Most of the points that I raised were very minor, however there is one
thing that I see as very problematic - the interval arithmetics
implementation:

For example we want to evaluate Integral(awful_expressions_of_x,
(x,0,y)) with respect to y.

lambdify will fail to produce a numeric value.
experimental_lambdify will succeed by using evalf (there are also
checks for complex number, for numpy functions and what not, but in
this case they are not important). Evalf is using mpmath for that
evaluation. But how will you force it to use mpi?

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