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,
the only digits displayed are those 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.
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