Numerical-precision control (NPC) is fundamental
to Mathematica's handling of numerical computations, ensuring
unrivaled trustworthiness of numerical results.
In specified output mode, you specify the output precision
and Mathematica automatically adjusts the working precision
through the computation sequence to reach this precision, assuming the
input precision was sufficient for this to be possible. Note that
working precision needed during a calculation is often much greater
than that justifiable at the end. In many cases, Mathematica
automatically takes this into account, stepping up precision
mid-calculation where necessary.
This technology is unique to Wolfram Research, with key algorithms developed in-house.
To achieve a set output precision in this solution to a nonlinear
differential equation, the precision during the calculation needs to
step up far beyond machine precision. Mathematica does this
automatically (red). Computing at machine precision gives a strikingly
different trajectory (gray).