Efficient Boys function evaluation using minimax approximation

We present an algorithm for efficient evaluation of Boys functions F0,…,Fkmax tailored to modern computing architectures, in particular graphical processing units, where maximum throughput is high and data movement is costly. The method combines rational minimax approximations with upward and downward recurrence relations. The non-negative real axis is partitioned into three regions, [0, ∞⟩ = A ∪ B ∪ C, where regions A and B are treated using rational minimax approximations and region C by an asymptotic approximation. This formulation avoids lookup tables and irregular memory access, making it well-suited for hardware with high maximum throughput and low latency. The rational minimax coefficients are generated using the rational Remez algorithm. For a target maximum absolute error of ɛtol = 5 × 10-14, the corresponding approximation regions and coefficients for Boys functions F0, …, F32 are provided in Appendix D.