Existing tile size selection approaches are tightly coupled with compiler transformation pipelines, often leading to inaccurate modeling of cache behavior and limited effectiveness for non-rectangular tile shapes. This article presents TileMind , a decoupled analytical model that combines compile-time and runtime information for tile size selection in affine programs. It introduces a transformation-aware pre-tiling step that enables the decoupled selector to remain consistent with compiler transformations while extracting compile-time metadata. The extracted metadata is then combined with profiled runtime characteristics to construct a richer yet tractable feasible domain, within which a nonlinear objective for tile size selection is formulated. This objective is subsequently transformed into a binary product linearization problem, with its nonlinear constraints also linearized for efficient optimization. Finally, an intra-tile optimization aligns computation with data layout to enhance data reuse within tiles. Across two multi-core Intel CPUs, TileMind achieves 1.49× (sequential) and 1.33× (parallel) mean speedups on twenty PolyBench kernels, and 2.08–3.54× speedups on three deep learning workloads over the state-of-the-art analytical model Pluto-tss . Compared with TVM’s latest autotuner MetaSchedule, TileMind delivers 1.35–1.46× mean speedups while reducing tuning overhead by 2–4 orders of magnitude. While demonstrating effectiveness on selecting tile sizes for non-rectangular tile shapes and compatibility with PPCG, Pluto, and TVM, we further provide proof-of-concept results on GPUs, illustrating the potential portability of TileMind across architectures.

Floating-point error detection is crucial in numerical computing, particularly for multi-parameter functions where even minor errors can propagate and significantly impact results. The sparse distribution of floating-point errors poses a significant detection challenge, as significant deviations are triggered by only rare inputs. Existing search algorithms face two major limitations: poor scalability for multi-parameter functions and insufficient utilization of floating-point representation characteristics. To address these challenges, we propose SDPS (Scalable Detection via Parallel Subdomain Search), a novel algorithm that combines adaptive domain partitioning with floating-point-specific heuristics. SDPS employs a multi-level error classification system and specialized point generation strategies, supported by efficient parallel processing through dynamic task allocation. Our comprehensive evaluation demonstrates that SDPS significantly outperforms state-of-the-art methods in both detection accuracy and computational efficiency, especially for multi-parameter functions where it effectively addresses the exponential growth of search space that limits existing approaches.

We introduce a floating-point (FP) error optimization approach called Arfa that partitions the domain D of an FP expression fe into regimes and rewrites fe in each regime where fe shows larger errors. First, Arfa seeks a rewrite substitution fo with lower errors across D, whose error distribution is plotted for effective regime inference. Next, Arfa generates an incomplete set of ordered rewrite candidates within each regime of interest, so that searching for the best rewrite substitutions is performed efficiently. Finally, Arfa selects the best rewrite substitution by inspecting the errors of top ranked rewrite candidates, with enhancing precision also considered. Experiments on 56 FPbench examples and four real-life programs show that Arfa not only reduces the maximum and average errors of fe by 4.73 and 2.08 bits on average (and up to 33 and 16 bits), but also exhibits lower errors, sometimes to a significant degree, than Herbie and NumOpt.

The MPFR and CR-LIBM math libraries are frequently utilized due to their ability to generate correctly rounded results for all double-precision inputs. However, it is worth noting that MPFR has a slower average performance, while CR-LIBM achieves correct rounding over two iterations, rendering it less stable. In addition, CR-LIBM has a poor performance in handling the worst-case of correct rounding. This paper implements a correctly rounded elementary function library called SCR-LIBM in double-precision, which is stable and efficient. Our key idea is to divide subdomains and use the low-degree Taylor polynomial to approximate the elementary function in each subdomain. We simulate the high-precision representation based on the double–double data format, and use the error-free transformation and Double-double algorithm to control the error in the process of polynomial approximation and output compensation. Our approach ensures that the elementary function is correctly rounded, without the need for redundant iterations. The experimental evaluation shows that the average performance of elementary functions implemented in SCR-LIBM is 8.534 times faster than that of MPFR, and 2.492 times faster than that of CR-LIBM when dealing with the worst-case of correct rounding. What’s more, our SCR-LIBM is more stable than CR-LIBM.

Existing search heuristics used to find input values that result in significant floating-point (FP) errors or small ranges that cover them are accompanied by severe constraints, complicating their implementation and restricting their general applicability. This paper introduces an error analysis tool called Eiffel to infer error-inducing input ranges instead of searching them. Given an FP expression with its domain$\mathcal{D}$, Eiffel first constructs an error data set by sampling values across a smaller domain$\mathcal{R}$and assembles these data into clusters. If more than two clusters are formed, Eiffel derives polynomial curves that best fit the bound coordinates of the error-inducing ranges in$\mathcal{R}$, extrapolating them to infer all target ranges of$\mathcal{D}$and reporting the maximal error. Otherwise, Eiffel simply returns the largest error across$\mathcal{R}$. Experimental results show that Eiffel exhibits a broader applicability than Atomu and$\mathbf{S}^{3}$FP by successfully detecting the errors of all 70 considered benchmarks while the two baselines only report errors for part of them. By taking as input the inferred ranges of Eiffel, Herbie obtains an average accuracy improvement of 3.35 bits and up to 53.3 bits.