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Matrix multiplication is one of the bottleneck computations for training the weights within deep neural networks. To speed up the training phase, we propose to use faster algorithms for matrix multiplication known as Arbitrary Precision Approximating (APA) algorithms. APA algorithms perform asymptotically fewer arithmetic operations than the classical algorithm, but they compute an approximate result with an error that can be made arbitrarily small in exact arithmetic. Practical APA algorithms provide significant reduction in computation time and still provide enough accuracy for many applications like neural network training. We demonstrate that APA algorithms can be efficiently implemented and parallelized for multicore CPUs to obtain up to 28% and 21% speedups over the fastest implementation of the classical algorithm using one core and 12 cores, respectively. Furthermore, using these algorithms to train a Multi-Layer Perceptron (MLP) network yields no significant reduction in the training or testing error. Our performance results on a large MLP network show overall sequential and multithreaded performance improvements of up to 25% and 13%, respectively. We also demonstrate up to 15% improvement when training the fully connected layers of the VGG-19 image classification network.