Program Verification in the presence of complex numbers, functions with branch cuts etc

In considering the reliability of numerical programs,
it is normal to “limit our study to the semantics
dealing with numerical precision” (Martel, 2005). On the other
hand, there is a great deal of work on the reliability of
programs that essentially ignores the numerics. The thesis
of this paper is that there is a class of problems that fall
between these two, which could be described as “does the lowlevel
arithmetic implement the high-level mathematics”. Many
of these problems arise because mathematics, particularly
the mathematics of the complex numbers, is more difficult
than expected: for example the complex function log is not
continuous, writing down a program to compute an inverse
function is more complicated than just solving an equation,
and many algebraic simplification rules are not universally
valid.
The good news is that these problems are theoretically
capable of being solved, and are practically close to being solved,
but not yet solved, in several real-world examples. However,
there is still a long way to go