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Searching Worst Cases of a One-Variable Function Using Lattice Reduction
March 2005 (vol. 54 no. 3)
pp. 340-346
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^{\frac{-d^2}{2d+1}} from a machine number in time O(N^{{\frac{d+1}{2d+1}}+\varepsilon}). For d=2, a detailed study improves on the O(N^{2/3+\varepsilon}) complexity from Lefèvre's algorithm to O(N^{4/7+\varepsilon}). 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(N^{1/2+\varepsilon}).

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Index Terms:
Computer arithmetic, multiple precision arithmetic, special function approximations.
Damien Stehl?, Vincent Lef?vre, Paul Zimmermann, "Searching Worst Cases of a One-Variable Function Using Lattice Reduction," IEEE Transactions on Computers, vol. 54, no. 3, pp. 340-346, March 2005, doi:10.1109/TC.2005.55
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