[Dylan16807]

> I wasn't trying to draw attention to comparisons for equality. Perhaps I should've used an arrow => instead

Then it's even less of issue. Yes if you convert to a float you get rounding, what did you expect when you introduced a float?

It's somewhat unintuitive but that's the nature of floating point.

> The real point is that you can get some non-intuitive answers from letting that numeric tower make conversion decisions for you. It's just a rule, and it's not an amazing rule.

But again, you can have the same kind of issue without bignums. It's not a tower problem it's a float problem.

[xscott]

I think I can summarize your point of view as, "any use of floats should be treated as undefined behavior, so you get what you deserve."

And that says nothing about whether implicit conversions are a good idea or not.

[Dylan16807]

The specific type of conversion is one I don't see as a big issue. The programmer deliberately decided to use an imprecise data type for the calculation.

But more importantly, I'm saying that the problematic rounding can occur even if your tower does not have both bigint and float. It can happen even if every layer can completely represent every value of the layer above it. Do you have any complaints that are unique to a tower that has both bigint and float, and don't apply to towers that only have float?

To elaborate on that, an implicit cast directly from a single bigint to a single float won't happen with the rules in the wikipedia article. You'd have to do something like bigint+float, which can have horrible rounding errors, but those horrible rounding errors are also present in float+float.

And you can even have these problems without a tower. So I don't see how the bigint and float scenario is an argument against towers.

[xscott]

From a distance, I kind of like Scheme, so I went and re-read the R5RS section on the topic. To me, the numeric tower (generalized) says:

   N < Z < Q < R < C < H

   ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ ⊂ ℍ

That's a nice statement about idealized sets of numeric values. So `integer?` implies `rational?` implies `real?` implies `complex?` implies `number?` in Scheme predicates and type conversions.

But no programming language can have "Reals" (they aren't computable), so floats are a common/useful approximation. And in actuality `bigint?` doesn't imply `floating?`, and `floating?` doesn't imply `bigint?`. Neither is a strict subset of the other, and because of this you can easily find examples where implicit conversion does something "questionable". You've made it about rounding errors, but I'm trying to criticize something about pretending they are subtypes/subsets. Claiming it's a tower and hand waving about exact/inexact doesn't make it a tower, and so I think implicit conversion for these is a poor choice.

You can have little subset relations for implicit conversions:

    float32?   implies  float64?
    float64?   implies  complex64?
    float32?   implies  complex32?
    complex32? implies  complex64?

    fixint?    implies  bigint?
    fixint?    implies  rational?
    bigint?    implies  rational?

Since this is supposedly in a discussion about JavaScript, maybe even:

    fixint?    implies  float64?

All of those relate true subsets for the collection of values they can represent, but it's not much of a tower. It's more a collection of DAGs.

[Dylan16807]

> But it's not much of a tower, and it's more of a collection DAGs.

> I'm trying to criticize something about pretending they are subtypes/subsets. Claiming it's a tower and hand waving about exact/inexact doesn't make it a tower

I thought we established right away that it's not a single tower. The description in the wikipedia page is two towers with links between them. (Or at least it's two if you don't waste effort on things like having both float64 and complex32.)

But I don't see any hand waving. The relationships and conversions are very clear. That's why I interpreted your complaint as being more about the specific operation. So with your correction, I need you to explain where you see hand-waving.

If you just don't like the name "Tower" for an implementation that has both bignums and floats then okay I agree I guess?

[xscott]

> I thought we established right away that it's not a single tower.

Where did we say that? The first picture on the Wikipedia page shows the tower as a linear stack of items from set theory. The Scheme predicates are named similarly. This is the appealing myth.

> The description in the wikipedia page is two towers with links between them.

Not on the page I'm seeing. Are you reading the English page? At the bottom, I see a tree of abstract types (sets).

This shows that you can traverse (Integer to Rational to Real) and (Float to Real) to find the common abstract type Real. But there isn't actually a Real type you can do operations with. You've got concrete BigInt and Float64, and even if Real is implemented as a C-style tagged-union of the two types, you still need to pick one or the other for doing operations like addition. Then the Scheme standard says stuff like, "try to be exact when you can, but inexact is ok sometimes". So all the set theory justification is out the window, and it's really just an ad hoc rule.

It's just not as elegant as it seems, and it gives an unsound justification to making implicit conversions.

> If you just don't like the name "Tower" then okay I agree I guess?

Please don't do that. I've tried to clarify details in response to your questions, but if you're just going to dismiss it with some snarky crap like that then you can go fuck yourself.

Reply if you want, but I'm guessing we're done here.

[Dylan16807]

> Where did we say that? The first picture on the Wikipedia page shows the tower as a linear stack of items from set theory. The Scheme predicates are named similarly. This is the appealing myth.

In the section where the wikipedia page talks about exact and inexact, the specific thing you were calling out, it says "Another common variation is to support both exact and inexact versions of the tower or parts of it; R7RS Scheme recommends but does not strictly require this of implementations. In this case, similar semantics are used to determine the permissibility of implicit coercion: inexactness is a contagious property of numbers,[6] and any numerical operation involving both exact and inexact values must yield inexact return values of at least the same precision as the most precise inexact number appearing in the expression, unless the precision is practically infinite (e.g. containing a detectable repetend), or unless it can be proven that the precision of the result of the operation is independent of the inexactness of any of its operands (for example, a series of multiplications where at least one multiplicand is 0).".

I want to especially highlight the phrase "exact and inexact versions of the tower or parts of it" Which I then reacted to by saying "The article describes a reasonable-sounding way to extend the tower with a second dimension of precision." Once you have two dimensions it's no longer a single tower. I thought that was the common ground that we were talking on, that if you use that method it's not a true tower anymore.

> Please don't do that. I've tried to clarify details in response to your questions, but if you're just going to dismiss it with some snarky crap like that then you can go fuck yourself.

That wasn't snark. I am really trying to understand your argument, because it looks like we've been talking about different things the entire time.

I had thought we established from the very start that the description on the wiki page wasn't actually a single tower. If you are still trying to convince me it's a more complicated graph, then I agree with you, and I don't understand how we got so far without that being clear. Sorry for sounding reductionist about it.

So please, honest question for clarification, do you object to the graph of number types described by that paragraph, do you object to using the word "tower" to talk about it, or do you object to both? Please don't get mad at me for asking, or think I'm trying to dismiss you.

And if someone builds a pure tower that goes int32, double, complex, quaternion, do you think that's inherently self-defeating because it can't live up to the promises of a tower? It doesn't have the issue of floats versus bignums; it's strict subsets all the way down.