标题： 快速混合精度实数评估 显示英文标题

标题： Fast Mixed-Precision Real Evaluation

摘要： 在计算数学、物理和数值分析中，高精度评估实值表达式是一项关键的基础工作。 一种典型的实现方法是以统一的精度评估整个表达式，然后将精度加倍直至获得足够准确的结果。 这种方法浪费资源：通常只有少数操作真正需要以高精度执行，而表达式的大部分内容可以更快地计算。 然而，到目前为止，这种非均匀精度分配的计算仍不可行。 我们提出了一种快速的新算法来推导这种精度分配。 该算法利用较低精度计算的结果，分析性地确定一种混合精度分配方案，从而得到足够准确的结果。 我们的实现，Reval，相比最先进的Sollya工具平均提速1.72倍，在最困难的输入点上提速可达5.21倍。 对使用和未使用精度调整的运算所用精度的检查表明，提速的原因在于大多数操作分配了更低的精度，尽管由非均匀精度分配带来的额外优化也起到了一定作用。 显示英文摘要

摘要： Evaluating real-valued expressions to high precision is a key building block in computational mathematics, physics, and numerics. A typical implementation evaluates the whole expression in a uniform precision, doubling that precision until a sufficiently-accurate result is achieved. This is wasteful: usually only a few operations really need to be performed at high precision, and the bulk of the expression could be computed much faster. However, such non-uniform precision assignments have, to date, been impractical to compute. We propose a fast new algorithm for deriving such precision assignments. The algorithm leverages results computed at lower precisions to analytically determine a mixed-precision assignment that will result in a sufficiently-accurate result. Our implementation, Reval, achieves an average speed-up of 1.72x compared to the state-of-the-art Sollya tool, with the speed-up increasing to 5.21x on the most difficult input points. An examination of the precisions used with and without precision tuning shows that the speed-up results from assigning lower precisions for the majority of operations, though additional optimizations enabled by the non-uniform precision assignments also play a role.