Find a formula

[AdolfoKub]

I encountered this problem and found it very difficult for me. Please help me.
Imagine a mysterious machine that transforms numbers in a peculiar way. After extensive experimentation, you've gathered the following data points:

Input (x) Output (y)
1 0.5403
2 1.0986
3 1.4142
4 1.6180
5 1.7725
6 1.8925
Your task is to decipher the hidden function formula that governs this machine. Here are some clues to guide you:

The function involves a combination of arithmetic operations and potentially non-standard mathematical functions.
The output values seem to approach a specific number as the input increases.
The graph of the function might reveal its shape and behavior.
Additional challenges:

Can you find a formula that accurately predicts the output for any input value, not just those in the table?

[Jerry]

A polynomial curve of degree five can be drawn through six points.

I greet You