From https://blog.justinmallone.com/sicp-12-procedures-and-the-processes-they-generate-part-1/

What exactly was the relevance of the golden ratio in understanding the inefficiency of the recursive fibonacci procedure? Quote of relevant material:

In fact, it is not hard to show that the number of times the procedure will compute

`(fib 1)`

or`(fib 0)`

(the number of leaves in the above tree, in general) is precisely Fib(n + 1). To get an idea of how bad this is, one can show that the value ofFib(n) grows exponentially withn. More precisely (see exercise 1.13),Fib(n)is the closest integer to \phi^5/\sqrt{n}, where …

From the book (for context):

Exercise 1.13.Prove that Fib(n) is the closest integer to \phi^n/\sqrt{5}, where \phi = (1+\sqrt{5})/2. Hint: let \psi = (1 - \sqrt{5})/2 Use induction and the definition of the Fibonacci numbers (see section 1.2.2) to prove that Fib(n) = (\phi^n - \psi^n)/\sqrt{5}.

I think the point of saying that Fib(n) \approx \phi^n/\sqrt{5} was to show specifically how Fib(n) grows exponentially with n and to ~foreshadow exercise 1.13.

Exercise 1.13 doesn’t seem *that* useful as a coding exercise, except insofar as rigorously analyzing the complexity of the `fib`

function, via Fib. That skill seems more relevant for like academics and ppl doing theoretical comp sci. That sort of thing doesn’t come up in day-to-day programming (though being able to analyze an algorithms complexity does; in this case the complexity is O(c^n) for some constant c; *edit: you don’t need to know what this means; I included it so that you could see the difference between more simplistic day-to-day complexity stuff and like in-depth analysis*)

IDK if that answers your q