Searching worst cases of a one-variable function using lattice reduction

We propose a new algorithm to find worst cases for the correct rounding of a mathematical function of one variable. We first reduce this problem to the real small value problem - i.e., for polynomials with real coefficients. Then, we show that this second problem can be solved efficiently by extending Coppersmith´s work on the integer small value problem - for polynomials with integer coefficients - using lattice reduction. For floating-point numbers with a mantissa less than N and a polynomial approximation of degree d, our algorithm finds all worst cases at distance less than N^-d2/2d+1 from a machine number in time O(N^(d+12d+1)+ε/). For d=2, a detailed study improves on the O(N²(3+ε)/) complexity from Lefevre´s algorithm to O(N⁴(7+ε)/). For larger d, our algorithm can be used to check that there exist no worst cases at distance less than N^-k in time O(N¹(2+ε)/).