A Faster Shortest Path Algorithm

Shortest paths is a very simple problem. There is a graph of vertices and (possibly directed) edges that connect them. Each edge has a real number weight. Starting from some vertex, for every other vertex in the graph you want to find the minimum total weight of a path, or report that it is unreachable.

In the version I considered, the graph was directed, I required exact answers and was given non-negative real weights for the edges.

I assume that real number weights can be compared and added. Any operations the shortest path algorithm does internally (such as counting how many nodes are visited or storing distances on intermediate vertices) counts towards running time.