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Formula-One Prompting: Adaptive Reasoning Through Equations For Applied Mathematics

Natapong Nitarach, Pittawat Taveekitworachai, Kunat Pipatanakul

TL;DR

Formula-One Prompting (F-1) introduces an equation-first prompting strategy that first formalizes governing equations from problem statements and then adaptively selects a solving approach (Direct, CoT, or PoT) within a single LLM call. Across four applied benchmarks and five models, F-1 achieves higher accuracy than both CoT ($+$5.76%) and PoT ($+$8.42%) on average, with the largest gains in FinanceMath ($+$13.30%) and physics problems in OlympiadBench ($+$2.55% vs. $+$0.44% for math). Ablation studies show equation formalization is the primary driver of improvement, and strategy selection accuracy on applied domains reaches $\sim$73%, approaching $81$–$84\%$ of an upper bound. The work demonstrates that explicitly encoding governing equations before solving makes adaptive reasoning more effective in applied mathematics, while maintaining single-call efficiency suitable for production systems.

Abstract

Prompting techniques such as Chain-of-Thought (CoT) and Program-of-Thought (PoT) improve LLM mathematical reasoning by structuring intermediate steps in natural language or code. However, applied mathematics problems in domains like finance, physics, and cryptography often require recalling or deriving governing equations, a step that current approaches do not explicitly leverage. We propose Formula-One Prompting (F-1), a two-phase approach that uses mathematical equations as an intermediate representation before adaptive solving. F-1 first formulates governing equations from problem descriptions, then selects a solving strategy among CoT, PoT, or direct computation based on the generated equations, all within a single LLM call. Results across five models and four benchmarks show F-1 outperforms CoT by +5.76% and PoT by +8.42% on average. Crucially, 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%). This demonstrates that F-1 is more effective than CoT in applied mathematics problems.

Formula-One Prompting: Adaptive Reasoning Through Equations For Applied Mathematics

TL;DR

Formula-One Prompting (F-1) introduces an equation-first prompting strategy that first formalizes governing equations from problem statements and then adaptively selects a solving approach (Direct, CoT, or PoT) within a single LLM call. Across four applied benchmarks and five models, F-1 achieves higher accuracy than both CoT (5.76%) and PoT (8.42%) on average, with the largest gains in FinanceMath (13.30%) and physics problems in OlympiadBench (2.55% vs. 0.44% for math). Ablation studies show equation formalization is the primary driver of improvement, and strategy selection accuracy on applied domains reaches 73%, approaching – of an upper bound. The work demonstrates that explicitly encoding governing equations before solving makes adaptive reasoning more effective in applied mathematics, while maintaining single-call efficiency suitable for production systems.

Abstract

Prompting techniques such as Chain-of-Thought (CoT) and Program-of-Thought (PoT) improve LLM mathematical reasoning by structuring intermediate steps in natural language or code. However, applied mathematics problems in domains like finance, physics, and cryptography often require recalling or deriving governing equations, a step that current approaches do not explicitly leverage. We propose Formula-One Prompting (F-1), a two-phase approach that uses mathematical equations as an intermediate representation before adaptive solving. F-1 first formulates governing equations from problem descriptions, then selects a solving strategy among CoT, PoT, or direct computation based on the generated equations, all within a single LLM call. Results across five models and four benchmarks show F-1 outperforms CoT by +5.76% and PoT by +8.42% on average. Crucially, 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%). This demonstrates that F-1 is more effective than CoT in applied mathematics problems.
Paper Structure (72 sections, 8 equations, 2 figures, 20 tables)

This paper contains 72 sections, 8 equations, 2 figures, 20 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Single-Call Prompt Engineering
  4. Multi-Call Prompt Engineering
  5. Summary
  6. Formula-One Prompting
  7. Phase 1: Problem Formalization
  8. Identify Givens
  9. Identify Targets
  10. Formulate Equations
  11. Phase 2: Adaptive Solving
  12. Prompt Design
  13. Experiments
  14. Datasets
  15. Evaluation Protocol
  16. ...and 57 more sections

Figures (2)

  • Figure 1: Comparison of prompting methods for mathematical reasoning. Traditional methods (Zero-shot, CoT, PoT) proceed directly to solving. F-1 adds an equation formalization phase before adaptively selecting the solving strategy, all within a single call.
  • Figure 2: Qualitative comparison. (Left) Physics: Baselines confused "independent of $x$" with "equals zero"; F-1's formalization prevented this. (Center) Finance: Only F-1 output correct decimal format (0.063). (Right) Cryptography: F-1 identified correct sub-hybrid argument while baselines used invalid reductions.