I cannot read Haskell so I was not sure how the factoring was done. Going through the example it seems like tmoertel was just factoring like terms.

My implementation doesn't actually do any factoring. Rather, it efficiently cancels all like terms before doing any multiplications or divisions. The idea is to replace expensive mathematical operations with inexpensive comparisons until I have picked all of the low-hanging fruit. If I pick enough fruit, there isn't enough left on the tree to make factoring worthwhile, and so I can just go ahead a multiply out the rest. (Although it might be interesting to emit the tree after my low-handing-fruit pass and process it with some of the other bits of code in this thread.)

The following shows how my code might compute 7!/(3!4!):

7!
-----
3! 4!

{ expand fac into series of multiplications }

 P(2,3,4,5,6,7)
-----------------
P(2,3) * P(2,3,4)

{ merge multiplied products }

 P(2,3,4,5,6,7)
-----------------
  P(2,2,3,3,4)

{ cancel like terms across division boundary }

 P(/,/,/,5,6,7)
-----------------
  P(/,2,/,3,/)

{ multiply and divide remaining terms }

5 * 6 * 7
---------
  2 * 3

{ result }

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[download]

In reality, the code does all of these steps in a parallel pipeline, which means I don't have to pay the price for large intermediate terms: bits of big terms are plucked off and used as they are produced lazily. This is an essentially free benefit of using Haskell.

Cheers,
Tom