# RECURSUM: Automated Code Generation for Recurrence Relations Exceeds Expert Optimization via LayeredCodegen

arXiv: https://arxiv.org/abs/2604.18585 · HTML: https://mlsanity.com/paper/2604.18585/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jan 2026.

## Built on by (most cited first, 1 of 1)

- [Density-Functional Excited-State Gradients and Nonadiabatic Couplings on a Consumer GPU from a Contraction-DAG](https://mlsanity.com/paper/2608.06536/) (2026)

## Abstract (arXiv)

Automated code generation can systematically exceed expert hand-optimization for recurrence relations-computational primitives ubiquitous in orthogonal polynomials, special functions, numerical integration, and molecular integral evaluation. We present RECURSUM, a Python-based domain-specific language generating optimized C++ for arbitrary recurrence relations via three backends: template metaprogramming for compile-time evaluation, a novel LayeredCodegen backend with architectural optimizations, and runtime loop-based evaluation. The DSL uses einsum-inspired notation to specify recurrences, validity constraints, and base cases in 10-30 lines of Python, generating 650+ lines of production C++. LayeredCodegen achieves 9.8x speedup over expert hand-written implementations and 1.9x over template metaprogramming for McMurchie-Davidson Hermite coefficients. Architecture analysis reveals three quantifiable effects: (1) zero-copy output parameters eliminate return-by-value overhead (70-80% of speedup), (2) guaranteed function inlining eliminates compiler-refused overhead (15-20%), (3) exact-sized stack buffers achieve 100% cache efficiency vs 27% for MAX-sized arrays (5-10%). We validate on 24 recurrence types spanning pure mathematics (Legendre, Chebyshev, Hermite, Laguerre polynomials), numerical analysis (Clenshaw, Golub-Welsch), and quantum chemistry (McMurchie-Davidson, Rys quadrature, Boys function). Production benchmarks show speedups propagate to complete algorithms, with generated code matching expert baselines within 3.3%. RECURSUM demonstrates that systematic code generation serves as the performance ceiling for recurrence algorithms. By eliminating the dual expertise barrier (domain knowledge + C++ metaprogramming), the framework democratizes high-performance scientific computing-establishing a paradigm where automated generation systematically exceeds manual optimization.