Closing the Performance Gap between Doubles and Rationals for Octagons

Octagons have enduring appeal because their domain operations are simple, readily mapping to for-loops which apply max, min and sum to the entries of a Difference Bound Matrix (DBM). In the quest for efficiency, arithmetic is often realised with double-precision floating-point, albeit at the cost of the certainty provided by arbitrary-precision rationals. In this paper we show how Compact DBMs (CoDBMs), which has recently been proposed as a memory refinement for DBMs, enable arithmetic calculation to be short-circuited in various domain operations. We also show how comparisons can be avoided by changing the tables which underpin CoDBMs. From the perspective of implementation, the optimisations are attractive because they too are conceptually simple, following the ethos of Octagons. Yet they halve the running time on rationals, putting CoDBMs on rationals on a par with DBMs on doubles.