# Formula-One Prompting: A Composable Equation-First Prefix for Applied Mathematics

arXiv: https://arxiv.org/abs/2601.19302 · HTML: https://mlsanity.com/paper/2601.19302/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 19 of its 29 references that are in the map.
- First posted on arXiv 27 Jan 2026.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 19 of 19)

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- [Challenging BIG-Bench Tasks and Whether Chain-of-Thought Can Solve Them](https://mlsanity.com/paper/2210.09261/) (2022)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Program of Thoughts Prompting: Disentangling Computation from Reasoning for Numerical Reasoning Tasks](https://mlsanity.com/paper/2211.12588/) (2022)
- [FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performance](https://mlsanity.com/paper/2305.05176/) (2023)
- [DataComp-LM: In search of the next generation of training sets for language models](https://mlsanity.com/paper/2406.11794/) (2024)
- [To CoT or not to CoT? Chain-of-thought helps mainly on math and symbolic reasoning](https://mlsanity.com/paper/2409.12183/) (2024)
- [OpenWebMath: An Open Dataset of High-Quality Mathematical Web Text](https://mlsanity.com/paper/2310.06786/) (2023)
- [Solving Math Word Problems by Combining Language Models With Symbolic Solvers](https://mlsanity.com/paper/2304.09102/) (2023)
- [Why think step by step? Reasoning emerges from the locality of experience](https://mlsanity.com/paper/2304.03843/) (2023)
- [How Well do LLMs Compress Their Own Chain-of-Thought? A Token Complexity Approach](https://mlsanity.com/paper/2503.01141/) (2025)
- [Formal Mathematical Reasoning: A New Frontier in AI](https://mlsanity.com/paper/2412.16075/) (2024)
- [FinanceMath: Knowledge-Intensive Math Reasoning in Finance Domains](https://mlsanity.com/paper/2311.09797/) (2023)
- [Plan, Verify and Switch: Integrated Reasoning with Diverse X-of-Thoughts](https://mlsanity.com/paper/2310.14628/) (2023)
- [Infini-gram mini: Exact n-gram Search at the Internet Scale with FM-Index](https://mlsanity.com/paper/2506.12229/) (2025)
- [Solving for X and Beyond: Can Large Language Models Solve Complex Math Problems with More-Than-Two Unknowns?](https://mlsanity.com/paper/2407.05134/) (2024)

## Abstract (arXiv)

This paper introduces Formula Prompting (FP) and Formula-One Prompting (F-1), two single-call methods that elicit governing equations before solving applied-math problems. Chain-of-Thought (CoT) and Program-of-Thought (PoT) prompting improve mathematical reasoning by eliciting reasoning traces or code-like structures learned during pretraining. This suggests a diagnostic question: which useful pretraining patterns remain under-elicited? Using infini-gram-mini, we scan 81.7 trillion pretraining tokens and find that, in curated corpora such as DataComp-LM, equation-centered language appears 121x more often than code and 3.79x more often than step-by-step narration, yet standard prompting methods do not explicitly elicit equation formulation. FP asks the model to formalize a problem's governing equations before solving; F-1 extends FP with a composable Phase 2 that selects Direct, CoT, or PoT-style solving in the same call. Across five reasoning models and four applied-math benchmarks (finance, physics, cryptography, competition math), F-1 outperforms CoT by 5.76 pp and PoT by 8.42 pp on average, with the largest gain of 13.30 pp on FinanceMath, while topping the accuracy-token efficiency frontier at only 68 prompt tokens of overhead. Variant ablations identify the equation-formalization prefix, not the strategy menu, as the primary driver: adding CoT or PoT on top of the prefix yields no further gain, and 73.3% of remaining failures occur downstream of a correct Phase-1 equation.