Large language models (LLMs) are getting smarter, but one key ingredient is holding them back from truly understanding long texts: precision. Specifically, the way computers represent numbers can significantly impact how well LLMs process information, especially in very long sequences. This problem becomes apparent when using a format called BFloat16, which is computationally efficient but sacrifices some numerical accuracy. Researchers discovered that BFloat16 disrupts a core component of LLMs called Rotary Positional Embedding (RoPE). RoPE helps LLMs understand the order of words, which is crucial for comprehending long passages. However, BFloat16 causes errors to accumulate, particularly impacting the very first word in a sequence, throwing off the LLM's understanding of the whole text. To combat this, researchers have developed a clever solution called AnchorAttention. This technique uses the first word as a consistent 'anchor' across multiple documents within a long text. This approach helps to stabilize the model's positional understanding and dramatically speeds up processing. Experiments show that AnchorAttention significantly boosts LLM performance on long-context benchmarks like RULER and LongBench, while maintaining performance on standard benchmarks like MMLU and HellaSwag. This breakthrough allows LLMs to tackle complex tasks like multi-document question answering, in-depth code analysis, and long-form content generation more effectively. The research also highlights the delicate balance between efficiency and precision in AI. While BFloat16 allows for faster computation, ensuring accurate numerical representation is crucial for LLM development. This work paves the way for even longer-context models capable of processing and understanding increasingly complex and information-rich tasks.