  We propose a non-commutative algorithm for multiplying 2x2 matrices using 7
coefficient products. This algorithm reaches simultaneously a better accuracy
in practice compared to previously known such fast algorithms, and a time
complexity bound with the best currently known leading term (obtained via
alternate basis sparsification). To build this algorithm, we consider matrix
and tensor norms bounds governing the stability and accuracy of numerical
matrix multiplication. First, we reduce those bounds by minimizing a growth
factor along the unique orbit of Strassen's 2x2-matrix multiplication tensor
decomposition. Second, we develop heuristics for minimizing the number of
operations required to realize a given bilinear formula, while further
improving its accuracy. Third, we perform an alternate basis sparsification
that improves on the time complexity constant and mostly preserves the overall
accuracy.
