  In this note, we introduce a new finite difference approximation called the
Black-Box Logarithmic Expansion Numerical Derivative (BLEND) algorithm, which
is based on a formal logarithmic expansion of the differentiation operator.
BLEND capitalizes on parallelization and provides derivative approximations of
arbitrarily precision, i.e., our analysis can be used to determine the number
of terms in the series expansion to guarantee a specified number of decimal
places of accuracy. Furthermore, in the vector setting, the complexity of the
resulting directional derivative is independent of the dimension of the
parameter.
