一句话：提出用 n+2 位 round-to-odd 多项式逼近，一次生成支持多种精度与全部 IEEE 舍入模式的正确舍入初等函数库。
主流浮点数学库不能对所有输入给出正确舍入结果；现有 CR-LIBM/RLIBM 只针对特定表示和单一舍入模式，换精度或舍入模式会因双重舍入出错。本文提出生成单个多项式逼近的方法：为 n 位表示生成 n+2 位、采用 round-to-odd 模式的多项式，并证明该逼近对 IEEE 五种舍入模式以及满足 |E|+1<k≤n 的多种 k 位表示均正确舍入。作者将多项式生成建模为线性规划，并针对 round-to-odd 模式改进生成算法。原型是首个用单个多项式逼近对所有输入、所有 IEEE 舍入模式产出正确舍入结果的 32 位浮点库，还支持 10 到 32 位配置，且比主流库更快。
Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode. Using such libraries for a representation with a new rounding mode or with different precision will result in wrong results due to double rounding. This paper proposes a novel method to generate a single polynomial approximation that produces correctly rounded results for all inputs for multiple rounding modes and multiple precision configurations. To generate a correctly rounded library for n -bits, our key idea is to generate a polynomial approximation for a representation with n +2-bits using the round-to-odd mode. We prove that the resulting polynomial approximation will produce correctly rounded results for all five rounding modes in the standard and for multiple representations with k -bits such that | E | +1 < k ≤ n , where | E | is the number of exponent bits in the representation. Similar to our prior work in the RLIBM project, we approximate the correctly rounded result when we generate the library with n +2-bits using the round-to-odd mode. We also generate polynomial approximations by structuring it as a linear programming problem but propose enhancements to polynomial generation to handle the round-to-odd mode. Our prototype is the first 32-bit float library that produces correctly rounded results with all rounding modes in the IEEE standard for all inputs with a single polynomial approximation. It also produces correctly rounded results for any FP configuration ranging from 10-bits to 32-bits while also being faster than mainstream libraries.
@article{paperbot1535,
title = {One polynomial approximation to produce correctly rounded results of an elementary function for multiple representations and rounding modes},
author = {Jay P. Lim and Santosh Nagarakatte},
journal = {Proceedings of the ACM on Programming Languages},
volume = {6},
number = {POPL},
year = {2022},
doi = {10.1145/3498664}
}