Performance improvements to existing software implementations.
In SymPy's Poly.discriminant(), we added direct quadratic and cubic formulas over polynomial coefficient domains, avoiding the expensive general resultant calculation. Complete calls ran 189× faster (geometric mean) across 13 benchmark cases, including 685× for a quadratic and 391× for a cubic.
In PersistenceLength, we replaced the quadratic bond-pair matrix with hybrid direct/FFT autocorrelation, reducing long-chain complexity from O(n²) time and memory to O(n log n) time and O(n) memory. The complete analysis ran 387× faster for four 8,192-bond chains across eight frames.
In Polylabel, we added a certified early exit for near-optimal label positions using enclosing-strip bounds, avoiding subdivision on thin polygons. Complete calls ran 13,700× faster (geometric mean) across 12 thin-polygon cases, with 15 control cases roughly unchanged.