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[Submitted on 3 Sep 2026] Title:Fast Polynomial Transcendentals for LLMs View a PDF of the paper titled Fast Polynomial Transcendentals for LLMs, by Robert Hu View PDF HTML (experimental) Abstract: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. Subjects: Machine Learning (cs.LG) Cite as: arXiv:2610.00049 [cs.LG] (or arXiv:2610.00049v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2610.00049 arXiv-issued DOI via DataCite Submission history From: Robert Hu [view email] [v1] Thu, 3 Sep 2026 17:18:10 UTC (404 KB) Full-text links: Access Paper: View a PDF of the paper titled Fast Polynomial Transcendentals for LLMs, by Robert Hu View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-10 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)