pysr_uncertainty

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Bootstrap confidence intervals for the numeric constants of a frozen expression, plus optional prediction bands on an x-grid. Typical flow: call pysr_run, pick an expression from the response (best_expression or a pareto_front entry), pass it back here with the same dataset to get CIs on its fit constants. Returns frequentist bootstrap confidence intervals, not Bayesian credible intervals — posterior inference over expression structures is an open research problem. This tool freezes the expression chosen by the caller and bootstraps only its numeric constants; uncertainty about *which* expression is correct is not quantified. Bootstrap semantics: - If y_sigma is supplied, uses parametric bootstrap (y_b = y + Normal(0, y_sigma)). CI reflects user-stated measurement noise. - Otherwise uses residual bootstrap: fit once, resample residuals. CI reflects estimated-from-residuals noise. Only Float constants in the expression become free parameters. Integers stay structural (the 2 in x**2 is a function-class choice, not a fit constant). Expressions with no Float constants (e.g. "x + y") will be rejected with a validation error. Expression grammar: the `expression` string is parsed by sympy. Accepted operators are the same set pysr_run emits: unary `sin`, `cos`, `tan`, `exp`, `log`, `log2`, `log10`, `sqrt`, `abs`, `sinh`, `cosh`, `tanh`; binary `+`, `-`, `*`, `/`, `^` (or `**`). Whitespace and parenthesization are free. Every free symbol in the expression must correspond to an entry in `feature_names` — an unrecognised symbol is silently treated as a fresh sympy Symbol and the fit will fail downstream rather than reject early. Parse failures (syntax errors, malformed operators) surface as tool errors. If `feature_names` is supplied, its length must equal the number of columns in `X`; a mismatch is rejected with a validation error. Pricing: always free, regardless of dataset size. This tool has no `payment` parameter and is never subject to the x402/Stripe gate. Large bootstrap jobs still count against the shared rate limit below, so budget `n_resamples` accordingly. Rate limit: 10 requests/hour per IP, 200/hour global, max queue depth 20 (shared with sindy_run and pysr_run).