For univariate polynomials f(x1), Horner scheme provides the fastest way to compute the value. For multivariate polynomials, several different version of Horner scheme are possible; it is not clear which of them is optimal. In this paper, we propose a greedy algorithm that will hopefully lead to good computation times.
A univariate Horner scheme has another advantage: if the value x1 is known with uncertainty, and we are interested in the resulting uncertainty in f(x1), then Horner scheme leads to a better estimate for this uncertainty than many other ways of computing f(x1). The second greedy algorithm that we propose tries to find the multivariate Horner scheme that leads to the best estimate for the uncertainty in f(x1,...,xn).