Worst Cases and Lattice Reduction

We propose a new algorithm to find worst cases for correct rounding of an analytic function. 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 [4, 5, 6]. For floating-point numbers with a mantissa less than , and a polynomial approximation of degree , our algorithm finds all worst cases at distance from a machine number in time . For , this improves on the complexity from Lefèvre’s algorithm [15, 16] to "! . We exhibit some new worst cases found using our algorithm, for double-extended and quadruple precision. For larger , our algorithm can be used to check that there exist no worst cases at distance $# % in time '&)( *,+ .