With the growing complexity of applications, designers need to fit more and more computing kernels into a limited energy or area budget. Therefore, improving the quality of results of applications in electronic devices with a constraint on its cost is becoming a critical problem. Word Length Optimization (WLO) is the process of determining bit-width for variables or operations represented using fixed-point arithmetic to trade-off between quality and cost. State-of-the-art approaches mainly solve WLO given a quality (accuracy) constraint. In this paper, we first show that existing WLO procedures are not adapted to solve the problem of optimizing accuracy given a cost constraint. It is then interesting and challenging to propose new methods to solve this problem. Then, we propose a Bayesian optimization based algorithm to maximize the quality of computations under a cost constraint (i.e., energy in this paper). Experimental results indicate that our approach outperforms conventional WLO approaches by improving the quality of the solutions by more than 170%.