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

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 methodology for generating polynomial approximations that can be used to implement correctly rounded math libraries. Existing methods produce polynomials that approximate the real value of an elementary function $f(x)$ and experience wrong results due to errors in the approximation and due to 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). This methodology 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. Our Bfloat16 library is 2.3$\times$ faster than the corresponding state-of-the-art while producing correct results for all inputs.