If you want the serious version of the idea instead of the LLM diarrhea, just go Jonathan Shewchuk’s robust predicates work: https://people.eecs.berkeley.edu/~jrs/papers/robustr.pdf from 1997.
For a library that implements just the two component version of this, commonly known as a double-double, for a mantissa of 107 bits and an exponent of 11, see: https://github.com/ncruces/dbldbl
If you want the serious version of the idea instead of the LLM diarrhea, just go Jonathan Shewchuk’s robust predicates work: https://people.eecs.berkeley.edu/~jrs/papers/robustr.pdf from 1997.