When Do Simple Formulas Beat Ensembles? A 31-Dataset Benchmark for Interpretable Binary Classification

When can a short algebraic formula replace a gradient-boosted ensemble for binary classification on tabular data? We answer this question empirically on 31 UCI/OpenML datasets using a unified protocol: each dataset is evaluated with 200 5-fold repeated stratified cross-validation (1,000 paired measurements), comparing depth-3 symbolic formulas against gradient boosting, random forest, and logistic regression under identical train/test splits. The search engine---Theory Radar---searches a bounded depth-3 formula family via GPU beam search and random-subspace exploration over 10 binary and 6 unary operators, evaluates each candidate with exact O(N N) sort-and-sweep F_1 optimization, and uses PLS projections to give three-term formulas implicit access to all d features. A meta-learned pruning technique discovers safe pruning rules from exhaustive micro-search, eliminating 88--99.6\ within the exhaustive micro-search space (extrapolation to deeper search is empirical). The benchmark results below use the unpruned beam search; pruning is evaluated separately as a search-efficiency contribution. On 9 of 31 datasets (29\ gradient boosting: Liver ( F_1 = +0.159), Haberman ( F_1 = +0.175), Transfusion ( F_1 = +0.118), Vertebral ( F_1 = +0.060), Diabetes ( F_1 = +0.032), Wine ( F_1 = +0.035), Heart ( F_1 = +0.028), Hepatitis ( F_1 = +0.021), and Breast~Cancer ( F_1 = +0.006). Three datasets are statistical ties, and 19 are ensemble wins. Winning datasets share a common profile: moderate sample size (N 800), a discriminative subspace captured by 2--3 PLS components, and medical/clinical provenance. We report all results honestly, including the 19 losses, and provide PLS loading analyses showing that winning formulas map to domain-interpretable feature combinations. Depth experiments to depth~9 with 5 -wider beams confirm that depth~3 provides the best generalization on 29 of 31 datasets. Code: available upon acceptance (anonymous review). Target venue: IEEE TAI. Author preprint deposited for archival and citation. Draft — pending author review.