The problem addressed in this paper is the computation of elementary functions (exponential, logarithm, trigonometric functions, hyperbolic functions and their reciprocals) in fixed precision, typically the computer single or double precision. The method proposed here combines Shift-and-Add algorithms and classical methods for the numerical integration of ODEs: it consists in performing the Shift-and-Add iteration until a point close enough to the argument is reached, thus only one step of Euler method or Runge-Kutta method is performed. This speeds up the computation while ensuring the desired accuracy is preserved. Time estimations on various processors are presented which illustrate the advantage of this hybrid method.