Reverse engineer numerical results to fractions of remarkable numbers?

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Numerical methods output decimal numbers that oftentimes result from the division of two (or more) numbers:

  • $1.5708... = \frac{\pi}{2}$
  • $0.3679... = \frac{1}{e}$
  • $0.7071... = \frac{\sqrt2}{2}$
  • $0.7501315 ... = \frac{{e}^{\frac{\pi}{12}}}{\sqrt{3}}$ (maybe this one is pushing it a little)

What is a good way to "reverse engineer" a decimal point number to the fraction it might have originated from?

Of course, maintaining a lookup table of values for most commonly encountered decimal numbers and their fractions of "remarkable numbers" is feasible to some extent, but is there a better way?

Side question: what is the proper jargon to succinctly describe this problem?