[swift-evolution] protocol-oriented integers (take 2)

[Brent Royal-Gordon]

> On Jan 30, 2017, at 11:31 AM, Max Moiseev <moiseev at apple.com> wrote:
> 
> doubleWidthDivide should not return a DoubleWidth<T> for two reasons:
> 1. The components of it’s return type are not high and low, but are quotient and remainder instead.
> 2. In DoubleWidth<T> high is T and low is T.Magnitude, which is not the case for quotient and remainder.

You're right about the return value; for `doubleWidthDivide(_:_:)`, I was thinking about changing the dividend. Specifically, I'm thinking we should change these to:

	static func doubleWidthMultiply(_ lhs: Self, _ rhs: Self) -> DoubleWidth<Self>
	static func doubleWidthDivide(_ lhs: DoubleWidth<Self>, _ rhs: Self) -> (quotient: Self, remainder: Self)

I'm also thinking a little bit about spelling of these operations. I'd *love* to be able to call them `*` and `/` and let the type system sort things out, but that would cause problems, especially for multiply (since the return value is the only thing different from a normal `*`). We could invent a new operator, but that would be a bit much. Could these be simply `multiply` and `divide`, and we'll permit the `DoubleWidth` parameter/return type to explain itself?

I'm also thinking the second parameter should be labeled `by`, since that's the way people talk about these operations. Applying both of these suggestions, we'd get:

	static func multiply(_ lhs: Self, by rhs: Self) -> DoubleWidth<Self>
	static func divide(_ lhs: DoubleWidth<Self>, by rhs: Self) -> (quotient: Self, remainder: Self)
	
	let x = Int.multiply(a, by: b)
	let (aʹ, r) = Int.divide(x, by: b)
	assert(a == aʹ)
	assert(r == 0)

Should the standard library provide extensions automatic definitions of multiplication and division in terms of their double-width equivalents?

	extension FixedWidthInteger {
		func multipliedWithOverflow(by other: Self) -> (partialValue: Self, overflow: ArithmeticOverflow) {
			let doubledResult = Self.multiply(self, by: other)
			let overflowed = doubledResult.high != (doubledResult < 0 ? -1 : 0)
			return (Self(bitPattern: doubledResult.lowerValue), overflowed ? .overflowed : .none)
		}
		
		func quotientAndRemainder(dividingBy other: Self) -> (quotient: Self, remainder: Self) {
			precondition(other != 0, "Divide by zero")
			return Self.divide(DoubleWidth(self), by: other)
		}
		
		func dividedWithOverflow(by other: Self) -> (partialValue: Self, overflow: ArithmeticOverflow) {
			guard other != 0 else { return (self, .overflowed) }
			
			let result = Self.divide(self, by: other)
			return (result.quotient, .none)
		}
		
		static func * (lhs: Self, rhs: Self) -> Self {
			let result = lhs.dividedWithOverflow(by: rhs)
			precondition(result.overflow == .none, "Multiplication overflowed")
			return result.partialValue
		}
		
		static func / (lhs: Self, rhs: Self) -> Self {
			let result = lhs.quotientAndRemainder(dividingBy: rhs)
			return result.quotient
		}
		
		static func % (lhs: Self, rhs: Self) -> Self {
			let result = lhs.quotientAndRemainder(dividingBy: rhs)
			return result.remainder
		}
}

Hmm...having actually written this out, I now have a couple of concerns:

1. There's a lot of jumping back and forth between instance methods and static methods. Can we standardize on just static methods? Or make sure that the user-facing interfaces are all either operators or instance methods?

2. There is no quotient-and-remainder-with-overflow, either regular or double-width. Can we do that?

3. "Overflow" is not really a good description of what's happening in division; the value is undefined, not overflowing. Is there a better way to express this?

4. For that matter, even non-fixed-width division can "overflow"; should that concept be hoisted higher up the protocol hierarchy?

5. For *that* matter, should we simply make these operations throw instead of returning a flag?

	enum ArithmeticError<NumberType: Arithmetic>: Error {
		// Are generic errors permitted?
		case overflow(partialValue: NumberType)
		case undefined
	}
	
	// Should these throwing definitions be higher up so that, when working with `Arithmetic` 
	// or similar types, you have an opportunity to handle errors instead of always trapping?
	protocol FixedWidthInteger: BinaryInteger {
		...
		func adding(_ other: Self) throws -> Self
		func subtracting(_ other: Self) throws -> Self
		func multiplied(by other: Self) throws -> Self
		func divided(by other: Self) throws -> Self
		...
	}

I'm *really* tempted to suggest adding throwing variants of the actual operators (strawman example: `^+`, `^-`, `^*`, `^/`, `^%`), but they may be too specialized to really justify that.

> Having said that, there is a solution for doubleWidthMultiply, that I think is worth trying:
> 
> enum DoubleWidth<T> {
>   case .parts(high: T, low: T.Magnitude)
> 
>   var high: T { switch self { case .parts(let high, _): return high } }
>   var low: T.Magnitude { switch self { case .parts(_, let low): return low } }
> }
> 
> This way it will be possible to both do pattern-matching on the result of doubleWidthMultiply, and use it as a whole, accessing r.high and r.low when needed.

This sounds like a good idea to me. (Unless we want to create a protocol for destructuring, but I assume that's out of scope.)

-- 
Brent Royal-Gordon
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