Formula-One Prompting: Equation-First Reasoning For Applied Mathematics

TL;DR

Formula-One Prompting (F-1) enhances reasoning in large language models by explicitly formulating equations as an intermediate step, leading to significant performance improvements across various benchmarks and domains.

Contribution

The paper introduces F-1, a novel two-phase prompting method that explicitly formulates equations before solving, improving reasoning accuracy over existing methods.

Findings

F-1 outperforms CoT by +5.76% and PoT by +8.42% on average.

F-1 achieves 88.3% success across 60 benchmark-model comparisons.

Largest gains observed in applied domains like FinanceMath and physics.

Abstract

LLMs encode vast mathematical knowledge including governing equations from pretraining on equation-rich corpora, yet existing prompting methods, including Chain-of-Thought (CoT) and Program-of-Thought (PoT), do not explicitly elicit equation formulation as a reasoning stage. We propose Formula-One Prompting (F-1), a single-call, two-phase approach that fills this equation gap by using mathematical equations as an intermediate representation before solving through natural flow reasoning. F-1 first formulates governing equations from problem descriptions; the model then naturally selects a solving strategy among CoT, PoT, or direct computation based on the formalized equation structure, without explicit routing rules. Results across five models and four benchmarks show F-1 outperforms CoT by +5.76% and PoT by +8.42% on average, winning 53 out of 60 benchmark-model comparisons (88.3%). Gains are largest in applied domains: +13.30% on FinanceMath over CoT, and within OlympiadBench, larger gains on physics (+2.55%) than pure math (+0.44%). Per-problem analysis confirms equation formalization is the primary driver.