  We present an optimized single-precision implementation of the Sparse
Approximate Matrix Multiply (\SpAMM{}) [M. Challacombe and N. Bock, arXiv {\bf
1011.3534} (2010)], a fast algorithm for matrix-matrix multiplication for
matrices with decay that achieves an $\mathcal{O} (n \log n)$ computational
complexity with respect to matrix dimension $n$. We find that the max norm of
the error achieved with a \SpAMM{} tolerance below $2 \times 10^{-8}$ is lower
than that of the single-precision {\tt SGEMM} for dense quantum chemical
matrices, while outperforming {\tt SGEMM} with a cross-over already for small
matrices ($n \sim 1000$). Relative to naive implementations of \SpAMM{} using
Intel's Math Kernel Library ({\tt MKL}) or AMD's Core Math Library ({\tt
ACML}), our optimized version is found to be significantly faster. Detailed
performance comparisons are made for quantum chemical matrices with differently
structured sub-blocks. Finally, we discuss the potential of improved hardware
prefetch to yield 2--3x speedups.
