A new automatic method to correct the first-order effect of floating point rounding errors on the result of a numerical algorithm is presented. A correcting term and a confidence threshold are computed using automatic differentiation, computation of elementary rounding error and running error analysis. Algorithms for which the accuracy of the result is not affected by higher order terms are identified. The correction is applied to the final result or to sensitive intermediate results. The properties and the efficiency of the method are illustrated with a sample numerical example.