  We introduce new efficient integral representations and methods for
evaluation of pdfs, cpds and quantiles of stable distributions. For wide
regions in the parameter space, absolute errors of order $10^{-15}$ can be
achieved in 0.005-0.1 msec (Matlab implementation), even when the index of the
distribution is small or close to 1. For the calculation of quantiles in wide
regions in the tails using the Newton or bisection method, it suffices to
precompute several hundred values of the characteristic exponent at points of
an appropriate grid (conformal principal components) and use these values in
formulas for cpdf and pdf, which require a fairly small number of elementary
operations. The methods of the paper are applicable to other classes of
integrals, especially highly oscillatory ones, and are typically faster than
the popular methods.

Numerical Analysis