expand(x**(a+b)) iff x != 0 or a & b both nonneg or nonpos

[smichr]

References to other Issues or PRs

Brief description of what is fixed or changed

fixes #23952 but required treating zoo/x as non-rational.

Other comments

This fixes the root issue (of the expansion to 0) but better handling needs to be implemented to get the original issue to be solved:

>>> import sympy as sp
>>> p, q = sp.symbols("p q", real=True, nonnegative=True)
>>> k1,k2 = sp.symbols("k1 k2", integer=True, nonnegative=True)
>>> n = sp.Symbol("n", integer=True, positive=True)
>>> expr = sp.Sum(sp.Abs(k1 - k2)*p**k1 *(1-q)**(n-k2), (k1,0,n), (k2,0,n))
>>> expr.subs(p,0).subs(q,1)
Sum(0**k1*0**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))
>>> _.doit()
...
sympy.polys.polyerrors.CoercionFailed: zoo is not in any domain

If the assumptions are omitted then

>>> S('''Sum(p**k1*(1 - q)**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))''')
Sum(p**k1*(1 - q)**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))
>>> ok=_;var('p q');ok.subs(p,0).subs(q,1)
(p, q)
Sum(0**k1*0**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))
>>> _.doit()
Sum(0**k1*0**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))

Release Notes

• core
  • x**(a + b) will not expand unless a and b are both nonnegative or nonpositive or x is not zero
  • expressions containing any illegal character as defined in numbers now gives expr.is_rational_function()->False

[sympy-bot]

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Here is what the release notes will look like:

• core

  • x**(a + b) will not expand unless a and b are both nonnegative or nonpositive or x is not zero (#23962 by @smichr)

  • expressions containing any illegal character as defined in numbers now gives expr.is_rational_function()->False (#23962 by @smichr)

This will be added to https://github.com/sympy/sympy/wiki/Release-Notes-for-1.12.

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#### Brief description of what is fixed or changed

fixes #23952 but required treating zoo/x as non-rational.

#### Other comments

This fixes the root issue (of the expansion to 0) but better handling needs to be implemented to get the original issue to be solved:
```python
>>> import sympy as sp
>>> p, q = sp.symbols("p q", real=True, nonnegative=True)
>>> k1,k2 = sp.symbols("k1 k2", integer=True, nonnegative=True)
>>> n = sp.Symbol("n", integer=True, positive=True)
>>> expr = sp.Sum(sp.Abs(k1 - k2)*p**k1 *(1-q)**(n-k2), (k1,0,n), (k2,0,n))
>>> expr.subs(p,0).subs(q,1)
Sum(0**k1*0**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))
>>> _.doit()
...
sympy.polys.polyerrors.CoercionFailed: zoo is not in any domain
```
If the assumptions are omitted then
```python
>>> S('''Sum(p**k1*(1 - q)**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))''')
Sum(p**k1*(1 - q)**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))
>>> ok=_;var('p q');ok.subs(p,0).subs(q,1)
(p, q)
Sum(0**k1*0**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))
>>> _.doit()
Sum(0**k1*0**(-k2 + n)*Abs(k1 - k2), (k1, 0, n), (k2, 0, n))
```
#### Release Notes

<!-- Write the release notes for this release below between the BEGIN and END
statements. The basic format is a bulleted list with the name of the subpackage
and the release note for this PR. For example:

* solvers
  * Added a new solver for logarithmic equations.

* functions
  * Fixed a bug with log of integers.

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<!-- BEGIN RELEASE NOTES -->
* core
    * `x**(a + b)` will not expand unless `a` and `b` are both nonnegative or nonpositive or `x` is not zero
    * expressions containing any illegal character as defined in numbers now gives `expr.is_rational_function()->False`
<!-- END RELEASE NOTES -->

Update

The release notes on the wiki have been updated.

[github-actions]

Benchmark results from GitHub Actions

Lower numbers are good, higher numbers are bad. A ratio less than 1
means a speed up and greater than 1 means a slowdown. Green lines
beginning with + are slowdowns (the PR is slower then master or
master is slower than the previous release). Red lines beginning
with - are speedups.

Significantly changed benchmark results (PR vs master)

Significantly changed benchmark results (master vs previous release)

       before           after         ratio
     [41d90958]       [ed600afb]
     <sympy-1.11.1^0>                 
-         995±2μs          628±3μs     0.63  solve.TimeSparseSystem.time_linear_eq_to_matrix(10)
-        2.86±0ms         1.17±0ms     0.41  solve.TimeSparseSystem.time_linear_eq_to_matrix(20)
-     5.71±0.02ms         1.71±0ms     0.30  solve.TimeSparseSystem.time_linear_eq_to_matrix(30)

Full benchmark results can be found as artifacts in GitHub Actions
(click on checks at the top of the PR).

[smichr]

Is there any reason we shouldn't do this?

[smichr]

I didn't think this would be so easy to fix. It is similar to not allowing x**2/x to become x unless x != 0 (something we don't prohibit). The power expanding is easier to control because it does not happen automatically. ping @asmeurer

[oscarbenjamin]

The title of this PR says expand(x**(a+b)) iff x != 0 or a + b != 0 but I don't understand why it's okay to expand if a + b != 0 even though x might be zero:

In [30]: e1 = x**(y + z)

In [31]: e2 = e1.expand()

In [32]: e1
Out[32]: 
 y + z
x     

In [33]: e2
Out[33]: 
 y  z
x ⋅x 

In [34]: e1.subs(x, 0).subs({y:1,z:-2})
Out[34]: zoo

In [35]: e2.subs(x, 0).subs({y:1,z:-2})
Out[35]: 0

[smichr]

@oscarbenjamin , to get your example to work you would have to have assumptions on symbols in a sum that would allow the assumptions system to determine that the exponent wasn't zero. The example you gave relies on one number being positive and the other negative; that will not evaluate as having a non-zero assumption. I think the nearest we could come is with nonnegative + positive which must be positive. When expanded this would give 0**nonnegative*0**positive which would correctly give 0. Can you think of an example with assumptions that would behave like the expression you gave?

[smichr]

answer:

>>> one = Symbol('1',integer=1,prime=0,odd=1,positive=1)
>>> two = Symbol('2',integer=1,prime=1,even=1)
>>> eq = two-one
>>> 0**(2-1)
0
>>> 0**eq
0**(-1 + 2)
>>> expand(_)
nan
>>> (x**eq).expand().subs(x,0)
nan

>>> S.Zero**-1
zoo
>>> 0**-eq
0**(1 - 2)
>>> expand(_)
nan
>>> (x**-eq).expand().subs(x,0)
nan

[oscarbenjamin]

to get your example to work you would have to have assumptions on symbols in a sum that would allow the assumptions system to determine that the exponent wasn't zero.

The question is under what conditions does the identity x**(a + b) == x**a * x**b hold?

The docs here claim that the identity is "always true":
https://docs.sympy.org/latest/tutorials/intro-tutorial/simplification.html#powers

I think that misses the possibility that x could be zero or infinite. If x is a nonzero finite complex number and a and b are finite complex numbers then all of x**a, x**b and x**(a + b) are well defined nonzero complex numbers and the identity holds. However if x is zero then we have
$$0^c = \begin{cases} 0 & c > 0 \\ 1 & c = 0 \\ \infty & c < 0 \\ \mathrm{undefined} & c \notin \mathbb{R} \end{cases}$$
Any situation where a + b, a and b are all nonnegative will be fine but otherwise we can have problems e.g.:

In [59]: e1 = x**(a + b)

In [60]: e2 = e1.expand()

In [61]: e1.subs(x, 0).subs({a:I, b:-I})
Out[61]: 1

In [62]: e2.subs(x, 0).subs({a:I, b:-I})
Out[62]: nan

In [67]: e1.subs(x, 0).subs({a:-1, b:2})
Out[67]: 0

In [68]: e2.subs(x, 0).subs({a:-1, b:2})
Out[68]: nan

In [69]: e1.subs(x, 0).subs({a:1, b:-2})
Out[69]: zoo

In [70]: e2.subs(x, 0).subs({a:1, b:-2})
Out[70]: 0

The last case is problematic because of the bogus evaluation of 0*x (#17224):

In [73]: e2.subs(x, 0).subs(b, -2)
Out[73]: 
 a    
0 ⋅zoo

In [74]: e2.subs(x, 0).subs(b, -2).subs(a, 1)
Out[74]: nan

In [75]: e2.subs(x, 0).subs(a, 1).subs(b, -2)
Out[75]: 0

[smichr]

I am not seeing why this is passing. I checked the Add._all_nonneg_or_nonpos and it gives None for y + 2 so why does the power expand?

    expand_power_exp(x**(y + 2))
Expected:
    x**(y + 2)
Got:
    x**2*x**y

[smichr]

I will squash to two commits and merge if there are no other concerns.

[oscarbenjamin]

Why have these changed?

[oscarbenjamin]

I think this mostly looks good but please don't self merge PRs.

[oscarbenjamin]

@mkoeppe after this PR the SAGE doctest is failing:

File "src/sage/calculus/calculus.py", line 1693, in sage.calculus.calculus.laplace
Failed example:
    laplace(t^n, t, s, algorithm='sympy')
Expected:
    (gamma(n + 1)/(s*s^n), 0, re(n) > -1)
Got:
    (s^(-n - 1)*gamma(n + 1), 0, re(n) > -1)

Is there a new SAGE ticket to track updating to SymPy 1.12 where a doctest like this could be updated?

[mkoeppe]

I've opened https://trac.sagemath.org/ticket/34489 for this

[oscarbenjamin]

So I guess the process is to update this line:

sympy/.github/workflows/ci-sage.yml

Line 58 in 8e65a42

SAGE_TICKET: 34118

and then push a commit to the trac ticket?

[oscarbenjamin]

I've opened a PR to update the ticket number: #24015

[mkoeppe]

Yes, that's right.