>>> float(1) ** float('inf')
1.0
>>> mpf(1) ** mpf('inf')
mpf('nan')

>>> float('nan') ** float(0)
1.0
>>> mpf('nan') ** mpf(0)
mpf('nan')

These questions have arised from
sympy/sympy#18500
sympy/sympy#18470
But the inconsistency of mpmath and python arithmetic itself is making the thoughts complicated.
Is it intended or is it a bug of mpmath?