Numerical evaluation of the interpolation accuracy of simple elementary functions

Comparison of the accuracy of the restoration of elementary functions by the values in the nodes was made for algorithms of low-degree piecewise-polynomial interpolation. The test results clearly demonstrate in graphical form the advantages and disadvantages of the widely used cubic interpolation splines.
The comparison revealed that, contrary to popular belief, the smoothness of the interpolant is not directly related to the accuracy of the approximation. In the 20 disparate examples considered, the piecewise quadratic interpolation is rarely and only slightly inferior in the form of the used classical cubic splines, often by orders of magnitude better than many of them.
In several examples the high interpolation error of simple functions on a fixed grid appears to be almost independent of the degree of the algorithm and the smoothness of the interpolant. The piecewise-linear interpolation unexpectally appeared the most accurate in one of examples.
A new problem is presented: to find a local interpolation algorithm, exactly restoring any rational functions of the second order.