Simplification using multiset_partitions()

gistfile1.py

In [1]: var('x1,x2,y1,y2,r1,r2')
Out[1]: (x₁, x₂, y₁, y₂, r₁, r₂)

In [2]: E=1/(2*x1**2 - 4*x1*x2 + 2*x2**2 + 2*y1**2 - 4*y1*y2 + 2*y2**2) * (-y1*r1**2 + y2*r1**2 + y1*r2**2 - y2*r2**2 + y1*x1**2 + y2*x1**2 - 2*x1*x2*y1 - 2*x1*x2*y2 + y1*x2**2 + y2*x2**2 + y1**3 - y2*y1**2 - y1*y2**2 + y2**3)

In [3]: n, d = E.as_numer_denom()

In [4]: _, D = factor(d).as_coeff_Mul()

In [5]: D
Out[5]: 
  2               2     2               2
x₁  - 2⋅x₁⋅x₂ + x₂  + y₁  - 2⋅y₁⋅y₂ + y₂ 

In [6]: from sympy.utilities.iterables import multiset_partitions

In [7]: %time P = [ factor(sum(f)) + factor(sum(g)) for f, g in multiset_partitions(D.args, 2) ]
CPU times: user 283 ms, sys: 0 ns, total: 283 ms
Wall time: 280 ms

In [8]: min([ p.count_ops() for p in P ])
Out[8]: 5

In [9]: [ p for p in P if p.count_ops() == _ ]
Out[9]: 
⎡         2            2⎤
⎣(x₁ - x₂)  + (y₁ - y₂) ⎦

In [10]: _[0]
Out[10]: 
         2            2
(x₁ - x₂)  + (y₁ - y₂)