This paper addresses the problem of provably efficient and practically good on-the-fly determinacy race detection in task parallel programs that use futures. Prior works determinacy race detection have mostly focused on either task parallel programs that follow a series-parallel dependence structure or ones with unrestricted use of futures that generate arbitrary dependences. In this work, we consider a restricted use of futures and show that it can be race detected more efficiently than general use of futures. Specifically, we present two algorithms: MultiBags and MultiBags+. MultiBags targets programs that use futures in a restricted fashion and runs in time $O(T_1 α(m,n))$, where $T_1$ is the sequential running time of the program, $α$ is the inverse Ackermann's function, $m$ is the total number of memory accesses, $n$ is the dynamic count of places at which parallelism is created. Since $α$ is a very slowly growing function (upper bounded by $4$ for all practical purposes), it can be treated as a close-to-constant overhead. MultiBags+ an extension of MultiBags that target programs with general use of futures. It runs in time $O((T_1+k^2)α(m,n))$ where $T_1$, $α$, $m$ and $n$ are defined as before, and $k$ is the number of future operations in the computation. We implemented both algorithms and empirically demonstrate their efficiency.