Say I have a number, that I know to a certain precision:
$$c = 1.46342 ...$$
where $...$ denote unknown digits. I want to find a simple formula connecting this number to a known mathematical constant. For example, it turns out that
$$c \approx \sqrt{\pi - 1}$$
In this example it so happens that the error is very small, but in reality the error could be larger. An algorithm should consider this error and try to minimize it, but without returning and over complicated formula. Obviously if the formula is complicated enough any number can be approximated to arbitrary precision, so a good algorithm should consider the trade-off between formula complexity and accuracy.
A related example that comes to mind is Mathematica's FindFormula. However this only attempts to find an approximate formula to fit a list of data points, which is close but not exactly what I want.
Is there such an algorithm / software / web service / etc. ?
Yes, Robert Munafo's RIES program may be the most well-known program that does this.
http://mrob.com/pub/ries/