Beschreibung:
<jats:p>
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
<jats:italic>n</jats:italic>
-bits, our key idea is to generate a polynomial approximation for a representation with
<jats:italic>n</jats:italic>
+2-bits using the
<jats:italic>round-to-odd</jats:italic>
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
<jats:italic>k</jats:italic>
-bits such that |
<jats:italic>E</jats:italic>
| +1 <
<jats:italic>k</jats:italic>
≤
<jats:italic>n</jats:italic>
, where |
<jats:italic>E</jats:italic>
| is the number of exponent bits in the representation. Similar to our prior work in the RLIBM project, we approximate the correctly rounded result when we generate the library with
<jats:italic>n</jats:italic>
+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.
</jats:p>