Context: I'm a secondary student working on a small exploration of the fast inverse square root algorithm. This algorithm uses a certain constant to "tune" the approximation to a small margin of error. Out of curiosity, I tried setting the value of the fast inverse square root equal to $\frac{1}{\sqrt{x}}$ for several values of $x$ in order to solve for a better constant that would completely eliminate the error at each point.
The issue: Plotting the results against $x$ leads to the graphs below. I've attached one version with a linear scale, and one with a logarithmic scale on the $x$-axis.
![- [Graph with linear scale;][2]](https://i.stack.imgur.com/RQ7Wa.png)
![- [Graph with semi-logarithmic scale.][3]](https://i.stack.imgur.com/F9jqa.png)
I am currently struggling to find a function that would produce this type of relationship. The process I used to generate the data involved some computer memory hacking (reinterpreting the bits of a float as a long and vice-versa), which is difficult to express mathematically.
However, the graph doesn't seem too outlandish. It has a repeating pattern, though the "period" quadruples each cycle. It has two distinct "peaks", with the low one occurring when $x$ is an odd power of $2$ and the high one on even powers of $2$.
I've done some tinkering but I don't feel that I've made any significant progress; $-|\cos{(\pi\cdot4^\frac{1}{x})}|$ has been my best guess and it's not remotely close. I would appreciate any help expressing this relationship in mathematical terms.
This is my first time asking, so I'll do my best to provide additional information and fix mistakes as needed.