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Low-precision formats have recently driven major breakthroughs in neural network (NN) training and inference by reducing the memory footprint of the NN models and improving the energy efficiency of the underlying hardware architectures. Narrow integer data types have been vastly investigated for NN inference and have successfully been pushed to the extreme of ternary and binary representations. In contrast, most training-oriented platforms use at least 16-bit floating-point (FP) formats. Lower-precision data types such as 8-bit FP formats and mixed-precision techniques have only recently been explored in hardware implementations. We present MiniFloat-NN, a RISC-V instruction set architecture extension for low-precision NN training, providing support for two 8-bit and two 16-bit FP formats and expanding operations. The extension includes sum-of-dot-product instructions that accumulate the result in a larger format and three-term additions in two variations: expanding and non-expanding. We implement an ExSdotp unit to efficiently support in hardware both instruction types. The fused nature of the ExSdotp module prevents precision losses generated by the non-associativity of two consecutive FP additions while saving around 30% of the area and critical path compared to a cascade of two expanding fused multiply-add units. We replicate the ExSdotp module in a SIMD wrapper and integrate it into an open-source floating-point unit, which, coupled to an open-source RISC-V core, lays the foundation for future scalable architectures targeting low-precision and mixed-precision NN training. A cluster containing eight extended cores sharing a scratchpad memory, implemented in 12 nm FinFET technology, achieves up to 575 GFLOPS/W when computing FP8-to-FP16 GEMMs at 0.8 V, 1.26 GHz.