> so your comments have only proven that some strings are different than others

Er... yes? What I'm trying to do here, is to provide an intuition for how IEEE754 bit patterns should be mentally modeled. They're byte-arrays that encode a semantic type, like a JSON document is a byte-array that encodes a semantic type.

And, just like `"0"` and `"0.0"` are different encodings of the same JSON-semantic Number, IEEE754 -0 and IEEE754 +0 are different encodings of the same mathematically-semantic number; namely, zero.

But, unlike with JSON — and I hope this should be obvious — there's no such thing as "decoding" IEEE754 to something else. The bit-patterns in IEEE754 just are... themselves. There's no more-canonical encoding to move them into that will erase the difference between IEEE754 -0 and +0, or between regular numbers and denormals, or between all the different NaNs. They're all distinct IEEE754 objects.

Which means that, unlike with JSON Numbers, where you just decode them to canonicalize them and can forget about the fact that there are multiple equally-canonical JSON encodings for any given decoded value... you can't just forget about there being multiple distinct IEEE754 objects with the same mathematical meaning.

You don't operate on JSON documents in their encoded form. But you do operate on IEEE754 bit-patterns in their encoded form. You don't turn them into "numbers" in some abstract sense, do math, and then turn them back into bit-patterns. They're bit patterns the whole time.

Every time the FPU does an IEEE754-math operation, it has to know what to do with those distinct equally-canonical encodings, applying logic to them such that they produce mathematically-semantically-correct — but potentially still bit-pattern distinct! — results. Your FPU isn't attempting to simulate mathematics over the field of the rationals (in the way that an arbitrary-precision decimal library would); it's doing specific IEEE754 operations that move an IEEE754-bit-pattern-typed register from containing a value representing one position on the IEEE754 lattice, to a value representing a different position on the IEEE754 lattice. Where some of those position-values of the lattice may be interpreted, by humans, to represent a mathematical number. (Non-uniquely, because -0 and denormals!)

There is no other domain to move the math into. You're stuck in encoded non-canonical bit-pattern land. And as such, it's helpful to be able to talk about the particular encoded non-canonical bit-patterns themselves, compare them for bit-pattern equality, and so forth. Because what those bit-patterns are can determine things like how long an operation takes, or how well a vector of said bit-patterns compresses, or whether external systems will be able to decode the results, or whether you'll still have any data at if you try to hard-cast your FP32 to an FP16.

(And don't get me started on the fact that computer "integers" aren't mathematical integers! At least IEEE754 is a standard, such that `float` means "position on the IEEE754 FP32 lattice" on every architecture targeted by programming languages with FP types, and when an arch doesn't have that support, it's shimmed in with a runtime IEEE754-correct FPU support library. The C machine-integer types — not just `int`, all of them — aren't even well-defined as being a position on a two's-complement ring vs a position on a signed-magnitude ring! Any program that ANDs two machine-integers together and expects a particular bit-pattern to result, is completely non-portable to many historical and esoteric architectures! Which comes up more than you'd think, because mathematical proof-assistants are effectively "esoteric architectures.")