> Your claim is fallacious, because a floating-point value denotes a range of the real number line, whereas you're insisting that, no, it stands for its literal value: the absolutely precise rational number that lies at the center of that range

See Misconception #2 of http://lipforge.ens-lyon.fr/www/crlibm/documents/cern.pdf.

Floating point numbers are _not_ intervals. If you read carefully any formal definition of floating point numbers (The IEEE standards, TAoCP Chapter 4, or Higham's _Accuracy and Stability of Numerical Algorithms_, to name just three possible references), you will see that floating point numbers by definition form an exact rational subset of the extended real line.

> All numbers in that range alias to the same double, yet differ wildly in their decimal digits beyond the fifteenth. That "long tail" of digits is a meaningless residue arising from the arbitrary difference between the chosen center-of-range point and the actual number it approximates.

See Misconceptions #1 and #2 on Kahan's list (https://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf).

You are not being clear about the set of real numbers a user may input and their ultimate representation as a floating point number. While it's true that many real numbers round to the same floating point number _f_, that's not the same as then saying that _f_ carries that interval around with it. The latter is false, since the intervals do _not_ propagate in floating point arithmetic. It's also false that the floating point number _f_ is the midpoint of the set of numbers that round to _f_; the precise set depends on the rounding mode and the granularity of the set of floating point numbers around _f_.