Quantization-aware training (QAT) has emerged as a key strategy for enabling efficient deep learning inference on resource-constrained platforms. Yet, most existing approaches rely on static, manually selected numeric formats—fixed-point or floating-point—and fixed bit-widths, limiting their adaptability and often requiring extensive design effort or architecture search. In this work, we introduce a novel QAT framework that breaks this rigidity by jointly learning, during training, both the numeric representation format and the associated bit-widths in an end-to-end differentiable manner. At the core of our method lies a unified parameterization that is capable of emulating both fixed- and floating-point arithmetic, paired with a bit-aware loss function that penalizes excessive precision in a hardware-aligned fashion. We demonstrate that our approach achieves state-of-the-art trade-offs between accuracy and compression on MNIST, CIFAR-10, and CIFAR-100, reducing average bit-widths to as low as 1.4 with minimal accuracy loss. Furthermore, FPGA implementation using Xilinx FINN confirms over $5\times$ LUT and $4\times$ BRAM savings. This is the first QAT method to unify numeric format learning with differentiable precision control, enabling highly deployable, precision-adaptive deep neural networks.