Correct rounding provides the best approximation of the exponential function by double-precision numbers. To obtain the correctly rounded exponential of some arguments, the exponential should be calculated with high accuracy. For small arguments, even higher accuracy is required. This article presents simple and very fast algorithms for small arguments. Yet another algorithm presented here demonstrates a good maximum execution time, which may be important for critical applications. This algorithm can be combined with some other already existing algorithms to achieve the best maximum and average execution times. All proposed algorithms calculate the correctly rounded exponential function for all rounding modes and use only double-precision arithmetic for computation. In the argument reduction step, precalculated tables are used. Test implementations of these algorithms were developed in C language and are portable. Full proofs are presented either in this article itself or in its appendices.