How can I see which transformations Simplify attempts?

Question

The documentation for Simplify[expr] says that it performs a sequence of algebraic and other transformations on expr, and returns the simplest form it finds. How can I see which transformations it applies?

Answer 1

I thought of this question while on the train but the solution appeared in my brain as soon as I got into work. All you need to do is create a ComplexityFunction that includes a side effect

f[x_] := (Print[x];
  LeafCount[x])

Simplify[TrigExpand[Tan[x + y]], ComplexityFunction -> f]

This gives the following output

(Cos[y] Sin[x])/(Cos[x] Cos[y]-Sin[x] Sin[y])+(Cos[x] Sin[y])/(Cos[x] Cos[y]-Sin[x] Sin[y])
(Cos[y] Sin[x]+Cos[x] Sin[y])/(Cos[x] Cos[y]-Sin[x] Sin[y])
(Cos[y] Sin[x]+Cos[x] Sin[y])/(Cos[x] Cos[y]-Sin[x] Sin[y])
(Cos[y] Sin[x]+Cos[x] Sin[y])/(Cos[x] Cos[y]-Sin[x] Sin[y])
(Cos[y] Sin[x])/(Cos[x] Cos[y]-Sin[x] Sin[y])+(Cos[x] Sin[y])/(Cos[x] Cos[y]-Sin[x] Sin[y])
(1/2 Csc[x] Sin[2 x] Sin[y]+1/2 Csc[y] Sin[x] Sin[2 y])/(-Sin[x] Sin[y]+1/4 Csc[x] Csc[y] Sin[2 x] Sin[2 y])
(Cos[y] Sin[x]+Cos[x] Sin[y])/(Cos[x] Cos[y]-Sin[x] Sin[y])
Tan[x+y]
Tan[x+y]
x+y
x+y
x+y
x+y
x+y
x+y
x+y  
x+y
x+y
x+y
x+y  
Tan[x+y]
Tan[x+y]
(Cos[y] Sin[x])/(Cos[x] Cos[y]-Sin[x] Sin[y])+(Cos[x] Sin[y])/(Cos[x] Cos[y]-Sin[x] Sin[y])
Tan[x+y]

Note that this shows every expression that the ComplexityFunction is applied to. Clearly x+y is not equivalent to Tan[x+y]

It is interesting to also apply this to FullSimplify to see all of the extra transformations that get applied

FullSimplify[TrigExpand[Tan[x + y]], ComplexityFunction -> f]