A Novel Approach to Generate Correctly Rounded Math Libraries for New Floating Point Representations

TL;DR

This paper introduces a new method for generating polynomial approximations that ensure correctly rounded results for elementary functions across various floating-point representations, improving accuracy and efficiency.

Contribution

It presents a linear programming-based approach to generate polynomials that produce correctly rounded results, addressing limitations of existing approximation methods.

Findings

Developed faster, correctly rounded implementations of elementary functions.

Guarantees correctness even with range reduction techniques.

Applicable to multiple new floating-point representations.

Abstract

Given the importance of floating-point~(FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and Posits). These representations do not have correctly rounded math libraries. Further, the use of existing FP libraries for these new representations can produce incorrect results. This paper proposes a novel approach for generating polynomial approximations that can be used to implement correctly rounded math libraries. Existing methods generate polynomials that approximate the real value of an elementary function $f(x)$ and produce wrong results due to approximation errors and rounding errors in the implementation. In contrast, our approach generates polynomials that approximate the correctly rounded value of $f(x)$ (i.e., the value of $f(x)$ rounded to the target representation). It provides more margin to identify efficient polynomials that produce correctly rounded results for all inputs. We frame the problem of generating efficient polynomials that produce correctly rounded results as a linear programming problem. Our approach guarantees that we produce the correct result even with range reduction techniques. Using our approach, we have developed correctly rounded, yet faster, implementations of elementary functions for multiple target representations.