Approximate computing allows tackling conflicting objectives, such as power and accuracy of computations. In this paper, we first describe how knowledge of stimuli's specific features can help in quantifying and improving power savings by means of approximate computing. We investigate FPGA implementations of several approximate circuits and compare their power consumption with non-approximating versions. In particular, we study approximate arithmetics and a clock-gate based technique called memoization. Moreover, we compare the accuracy of estimation techniques for power consumption evaluation versus real measurements under controlled environments. We also experimentally quantify the relationship between switching activity and power consumption. Two important results are concluded from our investigations: (1) Approximate arithmetics do not necessarily consume less power than conventional circuits, whereas memoization techniques can, in fact, reduce power consumption. (2) Simulation-based power evaluation for approximate FPGA implementations can reach fidelity values up to about 89% in input-dependent power characteristics. Yet, to evaluate absolute savings, measurements are required.