Assessing Large Language Models for Stabilizing Numerical Expressions in Scientific Software

Abstract

Scientific software relies on high-precision computation, yet finite floating-point representations can introduce precision errors that propagate in safety-critical domains. Despite the growing use of large language models (LLMs) in scientific applications, their reliability in handling floating-point numerical stability has not been systematically evaluated. This paper evaluates LLMs' reasoning on high-precision numerical computation through two numerical stabilization tasks: (1) detecting instability in numerical expressions by generating error-inducing inputs (detection), and (2) rewriting expressions to improve numerical stability (stabilization). Using popular numerical benchmarks, we assess six LLMs on nearly 2,470 numerical structures, including nested conditionals, high-precision literals, and multi-variable arithmetic. Our results show that LLMs are equally effective as state-of-the-art traditional approaches in detecting and stabilizing numerically unstable computations. More notably, LLMs outperform baseline methods precisely where the latter fail: in 17.4% (431) of expressions where the baseline does not improve accuracy, LLMs successfully stabilize 422 (97.9%) of them, and achieve greater stability than the baseline across 65.4% (1,615) of all expressions. However, LLMs struggle with control flow and high-precision literals, consistently removing such structures rather than reasoning about their numerical implications, whereas they perform substantially better on purely symbolic expressions. Together, these findings suggest that LLMs are effective at stabilizing expressions that classical techniques cannot, yet struggle when exact numerical magnitudes and control flow semantics must be precisely reasoned about, as such concrete patterns are rarely encountered during training.

Figures (6)

• Figure 1: Instability detection workflow
• Figure 2: Numerical stabilization workflow
• Figure 3: Temporal evolution of best-achieved relative error across models
• Figure 4: Average accuracy increase per model vs. baseline on expressions where rewrites improved, shown for zero-shot (OS) and few-shot (FS). Original expressions' average accuracy is 84%
• Figure 5: Average accuracy improvement per model grouped by number of variables. Each cluster of bars represents one model; bar color indicates the number of variables in the expression. Zero-shot (0S) and few-shot (FS) variants are shown side by side for each base model. Accuracy computation is explained in Section \ref{['s5.2:Approach']}