This paper proposes a novel approximate bfloat16 multiplier with on-the-fly adjustable accuracy for energy-efficient learning in deep neural networks. The size of the proposed multiplier is only 62% of the size of the exact bfloat16 multiplier. Furthermore, its energy footprint is up to five times smaller than the footprint of the exact bfloat multiplier. We demonstrate the advantages of the proposed multiplier in deep neural network learning,  where we successfully train the ResNet-20 network on the CIFAR-10 dataset from scratch.