Dijkstra Variant Problem

We have a directed graph G = (V, E) for a comm. network with each edge having a probability of not failing r(u, v) (defined as edge weight) which lies in interval [0, 1]. The probabilities are independent, so that from one vertex to another, if we multiply all probabilities, we get the the probability of the entire path not failing. I need an efficient algorithm to find a most reliable path from one given vertex to another given vertex (i.e., a path from the first vertex to the second that is least likely to fail). I am given that log(r · s) = log r + log s will be helpful.

My modified algorithm so far -:
DIJKSTRA-VARIANT (G, s, t)
1 for v in V:
2   val[v] ← ∞
3 A ← ∅
4 Q ← V to initialize Q with vertices in V.
5 val[s] ← 0
6 while Q is not ∅ and t is not in A
7   do x ← EXTRACT-MIN (Q)
8   A ← A ∪ {x}
9   for each vertex y ∈ Adj[x]
10      do if val[x] + p(x, y) < val[y]:
11         val[y] = val[x] + p(x, y)