Good points.
There are two kinds of use cases where I would want to calculate
things like
"sum(sqrt(p) for p in primes(1000))":
1. Just to have an idea what the result "value" is in RR, to some
accuracy.
2. Assume I have a complicated weird delicate expression of Gauss/
Kloosterman sums.
Which I could calculate numerically to the same accuracy, and it
is "value" from above to that accuracy.
Then I would want Sage to check for equality "rigorously" (this
probably cannot be done otherwise
than by constructing a number field of very high degree).
The example is not a good one (replace 1000 by 10, perhaps), but I
hope you can see what I mean anyway.
These are different use cases which require different optimizations,
even implementations.
However, both are valid, and ideally one could "automatically" go back
and forth in both directions.
The direction "falling down" from symbolic to numeric calculations (be
it complex or p-adic),
is by far the easier one, and I agree, that "smooth" behaviour here is
something that users expect.
Cheers,
gsw
P.S.:
Someday, one might be able to give to Sage some input like
"LHS of BSwD" equals "RHS of BSwD"
and Sage saying "True", and also being able to output the reduction
steps in some human-readable way.
After all, Pi already is Sage (we can calculate correctly "exp(Pi I)
is -1" already now).
And for number-theorists, Pi is nothing else but some "period",
just as the other non-algebraic terms involved in the BSwD
conjecture ...
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