A study demonstrates that greedy decoding from large language models is not precision-invariant, producing different outputs when using BF16 versus FP16 precision on identical hardware. Across six models and three benchmarks, 49-100% of prompts diverged, with single token flips cascading into larger trajectory changes. Selective FP32 recomputation at the language model head achieved +22-36 percentage point improvements in exact agreement with minimal latency overhead, though the mitigation only partially addresses the issue under certain conditions.