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The series expansion at the origin of the Airy function Ai(x) is alternatingand hence problematic to evaluate for x > 0 due to cancellation. Based on amethod recently proposed by Gawronski, M\"uller, and Reinhard, we exhibit twofunctions F and G, both with nonnegative Taylor expansions at the origin, suchthat Ai(x) = G(x)/F(x). The sums are now well-conditioned, but the Taylorcoefficients of G turn out to obey an ill-conditioned three-term recurrence. Weuse the classical Miller algorithm to overcome this issue. We bound all errorsand our implementation allows an arbitrary and certified accuracy, that can beused, e.g., for providing correct rounding in arbitrary precision.

Multiple precision evaluation of the Airy Ai function with reduced cancellation