Sub-Tasks:

• SR-13169 Sema: lift derivative registration restriction for protocol requirements
• SR-13168 SILGen: add SIL default witness table entries for default derivatives of protocol requirements
• SR-13167 Parse: add parsing/syntax support for @differentiable(default, ...)

Issue Description:

Overview

Default derivative implementations enables protocol requirements (like requirements from AdditiveArithmetic, FloatingPoint, ElementaryFunctions, etc) to be differentiable by default.

See "default derivatives and transposes" from the Differentiable Programming Manifesto for more info.

Example:

// In the standard library:
// public protocol AdditiveArithmetic: Equatable {
//   static func +(lhs: Self, rhs: Self) -> Self
//   ...
// }

extension AdditiveArithmetic where Self: Differentiable {
  @derivative(of: +)
  static func vjpAdd(_ lhs: Self, _ rhs: Self) ->
    (value: Self, pullback: (TangentVector) -> (TangentVector, TangentVector)) {
    return (lhs + rhs, { v in (v, v) })
  }
}

This is important for idiomatic protocol-oriented programming to avoid tons of code duplication. Without this feature, all conforming types of the protocol must:

• Define a @differentiable concrete implementation of the original protocol requirement, if none already exists (because a default implementation was used.)

• Register a derivative function for the concrete implementation.

Details

There are two cases to consider:

1. Non-@differentiable protocol requirement.

protocol P {
  func foo(_ x: Float) -> Float
}
extension P {
  @derivative(of: foo)
  func vjpFoo(_ x: Float) -> (value: Float, pullback: (Float) -> Float) {
    return (x, { $0 })
  }
}

2. @differentiable protocol requirement.

protocol P {
  @differentiable
  func foo(_ x: Float) -> Float
}
extension P {
  @derivative(of: foo)
  func vjpFoo(_ x: Float) -> (value: Float, pullback: (Float) -> Float) {
    return (x, { $0 })
  }
}

Supporting this may require lifting the current restriction that "all implementations of @differentiable protocol requirements must themselves be marked as @differentiable":

protocol AdditiveArithmetic: Equatable {
  static func +(lhs: Self, rhs: Self) -> Self
}

extension AdditiveArithmetic where Self: Differentiable {
  @derivative(of: +)
  static func vjpAdd(_ lhs: Self, _ rhs: Self) ->
    (value: Self, pullback: (TangentVector) -> (TangentVector, TangentVector)) {
    return (lhs + rhs, { v in (v, v) })
  }
}

struct Foo: AdditiveArithmetic {
  static func +(lhs: Self, rhs: Self) -> Self {
    lhs
  }
}

deriv.swift:13:8: error: type 'Foo' does not conform to protocol 'AdditiveArithmetic'
struct Foo: AdditiveArithmetic {
       ^
deriv.swift:14:15: note: candidate is missing attribute '@differentiable(wrt: (lhs, rhs) where Self : Differentiable)'
  static func +(lhs: Self, rhs: Self) -> Self {
              ^

SIL default witness table support may be needed.