A third problem is to minimize the "total" number of real multiplications and additions, sometimes called the "arithmetic complexity" (although in this context it is the exact count and not the asymptotic complexity that is being considered). Again, no tight lower bound has been proven. Since 1968, however, the lowest published count for power-of-two "N" was long achieved by the split-radix FFT algorithm, which requires formula_23 real multiplications and additions for "N" > 1. This was recently reduced to formula_24 (Johnson and Frigo, 2007; Lundy and Van Buskirk, 2007). A slightly larger count (but still better than split radix for "N" ≥ 256) was shown to be provably optimal for "N" ≤ 512 under additional restrictions on the possible algorithms (split-radix-like flowgraphs with unit-modulus multiplicative factors), by reduction to a satisfiability modulo theories problem solvable by brute force (Haynal & Haynal, 2011).