Are there any ready-made ways to do calculations on floating point numbers with an error at each stage and a total error?

Question

Mathematics

Are there any ready-made solutions (libraries, approaches, techniques, etc.) for performing rigorous calculations? So that each number is stored as a pair of two numbers, the number itself and a known error. And further, on all arithmetic operations, this error would be taken into account and also calculated for the result? So that you could do sqrt(2)*sqrt(2) and get the result [2.0000000000000004, "so many decimal places are correct, let's say +- 0.000000000001"]