curve_fit

Fit curve(s) to paired (x, y) data and report each fitted equation with its error.

Reach for curve_fit when you have observations and want an EQUATION that describes them; use calculate to evaluate an equation you already have, or solver to find the input that drives a known equation to a target. For example, x=[1, 1.5, 2], y=[2, 5.8, 8.9] comes back with the line 6.9*x - 4.78… ranked first by its residual error, so you can paste that equation straight into calculate to predict new points.

curve_fit takes the observations as two equal-length lists — x and y — and estimates, for every curve form it knows, the parameter values that best match the data in the least-squares sense, reporting each form's fitted equation and the residual error (the sum of squared residuals). The curve library holds the straight line a*x + b, the quadratic a*x**2 + b*x + c, the cubic a*x**3 + b*x**2 + c*x + d, the power law a*x**b, the exponential a*exp(b*x), the logarithm a + b*ln(x), the square root a*sqrt(x) + b, the reciprocal a/x + b, the sinusoid a*sin(b*x + c) + d, the gaussian a*exp(-(x-b)**2/(2*c**2)), the saturation x/(a*x + b) (Michaelis-Menten) and the hyperbolic 1/(a*x + b) (44.2.1-44.2.12, the complete library). Every form but the sinusoid is fitted in CLOSED FORM — the polynomials and the affine logarithm/square-root/reciprocal forms by the exact normal equations, the power and exponential laws by a log-linearisation (ln y = ln a + b·ln x for the power, ln y = ln a + b·x for the exponential), the gaussian by Caruana's method (logs to the quadratic ln y = p2·x**2 + p1·x + p0, fitted by the normal equations, then the peak/centre/width are recovered from its coefficients), and the saturation and hyperbolic by a reciprocal-line transform (the double-reciprocal Lineweaver-Burk line 1/y = a + b·(1/x) for the saturation, the line 1/y = a·x + b for the hyperbolic). The sinusoid alone has no closed-form solution, so it is fitted by an ITERATIVE frequency search: it is non-linear only in the frequency b, so for each candidate b the best amplitude/phase/offset is a linear sub-fit, and a coarse scan over a data-derived frequency range (then a golden-section refinement) picks the b minimising the residual. A form that cannot fit the data is dropped, not fatal: the power and logarithm need x > 0, the square root x >= 0, the reciprocal x != 0, the exponential, power and gaussian y > 0 (and the gaussian needs bell-shaped, downward-parabola data), the saturation x != 0 and y != 0, the hyperbolic y != 0; the power, exponential, logarithm, square-root, sinusoid and gaussian forms also drop in rational mode, which cannot represent their irrational logs/roots/sines/exp (the saturation and hyperbolic, being pure reciprocals, stay exact in rational); and a polynomial needs enough distinct x (the sinusoid four). The forms that fit are ranked by their residual error — least error first — and only the best three are returned (fewer when fewer forms fit).

mode and min_fixed_point_precision behave as in calculate — the whole fit runs in that numeric type, so the parameters and error are exact in rational, rounded at the scale in fixed-point, and native doubles in floating-point. As in the solver, a fixed-point fit at scale 0 would round every fitted parameter to a whole number, so when min_fixed_point_precision is omitted it DEFAULTS to 9 (sub-unit resolution); pass an explicit value for more/fewer decimals. Complex mode is not supported (the fit is real-valued).

Returns a dict: fits is the ranked list of fitted forms (best first, at most three), each {form, equation, parameters, fit_error, fit_error_hex_dump, exact, precision} — form names the curve, equation is the model with its parameters substituted (over the variable x, so it can be pasted into calculate), parameters is a list of {name, value, value_hex_dump} for each fitted coefficient (each value annotated with its own precision verdict), and fit_error is the residual error annotated the same way with its hex dump; that fit's own exact and precision describe its error. mode is the resolved numeric type, and the top-level exact / precision describe the BEST fit's error (the headline number), as in solver. On any failure — a bad mode/precision, mismatched or too-short data, or data that cannot support the form (every x equal, so a line has no slope) — fits/mode/exact/precision are null and error carries the message; on success error is null.