M dependence of the errors of the ENP5KE algorithm for solving KE in double precision for e = 0.999 (left) and e = 1 − ϵ (right), when the input accuracy (tol) is set to the level 3 × 10⁻⁹ rad (top), 3 × 10⁻¹² rad (center),and 3 × 10⁻¹⁵ (bottom). For these values of e, the solution is obtained using the piecewise quintic polynomial for 0.0045 rad ≤ M ≤ 2π − 0.0045 rad, and with breakpoint bracketing followed by continuous bisection root search in the intervals M < 0.0045 rad and M > 2π − 0.0045 rad.