What is this feature of floating point?

Question

Real Close to the Machine: Floating Point in D

https://dlang.org/articles/d-floating-point.html

says

Useful relations for a floating point type F, where x and y are of type F

...

• x>0 if and only if 1/(1/x) > 0; x<0 if and only if 1/(1/x) < 0.

what is the meaning of this sentence?

Answer 1

In the text you're quoting, we're looking at how the representation is symmetric around 1, and that the rounding doesn't break this. That is, for any number 0 < x < 1, there's a corresponding number 1 < y < ∞, such that y = 1/x and 1/y = x. That's the first half - the second is simply the same for negative numbers: 0 > x > -1 and -1 > y > -∞.

It may not be immediately obvious how this can be a problem, but consider x = 3. y must then be 1/3 = 0.333.... With a limited precision of 3 decimal digits, 1/y would then be 3.003003003.... IEEE 754 defines how this should work, and says that the rounding should ensure 1/(1/x) should be equal to x, and thus that the result should be 3, even if there are rounding errors in both 1/x and 1/y - they should cancel each other out.

Older floating-point systems weren't as well-behaved as IEEE 754. I'm not sure if any of them weren't symmetric around 1, but that's certainly within the realm of possibility.