A closed-form expression for the n-th Fibonacci number (discovered by Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli) is shown below. φ is the golden ratio.

A tiny-C program which uses this formula via a "plugin" at

http://primepuzzle.com/not.just.tiny.c/pifib.c

is at

http://primepuzzle.com/not.just.tiny.c/fibpi.tc

D:\my-git-repo>tc fibpi.tc

*** TINY-C VERSION 1.0, COPYRIGHT
1977, T A GIBSON *** This C version copyright 2017, T A Gibson

fibpi.tc - lrb - 6/25/21

enter a number < 40 39

fib(40) = 102334155

fib(39) = 63245986

ratio of these two Fibonacci numbers is 1.61803398874989473640

done

D:\my-git-repo>tc fibpi.tc

***  TINY-C VERSION 1.0,  COPYRIGHT 1977, T A GIBSON  ***
        This C version copyright 2017, T A Gibson

fibpi.tc - lrb - 6/25/21

enter a number < 40 20

fib(21) = 10946

fib(20) = 6765

ratio of these two Fibonacci numbers is 1.61803399852180339985

done

D:\my-git-repo>

For an interesting and creative discussion of how this ratio appears
in sunflowers Chip Bradley posted the following.

http://primepuzzle.com/not.just.tiny.c/sunflower.whorls.html