Compression of deep neural networks has become a necessary stage for optimizing model inference on resource-constrained hardware. This paper presents FITCompress, a method for unifying layer-wise mixed precision quantization and pruning under a single heuristic, as an alternative to neural architecture search and Bayesian-based techniques. FITCompress combines the Fisher Information Metric, and path planning through compression space, to pick optimal configurations given size and operation constraints with single-shot fine-tuning. Experiments on ImageNet validate the method and show that our approach yields a better trade-off between accuracy and efficiency when compared to the baselines. Besides computer vision benchmarks, we experiment with the BERT model on a language understanding task, paving the way towards its optimal compression.