Differentiate erf and retain it in SQL programs - #256

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🌀 Claude: Thank you, this is a clean, useful change, and I'd merge it.

Strengths

• It's small and correct: the derivative 2/sqrt(pi) · exp(-x²) goes through the existing chain rule, and the reader and writer keep erf in the generated programs.
• The test checks the composed first and second derivatives, not just one value.
• It helps beyond nanoGPT. xarray-sql's Samudra demo (Demo 4: Samudra 2 in SQL, checked against Samudra's PyTorch (draft) xarray-sql#300) needs exact GELU too. Neither DataFusion nor DuckDB has erf, so it approximates erf in SQL with a polynomial (Abramowitz and Stegun 7.1.26). With this rule, an engine that registers erf gets its exact derivative.

Before merging

• Copyright header. OPTIMIZATION.md says SPDX-FileCopyrightText: 2026 OpenAI. It should follow ddx's own convention for its files; the tool that wrote it likely added that line.
• OPTIMIZATION.md. I'd keep this out of the repository root, or fold the erf part into ddx's docs on supported functions. The rest describes downstream training work rather than ddx.

🔥 Claude (torch-sql): One more thing exact GELU will need, from a spike in torch-sql (a prototype that runs PyTorch models as SQL on XQL).

Samudra's activation, CappedGELU, is clamp(gelu(x), max=10) with the exact, erf-based GELU. Neither engine has an erf function: DataFusion 54 and DuckDB 2.0 both reject it. So besides the rule in this PR, an erf UDF has to exist in each engine before a query can call it, either registered by ddxdb or upstream.

Until then, torch-sql can use GELU's tanh form, which differs from the exact one by at most 4.7e-4. Through index tables, ddx's gradient of that form, with the cap written as LEAST(…, 10), matches PyTorch to 3e-14 with no new rule.