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A precision-scalable processor for low-power ConvNets or convolutional neural networks is implemented in a 40-nm CMOS technology. To minimize energy consumption while maintaining throughput, this paper is the first to implement dynamic precision and energy scaling and exploit the sparsity of convolutions in a dedicated processor architecture. The processor's 256 parallel processing units achieve a peak 102 GOPS running at 204 MHz and 1.1 V. It is fully C-programmable through a custom generated compiler and consumes 25-287 mW at 204 MHz and a scaling efficiency between 0.3 and 2.7 effective TOPS/W. It achieves 47 frames/s on the convolutional layers of the AlexNet benchmark, consuming only 76 mW. This system hereby outperforms the state-of-the-art up to five times in energy efficiency.