If
and
are relative errors in the argument and result, respectively, then in principle
, so that the relative error in the argument,
, is amplified by a factor
in the result.
Near
this factor behaves as
and hence the accuracy is largely determined by the machine precision. Also for large negative
, where the factor is
, accuracy is mainly limited by machine precision. However, for large positive
, the factor becomes
and to an extent relative accuracy is necessarily lost. The absolute accuracy
is given by
so absolute accuracy is guaranteed for all
.