error propagation bounds

This paper addresses the forward error analysis of sum-product algorithms under stochastic rounding (SR). We propose a probabilistic error bounding method grounded in martingale theory. Our key contributions are threefold: (1) We introduce the first automated martingale construction framework tailored to multilinear computational structures—encompassing addition, subtraction, multiplication, and intermediate result reuse; (2) We extend SR error analysis to algorithms with structural reuse, notably Karatsuba polynomial multiplication—previously unaddressed in SR literature; (3) Leveraging the Azuma–Hoeffding inequality, we derive a tight probabilistic error bound of $O(sqrt{n},u)$, markedly improving upon the classical worst-case bound $O(n,u)$. Our framework uniformly recovers known error guarantees for pairwise summation and Horner’s method, and—crucially—provides the first rigorous SR error guarantee for Karatsuba multiplication.

Existing approaches lack the capability to perform automated backward error analysis for numerical programs, making it difficult to verify their backward stability. This work proposes a formal framework that generalizes the definition of backward stability, introduces the category Shel to model stable numerical computations, and develops the tool eggshel to automatically synthesize error bounds. The framework incorporates a novel, composable, and flexible notion of stability, integrating category theory, formal verification, and symbolic reasoning to automatically search for stability proofs within subcategories of Shel. Notably, eggshel is the first tool capable of automating the analysis of programs with variable reuse, successfully generating backward error bounds for several numerical programs previously beyond the reach of existing methods, while providing formal correctness guarantees.

This work addresses the problem of deriving provably tight floating-point rounding error bounds for numerical programs featuring conditional branches, no loops, and mixed-precision arithmetic. Methodologically, it unifies the modeling of conditional control flow and precision heterogeneity via two novel quantitative metrics—“instability jumps” and “window width”—and integrates interval arithmetic, abstract interpretation, and precision-aware semantic modeling, augmented with abstraction-guided global optimization. Its key contribution is the first formal framework enabling joint, compositional analysis of conditional branching and mixed precision, achieving both high bound tightness and practical analysis efficiency. Experimental evaluation on standard benchmarks demonstrates significantly tighter error bounds compared to prior approaches. Furthermore, the framework successfully guides precision configuration—e.g., step size and search direction—in the conjugate gradient method, empirically validating its utility in supporting design-time trade-offs among accuracy, error bounds, and computational efficiency.

This work addresses the automatic verification of expected output bounds for probabilistic programs featuring general loops, continuous distributions, and conditional branching—where the integral semantics induced by continuous sampling impede conventional invariant-based reasoning. We propose a Riemann-sum-based approximation of the expected semantics, transforming integral bounds into quantitative invariants expressible in SMT logic. This constitutes the first systematic integration of Riemann integration into probabilistic program verification, accompanied by formal convergence guarantees for the approximation and a proof that the verification problem is coRE-complete. We implement a prototype within the Caesar verification framework, supporting intermediate-language encoding and SMT-driven inference; it successfully verifies multiple benchmarks involving continuous sampling and loops. Our approach bridges discrete program verifiers with continuous probabilistic analysis, enabling existing discrete verification tools to scale to programs with continuous distributions.

The “black-box” nature of large language models hinders rigorous performance verification. Method: We propose the first framework unifying mechanistic interpretability with formal performance verification: via weight-level mechanistic reverse-engineering, we decompose small-scale Transformer behavior on Max-of-K tasks into human-understandable algorithms and generate compact, machine-verifiable mathematical proofs (e.g., accuracy lower bounds). Contribution/Results: This establishes the first end-to-end closed loop from mechanistic understanding to formal proof. We discover that proof length positively correlates with both mechanistic insight depth and bound tightness, and identify “structural deficiency errors”—gaps between inferred mechanisms and true computational structure—as the key bottleneck limiting proof conciseness and fidelity. Validated across 151 random seeds and 4 values of K, our framework constructs 102 distinct strategies; empirical results confirm that shorter proofs reflect deeper mechanistic understanding, while higher-fidelity interpretations yield tighter performance bounds.

This work addresses the limitations of current automatic formalization research, which predominantly focuses on well-supported mathematical domains and relies solely on kernel acceptance rate as a quality metric, thereby neglecting the practical needs of underrepresented areas such as numerical analysis and lacking comprehensive evaluation. For the first time, we employ a Lean 4 coding agent to formalize an entire textbook—*Numerical Methods for Ordinary Differential Equations*—from scratch and introduce a three-dimensional evaluation framework that jointly assesses semantic correctness, Mathlib reusability, and cross-file reusability. Through LLM-as-judge, semantic validation, and dependency analysis, we uncover pervasive issues in existing systems, including incomplete statements and weakened assumptions, demonstrating that kernel acceptance rate substantially overestimates formalization quality. Our approach establishes a reproducible, multidimensional auditing paradigm for trustworthy automated formalization.

This work addresses the longstanding challenge of reconciling theoretical correctness with practical efficiency by introducing Algorithmist, a multi-agent autonomous research system built upon GitHub Copilot. Through an iterative research-review cycle, Algorithmist collaboratively performs algorithm design, formal verification, proof-guided code generation, and consistency validation. The system establishes a scalable paradigm for provably correct algorithm synthesis by integrating large language models, structured natural-language proof representations, and formal verification techniques to generate algorithms tailored to specific datasets and deployment scenarios. In applications to privacy-preserving data analysis and clustering tasks, Algorithmist automatically produces novel algorithms that simultaneously offer rigorous theoretical guarantees and strong empirical performance, uncovers previously overlooked proof flaws in existing work, and achieves state-of-the-art results in several settings.

Floating-point round-off errors are ubiquitous in numerically intensive programs arising in fields such as scientific computing and optimization. As floating-point errors potentially lead to unexpected and catastrophic program failures, one must derive guaranteed round-off thresholds to ensure the correctness of these programs. However, deterministic round-off thresholds tend to be too conservative to be usable in practice, since they often involve large round-off errors that occur with small probability. Probabilistic thresholds relax deterministic ones by specifying that the probability of the round-off error exceeding a threshold is below a given confidence. In this work, we propose a novel approach to probabilistic round-off analysis, by applying concentration inequalities over the Taylor expansion from FPTaylor (TOPLAS 2018). A major obstacle in applying concentration inequalities is that the Taylor expansion involves absolute value operators that make the calculation of the expected values of the first order partial differential terms difficult. Our first step to overcome this obstacle is a sound over-approximation that removes the absolute value operators in polynomial expressions. Then, we show how to handle fractional expressions by a transformation into polynomial case. Finally, we show how to improve our approach with range partitioning. Our approach is scalable since the key computational part is the calculation of expected values of polynomial expressions with independent variables, for which the linear and independence properties of expectation boost the computation. Experimental results show that our approach is orders of magnitude more time efficient, while producing thresholds with comparable precision against the state of the art.

This work addresses the lack of machine-verifiable formalizations of line search methods in nonlinear optimization, which has hindered algorithmic reliability. Within the Lean 4 theorem prover, it presents the first systematic formalization of several classical line search criteria—including Armijo, Goldstein, Wolfe, and their nonmonotone variants—alongside rigorous definitions of gradient descent, descent directions, and backtracking step-size selection. The study fully verifies the Zoutendijk convergence theorem within this framework, thereby establishing a comprehensive formal foundation for line search theory. This contribution significantly enhances the verifiability and trustworthiness of nonlinear optimization algorithms through mechanized mathematical reasoning.

This study investigates whether artificial intelligence can effectively contribute to creative mathematical research under rigorous human supervision. By establishing a human–AI collaborative framework that integrates symbolic algebra manipulation, automated proof exploration, semantic synthesis of mathematical literature, and LaTeX-based formalization—augmented by human mathematical intuition and verification—the work systematically discovers and proves novel error representations and bounds for Hermite quadrature formulas. The research presents the first fully documented, high-transparency account of an end-to-end human–AI co-discovery process in mathematics, elucidating effective collaboration patterns and failure modes. It thereby establishes a viable pathway and validation protocol for AI-assisted mathematical inquiry, extends classical error theory, and underscores the irreplaceable role of human domain expertise in advanced mathematical discovery.