polynom division with refine

[TSlivede]

I tried to do some simplification (on live.sympy.org) with refine, but couldn't get it to work. It seems to boil down to this:

refine((1-x**2)/(1-x),Q.real(x))

gives back (1-x**2)/(1-x),
while

simplify((1-x**2)/(1-x))

correctly simplifies it to x+1.

Maybe I'm just not understanding this correctly, as I'm relatively new to sympy - but shouldn't refine also simplify that expression?

I want to use refine because I actually have expressions of the form

refine(sqrt(1-x**2)/sqrt(1-x), Q.positive(1-x) & Q.positive(1-x**2))

where I don't think I can use simplify, because I can't tell sympy, that the root is positive.

[oscarbenjamin]

You can do this if x is positive using factor:

In [21]: x = Symbol('x', positive=True)                                                                                           

In [22]: sqrt(1-x**2)/sqrt(1-x)                                                                                                   
Out[22]: 
   ________
  ╱      2 
╲╱  1 - x  
───────────
   _______ 
 ╲╱ 1 - x  

In [23]: factor(_)                                                                                                                
Out[23]: 
  _______
╲╱ x + 1 

I'm not sure why x being positive makes this work.

In general I think there should be a function for doing sqrt(a)/sqrt(b) -> sqrt(a/b). Maybe powsimp force=True should do it. This works for arbitrary y:

In [38]: e = x**y/z**y                                                                                                            

In [39]: e                                                                                                                        
Out[39]: 
 y  -y
x ⋅z  

In [40]: powsimp(e)                                                                                                               
Out[40]: 
 y  -y
x ⋅z  

In [41]: powsimp(e, force=True)                                                                                                   
Out[41]: 
   y
⎛x⎞ 
⎜─⎟ 
⎝z⎠ 

However if y = 1/2 then it fails:

In [42]: e2 = e.subs(y, S(1)/2)                                                                                                   

In [43]: e2                                                                                                                       
Out[43]: 
√x
──
√z

In [45]: powsimp(e2)                                                                                                              
Out[45]: 
√x
──
√z

In [46]: powsimp(e2, force=True)                                                                                                  
Out[46]: 
√x
──
√z

[oscarbenjamin]

It seems that powsimp calls itself recursively and the difference above comes from

In [8]: powsimp(z**-S.Half, force=True)                                                                                                        
Out[8]: 
1 
──
√z

In [9]: powsimp(z**-y, force=True)                                                                                                             
Out[9]: 
   y
⎛1⎞ 
⎜─⎟ 
⎝z⎠

Maybe it should be changed so that we have:

In [8]: powsimp(z**-S.Half, force=True)                                                                                                        
Out[8]: 
    ___
   ╱ 1 
  ╱  ─ 
╲╱   z 

I.e. powsimp(1/sqrt(z), force=True) -> sqrt(1/z).

[oscarbenjamin]

Looks like the difference is here:

sympy/sympy/simplify/powsimp.py

Lines 395 to 397 in 58e7820

	exp_c, exp_t = e.as_coeff_Mul(rational=True)
	if exp_c is not S.One and exp_t is not S.One:
	c_powers[i] = [Pow(b, exp_c), exp_t]

since we have:

In [1]: (-y).as_coeff_Mul()                                                                                                                    
Out[1]: (-1, y)

In [2]: (-S.Half).as_coeff_Mul()                                                                                                               
Out[2]: (-1/2, 1)

[oscarbenjamin]

This makes it possible to use powsimp:

diff --git a/sympy/simplify/powsimp.py b/sympy/simplify/powsimp.py
index baed0aac83..e0208e17a5 100644
--- a/sympy/simplify/powsimp.py
+++ b/sympy/simplify/powsimp.py
@@ -393,6 +393,8 @@ def update(b):
             if not (all(x.is_nonnegative for x in b.as_numer_denom()) or e.is_integer or force or b.is_polar):
                 continue
             exp_c, exp_t = e.as_coeff_Mul(rational=True)
+            if exp_t is S.One and exp_c.is_negative:
+                exp_c, exp_t = -exp_t, -exp_c
             if exp_c is not S.One and exp_t is not S.One:
                 c_powers[i] = [Pow(b, exp_c), exp_t]

Then we have:

In [2]: powsimp(sqrt(1-x**2)/sqrt(1-x), force=True)                                                                                            
Out[2]: 
     ________
    ╱      2 
   ╱  1 - x  
  ╱   ────── 
╲╱    1 - x  

However I don't see how to simplify it from there...

In [3]: factor(_)                                                                                                                              
Out[3]: 
    ___________________
   ╱ -(x - 1)⋅(x + 1)  
  ╱  ───────────────── 
╲╱         1 - x       

In [4]: cancel(_)                                                                                                                              
Out[4]: 
     _________________
    ╱      2          
   ╱      x       1   
  ╱   - ───── + ───── 
╲╱      1 - x   1 - x 

In [6]: simplify(_)                                                                                                                            
Out[6]: 
     ________
    ╱  2     
   ╱  x  - 1 
  ╱   ────── 
╲╱    x - 1  

[oscarbenjamin]

Okay here it is (using the diff above):

In [1]: powsimp(sqrt(1-x**2)/sqrt(1-x), force=True)                                                                                            
Out[1]: 
     ________
    ╱      2 
   ╱  1 - x  
  ╱   ────── 
╲╱    1 - x  

In [2]: factor(_, deep=True)                                                                                                                   
Out[2]: 
  _______
╲╱ x + 1 

If anyone wants to add that diff (plus tests) as a PR then go ahead.

[arun-y99]

I'll add the diff and tests

[espatatis]

Can I get this issue assigned ?

[oscarbenjamin]

We don't assign issues but you can work on this if you want to. There already is a PR #17666 which you could use as a starting point.

[AnjaliJha167]

If no one is working on this issue and the issue is still open. Can I start working on it?
Also, it would be great, if anyone could guide me as to where to start from

[oscarbenjamin]

I would start by looking at the two existing PRs (#18031 and #17666) to see why they didn't work. Apparently this isn't as easy as I first thought.

[czgdp1807]

The issue still persists on master. Open for work.