That square root estimation is new to me and freaky accurate. The error bounces and is generally decreasing. It's also always an underestimate. Here are the first six local maxima of errors.

    2: 0.0809
    6: 0.0495
    12: 0.0355
    20: 0.0277
    30: 0.0227
    42: 0.0192

And there's an obvious pattern there. Interesting stuff.