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 (orange). Computing at machine precision gives a strikingly different trajectory (gray).

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.