acb_calc_integrate_gl_auto_deg

With \Delta = (b - a)/2 and m = (a + b)/2, the integral is \int_{-1}^{1} g with g(t) = \Delta f(\Delta t + m). If g is holomorphic and |g| \le M inside the ellipse with foci \pm 1 and semi-axes X and Y = \sqrt{X^2 - 1}, \rho = X + Y, the error of the n-point Gauss–Legendre rule is at most

a slightly weakened form of a bound of Trefethen [Tre2008]. M is obtained by evaluating f with order 1 on the rectangle [\pm X] + [\pm Y] i mapped to the segment. The function tries X = 2, 4, 8, 32, 512, \ldots (doubling the exponent) as long as M is finite, and for each X the smallest degree n from the sequence 1, 2, 4, 6, 8, 12, 16, 22, 32, \ldots (steps of about \sqrt{2}) that meets the tolerance; the smallest n found is used. The nodes and weights are computed by arb_hypgeom_legendre_p_ui_root [JM2018] (Roots of Legendre polynomials and Gauss–Legendre nodes) and cached (per thread), and the evaluations are distributed over threads when flint_get_num_threads() is greater than one.