Efficient Boys function evaluation using minimax approximation

Abstract

We present an algorithm for efficient evaluation of Boys functions $F_0,\dots,F_{k_\mathrm{max}}$ tailored to modern computing architectures, in particular graphical processing units (GPUs), 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, $[0,\infty\rangle = A\cup B\cup C$, 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 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 $\varepsilon_\mathrm{tol} = 5\cdot10^{-14}$, the corresponding approximation regions and coefficients for Boys functions $F_0,\dots,F_{32}$ are provided in the appendix.