Floating point visually explained (2017)
Why is everyone complaining about people finding floats hard? Sure, scientific notation is easy to grasp, but you can't honestly tell me that it's trivial AFTER you consider rounding modes, subnormals, etc. Maybe if hardware had infinite precision like the real numbers nobody would be complaining ;)One thing I dislike in discussions about floats is this incessant focus on the binary representation. The representation is NOT the essence of the number. Sure it matters if you are a hardware guy, or need to work with serialized floats, or some NaN-boxing trickery, but you can get a perfectly good understanding of binary floats by playing around with the key parameters of a floating point number system:- precision = how many bits you have available- exponent range = lower/upper bounds for exponent- radix = 2 for binary floatsConsider listing out all possible floats given precision=3, exponents from -1 to 1, radix=2. See what happens when you have a real number that needs more than 3 bits of precision. What is the maximum rounding error using different rounding strategies? Then move on to subnormals and see how that adds a can of worms to underflow scenarios that you don't see in digital integer arithmetic. For anyone interested in a short book covering all this, I would recommend "Numerical Computing with IEEE Floating Point Arithmetic" by Overton [1].[1]: https://cs.nyu.edu/~overton/book/index.html