I have a paper where I am trying to show that the growth of a certain function is exponential of the order $a^n$. I would like to compute $a$, at least approximately. The base $a$ satisfies a very complicated formula. I do considerable amount of completely rigorous analysis to prove that this formula works, and this is really the interesting part of the paper.

This formula cannot be evaluated exactly, but there is a complicated method using numerical analysis techniques (numerical integration, root-finding, numerical optimization, and so forth) to compute it approximately.

I can use an off-the-shelf numerical package to compute $a$, seemingly to high precision. However, to get a rigorous bound, I would have to go through the entire recipe for computing $a$, showing that all the relevant functions are sufficiently smooth, all of the relevant local minima are in fact the global minima, the functions have the right concavity, the number of digits of accuracy is sufficiently large in each step, and so on. This would be extremely difficult, tedious, and frankly unenlightening --- if you look at a graph of the function it is clear that it has the right smoothness, and I don't want to waste a huge amount of space proving it has these properties.

Is it OK if I just numerically solve for $a$ without proving that my numerical solution is sufficiently accurate? Does it matter that this paper is in the subject of computer science, not pure math? Is there any better alternative?

Thanks for any advice