[David Mertz <mertz@gnosis.cx>]
>> For example, this can be true (even without reaching inf):
>>
>> >>> x.is_integer()
>> True
>> >>> (math.sqrt(x**2)).is_integer()
>> False

[Mark Dickinson <dickinsm@gmail.com> ]
> If you have a moment to share it, I'd be interested to know what value of
> `x` you used to achieve this, and what system you were on. This can't happen
> under IEEE 754 arithmetic.

I expect it might happen under one of the directed rounding modes
(like "to +infinity").

But under 754 binary round-nearest/even arithmetic, it's been formally
proved that sqrt(x*x) == x exactly for all non-negative finite x such
that x*x neither overflows nor underflows (and .as_integer() has
nothing to do with that very strong result):

    https://hal.inria.fr/hal-01148409/document

OTOH, the paper notes that it's not necessarily true for IEEE decimal
arithmetic; e.g.,

>>> import decimal
>>> decimal.getcontext().prec = 4
>>> (decimal.Decimal("31.66") ** 2).sqrt()  # result is 1 ulp smaller
Decimal('31.65')

>>> decimal.getcontext().prec = 5
>>> (decimal.Decimal("31.660") ** 2).sqrt() # result is 1 ulp larger
Decimal('31.661')