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Convolutional neural networks (CNNs) achieve state-of-the-art performance at the cost of becoming deeper and larger. Although quantization (both fixed-point and floating-point) has proven effective for reducing storage and memory access, two challenges -- 1) accuracy loss caused by quantization without calibration, fine-tuning or re-training for deep CNNs and 2) hardware inefficiency caused by floating-point quantization -- prevent processors from completely leveraging the benefits. In this paper, we propose a low-precision floating-point quantization oriented processor, named Phoenix, to address the above challenges. We primarily have three key observations: 1) 8-bit floating-point quantization incurs less error than 8-bit fixed-point quantization; 2) without using any calibration, fine-tuning or re-training techniques, normalization before quantization further reduces accuracy degradation; 3) 8-bit floating-point multiplier achieves higher hardware efficiency than 8-bit fixed-point multiplier if the full-precision product is applied. Based on these key observations, we propose a normalization-oriented 8-bit floating-point quantization method to reduce storage and memory access with negligible accuracy loss (within 0.5%/0.3% for top-1/top-5 accuracy, respectively). We further design a hardware processor to address the hardware inefficiency caused by floating-point multiplier. Compared with a state-of-the-art accelerator, Phoenix is 3.32x and 7.45x better in performance with the same core area for AlexNet and VGG16, respectively.