Abstract
Our RLibm project has recently proposed methods to generate a single implementation for an elementary
function that produces correctly rounded results for multiple rounding modes and representations with
up to 32-bits. They are appealing for developing fast reference libraries without double rounding issues.
The key insight is to build polynomial approximations that produce the correctly rounded result for a
representation with two additional bits when compared to the largest target representation and with the
“non-standard” round-to-odd rounding mode, which makes double rounding the RLibm math library result to
any smaller target representation innocuous. The resulting approximations generated by the RLibm approach
are implemented with machine supported floating-point operations with the round-to-nearest rounding mode.
When an application uses a rounding mode other than the round-to-nearest mode, the RLibm math library
saves the application’s rounding mode, changes the system’s rounding mode to round-to-nearest, computes
the correctly rounded result, and restores the application’s rounding mode. This frequent change of rounding
modes has a performance cost.
This paper proposes two new methods, which we call rounding-invariant outputs and rounding-invariant
input bounds, to avoid the frequent changes to the rounding mode and the dependence on the round-to-nearest
mode. First, our new rounding-invariant outputs method proposes using the round-to-zero rounding mode
to implement RLibm’s polynomial approximations. We propose fast, error-free transformations to emulate
a round-to-zero result from any standard rounding mode without changing the rounding mode. Second,
our rounding-invariant input bounds method factors any rounding error due to different rounding modes
using interval bounds in the RLibm pipeline. Both methods make a different set of trade-offs and improve the
performance of resulting libraries by more than 2×.