Group invariants are used in high energy physics to define quantum field theory interactions. In this paper, we are presenting the parallel algebraic computation of special invariants called symplectic and even focusing on one particular invariant that finds recent interest in physics. Our results will export to other invariants. The cost of performing basic computations on the multivariate polynomials involved evolves during the computation, as the polynomials get larger or with an increasing number of terms. However, in some cases, they stay small. Traditionally, high-performance software is optimized by running it on a smaller data set in order to use profiling information to set some tuning parameters. Since the (communication and computation) costs evolve during the computation, the first iterations of the computation might not be representative of the rest of the computation and this approach cannot be applied in this case. To cope with this evolution, we are presenting an approach to get performance data and tune the algorithm during the execution.