alt Hacker NewsThat does make sense, because a half of all available fp numbers are less than 1 in their magnitude. In particular there should be a plenty of numbers x such that |x| << 1 so x + 1 ~= 1; in fact, the proportion should be just shy of 50%.
Replies
That is indeed one of the problems with IEEE floats. There are only 10^80 atoms in the universe, and a Planck length is 1^-60th of the radius of the universe. But 64-bit floats have an absurd range of over 10^±300! Worse than that, notice that there are as many bit patterns in the never-used range between 10^300 and 10^301 as there are in the super-important range between 1 and 10! Super wasteful. Not to mention the quadrillions of values reserved to represent "NaN"...
This is one of the problems that alternative formats such as the Posit aim to solve. It's quite interesting: I've got an implementation in rust here if you want to play with it https://github.com/andrepd/posit-rust
Not really. 1+x/2, however, would be a good approximation to sqrt(1+x) for small (in absolute value) x.
But I guess using the density distribution of floating points is rarely useful in a problem. Your actual distribution will almost surely be way different. Imo, the tool presented here should provide a way to manually provide a custom density function (with some common presets like uniform and normal distributions).