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              <p>The series expansion at the origin of the Airy function Ai(x) is alternating and hence problematic to evaluate for x &gt; 0 due to cancellation. Based on a method recently proposed by Gawronski, Müller, and Reinhard, we exhibit two functions F and G, both with nonnegative Taylor expansions at the origin, such that Ai(x) = G(x)/F(x). The sums are now well-conditioned, but the Taylor coefficients of G turn out to obey an ill-conditioned three-term recurrence. We use the classical Miller algorithm to overcome this issue. We bound all errors and our implementation allows an arbitrary and certified accuracy, that can be used, e.g., for providing correct rounding in arbitrary precision.</p>
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              <p>Le développement en série à l'origine de la fonction d'Airy Ai(x) est alterné, ce qui rend son évaluation numérique pour x &gt; 0 problématique en raison de phénomènes de compensation numérique. Sur la base d'une méthode proposée récemment par Gawronski, Müller et Reinhard, nous exhibons deux fonctions F et G, toutes deux avec un développement de Taylor à l'origine à coefficients positifs, telles que Ai(x)=G(x)/F(x). Les sommes sont alors bien conditionnées, mais il s'avère que les coefficients de Taylor de G obéissent à une récurrence à trois termes mal conditionnée. Nous utilisons la méthode classique de Miller pour surmonter cette difficulté. Nous bornons toutes les erreurs commises, et notre implémentation fonctionne de manière garantie à précision arbitraire, ce qui permet par exemple de l'utiliser pour fournir un arrondi correct à précision arbitraire.</p>
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