The use of low-precision fixed-point arithmetic along with stochastic rounding has been proposed as a promising alternative to the commonly used 32-bit floating point arithmetic to enhance training neural networks training in terms of performance and energy efficiency. In the first part of this paper, the behaviour of the 12-bit fixed-point arithmetic when training a convolutional neural network with the CIFAR-10 dataset is analysed, showing that such arithmetic is not the most appropriate for the training phase. After that, the paper presents and evaluates, under the same conditions, alternative low-precision arithmetics, starting with the 12-bit floating-point arithmetic. These two representations are then leveraged using local scaling in order to increase accuracy and get closer to the baseline 32-bit floating-point arithmetic. Finally, the paper introduces a simplified model in which both the outputs and the gradients of the neural networks are constrained to power-of-two values, just using 7 bits for their representation. The evaluation demonstrates a minimal loss in accuracy for the proposed Power-of-Two neural network, avoiding the use of multiplications and divisions and thereby, significantly reducing the training time as well as the energy consumption and memory requirements during the training and inference phases.