&gt; Your claim is fallacious, because a floating-point value denotes a range of the real number line, whereas you&#x27;re insisting that, no, it stands for its literal value: the absolutely precise rational number that lies at the center of that range<p>See Misconception #2 of <a href="http:&#x2F;&#x2F;lipforge.ens-lyon.fr&#x2F;www&#x2F;crlibm&#x2F;documents&#x2F;cern.pdf" rel="nofollow">http:&#x2F;&#x2F;lipforge.ens-lyon.fr&#x2F;www&#x2F;crlibm&#x2F;documents&#x2F;cern.pdf</a>.<p>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&#x27;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.<p>&gt; All numbers in that range alias to the same double, yet differ wildly in their decimal digits beyond the fifteenth. That &quot;long tail&quot; of digits is a meaningless residue arising from the arbitrary difference between the chosen center-of-range point and the actual number it approximates.<p>See Misconceptions #1 and #2 on Kahan&#x27;s list (<a href="https:&#x2F;&#x2F;www.cs.berkeley.edu&#x2F;~wkahan&#x2F;JAVAhurt.pdf" rel="nofollow">https:&#x2F;&#x2F;www.cs.berkeley.edu&#x2F;~wkahan&#x2F;JAVAhurt.pdf</a>).<p>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&#x27;s true that many real numbers round to the same floating point number _f_, that&#x27;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&#x27;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_.