SHF:Small:Techniques for Generating Correctly Rounded Math Libraries

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Every programming language needs math libraries, which provide implementations of elementary functions for the floating-point representation and its variants. This project aims to develop correctly rounded math libraries for a wide range of representations that approximate real numbers. This project's novelty lies in creating polynomial approximations that produce the correctly rounded value of an elementary function f(x) (i.e., the value of f(x) rounded to the target representation) rather than the real value of f(x). It provides more margin to identify correct polynomials while generating efficient polynomials. This project's impacts are in advancing the state-of-the-art in approximating elementary functions for a large number of data types while allowing domain scientists to experiment with both precision and dynamic range of their data types. It has the potential to influence committees on various standards to mandate correctly rounded results for existing and new representations. This project also will educate practitioners, graduate and undergraduate students on foundational abstractions in computing.This project structures the task of generating efficient polynomial approximations that produce correctly rounded results as a linear-programming problem. Specifically, it proposes counterexample-guided polynomial generation for generating correct results for all inputs with large data types. To generate implementations with good performance, it proposes the generation of piecewise polynomials. It also explores new range-reduction techniques that are amenable to the linear-programming formulation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.