  We demonstrate that, even when there are moderate overlaps in the inputs of
sloppy or accurate double-word addition algorithms in the QD library, these
algorithms still guarantee error bounds of $O(u^2(|a|+|b|))$ in faithful
rounding. Furthermore, the accurate algorithm can achieve a relative error
bound of $O(u^2)$ in the presence of moderate overlaps in the inputs when
rounding function is round-to-nearest. The relative error bound also holds in
directed rounding, but certain additional conditions are required.
Consequently, in double-word multiplication and addition operations, we can
safely omit the normalization step of double-word multiplication and replace
the accurate addition algorithm with the sloppy one. Numerical experiments
confirm that this approach nearly doubles the performance of double-word
multiplication and addition operations, with negligible precision costs.
Moreover, in directed rounding mode, the signs of the errors of the two
algorithms are consistent with the rounding direction, even in the presence of
input overlap. This allows us to avoid changing the rounding mode in interval
arithmetic. We also prove that the relative error bound of the sloppy addition
algorithm exceeds $3u^2$ if and only if the input meets the condition of
Sterbenz's Lemma when rounding to nearest. These findings suggest that the two
addition algorithms are more robust than previously believed.
