Identification of hardest-to-round points for elementary functions across precisions

Identify hardest-to-round input values for elementary mathematical functions (e.g., exp, log, sin, cos, tan) across all relevant input and output precisions in binary floating-point arithmetic to enable provable correct rounding guarantees.

Background

Hardest-to-round points are inputs where the exact function value lies extremely close to a rounding breakpoint, necessitating higher internal precision to determine the correct rounded result. Knowledge of these points enables implementations to target accuracy and ensure correct rounding.

The authors note that while some hardest-to-round points have been discovered for certain precisions, a general catalogue for all functions and input/output precisions is still lacking, limiting the ability to guarantee optimal accuracy broadly.