  This work establishes the existence of addition theorems and double-angle
formulas for Ck real scalar functions. Moreover, we determine necessary and
sufficient conditions for a bivariate function to be an addition formula for a
Ck real function. The double-angle formulas allow us to generate a duplication
algorithm, which can be used as an alternative to the classical numerical
methods to obtain an approximation for the solution of an ordinary differential
equation. We demonstrate that this algorithm converges uniformly in any compact
domain contained in the maximal domain of that solution. Finally, we carry out
some numerical simulations showing a good performance of the duplication
algorithm when compared with standard numerical methods
