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The free Eureqa program has recently received extensive press praise. A representative quote is "There are very clever 'thinking machines' in existence today, such as Watson, the IBM computer that conquered Jeopardy! last year. But next to Eureqa, Watson is merely a glorified search engine."The program is designed to work with noisy experimental data, searching for then returning a set of result expressions that attempt to optimally trade off conciseness with accuracy.However, if the data is generated from a formula for which there exists more concise equivalent formulas, sometimes some of the candidate Eureqa expressions are one or more of those more concise equivalents expressions. If not, perhaps one or more of the returned Eureqa expressions might be a sufficiently accurate approximation that is more concise than the given formula. Moreover, when there is no known closed form expression, the data points can be generated by numerical methods, enabling Eureqa to find expressions that concisely fit those data points with sufficient accuracy. In contrast to typical regression software, the user does not have to explicitly or implicitly provide a specific expression or class of expressions containing unknown constants for the software to determine.Is Eureqa useful enough in these regards to provide an additional tool for experimental mathematics, computer algebra users and numerical analysts? Yes, if used carefully. Can computer algebra and numerical methods help Eureqa? Definitely.

The Derive computer-algebra program has Expand as one of the menu choices: The user is prompted for successively less main expansion variables, which can be all of the variables or any proper subset. It is clear how to proceed when the expression is a polynomial: Fully distribute with respect to all expansion variables, but collect as coefficient polynomials all terms that share the same exponents for the expansion variables. Derive uses a partially factored form, so the collected coefficient polynomials can be fortuitously partially factored. For rational expressions the expand function does partial fraction expansion because it is the most useful kind of rational expansion. However, most other computer algebra systems and examples in the literature focus on partial fraction expansion with respect to only one variable, where any other variables are considered mere parameters. For consistency with multivariate polynomial expansion, we wanted a useful and well-defined meaning for multivariate partial fraction expansion. This paper provides such a definition and a corresponding algorithm.

Most of us have taken the exact rational and approximate numbers in our computer algebra systems for granted for a long time, not thinking to ask if they could be significantly better. With exact rational arithmetic and adjustable-precision floating-point arithmetic to precision limited only by the total computer memory or our patience, what more could we want for such numbers? It turns out that there is much more that can be done that permits us to obtain exact results more often, more intelligible results, approximate results guaranteed to have requested error bounds, and recovery of exact results from approximate ones.

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