A wider state, not wider arithmetic, for single-precision time integration
Abstract
Single precision (FP32) halves the memory and data movement of double precision (FP64), and on accelerators is often several times faster. Its rounding errors are harmless except in the stored state. A modified equation for Runge-Kutta methods in Shu-Osher form and linear multistep methods separates the rounding of the stored state, a forcing proportional to 1/Δt, from truncation and from right-hand-side rounding, which sets a Δt-independent floor. We bound this forcing for strong-stability-preserving schemes and identify two mechanisms that keep its sign fixed, so it accumulates as a bias: past a turnaround step, refining the step increases the error, in FP64 as in FP32. For a model oscillator the analysis predicts the floor and the FP32-to-FP64 ratios of turnaround step and minimum error within 30%. Our remedy needs no FP64 arithmetic: each state variable is stored as the unevaluated sum of two FP32 numbers, updated with error-free transformations, while every right-hand-side evaluation stays in FP32. Unlike compensated summation, it needs no rewriting of the scheme. In model problems from geophysical flow, mechanical dynamics and celestial mechanics, with five Runge-Kutta and multistep schemes, it restores design-order convergence to the floor. In large-eddy simulations of a high-lift airplane wing on three refined grids, up to 544 million cells, with time steps Δt, Δt/2 and Δt/4, FP32 overpredicts the lift by 2.1, 4.5 and 7.7%, while the two-float lift agrees with FP64 within 0.3% on every grid, and on graphics processing units the production solver runs 1.6 to 2.7 times faster than FP64.
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BibTeX
@misc{barad2026wider,
title = {{A wider state, not wider arithmetic, for single-precision time integration}},
author = {Michael F. Barad},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05365},
eprint = {2611.05365},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05365},
}
RIS
TY - UNPB
TI - A wider state, not wider arithmetic, for single-precision time integration
AU - Michael F. Barad
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05365
UR - https://arxiv.org/abs/2611.05365
AB - Single precision (FP32) halves the memory and data movement of double precision (FP64), and on accelerators is often several times faster. Its rounding errors are harmless except in the stored state. A modified equation for Runge-Kutta methods in Shu-Osher form and linear multistep methods separates the rounding of the stored state, a forcing proportional to 1/Δt, from truncation and from right-hand-side rounding, which sets a Δt-independent floor. We bound this forcing for strong-stability-preserving schemes and identify two mechanisms that keep its sign fixed, so it accumulates as a bias: past a turnaround step, refining the step increases the error, in FP64 as in FP32. For a model oscillator the analysis predicts the floor and the FP32-to-FP64 ratios of turnaround step and minimum error within 30%. Our remedy needs no FP64 arithmetic: each state variable is stored as the unevaluated sum of two FP32 numbers, updated with error-free transformations, while every right-hand-side evaluation stays in FP32. Unlike compensated summation, it needs no rewriting of the scheme. In model problems from geophysical flow, mechanical dynamics and celestial mechanics, with five Runge-Kutta and multistep schemes, it restores design-order convergence to the floor. In large-eddy simulations of a high-lift airplane wing on three refined grids, up to 544 million cells, with time steps Δt, Δt/2 and Δt/4, FP32 overpredicts the lift by 2.1, 4.5 and 7.7%, while the two-float lift agrees with FP64 within 0.3% on every grid, and on graphics processing units the production solver runs 1.6 to 2.7 times faster than FP64.
ER -