  Numerically obtaining the inverse of a function is a common task for many
scientific problems, often solved using a Newton iteration method. Here we
describe an alternative scheme, based on switching variables followed by spline
interpolation, which can be applied to monotonic functions under very general
conditions. To optimize the algorithm, we designed a specific ultra-fast spline
routine. We also derive analytically the theoretical errors of the method and
test it on examples that are of interest in physics. In particular, we compute
the real branch of Lambert's $W(y)$ function, which is defined as the inverse
of $x \exp(x)$, and we solve Kepler's equation. In all cases, our predictions
for the theoretical errors are in excellent agreement with our numerical
results, and are smaller than what could be expected from the general error
analysis of spline interpolation by many orders of magnitude, namely by an
astonishing $3\times 10^{-22}$ factor for the computation of $W$ in the range
$W(y)\in [0,10]$, and by a factor $2\times 10^{-4}$ for Kepler's problem. In
our tests, this scheme is much faster than Newton-Raphson method, by a factor
in the range $10^{-4}$ to $10^{-3}$ for the execution time in the examples,
when the values of the inverse function over an entire interval or for a large
number of points are requested. For Kepler's equation and tolerance $10^{-6}$
rad, the algorithm outperforms Newton's method for all values of the number of
points $N\ge 2$.
