Floating-point arithmetic is prone to accuracy problems due to the round-off errors. The combination of the round-off errors and of the out of order execution of arithmetic operations due to the scheduling of parallel tasks, introduces additional numerical accuracy issues. In this article, we address the problem of improving the numerical accuracy and reproducibility of summation operators. We propose two efficient parallel algorithms for summing n floating-point numbers. The first objective of our algorithms is to obtain an accurate result without increasing the linear complexity of the naive algorithm. The second objective is to improve the reproducibility of the summations compared to those computed by the naive algorithm and this regardless of the number of processors used for the computations.