Fuzzy simplification of non-numeric expressions containing some intervals and/or floating point numbers

Abstract

This article describes a Mathematica package that improves simplification of general non-numeric expressions containing any mixture of Gaussian rational numbers, symbolic constants, machine and arbitrary-precision floating-point numbers, together with intervals having any mixture of such endpoints. Such generalized numbers are not automatically all converted to floats or to intervals. Expressions can be multivariate and non-polynomial. Techniques include: • Recognition and unification of approximately similar and approximately proportional factors and terms. • The option of infinity-norm normalization that is more robust than monic normalization. • The option of a unit-normal quasi-primitive normalization that uses large rational approximate common divisors of mixed number types and intervals to nicely normalize sums. • A polynomial division algorithm tolerant of terms with coefficients that are float zeros or intervals containing 0. • The ability to round or underflow negligible terms while satisfying the inclusion property of interval arithmetic. The package and a more detailed version of this article will be posted at arXiv.org.