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For large $N$, we consider the ordinary continued fraction of $x=p/q$ with$1\le p\le q\le N$, or, equivalently, Euclid's gcd algorithm for two integers$1\le p\le q\le N$, putting the uniform distribution on the set of $p$ and$q$s. We study the distribution of the total cost of execution of the algorithmfor an additive cost function $c$ on the set $\mathbb{Z}_+^*$ of possibledigits, asymptotically for $N\to\infty$. If $c$ is nonlattice and satisfiesmild growth conditions, the local limit theorem was proved previously by thesecond named author. Introducing diophantine conditions on the cost, we areable to control the speed of convergence in the local limit theorem. We useprevious estimates of the first author and Vall\'{e}e, and we adapt to oursetting bounds of Dolgopyat and Melbourne on transfer operators. Ourdiophantine condition is generic (with respect to Lebesgue measure). For smoothenough observables (depending on the diophantine condition) we attain theoptimal speed.

A local limit theorem with speed of convergence for Euclidean algorithms and diophantine costs