Making Interval Arithmetic Robust to Overflow

In theory, interval arithmetic at high precision can compute mathematical expressions to any required accuracy. An arbitrary precision library like MPFR can thus be used to evaluate real-valued expressions. In practice, however, MPFR’s maximum representable exponent means some inputs cannot be evaluated to the required accuracy. This paper introduces movability flags, which soundly detect the majority of these inputs. Movability flags are set when overflow occurs and track whether recomputing at higher precision could possibly result in narrow intervals. In our tests on 481 expressions, movability flags detect 81.0% of unsamplable inputs. Compared to Mathematica, our movability-flag-enhanced interval arithmetic library resolves 60.3% more challenging inputs, returns 7.6× fewer completely indeterminate results, and avoids 64 cases of fatal error.