Fast Polynomial Transcendentals for LLMs

Graphics processing unit (GPU) generations scale matrix, special-function, and memory pipelines at different rates, so kernel bottlenecks move as hardware evolves. FlashAttention-4 exposed this imbalance inside attention on NVIDIA Blackwell. We test whether short polynomial programs can accelerate other special-function-unit (SFU) operations in large language models (LLMs). We first compare native PyTorch evaluation with packed fused multiply--add (FMA) programs in an isolated IEEE binary16 (FP16) sweep spanning L2-resident and high-bandwidth-memory (HBM)-resident working sets. We then replace native sigmoid, tanh, and sigmoid linear unit (SiLU) with degree-3 or degree-4 bfloat16 (BF16) programs in four GB200 integration tasks: dense SiLU, tanh-softcapped attention, sigmoid attention, and routed-expert Swish-gated linear unit (SwiGLU). The programs combine analytical symmetry, target-format rounding, and packed arithmetic inside consuming kernels. The isolated paths improve by 1.19--2.19x in L2 and 1.00--1.70x in HBM. The dense-SiLU, tanh-softcapped-attention, and routed-expert substitutions improve complete training-step throughput by 2.7%, 2.9%, and 8.0%, respectively. The sigmoid-attention substitution improves complete-attention forward by 7.4% and the complete GPU step by 0.3%. Same-checkpoint open-weight ablations and one paired pre-training comparison per task extend the evaluation to model behavior. At common horizons near 100 billion tokens, the final smoothed training-loss differences (polynomial minus native) range from -0.107 to +0.079 across the four tasks.