Can Sympy simplify a rational expression by collecting multiple terms?

Question

Given a rational expression E such as the one below, I'm looking to use Sympy to simplify it to something that looks like F (defined in the second block of Python code below):

import sympy as sp

a, b, c, d, n, t, A, B, C = sp.symbols('a, b, c, d, n, t, A, B, C', real = True)

E = n/(c-b) * ( B - (c-b)/(c-a)*A - (b-a)/(c-a)*B ) * (c-t)/(c-b) + n/(c-b) * ( (d-c)/(d-b)*B + (c-b)/(d-b)*C - B ) * (t-b)/(c-b)

print(sp.pretty( E ))
print(sp.pretty( E.simplify() ))

This prints

           ⎛     B⋅(-c + d)   C⋅(-b + c)⎞             ⎛  A⋅(-b + c)   B⋅(-a + b)    ⎞
n⋅(-b + t)⋅⎜-B + ────────── + ──────────⎟   n⋅(c - t)⋅⎜- ────────── - ────────── + B⎟
           ⎝       -b + d       -b + d  ⎠             ⎝    -a + c       -a + c      ⎠
───────────────────────────────────────── + ─────────────────────────────────────────
                        2                                           2                
                (-b + c)                                    (-b + c)
                
                
-n⋅((a - c)⋅(b - t)⋅(-B⋅(b - d) + B⋅(c - d) + C⋅(b - c)) + (b - d)⋅(c - t)⋅(A⋅(b - c) + B⋅(a - b) - B⋅(a - c))) 
────────────────────────────────────────────────────────────────────────────────────────────────────────────────
                                                           2                                                    
                                            (a - c)⋅(b - c) ⋅(b - d)

However, the expression can be — manually — simplified further, the result of which I've labeled F:

F = n/(c-a) * (B - A) * (c-t)/(c-b) + n/(d-b) * (C - B) * (t-b)/(c-b)

print(sp.pretty( F ))
print((F-E).simplify())

This outputs

n⋅(-A + B)⋅(c - t)   n⋅(-B + C)⋅(-b + t)
────────────────── + ───────────────────
(-a + c)⋅(-b + c)     (-b + c)⋅(-b + d) 


0

I've looked into various options including factor(), collect() and apart(), but none of these seem to yield expressions that have the same structure as F. Any pointers on how to proceed?

Additionally, I wondered whether Sympy's pretty print function can be tweaked somehow to

Keep the original order of the variables in both the numerator and denominator (e.g. B - A instead of -A + B). Currently the order is flipped in most cases, which looks rather ugly with the leading minus signs.
Show composite fractions as products of simple fractions (e.g. a/b c/d instead of ac/bd), though in certain cases it can of course be ambiguous where/how to "split" such composite fractions.

user_9450991 · Answer

The situation here is that you have an Add of two terms. Each term separately can be simplified using factor but the factors to cancel are different for each so calling factor on the whole Add fails to find the possible cancellation.

With that in mind we need to be careful to process the terms of the Add independently which we can do by accessing .args:

In [122]: E.func(*(factor(term) for term in E.args))
Out[122]: 
n⋅(A - B)⋅(-c + t)   n⋅(B - C)⋅(-b + t)
────────────────── - ──────────────────
 (a - c)⋅(b - c)      (b - c)⋅(b - d)

The order of the variables is actually determined by the printer when displaying the expression and is not necessarily the same as the internal order of the args or necessarily the order used when you created the expression. A call to signsimp can normalise the minus signs in the expression though

In [123]: signsimp(_)
Out[123]: 
  n⋅(A - B)⋅(c - t)   n⋅(B - C)⋅(b - t)
- ───────────────── + ─────────────────
   (a - c)⋅(b - c)     (b - c)⋅(b - d)

Can Sympy simplify a rational expression by collecting multiple terms?

1 Answers