On Faster Integer Calculations Using Non-arithmetic Primitives
UC '08 Proceedings of the 7th international conference on Unconventional Computing
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The authors prove upper and lower bounds for approximately computing the square root using a given set of operations. The bounds are extended to hold for approximating the kth root, for any fixed k. Several tools from approximation theory are used to prove the lower bound. These include Markoff inequality, Chebyshev polynomials, and a theorem that relates the degree of a rational function to its deviation from the approximated function over a given interval. The lower bound can be generalized to other algebraic functions. The upper bound can be generalized to obtain an O(1)-step straight-line program for evaluating any rational function with integer coefficients at a given integer point.