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The substantial computational and memory demands of Large Language Models (LLMs) present barriers to their deployment. Block Floating Point (BFP) has been instrumental in accelerating linear operations, which are fundamental to LLM workloads. However, as the sequence length of LLMs increases, nonlinear operations have increasingly become performance bottlenecks, with Attention being a typical example due to its computational complexity scaling quadratically with input length. These nonlinear operations continue to be predominantly executed using inefficient floating-point formats, which renders the system challenging to optimize software efficiency and hardware overhead. In this paper, we delve into the limitations and potential of applying BFP to nonlinear operations. Given our findings, we introduce a novel hardware-software co-design framework (DB-Attn), including: (i) DBFP, an advanced BFP version, overcomes nonlinear operation challenges with a pivot-focus strategy for diverse data and an adaptive grouping strategy for flexible exponent sharing. (ii) DH-LUT, a novel lookup table algorithm dedicated to accelerating nonlinear operations with DBFP format. (iii) An RTL-level DBFP-based engine is implemented to support DB-Attn, applicable to FPGA and ASIC. Results show that DB-Attn provides significant performance improvements with negligible accuracy loss, achieving 74% GPU speedup on Softmax of LLaMA and 10x low-overhead performance improvement over SOTA ASIC designs.