A novel computationally efficient and very accurate method for the calculation of error probabilities in systems with interference and Gaussian noise is presented. The main idea is to approximate the natural logarithm of the Q-function by a truncated version of its Taylor series. As a result, Q(x) can be expressed as a finite product of exponential functions. This enables us to find true and approximate upper bounds for the probability of error by evaluating exponential moments of the interference provided that the individual interference components are mutually independent. The described method is very accurate. In numerous examples, the relative errors between the true probability of error and the approximations did not exceed 1% for systems with an "open eye" and 100% when the eye was closed. Additionally, the method is very effective and easy to use, outperforming most published methods in both the number of computations required and simplicity.