Implement Mathematica's capability of rationalizing machine reals
The continued fraction algorithm is easy enough to implement. The first step is to compute the continued fraction of the input $x = [c_0;c_1,\ldots]$. You start with $x_0 = x$, and use the recurrence $c_i = \lfloor x_i \rfloor$, $x_{i+1} = 1/(x_i - c_i)$. You stop when $x_i - c_i$ is "small enough". The next step is to compute the convergent of the continued fraction. You can find the appropriate formulas on the web. In practice, both steps are performed at the same time.

There are some issues with numerical stability, and that might make it difficult to match the performance of Mathematica, but for simple cases it works well enough.

The resulting rational number is not actually equivalent to the floating point number - there's probably some small error. In fact, the rational approximation may be more accurate in some cases, and off the mark in others. However, floating point numbers are secretly rational numbers - they are of the form $\pm 2^x M$, where $x$ is the integer exponent (positive, zero or negative) and $M$, the mantissa, is a rational number between $0$ and $1$ (or $0$ and $2$). In fact, it's always of the form $A/2^B$, where $B$ is the length of the mantissa and $A$ is some non-negative integer. If you really care for it, you could decode a real number into this representation, and thus obtain a truly equivalent rational number.