Ai Scientist

These are just starting points — the space of algorithmic approaches is vast. New search strategies, heuristics, and acceleration methods should be continually explored.

Machine Learning Approaches

1. Reinforcement Learning (RL): Frame the search for polynomial formulas as an RL game where matrix multiplications must align toward a high-accuracy representation of target constants.
2. Neural-guided search: Use neural networks to predict promising regions of the polynomial coefficient space.
3. Pattern recognition in formula databases: Use ML to identify structural patterns across known formulas that suggest new search directions.

Computational Acceleration

1. Write C extensions for hot loops. Modular matrix products, polynomial evaluation, and GCD computation are common bottlenecks. Use __int128 for 128-bit intermediate products. A C inner loop gives 10–100× over Python.
2. Binary splitting: $O(N \log N)$ multiplications for computing products of holonomic sequences, vs $O(N)$ sequential. Choose based on actual profiling.
3. Multiprocessing: Use multiprocessing with freeze_support() on Windows. Spawn-based pools only. Each worker should be independent with zero communication when possible.
4. Batch operations: The ramanujantools library supports batched walk/limit computation — use it instead of single-iteration loops.
5. gmpy2: For big-integer-heavy workloads, gmpy2 wraps GMP and provides 3–10× over Python int for large numbers.

Symbolic Computation for Discovery

• SymPy: Use for polynomial algebra, recurrence solving, Gröbner bases, series manipulation. The workhorse for algorithmic math.
• Mathematica / Wolfram Language: For hypergeometric simplification, creative telescoping, Zeilberger's algorithm, and tasks where SymPy is insufficient.
• RISC packages (JKU Linz): Specialized Mathematica packages:
  • Guess: Finds recurrences from sequence data
  • HolonomicFunctions: Zeilberger's algorithm, creative telescoping
  • Sigma, EvaluateMultiSums: Advanced summation
  • Access may require a Mathematica license — ask the team for access
• ramanujantools: The group's own library — PCF, CMF, LinearRecurrence, limit computation, asymptotics. Install via pip install ramanujantools.
• ASyMOB (Algebraic Symbolic Mathematical Operations Benchmark): From RamanujanMachine/ASyMOB — useful for benchmarking symbolic computation approaches.
• Always tell the team when external tools would help. Special access to commercial/academic tools is available.

Guardrails

• Run code to verify every claim. Never trust a formula without computing it numerically to high precision (100+ digits via mpmath).
• Benchmark before committing to any approach. Pure-Python sequential can outperform parallel implementations due to IPC overhead for small N. The crossover point matters — measure it.
• Profile first, optimize second. Use cProfile, line_profiler, or timeit to identify actual bottlenecks before writing C or rewriting algorithms.
• Measure wall-clock time of computations at realistic scale. Report timing alongside results.
• Use high-precision arithmetic. Set mpmath.mp.dps to at least 2× the number of digits you need. Never use Python float for verification of mathematical formulas.
• Cross-validate new methods against existing library output on shared benchmarks before trusting them.

Performance Engineering Principles