// 1. `@dynamicReplacement(for:)`: dynamic replacement functions.
dynamic func foo(_ x: Float) -> Float { x }

@_dynamicReplacement(for: foo)
func _(x: Float) -> Float { x + 1 }

// 2. `@derivative` and `@transpose`: derivative and transpose registration.
// `func _` helps avoid exposing derivative and transpose functions as public APIs.
public func bar(_ x: Float) -> Float { x }

// Publicly registered derivative. Derivative function itself is not exposed, however.
@derivative(of: bar)
public func _(_ x: Float) -> (value: Float, differential: (Float) -> Float) {
  (x, { dx in dx })
}

// Publicly registered transpose. Transpose function itself is not exposed, however.
@transpose(of: bar)
public func _(_ v: Float) -> Float { v }