Based on that we present an algorithm for calculating the rounded-to-nearest result of s := p i for a given vector of floatingpoint numbers p i , as well as algorithms for directed rounding.  ...  Our algorithms are fast in terms of measured computing time because they allow good instruction-level parallelism, they neither require special operations such as access to mantissa or exponent, they contain  ...  In Part I [22] of this paper we gave a fast algorithm to compute a faithfully rounded result of p i . Our methods are based on error-free transformations.  ...