@Article{Chevillard2012,
  author = {Chevillard, S.},
  title = {The functions $\erf$ and $\erfc$ computed with arbitrary precision and explicit error bounds},
  journal = {Information and Computation},
  year = {2012},
  volume = {216},
  pages = {72 -- 95},
  note = {Special Issue: 8th Conference on Real Numbers and Computers},
  abstract = {The error function $\erf$ is a special function. It is widely used
	in statistical computations for instance, where it is also known
	as the standard normal cumulative probability. The complementary
	error function is defined as $\erfc(x)=\erf(x)-1$.
	
	
	In this paper, the computation of $\erf(x)$ and $\erfc(x)$ in arbitrary
	precision is detailed: our algorithms take as input a target precision
	$t'$ and deliver approximate values of $\erf(x)$ or $\erfc(x)$ with
	a relative error bounded by $2^{-t'}$.
	
	
	We study three different algorithms for evaluating $\erf$ and $\erfc$.
	These algorithms are completely detailed. In particular, the determination
	of the order of truncation, the analysis of roundoff errors and the
	way of choosing the working precision are presented.
	
	
	We implemented the three algorithms and studied experimentally what
	is the best algorithm to use in function of the point $x$ and the
	target precision $t'$.},
  keywords = {error function, complementary error function, $\erf$, $\erfc$, floating-point arithmetic, arbitrary precision, multiple precision}
}