Redundant computations appear during the execution of real programs. Multiple factors contribute to these unnecessary computations, such as repetitive inputs and patterns, calling functions with the same parameters or bad programming habits. Compilers minimize non useful code with static analysis. However, redundant execution might be dynamic and there are no current approaches to reduce these inefficiencies. Additionally, many algorithms can be computed with different levels of accuracy. Approximate computing exploits this fact to reduce execution time at the cost of slightly less accurate results. In this case, expert developers determine the desired tradeoff between performance and accuracy for each application. In this paper, we present Approximate Task Memoization (ATM), a novel approach in the runtime system that transparently exploits both dynamic redundancy and approximation at the task granularity of a parallel application. Memoization of previous task executions allows predicting the results of future tasks without having to execute them and without losing accuracy. To further increase performance improvements, the runtime system can memoize similar tasks, which leads to task approximate computing. By defining how to measure task similarity and correctness, we present an adaptive algorithm in the runtime system that automatically decides if task approximation is beneficial or not. When evaluated on a real 8-core processor with applications from different domains (financial analysis, stencil-computation, machine-learning and linear-algebra), ATM achieves a 1.4x average speedup when only applying memoization techniques. When adding task approximation, ATM achieves a 2.5x average speedup with an average 0.7% accuracy loss (maximum of 3.2%).