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Everyone uses math libraries and they are seldom correct. Mainstream math libraries produce incorrect results for millions of inputs. In the RLIBM project, our goal is to build a collection of correctly rounded math libraries for multiple representations ( e.g., 32-bit float, posits, bfloat16, tensorfloat32) for multiple rounding modes.
This project makes a case for approximating the correctly rounded result of an elementary function rather than the real value of an elementary function. When we approximate the correctly rounded result, there is an interval of real values around the correctly rounded result such that producing a real value in this interval rounds to the correct result. This interval is the freedom that the polynomial approximation has for an input, which is larger than the freedom with prior approaches (i.e., mini-max approaches). Hence, the RLIBM approach has more margin to generate correct, yet, efficient polynomials.
Using these intervals, we structure the problem of generating polynomial approximations that produce correctly rounded results for all inputs as a linear programming problem. We have developed correctly rounded implementations of elementary functions for multiple representations: 32-bit floating point, 32-bit posits, 16-bit posits, bfloat16, and tensorfloat32.
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