FullSimplify wont simplify to expression with lower SimplifyCount

Question

I have been trying without success to get FullSimplify to work on some trigonometric expressions. It refuses to convert them to forms that are clearly much simpler, as measured by SimplifyCount or LeafCount

I'd really appreciate if someone can explain what is going on here (and hopefully how to tackle it)

Here is a simple case:

Test1 = (11 + 4 Cos[2 a] + Cos[4 a])/8
Test2 = 1 + Cos[a]^4

FullSimplify[Test1 == Test2]
 True

SimplifyCount /@ {Test1, Test2}
 {17, 6}

LeafCount /@ {Test1, Test2}
 {16, 6}

As far as I understand (please correct me if I'm wrong), FullSimplify works by choosing the expression with the lowest ComplexityFunction value, and the default ComplexityFunction is very similar to SimplifyCount or LeafCount. So it should simplify Test1 to Test2. However...

FullSimplify[Test1]
1/8 (11 + 4 Cos[2 a] + Cos[4 a])

Again, as far as I understand, this can only happen if the specific transformation needed to go from Test1 to Test2 is not used by Mathematica, and has to be 'taught' using TransformationFunctions. But in this case only the most basic trigonometric identity is needed, which Mathematica surely must know (e.g. Cos[2a] = 2 Cos[a]^2 - 1 ). I did anyway try to implement this as a new TransformationFunction, eg

CosTrans[expr_] := expr /. Cos[2 a_] :> 2 Cos[a]^2 - 1;

But nothing changed.

(It could of course just be that my implementation is wrong; I'm not too clear on the appropriate syntax for new transformation functions)

Anyway, thanks to anyone that can point me in the right direction.

Answer 1

The problem is that the number of trigonometric transformations that Mathematica could try is rather large and grows with every step. This problem is compounded with the problem that the LeafCount does not always go down with every step. Let me illustrate this with a potential series of steps that Mathematica could have taken to arrive at the solution.

Test1 // LeafCount

16

step1 = (Test1 // TrigExpand) /. {Cos -> cos, Sin -> sin}

11/8 + cos[a]^2/2 + cos[a]^4/8 - sin[a]^2/2 - 3/4 cos[a]^2 sin[a]^2 + sin[a]^4/8

(I replaced Cos and Sin with inert functions to prevent Mathematica from performing any simplifications I do not want)

step1 // LeafCount

48

step2 = step1 // Simplify

1/8 (11 + cos[a]^4 - 4 sin[a]^2 + sin[a]^4 + cos[a]^2 (4 - 6 sin[a]^2))

step2 // LeafCount

33

step3 = step2 /. sin[a]^2 -> (1 - cos[a]^2)

1/8 (11 + cos[a]^4 - 4 (1 - cos[a]^2) + cos[a]^2 (4 - 6 (1 - cos[a]^2)) + sin[a]^4)

step3 // LeafCount

41

step4 = step3 // Simplify

1/8 (7 + 2 cos[a]^2 + 7 cos[a]^4 + sin[a]^4)

step4 // LeafCount

22

step5 = step4 /. sin[a]^4 -> (1 - cos[a]^2)^2

1/8 (7 + 2 cos[a]^2 + 7 cos[a]^4 + (1 - cos[a]^2)^2)

step5 // LeafCount

28

step6 = step5 // Simplify

1 + cos[a]^4

As you can see, along the route the LeafCount went up three times, and the first five steps all took you to a higher count than your starting value, so it may not really be clear at any moment that you're actually closing in on your goal. Mathematica must at some time conclude that further continuation may be fruitless although the solution may be just beyond the horizon.