These two simulations of
light bouncing among cylindrical mirrors show the importance of
using significance arithmetic. Compare the machine precision (blue)
and significance arithmetic (orange) paths. After fewer than 20
reflections, the blue path is highly inaccurate. For the orange path,
Mathematica was started with 120 digits of precision.
Because Mathematica uses significance arithmetic throughout, it
was able to report automatically that 62 digits of precision remained at the
end.
Significance arithmetic, a basis for Mathematica's
high-precision arithmetic, is a powerful technique that offers many
advantages over fixed precision (such as the floating-point or integer
arithmetic used by purely numerical technical software).
It not only keeps track of numerical results, but also uses error
propagation to track their accuracy. In this way, numerical
computations can carry accuracy and precision information with them,
returning in the end a numerical quantity together with its estimated
uncertainty.
This means that with Mathematica, unlike other software, you
always get both a numerical result and the knowledge of to how many
digits it is correct.
