# [swift-evolution] [Draft][Proposal] Formalized Ordering

Fri Jul 22 21:09:35 CDT 2016

```> 1. NaN != NaN and +0 == -0 [what the traditional comparison operators are constrained to do]
> 2. NaN == NaN, +0 == -0, and the same number encoded different ways compare equal
> 3. NaN == NaN, +0 != -0, and the same number encoded different ways compare not equal

Though it seems super confusing that a language have THREE ways to compare values, that does almost seem necessary here. Do we actually need an operator that performs #3? I understand that that is equality under total ordering, but couldn't users just write `(a <=> b) == .same` if they want that?

> On Jul 22, 2016, at 7:04 PM, Xiaodi Wu <xiaodi.wu at gmail.com> wrote:
>
> On Fri, Jul 22, 2016 at 8:57 PM, Matthew Johnson <matthew at anandabits.com <mailto:matthew at anandabits.com>> wrote:
>
>> On Jul 22, 2016, at 8:54 PM, Xiaodi Wu via swift-evolution <swift-evolution at swift.org <mailto:swift-evolution at swift.org>> wrote:
>>
>> On Fri, Jul 22, 2016 at 8:52 PM, Jaden Geller via swift-evolution <swift-evolution at swift.org <mailto:swift-evolution at swift.org>> wrote:
>> "The totalOrder predicate will order these cases, and it also distinguishes between different representations of NaNs and between the same decimal floating point number encoded in different ways."
>> - [Wikipedia](https://en.wikipedia.org/wiki/IEEE_floating_point#Total-ordering_predicate <https://en.wikipedia.org/wiki/IEEE_floating_point#Total-ordering_predicate>)
>>
>> Sounds like `===` should not return `true` for zeros of different signs, then.
>>
>> Fair enough; the result of that will be, as Pyry noted above, that:
>>
>> ```
>> [-0.0, 1.0, .nan, 0.0].firstIndex(of: 0.0) //=> 3, not 0
>> ```
>
> Maybe we need floating point specific implementations of some algorithms to resolve this problem?
>
> It doesn’t seem like there is a way to provide the semantics required by generic algorithms and still provide the expected behavior for floating point values.
>
> Well, what I'm trying to say is that generic algorithms such as `index(of:)` require only an equivalence relation. For floating point types, there are three ways to slice it:
>
> 1. NaN != NaN and +0 == -0 [what the traditional comparison operators are constrained to do]
> 2. NaN == NaN, +0 == -0, and the same number encoded different ways compare equal
> 3. NaN == NaN, +0 != -0, and the same number encoded different ways compare not equal
>
> Both #2 and #3 can fall out of valid equivalence relations; if `===` behaved like #2 for FloatingPoint types, then generic algorithms work just fine. If we insist on using a total ordering defined by `<=>` all the time, then we've got problems.
>
>
>
>>
>>> On Jul 22, 2016, at 6:48 PM, Dave Abrahams via swift-evolution <swift-evolution at swift.org <mailto:swift-evolution at swift.org>> wrote:
>>>
>>>
>>> on Fri Jul 22 2016, Jaden Geller <swift-evolution at swift.org <mailto:swift-evolution at swift.org>> wrote:
>>>
>>>>> For floating point, I'd hope that `a === b` if `(a <=> b) == .same`
>>>>> *but not iff*. This is to satisfy IEEE 754: "Comparisons shall
>>>>> ignore the sign of zero (so +0 = −0)".
>>>>
>>>> I don't see why both `(+0) === (-0)` and `(+0) <=> (-0)` can't return
>>>> `true` and `.same`, respectively. This doesn't break the total
>>>> ordering of values. `===` doesn't do raw memory comparison. They're
>>>> "identical", so it ought to return `true`.
>>>
>>> It ought to do whatever IEEE-754 specifies that its total ordering test
>>> does.  That is, IEEE-754 gets to decide whether the difference between
>>> +0 and -0 is “essential” to IEEE-754 floating point types, or not.
>>>
>>>>
>>>>> On Jul 22, 2016, at 6:37 PM, Xiaodi Wu via swift-evolution
>>>>> <swift-evolution at swift.org <mailto:swift-evolution at swift.org>> wrote:
>>>>>
>>>>> On Fri, Jul 22, 2016 at 8:20 PM, Dave Abrahams via swift-evolution
>>>>> <swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>
>>>>> wrote:
>>>>>
>>>>> on Fri Jul 22 2016, Daniel Duan <daniel-AT-duan.org <http://daniel-at-duan.org/>> wrote:
>>>>>
>>>>>>> On Jul 22, 2016, at 3:00 PM, Dave Abrahams via swift-evolution
>>>>>>> <swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>
>>>>>>> wrote:
>>>>>>>
>>>>>>>
>>>>>>> on Fri Jul 22 2016, Daniel Duan
>>>>>>> <swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>>
>>>>>>> wrote:
>>>>>>>
>>>>>>
>>>>>>>>> On Jul 22, 2016, at 11:05 AM, Dave Abrahams via swift-evolution
>>>>>>>>> <swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>>
>>>>>>>>> wrote:
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> on Thu Jul 21 2016, Duan
>>>>>>>>
>>>>>>>>> <swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>>>
>>>>>>>>> wrote:
>>>>>>>>>
>>>>>>>>>> Great proposal. I want to second that areSame may mislead user to
>>>>>>>>>> think this is about identity.
>>>>>>>>>>
>>>>>>>>>> I like areEquivalent() but there may be better names.
>>>>>>>>>
>>>>>>>>> It really *is* about identity as I posted in a previous message.  But
>>>>>>>>> that doesn't change the fact that areEquivalent might be a better name.
>>>>>>>>> It's one of the things we considered; it just seemed long for no real
>>>>>>>>> benefit.
>>>>>>>>>
>>>>>>>>
>>>>>>>> If the addresses of the arguments aren’t being used, then we don’t consider
>>>>>>>> them part of their *identity*. I can follow this logic. My fear is most users
>>>>>>>> won’t make this leap on their own and get the same initial impression as I did.
>>>>>>>> It's entirely possible this fear is unfounded. Some educated bikesheding
>>>>>>>> wouldn't hurt here IMO :)
>>>>>>>
>>>>>>> Well, it's still a very real question whether we ought to have the
>>>>>>> additional API surface implied by areSame, or wether we should collapse
>>>>>>> it with ===.
>>>>>>>
>>>>>>
>>>>>> To spell this out (because I had to think about it for a second): === will be derived from
>>>>>> <=>,
>>>>>> but also becomes default implementation for ==, which remains open for
>>>>>> customization.
>>>>>
>>>>> I was imagining roughly this (untested):
>>>>>
>>>>>  /// Two references are identical if they refer to the same
>>>>>  /// instance.
>>>>>  ///
>>>>>  /// - Note: Classes with a more-refined notion of “identical”
>>>>>  ///   should conform to `Identifiable` and implement `===`.
>>>>>  func ===(lhs: AnyObject, rhs: AnyObject) -> Bool {
>>>>>    ObjectIdentifier(lhs) == ObjectIdentifier(rhs)
>>>>>  }
>>>>>
>>>>>  /// Supports testing that two values of `Self` are identical
>>>>>  ///
>>>>>  /// If `a` and `b` are of type `Self`, `a === b` means that
>>>>>  /// `a` and `b` are interchangeable in most code.  A conforming
>>>>>  /// type can document that specific observable characteristics
>>>>>  /// (such as the `capacity` of an `Array`) are inessential and
>>>>>  /// thus not to be considered as part of the interchangeability
>>>>>  /// guarantee.
>>>>>  ///
>>>>>  /// - Requires: `===` induces an equivalence relation over
>>>>>  ///   instances.
>>>>>  /// - Note: conforming types will gain an `==` operator that
>>>>>  ///   forwards to `===`.
>>>>>  /// - Note: Types that require domain-specific `==`
>>>>>  ///   implementations with different semantics (e.g. floating
>>>>>  ///   point) should define a more-specific overload of `==`,
>>>>>  ///   which will be used in contexts where the static type is
>>>>>  ///   known to the compiler.
>>>>>  /// - Note: Generic code should usually use `==` to compare
>>>>>  ///   conforming instances; that will always dispatch to `===`
>>>>>  ///   and will be unaffected by more specific overloads of
>>>>>  ///   `==`.
>>>>>  protocol Identifiable { // née Equatable name is negotiable
>>>>>    func ===(_: Self, _: aSelf) -> Bool
>>>>>  }
>>>>>
>>>>>  /// Default definition of `==` for Identifiable types.
>>>>>  func ==<T: Identifiable>(lhs: T, rhs: T) -> Bool {
>>>>>    return lhs === rhs
>>>>>  }
>>>>>
>>>>>  /// Conforming types have a default total ordering.
>>>>>  ///
>>>>>  /// If `a` and `b` are of type `Self`, `a <=> b` means that
>>>>>  /// `a` and `b` are interchangeable in most code.  A conforming
>>>>>  /// type can document that specific observable characteristics
>>>>>  /// (such as the `capacity` of an `Array`) are inessential and
>>>>>  /// thus not to be considered as part of the interchangeability
>>>>>  /// guarantee.
>>>>>  ///
>>>>>  /// - Requires: `<=>` induces a total ordering over
>>>>>  ///   instances.
>>>>>  /// - Requires: the semantics of `<=>` are  consistent with
>>>>>  ///   those of `===`.  That is, `(a <=> b) == .equivalent`
>>>>>  ///   iff `a === b`.
>>>>>
>>>>> For floating point, I'd hope that `a === b` if `(a <=> b) == .same` *but not iff*. This is to satisfy IEEE 754: "Comparisons shall ignore the sign of zero (so +0 = −0)".
>>>>>
>>>>>  /// - Note: conforming types will gain `<`, `<=`, `>`, and `>=`
>>>>>  ///   operators defined in terms of `<=>`.
>>>>>  /// - Note: Types that require domain-specific `<`, etc.
>>>>>  ///   implementations with different semantics (e.g. floating
>>>>>  ///   point) should define more-specific overloads of those
>>>>>  ///   operators, which will be used in contexts where the
>>>>>  ///   static type is known to the compiler.
>>>>>  /// - Note: Generic code can freely use `<=>` or the traditional
>>>>>  ///   comparison operators to compare conforming instances;
>>>>>  ///   the result will always be supplied by `<=>`
>>>>>  ///   and will be unaffected by more specific overloads of
>>>>>  ///   the other operators.
>>>>>  protocol Comparable : Identifiable {
>>>>>    func <=> (lhs: Self, rhs: Self) -> Ordering
>>>>>  }
>>>>>
>>>>>  /// Default implementations of `<`, `<=`, `>`, and `>=`.
>>>>>  extension Comparable {
>>>>>    static func <(lhs: Self, rhs: Self) -> Bool {
>>>>>      return (lhs <=> rhs) == .ascending
>>>>>    }
>>>>>    static func <=(lhs: Self, rhs: Self) -> Bool {
>>>>>      return (rhs <=> lhs) != .ascending
>>>>>    }
>>>>>    static func >(lhs: Self, rhs: Self) -> Bool {
>>>>>      return (lhs <=> rhs) == .descending
>>>>>    }
>>>>>    static func >=(lhs: Self, rhs: Self) -> Bool {
>>>>>      return (rhs <=> lhs) != .descending
>>>>>    }
>>>>>  }
>>>>>
>>>>>> I like this idea. If we keep === as a separate thing, now users have 3 “opportunities” to define
>>>>>> equality. The must be few, if any, use cases for this.
>>>>>>
>>>>>> Would love to see if anyone on the list can give us an example. Otherwise we should make
>>>>>> areSame === again™!
>>>>>>
>>>>>>>>
>>>>>>>>>> Daniel Duan
>>>>>>>>>> Sent from my iPhone
>>>>>>>>>>
>>>>>>>>>>> On Jul 21, 2016, at 6:32 PM, Robert Widmann via swift-evolution
>>>>>>>>>>> <swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>
>>>>>>>>>>> wrote:
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>> On Jul 21, 2016, at 6:19 PM, Xiaodi Wu
>>>>>>>>>>>> <xiaodi.wu at gmail.com <mailto:xiaodi.wu at gmail.com>
>>>>>>>>>>>> <mailto:xiaodi.wu at gmail.com <mailto:xiaodi.wu at gmail.com>>>
>>>>>>>>>>>> wrote:
>>>>>>>>>>>>
>>>>>>>>>>>> This is nice. Is `areSame()` being proposed because static `==` is
>>>>>>>>>>>> the status quo and you're trying to make the point that `==` in the
>>>>>>>>>>>> future need not guarantee the same semantics?
>>>>>>>>>>>
>>>>>>>>>>> Yep!  Equivalence and equality are strictly very different things.
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Nit: I think the more common term in stdlib would be
>>>>>>>>>>>> `areEquivalent()`. Do you think `same` in that context (independent
>>>>>>>>>>>> of the word "ordering") might erroneously suggest identity?
>>>>>>>>>>>
>>>>>>>>>>> There is room for improvement here.  Keep ‘em coming.
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>> On Thu, Jul 21, 2016 at 8:11 PM, Robert Widmann via
>>>>>>>>>>>>> swift-evolution
>>>>>>>>>>>>> <swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>
>>>>>>>>>>>>> wrote:
>>>>>>>>>>>>> Hello Swift Community,
>>>>>>>>>>>>>
>>>>>>>>>>>>> Harlan Haskins, Jaden Geller, and I have been working on a
>>>>>>>>>>>>> proposal to clean up the semantics of ordering relations in the
>>>>>>>>>>>>> standard library.  We have a draft that you can get as a gist.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Cheers,
>>>>>>>>>>>>>
>>>>>>>>>>>>> ~Robert Widmann
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> _______________________________________________
>>>>>>>>>>>>> swift-evolution mailing list
>>>>>>>>>>>>> swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>>>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> _______________________________________________
>>>>>>>>>>> swift-evolution mailing list
>>>>>>>>>>> swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>>>>>> _______________________________________________
>>>>>>>>>> swift-evolution mailing list
>>>>>>>>>> swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> --
>>>>>>>>> Dave
>>>>>>>>>
>>>>>>>>> _______________________________________________
>>>>>>>>> swift-evolution mailing list
>>>>>>>>> swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>>
>>>>>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>>
>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>>>
>>>>>>>> _______________________________________________
>>>>>>>> swift-evolution mailing list
>>>>>>>> swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>
>>>>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>>
>>>>>>>>
>>>>>>>
>>>>>>> --
>>>>>>> Dave
>>>>>>>
>>>>>>> _______________________________________________
>>>>>>> swift-evolution mailing list
>>>>>>> swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>
>>>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>>
>>>>>>> https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>
>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>
>>>>>>> <https://lists.swift.org/mailman/listinfo/swift-evolution <https://lists.swift.org/mailman/listinfo/swift-evolution>>>
>>>>>
>>>>> --
>>>>> Dave
>>>>> _______________________________________________
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>>>>> <mailto:swift-evolution at swift.org <mailto:swift-evolution at swift.org>>
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>>>>>
>>>>> _______________________________________________
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>>>> _______________________________________________
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>>>>
>>>
>>> --
>>> Dave
>>>
>>> _______________________________________________
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>>
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