We introduce a task-based programming model and runtime system that exploit the observation that not all parts of a program are equally significant for the accuracy of the end-result, in order to trade off the quality of program outputs for increased energy-efficiency. This is done in a structured and flexible way, allowing for easy exploitation of different points in the quality/energy space, without adversely affecting application performance. The runtime system can apply a number of different policies to decide whether it will execute less-significant tasks accurately or approximately. The experimental evaluation indicates that our system can achieve an energy reduction of up to 83% compared with a fully accurate execution and up to 35% compared with an approximate version employing loop perforation. At the same time, our approach always results in graceful quality degradation.