Automatically convert recursion to iteration

Question

I have seen these questions:

• Can and do compilers convert recursive logic to equivalent non-recursive logic?
• Way to go from recursion to iteration
• Do any programming language compilers automatically convert recursion to iteration?

These all mention tail-call optimization. I am wondering though if, given a function such as what follows, it is possible to automatically "flatten" it from recursive to iterative, using something other than tail-call optimization (which I don't think applies in this case, and I wouldn't want to place that restriction otherwise).

a() {
  while x < 3 {
    b()
  }
}

b() {
  c()
}

c() {
  d()
}

d() {
  a()
}

Notice d calls a and starts the recursion over, but assume that it won't recurse forever, just a few levels deep (this is just pseudocode). This is different than the fibonacci example which just has 1 function recursively calling itself. I'm wondering if any research has been done on the topic of automatically converting complex recursive functions like this into iterative ones.