An approach to generate correctly rounded math libraries for new floating point variants

Abstract

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 approach for generating polynomial approximations that can be used to implement correctly rounded math libraries. Existing methods generate polynomials that approximate the real value of an elementary function 𝑓 (𝑥) and produce wrong results due to approximation errors and 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). It 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. Using our approach, we have developed correctly rounded, yet faster, implementations of elementary functions for multiple target representations.

High performance correctly rounded math libraries for 32-bit floating point representations

“…Our goal is to generate polynomial approximations for elementary functions ( ) that produce the correctly rounded result for all inputs in 32-bit target representations T. Similar to our prior work on RLibm [31,32], we approximate the correctly rounded result rather than the real value of ( ). We extend it in three main directions.…”

“…Once we have the reduced input and the reduced intervals, we structure the problem of generating polynomials as a linear programming problem similar to our prior work on RLibm [31,32]. Even after range reduction and creation of sub-domains for the generation of piecewise polynomials, we need to generate a polynomial approximation when there are several million reduced inputs and reduced intervals in the context of 32-bit types.…”

“…Numerous groups have developed correctly rounded elementary functions [7,24]. Some correctly rounded libraries for FP are IBM LibUltim [49], Sun Microsystem's LibMCR, CR-LIBM [13], MPFR math library [15], and RLIBM [31,32]. CR-LIBM is a correctly rounded double library developed using Sollya [9], which generates mini-max polynomials to approximate elementary functions [4,5].…”

“…As approximating a polynomial in a small domain [ , ] is much easier, the input domain of the function is reduced using range reduction [12,31,45]. The approximated result is adjusted to produce the result for the original input (i.e., output compensation).…”

“…RLibm. Our RLibm approach [31,32] generates polynomials that approximate the correctly rounded result rather than the real value of the elementary function. The generation of the polynomial considers errors in polynomial approximation and numerical errors in polynomial evaluation, range reduction, and output compensation to produce the correctly rounded output for all inputs.…”