Suitable choice for moderate-size square matrix multiplication?

The problem is to find $C = AB$, where $A$ and $B$ are $n \times n$ matrices that may be sparse. Let $n$ be around 1000. The elements of $A$ and $B$ are real values, though, for practicality's sake, let's say we use floating-point numbers. Ideally, the approach would be fast enough for small $n$ and, of course, should not require size ($n$ value) of ~$10 ^ 6$ to break even relative to brute-force cubic time approach. The lower the running time, the better. Given ties, the lower the coefficient for running time, the better. Given ties again, lower space requirement is desired. Algebraic or combinatorial approaches are acceptable.

Then, what is a suitable algorithm choice for this moderate-size-input square matrix product problem?

Dumas and Pan recently wrote a non-asymptotic survey of fast matrix multiplication in practice, which hopefully answers your questions. They concentrate on matrices of order at most a million, and list all relevant algorithms.

StackExchange Computer Science Q#67478, answer score: 7