This animation shows how the output precision of a computation can vary widely even for a simple function where the input precision is being kept constant. Mathematica takes the effect into account to ensure accurate results.

Numerical-precision control (NPC) is fundamental to Mathematica's handling of numerical computations, ensuring unrivaled trustworthiness of numerical results.

In specified input mode, you give Mathematica input to a certain precision, and Mathematica adjusts subsequent computations to preserve as much precision as possible. At the end, digits are only displayed that can be justified given the input precision and the sequence of computations performed. For certain computations the output precision exceeds the input precision.

Without precision control, even simple numerical computations can return vastly inaccurate results. NPC technology uses advanced heuristics and precision tracking--switching to higher-precision computations, adjusting the step size in iterative computations, or changing algorithms on the fly when it expects a different approach to give a more accurate answer.