The edit distance is a way of quantifying how similar two strings are to one another by counting the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. In this paper we study the computational problem of computing the edit distance between a pair of strings where their distance is bounded by a parameter $k\ll n$. We present two streaming algorithms for computing edit distance: One runs in time $O(n+k^2)$ and the other $n+O(k^3)$. By writing $n+O(k^3)$ we want to emphasize that the number of operations per an input symbol is a small constant. In particular, the running time does not depend on the alphabet size, and the algorithm should be easy to implement. Previously a streaming algorithm with running time $O(n+k^4)$ was given in the paper by the current authors (STOC'16). The best off-line algorithm runs in time $O(n+k^2)$ (Landau et al., 1998) which is known to be optimal under the Strong Exponential Time Hypothesis.