In this paper, the computation of erf(x) in arbitrary precision is detailed. A feature of our implementation is correct rounding: the returned result is the exact result (as if it were computed with infinite precision) rounded according to the specified rounding mode. The algorithm that computes the correctly rounded value of erf(x) for any argument x is detailed in this paper. In particular, the choice of the approximation formula, the determination of the truncation rank and of the computing precision are presented. When the current truncation rank and computing precision do not suffice to determine the correctly rounded value, they are increased and the computation is done again. The optimal strategy to adapt them is given in this paper. Finally, timings on some experiments are given, and the implementation of the complementary error function erfc is then sketched.