RLIBM-ALL: A Novel Polynomial Approximation Method to Produce Correctly Rounded Results for Multiple Representations and Rounding Modes

TL;DR

This paper introduces RLIBM-ALL, a polynomial approximation method that ensures correctly rounded floating point results across multiple representations and rounding modes, improving accuracy and efficiency over existing libraries.

Contribution

It presents a novel polynomial approximation technique that guarantees correct rounding for various floating point formats and modes, extending prior work to multiple precisions and configurations.

Findings

First 32-bit float library with correct rounding for all modes

Supports multiple precisions from 10-bit to 32-bit

Faster than mainstream floating point libraries

Abstract

Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode. Using such libraries for a representation with a new rounding mode or with different precision will result in wrong results due to double rounding. This paper proposes a novel method to generate a single polynomial approximation that produces correctly rounded results for all inputs for multiple rounding modes and multiple precision configurations. To generate a correctly rounded library for $n$-bits, our key idea is to generate such a polynomial approximation for a representation with $n+2$-bits using the \emph{round-to-odd} mode. We prove that the resulting polynomial approximation will produce correctly rounded results for all five rounding modes in the standard and for multiple representations with $k$-bits such that $|E| +1 < k \leq n$, where $|E|$ is the number of exponent bits in the representation. Building on our prior work in the RLIBM project, we also approximate the correctly rounded result when we generate the library with $n+2$-bits using the round-to-odd mode. We also generate polynomial approximations by structuring it as a linear programming problem but propose enhancements to polynomial generation to handle the round-to-odd mode. Our prototype is the first 32-bit float library that produces correctly rounded results with all rounding modes in the IEEE standard for all inputs with a single polynomial approximation. It also produces correctly rounded results for any FP configuration ranging from 10-bits to 32-bits while also being faster than mainstream libraries.