Fixed point arithmetic with a dynamic scale is presented along the way motivating floating point in probably every computer architecture class. It’s a floating point.

This guy needs to open a book. I recommend Nick Higham’s _Accuracy and Stability of Numerical Algorithms_.

x = m * 2^p

where m is an integer(mantissa) and p is an integer (point position)

Well this is clearly wrong, take x = 1/3 for example

1/3 = m * 2^p

m = 1 / (3 * 2^p), where m is an integer, doesn't hold true for any integer p.

If the author had read first 2 pages of

https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.h...

they could have avoided the embarrassment.

In DPA the mantissa just overflows (silently in the C implementation) and then what?

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.

> Every error compounds.

https://en.wikipedia.org/wiki/Kahan_summation_algorithm

Again and again people try to "fix" IEEE floating points without realizing that they are trying to do something akin to creating free energy. Whenever you start out with a project like this you have to start by asking yourself what desirable property you are willing to let go of. Not presenting that flaw makes the whole thing look either dishonest or uninformed.

>Any real number x can be represented as

This statement is just false. I do not know why you would start out by making such a basic error. The real numbers are uncountable you can not represent all real numbers by a pair of integers. This is basic real analysis

>The mathematics is elementary. The impact is revolutionary.

???

>Special thanks to everyone who said "that's just how it works" - you motivated me to prove otherwise.

Maybe the people know better than you? It is after all a mathematical theorem.

I think it's great to experiment with improving fundamental algorithms, but not to make misleading claims about your results.