How do I prevent this precision exception?

Question

I have the following as the first step to a sequence.

x = 2 - GoldenRatio;
Ceiling[x + x^(1/2)]

It gets a precision exception. The value is correct, but I would like to prevent the exception. (If we subtract 0.000001 from x, everything is fine.)

Is there a way to limit the precision? For one step only?

Edit: the sequence where 2-GoldenRatio is the seed.

x = 2 - GoldenRatio; t = Table[(x = Ceiling[x + x^(1/2)]), {2000}];

If ceiling is removed the first step works fine. However, I need the ceiling for alternate steps.

Edit2: $0 < x \leq 2-\phi$ where $\phi$ is the golden ratio. $\textit{When x = 2 - $\phi$, the first step equals $1$}$.

Answer 1

Does this get you what you need:

x = N[2 - GoldenRatio]
Ceiling[x + x^(1/2)]

0.381966
1

This doesn't get a precision exception.

This as per the update of your question also works without getting an error.

x = N[2 - GoldenRatio];
t = Table[(x = Ceiling[x + x^(1/2)]), {2000}];

Answer 2

Perhaps you could make a value inexact:

x = 2` - GoldenRatio;
x + x^(1/2)

1.

Or use FullSimplify:

x = 2 - GoldenRatio;
x + x^(1/2) // FullSimplify

1