the soundness of finite arithmetic expression DAGs follows from Outward-rounded operations are inclusion-isotone and enclose DAG evaluations. Real elementary function ranges use Certified real elementary-function evaluations enclose their ranges, and complex exponential and principal-logarithm ranges use Certified complex exponential and principal logarithm enclose their ranges. An operation that returns a non-finite status supplies no finite enclosure claim.

A finite-budget evaluator receives a branch input enclosure wholly within the stated domain, absolute tolerance ${\epsilon }_a>0$, optional relative tolerance ${\epsilon }_r>0$, finite integer work and precision caps, and the fixed format and truncation parameters required by the backend. Its result has one of four statuses:

• success: the returned enclosure contains the exact requested value and its radius is at most ${\epsilon }_a$. If $0\mathrm{\notin }B$ and relative tolerance was requested, the evaluator may also report relative_certified: true only when $\mathrm{rad}(B)\mathrm{/}d(B,0)\le {\epsilon }_r$.
• inconclusive: the procedure exhausted a finite cap before its requested enclosure test passed.
• domain: the input is outside the stated branch domain, violates a stated parameter condition, or lies on a branch cut excluded by that branch.
• range: a certified finite enclosure cannot be represented in the requested output format; the procedure returns no unsound finite interval.

If a success enclosure contains $0$, it still supplies the stated absolute bound, but it carries no relative-accuracy claim. The work and precision caps do not imply that success occurs for every input.