The control of error during numerical evaluation of an expression is a crucial problem. A natural solution is to use interval arithmetic during calculations. For this reason, we need algorithms able to efficiently determine the image of an interval by the elementary functions. Adapting classical algorithms to interval arithmetic leads to a large overestimation of the result. In this paper, we propose a method based on Taylor approximation which evaluates functions on the bounds of the interval and deduces the resulting image. The computation of each bound in done with the Taylor approximation evaluated by the Smith scheme. Indeed, polynomial approximation seems to be the most adapted to our range of precision (several hundreds of digits). This approach is better than the classical algorithm using the Horner scheme and than the other polynomial approximations (e.g. minimax and Chebyshev polynomials) which are more precise but also more complicated. Furthermore, we present a rigorous evaluation of the error. Thus, the desired accuracy of the result is given, even on the numerically instable points (big number, (pi) /2 ...). For the computation of a point, our algorithm implemented with GMP is up to 2 times faster than the corresponding computation with softwares like MAPLE or MUPAD which don't even guarantee the relative precision. We show that the control of the error is not costly. And since the Taylor approximation is just a bit less precise but far more simple than the other approximations, it can be rapidly determined and, all in all, is the most efficient.