Simplification patterns can easily be written for algebraic expressions, e.g.:
(- x (- y)) ---> (+ x y) (- (- x)) ---> x (- (+ x y) (+ z y)) ---> (- x z)
Implementing many such patterns with Lisp code to test for each case would be difficult and result in code that was large, error-prone, hard to change, and yet possibly still inadequate. It would be better to use the patterns directly.