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

1. Problem Statement

Large Language Models (LLMs) possess vast mathematical knowledge encoded during pretraining on equation-rich corpora (e.g., LaTeX, scientific papers). However, existing prompting methods fail to explicitly leverage this knowledge in a structured way:

• Chain-of-Thought (CoT): Relies on step-by-step natural language reasoning. While effective for arithmetic, it often introduces cascading errors in complex derivations and fails to explicitly recall governing domain equations.
• Program-of-Thought (PoT): Generates executable code for computation but struggles with conceptual reasoning and problems that cannot be easily expressed programmatically.
• The "Equation Gap": Current methods proceed directly from problem statement to solution steps without an intermediate stage of equation formulation. This causes models to miss domain-specific constraints (e.g., Newton's laws in physics, compound interest in finance) that are central to applied mathematics.
• Limitations of Multi-Call Methods: Approaches that use multiple LLM calls or external routers (e.g., Tree of Thoughts, Adaptive-Solver) are computationally expensive and impractical for production systems.

2. Methodology: Formula-One Prompting (F-1)

The authors propose Formula-One Prompting (F-1), a single-call, two-phase approach designed to bridge the equation gap. F-1 forces the model to formalize governing equations before selecting a solving strategy.

Phase 1: Equation Formalization

The model is instructed to extract the problem structure before solving:

1. Identify Givens: Extract explicit values, conditions, and constraints.
2. Identify Targets: Clarify the goal (what needs to be found or proved).
3. Formulate Equations: Express the mathematical relationships connecting givens to targets. This forces the model to internalize the underlying structure (e.g., writing out the Lorentz transformation or Grinold-Kroner model) before attempting a solution.

Phase 2: Natural Flow Reasoning

Once equations are formalized, the model naturally selects a solving strategy based on the equation structure, without explicit routing rules or decision trees:

• Direct Calculation: Used for simple substitutions.
• Chain-of-Thought (CoT): Used for multi-step symbolic derivations or logical proofs.
• Program-of-Thought (PoT): Used for iterative, recursive, or complex numerical computations.
• Verification: The model verifies the solution against the formalized equations before outputting the final answer.

Key Innovation: Unlike methods that hard-code routing logic, F-1 relies on the model's internal reasoning flow to determine the best strategy after the equation structure is established. This occurs within a single API call, maintaining low latency and cost.

3. Key Contributions

1. F-1 Prompting Framework: Introduces a unified, single-call prompt that uses mathematical equations as a structured intermediate representation to guide reasoning.
2. Empirical Validation of Equation Formalization: Through per-problem analysis, the authors demonstrate that the act of writing equations is the primary driver of performance, flipping 168 problems from incorrect to correct across 1,916 problems compared to CoT.
3. Adaptive Strategy Selection: Shows that F-1 achieves ~73% accuracy in selecting the optimal solving strategy (CoT vs. PoT vs. Direct) for applied domains, reaching 81–84% of the theoretical upper bound.
4. Efficiency: F-1 maintains single-call efficiency while outperforming multi-call methods, with a minimal token overhead (+68 tokens vs. Zero-Shot).

4. Experimental Results

The study evaluated F-1 across five models (GPT-5, Gemini 2.5 Pro, DeepSeek-V3.1, Qwen3-235B, Qwen3-30B) and four benchmarks focused on applied mathematics:

• Benchmarks: IMO-Bench (Competition Math), OlympiadBench (Math + Physics), FinanceMath (Applied Finance), and AICrypto (Cryptography).
• Overall Performance:
  • F-1 outperforms CoT by +5.76% and PoT by +8.42% on average.
  • F-1 wins 53 out of 60 (88.3%) benchmark-model comparisons.
• Domain-Specific Gains:
  • FinanceMath: +13.30% improvement over CoT.
  • OlympiadBench (Physics): +2.55% improvement over CoT (vs. +0.44% for pure math), confirming F-1's strength in applied domains.
  • AICrypto: +7.24% improvement over CoT.
• Strategy Adaptation: F-1 naturally favors PoT for computation-heavy FinanceMath problems (31.4% usage) and CoT/Hybrid for proof-oriented Cryptography and IMO problems, without explicit instructions to do so.

5. Significance and Impact

• Bridging the Gap: F-1 effectively unlocks the "equation knowledge" latent in LLMs that standard prompting methods fail to access. It aligns the model's reasoning process with how human experts solve applied problems (formulate first, solve second).
• Production Viability: By achieving superior results with a single call and minimal token overhead, F-1 is highly suitable for real-world deployment where latency and cost are critical constraints.
• Error Reduction: The method significantly reduces specific error types, such as semantic confusion (e.g., confusing "independent of x" with "equals zero") and format mismatches (e.g., outputting percentages instead of decimals), by enforcing a structured formalization step.
• Future Directions: The authors suggest that F-1 could be combined with multi-call mechanisms for even harder problems and that domain-specific equation templates could further enhance performance.

In conclusion, Formula-One Prompting represents a paradigm shift in mathematical reasoning for LLMs, moving from purely narrative or code-based reasoning to an equation-first approach that significantly boosts accuracy in applied scientific and financial domains.