Tartan (TRT), a hardware accelerator for inference with Deep Neural Networks (DNNs), is presented and evaluated on Convolutional Neural Networks. TRT exploits the variable per layer precision requirements of DNNs to deliver execution time that is proportional to the precision p in bits used per layer for convolutional and fully-connected layers. Prior art has demonstrated an accelerator with the same execution performance only for convolutional layers. Experiments on image classification CNNs show that on average across all networks studied, TRT outperforms a state-of-the-art bit-parallel accelerator by 1:90x without any loss in accuracy while it is 1:17x more energy efficient. TRT requires no network retraining while it enables trading off accuracy for additional improvements in execution performance and energy efficiency. For example, if a 1% relative loss in accuracy is acceptable, TRT is on average 2:04x faster and 1:25x more energy efficient than a conventional bit-parallel accelerator. A Tartan configuration that processes 2-bits at time, requires less area than the 1-bit configuration, improves efficiency to 1:24x over the bit-parallel baseline while being 73% faster for convolutional layers and 60% faster for fully-connected layers is also presented.