gh-108885: Use subtests for doctest examples run by unittest

[serhiy-storchaka]

Run each example as a subtest in unit tests synthesized by doctest.DocFileSuite() and doctest.DocTestSuite().

• Issue: Better integration of doctest with unittest and libregrtest #108885

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📚 Documentation preview 📚: https://cpython-previews--134890.org.readthedocs.build/

[iritkatriel]

We don't need to use sys here:

except BaseException as e:
    exc_info = type(e), e, e.__traceback__.tb_next

[iritkatriel]

Alternatively, save just the exception here and access its parts below where you need them.

[serhiy-storchaka]

It does not work, because a traceback here is a not a real traceback. You cannot set it as the part of the exception. Fortunately, all traceback formatting functions and TestCase.addSubTest() work with the exc_info triple.

[serhiy-storchaka]

Well, here it is a real traceback, but in other place we use a mock traceback, so it is easier to use the same pattern here.

[iritkatriel]

Yeah, I meant that you extract it from the exception object when you need it below (in this function) and pass that to the internal functions that want the triplet. Just get it from the exception and not from sys.exc_info(). It's the same thing.

[serhiy-storchaka]

This line is actually a part of #134858.

[serhiy-storchaka]

I applied your suggestion there.

[serhiy-storchaka]

Before merging this PR I am going to create and merge a PR for documentation fixes, because the current documentation has some errors.