ceil() of algebraic numbers is buggy

[dimpase]

Steps To Reproduce

sage: x = QQbar(sqrt(8))
....: ceil(x-2*QQbar(sqrt(2)))
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Cell In[41], line 2
      1 x = QQbar(sqrt(Integer(8)))
----> 2 ceil(x-Integer(2)*QQbar(sqrt(Integer(2))))

File /mnt/opt/Sage/sage-clang/src/sage/functions/other.py:433, in Function_ceil.__call__(self, x, **kwds)
    416 def __call__(self, x, **kwds):
    417     """
    418     Allow an object of this class to behave like a function. If
    419     ``ceil`` is an instance of this class, we can do ``ceil(n)`` to get
   (...)
    431         -2
    432     """
--> 433     return _eval_floor_ceil(self, x, "ceil", **kwds)

File /mnt/opt/Sage/sage-clang/src/sage/functions/other.py:297, in _eval_floor_ceil(self, x, method, bits, **kwds)
    293         continue
    295     bits *= 2
--> 297 raise ValueError("cannot compute {}({!r}) using {} bits of precision".format(method, x, RIF.precision()))

ValueError: cannot compute ceil(0.?e-307) using 512 bits of precision

Documentation of ceil() says:

3. If none of the above work, Sage returns a "Expression" object.

This is contradicting next pieces of the docs:

   Small numbers which are extremely close to an integer are hard to
   deal with:

      sage: ceil((33^100 + 1)^(1/100))
      Traceback (most recent call last):
      ...
      ValueError: cannot compute ceil(...) using 256 bits of precision

The latter is not an Expression.

Worse, increasing the precision does not seem to have an effect:

ValueError: cannot compute ceil(0.?e-157825) using 262144 bits of precision

Is this a bug in RIF?

This was reported to me by @Ilia-Smilga

Expected Behavior

it should work

Actual Behavior

it doesn't

Additional Information

Note that in this case, one can immediately see that we're computing ceil of an algebraic number 0:

sage: n=t-2*QQbar(sqrt(2))
sage: n.minpoly()
x

The latter can be used as follows:

sage: f=n.minpoly()
sage: f.variables()[0]==f
True
sage: f.variables()
(x,)
sage: f.variables()[0]==f
True

Environment

current Sage beta

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[dimpase]

@kwankyu @ekwankyu @mezzarobba

[dimpase]

Should we implement ceil for algebraic numbers via the minpoly trick as above ?

[Ilia-Smilga]

Note that there is already some code written to handle algebraic numbers that appear almost indistinguishable from 0 - namely the functions bool(self) and sign(self) in the QQbar module ( https://github.com/sagemath/sage/blob/develop/src/sage/rings/qqbar.py ). So ceil() should maybe call one of these two functions.

[mezzarobba]

There is a (slow but) working implementation of ceil() for (real) algebraic numbers:

sage: x = AA(sqrt(8))
sage: x.ceil()
3

The issue here appears to be with the symbolic function ceil.

Related issues: #22079, #25827, #12121, #18333.