When repeatedly evaluating x = 11 x - 7, precision is
lost at each step. This iteration is evaluated from a starting value
of 0.7 (with 16-digit precision) in both Mathematica and a
non-Mathematica system, and should result in 0.7 at each
step. Mathematica tracks the precision loss, outputting only
justifiable digits; the non-Mathematica system does not, leading to
a highly misleading result.
Any numerical package or programming language can give numerical
results up to machine precision, but only Mathematica will also
automatically track and communicate how many digits of the result are
accurate--using numerical-precision tracking (NPT) technology. This
gives almost complete protection from numerical errors, be they
round-off errors or from badly conditioned systems.