[AutoDiff upstream] Add the _Differentiable protocol.

The `_Differentiable` protocol generalizes all types that work with
differentiation. It is a core piece of the differentiable programming
project. Other parts depending on the `_Differentiable` protocol (e.g.
attribute type-checking) will be upstreamed later.

Author: dan-zheng

stdlib/public/core/CMakeLists.txt

@@ -204,6 +204,7 @@ set(SWIFTLIB_SOURCES
   Availability.swift
   CollectionDifference.swift
   CollectionOfOne.swift
+  Differentiable.swift
   Diffing.swift
   Mirror.swift
   PlaygroundDisplay.swift

stdlib/public/core/Differentiable.swift

@@ -0,0 +1,44 @@
+//===--- Differentiable.swift ---------------------------------*- swift -*-===//
+//
+// This source file is part of the Swift.org open source project
+//
+// Copyright (c) 2014 - 2019 Apple Inc. and the Swift project authors
+// Licensed under Apache License v2.0 with Runtime Library Exception
+//
+// See https://swift.org/LICENSE.txt for license information
+// See https://swift.org/CONTRIBUTORS.txt for the list of Swift project authors
+//
+//===----------------------------------------------------------------------===//
+//
+// This file defines the _Differentiable protocol, used by the experimental
+// differentiable programming project. This API is not stable and subject to
+// change.
+//
+// Please see forum discussion for more information about the differentiable
+// programming project:
+// https://forums.swift.org/t/differentiable-programming-mega-proposal/28547
+//
+//===----------------------------------------------------------------------===//
+
+/// A type that mathematically represents a differentiable manifold whose
+/// tangent spaces are finite-dimensional.
+public protocol _Differentiable {
+  /// A type representing a differentiable value's derivatives.
+  ///
+  /// Mathematically, this is equivalent to the tangent bundle of the
+  /// differentiable manifold represented by the differentiable type.
+  associatedtype TangentVector: _Differentiable & AdditiveArithmetic
+    where TangentVector.TangentVector == TangentVector
+
+  /// Moves `self` along the given direction. In Riemannian geometry, this is
+  /// equivalent to exponential map, which moves `self` on the geodesic surface
+  /// along the given tangent vector.
+  mutating func move(along direction: TangentVector)
+}
+
+public extension _Differentiable where TangentVector == Self {
+  @_alwaysEmitIntoClient
+  mutating func move(along direction: TangentVector) {
+    self += direction
+  }
+}

stdlib/public/core/GroupInfo.json

@@ -227,5 +227,8 @@
   ],
   "Result": [
     "Result.swift"
+  ],
+  "DifferentiableProgramming": [
+    "Differentiable.swift"
   ]
 }