Algorithm for comparing the size of extremely large numbers Is there a simple algorithm to decide which of the numbers
$$a \uparrow ^b c \text{ and } d \uparrow ^e f$$
is the bigger one ?
Using the hyperoperation, the numbers can be denoted with
$$H_{b+2}(a,c)\text{ and } H_{e+2}(d,f)$$
I tried using the recursive definition of $H$
$$H_n(a,b) = H_{n-1}(a,H_n(a,b-1))$$ 
and induction to get useful properties, but without substantial success.
If the given numbers are very large, the following heuristic should give the
correct result in many cases :
If $b>e$, then the first number is bigger.
If $b=e$ and $c>f$, then the first number is bigger.
If $b=e$ and $c=f$, it is trivial to compare the numbers.
Of course, this heuristic cannot hold in all cases.
Any ideas?
 A: Since this question has no answers, I will give it my best shot and hope someone can improve on this.
I will start off by noting that googology is a good place if you are really interested in very very VERY large numbers.  It might also be able to help you a little bit.
To answer your question, I will use a few ideas...
Imagine comparing the following numbers:$$N_1<N_2$$
It is given that $N_2$ is larger, but we don't know how to prove it.
One method is to find $A,B$, such that we have:
$$N_1<A<B<N_2$$
Basically, try to find the middle ground.
Another idea of mine is to compare growth rates:
$$a\uparrow^bc\text{ and }d\uparrow^ef$$
Well, how do I compare this?  Start with $b,e$.
If $b$ or $e$ is sufficiently larger than the other, then it ultimately determines how big the number will be.
Next in importance is $c$ and $d$.  These determine how many times you will be repeating a certain process, so the larger, the bigger your number will be.
Ultimately, it comes down to the following:
$$f(a,b,c)=a\uparrow^bc$$
Start with $a,b,c=3$.  Then try $f(4,3,3),f(3,4,3),f(3,3,4)$.  Which one is biggest?  Use this information to determine how each number affects the final result and which one has a larger effect.
