Significant digits question. compute $\sin x$

Question

Why if $x$ is a machine number on a $32$-bit computer that satisfies the inequality $x > \pi 2^{25}$, then $\sin x$ can always be computed with no significant digits?

Thank you.

Answer 1

You need to assess the difference between two floating point numbers at that range. The difference will be large enough that within the range that one number represents, the $\sin$ function changes greatly.