[AutoDiff] Move stdlib sources to stdlib/public/core/Differentiation.

include/swift/AST/KnownIdentifiers.def

@@ -53,6 +53,7 @@ IDENTIFIER(decode)
 IDENTIFIER(decodeIfPresent)
 IDENTIFIER(Decoder)
 IDENTIFIER(decoder)
+IDENTIFIER_(Differentiation)
 IDENTIFIER(dynamicallyCall)
 IDENTIFIER(dynamicMember)
 IDENTIFIER(Element)

include/swift/AST/KnownProtocols.def

@@ -93,7 +93,7 @@ PROTOCOL_(DestructorSafeContainer)
 
 PROTOCOL(StringInterpolationProtocol)
 
-PROTOCOL_(Differentiable)
+PROTOCOL(Differentiable)
 
 EXPRESSIBLE_BY_LITERAL_PROTOCOL(ExpressibleByArrayLiteral, "Array", false)
 EXPRESSIBLE_BY_LITERAL_PROTOCOL(ExpressibleByBooleanLiteral, "BooleanLiteralType", true)

stdlib/public/CMakeLists.txt

@@ -63,6 +63,12 @@ if(SWIFT_BUILD_STDLIB)
   add_subdirectory(SwiftOnoneSupport)
 endif()
 
+# Build differentiable programming support library only if enabled.
+if(SWIFT_ENABLE_EXPERIMENTAL_DIFFERENTIABLE_PROGRAMMING AND SWIFT_BUILD_STDLIB)
+  message(STATUS "Building Swift differentiable programming support library.")
+  add_subdirectory(Differentiation)
+endif()
+
 # SWIFT_ENABLE_TENSORFLOW
 if(SWIFT_BUILD_STDLIB)
   add_subdirectory(Python)

stdlib/public/Differentiation/CMakeLists.txt

@@ -0,0 +1,28 @@
+#===--- CMakeLists.txt - Differentiable programming support library ------===#
+#
+# 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
+#
+#===----------------------------------------------------------------------===#
+
+# SWIFT_ENABLE_TENSORFLOW
+# NOTE: A non-empty `_Differentiation` module is currently created only on
+# master branch, not on tensorflow branch.
+#
+# Instead, on tensorflow branch, the differentiation-related Swift source files
+# in this directory are directly built as part of swiftCore: see
+# stdlib/public/core/CMakeLists.txt. The `_Differentiation` module is created
+# empty to avoid `#if canImport(_Differentiation)` guards in tests.
+# SWIFT_ENABLE_TENSORFLOW END
+
+add_swift_target_library(swift_Differentiation ${SWIFT_STDLIB_LIBRARY_BUILD_TYPES} IS_STDLIB
+  Empty.swift
+
+  SWIFT_COMPILE_FLAGS ${SWIFT_STANDARD_LIBRARY_SWIFT_FLAGS}
+  LINK_FLAGS "${SWIFT_RUNTIME_SWIFT_LINK_FLAGS}"
+  INSTALL_IN_COMPONENT stdlib)

stdlib/public/core/Differentiable.swift → stdlib/public/Differentiation/Differentiable.swift

@@ -11,20 +11,23 @@
 //===----------------------------------------------------------------------===//
 //
 // This file defines the Differentiable protocol, used by the experimental
-// differentiable programming project. Please see forum discussion for more
-// information:
+// 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 {
+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
+  associatedtype TangentVector: Differentiable & AdditiveArithmetic
     where TangentVector.TangentVector == TangentVector
 
   /// Moves `self` along the given direction. In Riemannian geometry, this is
@@ -44,15 +47,18 @@ public protocol _Differentiable {
       this property
       """)
   var zeroTangentVector: TangentVector { get }
+  // SWIFT_ENABLE_TENSORFLOW END
 }
 
-public extension _Differentiable where TangentVector == Self {
+public extension Differentiable where TangentVector == Self {
+  @_alwaysEmitIntoClient
   mutating func move(along direction: TangentVector) {
     self += direction
   }
 }
 
-public extension _Differentiable {
+// SWIFT_ENABLE_TENSORFLOW
+public extension Differentiable {
   // This is a temporary solution that allows us to add `zeroTangentVector`
   // without implementing derived conformances. This property is marked
   // unavailable because it will produce incorrect results when tangent vectors
@@ -61,6 +67,4 @@ public extension _Differentiable {
   // implementation.
   var zeroTangentVector: TangentVector { .zero }
 }
-
-// SWIFT_ENABLE_TENSORFLOW
-public typealias Differentiable = _Differentiable
+// SWIFT_ENABLE_TENSORFLOW END

stdlib/public/Differentiation/Empty.swift

@@ -0,0 +1,3 @@
+// SWIFT_ENABLE_TENSORFLOW
+// Empty Swift file, only for tensorflow branch.
+// See explanation in stdlib/public/Differentiation/CMakeLists.txt.

stdlib/public/core/CMakeLists.txt

@@ -197,15 +197,16 @@ set(SWIFTLIB_ESSENTIAL_GYB_SOURCES
   UnsafeRawBufferPointer.swift.gyb
   )
 
+# SWIFT_ENABLE_TENSORFLOW
 # Compile differentiable programming sources only if enabled.
 set(SWIFTLIB_DIFFERENTIABLE_PROGRAMMING_SOURCES)
 if(SWIFT_ENABLE_EXPERIMENTAL_DIFFERENTIABLE_PROGRAMMING)
   list(APPEND SWIFTLIB_DIFFERENTIABLE_PROGRAMMING_SOURCES
-    Differentiable.swift
-    # SWIFT_ENABLE_TENSORFLOW
-    DifferentiationSupport.swift)
+    ../Differentiation/Differentiable.swift
+    ../Differentiation/DifferentiationSupport.swift)
   message(STATUS "Differentiable programming standard library additions enabled.")
 endif()
+# SWIFT_ENABLE_TENSORFLOW END
 
 # The complete list of sources in the core standard library.  Includes
 # all the essential sources listed above.
@@ -225,7 +226,9 @@ set(SWIFTLIB_SOURCES
   VarArgs.swift
   Zip.swift
   "${SWIFT_SOURCE_DIR}/stdlib/linker-support/magic-symbols-for-install-name.c"
+  # SWIFT_ENABLE_TENSORFLOW
   ${SWIFTLIB_DIFFERENTIABLE_PROGRAMMING_SOURCES}
+  # SWIFT_ENABLE_TENSORFLOW END
   )
 
 set(SWIFTLIB_GYB_SOURCES

stdlib/public/core/SIMDVectorTypes.swift.gyb

@@ -192,12 +192,17 @@ extension SIMD${n} where Scalar: BinaryFloatingPoint {
 // SWIFT_ENABLE_TENSORFLOW
 extension SIMD${n} : AdditiveArithmetic where Scalar : FloatingPoint {}
 
-extension SIMD${n} : Differentiable & EuclideanDifferentiable
-  where Scalar : EuclideanDifferentiable & BinaryFloatingPoint,
+extension SIMD${n} : Differentiable
+  where Scalar : Differentiable & BinaryFloatingPoint,
         Scalar.TangentVector : BinaryFloatingPoint {
   public typealias TangentVector = SIMD${n}
 }
 
+extension SIMD${n} : EuclideanDifferentiable
+  where Scalar : EuclideanDifferentiable & BinaryFloatingPoint,
+        Scalar.TangentVector : BinaryFloatingPoint {
+}
+
 extension SIMD${n}
   where Scalar : EuclideanDifferentiable & BinaryFloatingPoint,
         Scalar.TangentVector : BinaryFloatingPoint {

test/AutoDiff/derived_differentiable.swift

@@ -44,7 +44,7 @@ struct TestNoDerivative : EuclideanDifferentiable {
 // CHECK-AST:         var w: Float
 // CHECK-AST:         @noDerivative internal var technicallyDifferentiable: Float
 // CHECK-AST:         internal init(w: Float, technicallyDifferentiable: Float)
-// CHECK-AST:         internal struct TangentVector : _Differentiable, AdditiveArithmetic, ElementaryFunctions
+// CHECK-AST:         internal struct TangentVector : Differentiable, AdditiveArithmetic, ElementaryFunctions
 // CHECK-AST:           internal typealias TangentVector = TestNoDerivative.TangentVector
 // CHECK-AST:         internal var differentiableVectorView: TestNoDerivative.TangentVector { get }
 
@@ -57,7 +57,7 @@ struct TestPointwiseMultiplicative : Differentiable {
 // CHECK-AST:         var w: PointwiseMultiplicativeDummy
 // CHECK-AST:         @noDerivative internal var technicallyDifferentiable: PointwiseMultiplicativeDummy
 // CHECK-AST:         internal init(w: PointwiseMultiplicativeDummy, technicallyDifferentiable: PointwiseMultiplicativeDummy)
-// CHECK-AST:         internal struct TangentVector : _Differentiable, AdditiveArithmetic, PointwiseMultiplicative
+// CHECK-AST:         internal struct TangentVector : Differentiable, AdditiveArithmetic, PointwiseMultiplicative
 // CHECK-AST:           internal typealias TangentVector = TestPointwiseMultiplicative.TangentVector
 
 
@@ -70,14 +70,14 @@ struct TestKeyPathIterable : Differentiable, KeyPathIterable {
 // CHECK-AST:         var w: Float
 // CHECK-AST:         @noDerivative internal var technicallyDifferentiable: Float
 // CHECK-AST:         internal init(w: Float, technicallyDifferentiable: Float)
-// CHECK-AST:         internal struct TangentVector : _Differentiable, AdditiveArithmetic, ElementaryFunctions, VectorProtocol, KeyPathIterable
+// CHECK-AST:         internal struct TangentVector : Differentiable, AdditiveArithmetic, ElementaryFunctions, VectorProtocol, KeyPathIterable
 // CHECK-AST:           internal typealias TangentVector = TestKeyPathIterable.TangentVector
 
 struct GenericTanMember<T : Differentiable> : Differentiable, AdditiveArithmetic {
   var x: T.TangentVector
 }
 
-// CHECK-AST-LABEL: internal struct GenericTanMember<T> : Differentiable, AdditiveArithmetic where T : _Differentiable
+// CHECK-AST-LABEL: internal struct GenericTanMember<T> : Differentiable, AdditiveArithmetic where T : Differentiable
 // CHECK-AST:   internal var x: T.TangentVector
 // CHECK-AST:   internal init(x: T.TangentVector)
 // CHECK-AST:   internal typealias TangentVector = GenericTanMember<T>
@@ -92,7 +92,7 @@ public struct ConditionallyDifferentiable<T> {
 extension ConditionallyDifferentiable : Differentiable where T : Differentiable {}
 
 // CHECK-AST-LABEL: public struct ConditionallyDifferentiable<T> {
-// CHECK-AST:         @differentiable(wrt: self where T : _Differentiable)
+// CHECK-AST:         @differentiable(wrt: self where T : Differentiable)
 // CHECK-AST:         public var x: T
 // CHECK-AST:         internal init(x: T)
 // CHECK-AST:       }
@@ -121,7 +121,7 @@ final class AdditiveArithmeticClass<T : AdditiveArithmetic & Differentiable> : A
   }
 }
 
-// CHECK-AST-LABEL: final internal class AdditiveArithmeticClass<T> : AdditiveArithmetic, Differentiable where T : AdditiveArithmetic, T : _Differentiable {
+// CHECK-AST-LABEL: final internal class AdditiveArithmeticClass<T> : AdditiveArithmetic, Differentiable where T : AdditiveArithmetic, T : Differentiable {
 // CHECK-AST:         final internal var x: T, y: T
-// CHECK-AST:         internal struct TangentVector : _Differentiable, AdditiveArithmetic
+// CHECK-AST:         internal struct TangentVector : Differentiable, AdditiveArithmetic
 // CHECK-AST:       }

test/AutoDiff/differentiability_witness_function_inst.sil

@@ -76,8 +76,8 @@ bb0:
 // CHECK:   {{%.*}} = differentiability_witness_function [vjp] [parameters 0 1] [results 0] @foo : $@convention(thin) (Float, Float, Float) -> Float
 // CHECK:   {{%.*}} = differentiability_witness_function [jvp] [parameters 0] [results 0] @bar : $@convention(thin) (Float, Float, Float) -> (Float, Float)
 // CHECK:   {{%.*}} = differentiability_witness_function [vjp] [parameters 0 1] [results 0 1] @bar : $@convention(thin) (Float, Float, Float) -> (Float, Float)
-// CHECK:   {{%.*}} = differentiability_witness_function [jvp] [parameters 0] [results 0] <τ_0_0 where τ_0_0 : _Differentiable> @generic : $@convention(thin) <τ_0_0> (@in_guaranteed τ_0_0, Float) -> @out τ_0_0
-// CHECK:   {{%.*}} = differentiability_witness_function [vjp] [parameters 0 1] [results 0] <τ_0_0 where τ_0_0 : AdditiveArithmetic, τ_0_0 : _Differentiable> @generic : $@convention(thin) <τ_0_0> (@in_guaranteed τ_0_0, Float) -> @out τ_0_0
+// CHECK:   {{%.*}} = differentiability_witness_function [jvp] [parameters 0] [results 0] <τ_0_0 where τ_0_0 : Differentiable> @generic : $@convention(thin) <τ_0_0> (@in_guaranteed τ_0_0, Float) -> @out τ_0_0
+// CHECK:   {{%.*}} = differentiability_witness_function [vjp] [parameters 0 1] [results 0] <τ_0_0 where τ_0_0 : AdditiveArithmetic, τ_0_0 : Differentiable> @generic : $@convention(thin) <τ_0_0> (@in_guaranteed τ_0_0, Float) -> @out τ_0_0
 // CHECK: }
 
 // CHECK-LABEL: sil @test_transpose_witnesses : $@convention(thin) () -> () {
@@ -86,7 +86,7 @@ bb0:
 // CHECK:   {{%.*}} = differentiability_witness_function [transpose] [parameters 0 1] [results 0] @foo : $@convention(thin) (Float, Float, Float) -> Float
 // CHECK:   {{%.*}} = differentiability_witness_function [transpose] [parameters 0] [results 0] @bar : $@convention(thin) (Float, Float, Float) -> (Float, Float)
 // CHECK:   {{%.*}} = differentiability_witness_function [transpose] [parameters 0 1] [results 0 1] @bar : $@convention(thin) (Float, Float, Float) -> (Float, Float)
-// CHECK:   {{%.*}} = differentiability_witness_function [transpose] [parameters 0] [results 0] <τ_0_0 where τ_0_0 : _Differentiable> @generic : $@convention(thin) <τ_0_0> (@in_guaranteed τ_0_0, Float) -> @out τ_0_0
-// CHECK:   {{%.*}} = differentiability_witness_function [transpose] [parameters 0 1] [results 0] <τ_0_0 where τ_0_0 : _Differentiable, τ_0_0 == τ_0_0.TangentVector> @generic : $@convention(thin) <τ_0_0> (@in_guaranteed τ_0_0, Float) -> @out τ_0_0
+// CHECK:   {{%.*}} = differentiability_witness_function [transpose] [parameters 0] [results 0] <τ_0_0 where τ_0_0 : Differentiable> @generic : $@convention(thin) <τ_0_0> (@in_guaranteed τ_0_0, Float) -> @out τ_0_0
+// CHECK:   {{%.*}} = differentiability_witness_function [transpose] [parameters 0 1] [results 0] <τ_0_0 where τ_0_0 : Differentiable, τ_0_0 == τ_0_0.TangentVector> @generic : $@convention(thin) <τ_0_0> (@in_guaranteed τ_0_0, Float) -> @out τ_0_0
 // CHECK:   return undef : $()
 // CHECK: }

test/AutoDiff/differentiable_attr_silgen.swift

@@ -30,7 +30,7 @@ public func foo_indir_ret<T: Differentiable>(_ x: Float, _ y: T) -> T {
   return y
 }
 
-// CHECK-SIL-LABEL: sil [differentiable source 0 wrt 0, 1 vjp @AD__foo_indir_ret__vjp_src_0_wrt_0_1] [ossa] @foo_indir_ret : $@convention(thin) <T where T : _Differentiable> (Float, @in_guaranteed T) -> @out T {
+// CHECK-SIL-LABEL: sil [differentiable source 0 wrt 0, 1 vjp @AD__foo_indir_ret__vjp_src_0_wrt_0_1] [ossa] @foo_indir_ret : $@convention(thin) <T where T : Differentiable> (Float, @in_guaranteed T) -> @out T {
 // CHECK-SIL: bb0(%0 : $*T, %1 : $Float, %2 : $*T):
 
 @_silgen_name("dfoo_indir_ret")
@@ -101,7 +101,7 @@ struct DiffComputedProp : Differentiable & AdditiveArithmetic {
 // Check that `@differentiable` attribute is transferred from computed property
 // storage declaration to getter accessor.
 
-// CHECK-AST: struct DiffComputedProp : _Differentiable & AdditiveArithmetic {
+// CHECK-AST: struct DiffComputedProp : AdditiveArithmetic & Differentiable {
 // CHECK-AST-NEXT:   var computedProp: Float { get }
 // CHECK-AST: }
 

test/AutoDiff/differentiable_attr_type_checking.swift

@@ -765,7 +765,7 @@ struct TF_521<T: FloatingPoint> {
     self.imaginary = imaginary
   }
 }
-// expected-error @+2 {{type 'TF_521<T>' does not conform to protocol '_Differentiable'}}
+// expected-error @+2 {{type 'TF_521<T>' does not conform to protocol 'Differentiable'}}
 // expected-note @+1 {{do you want to add protocol stubs}}
 extension TF_521: Differentiable where T: Differentiable {
   // expected-note @+1 {{possibly intended match 'TF_521<T>.TangentVector' does not conform to 'AdditiveArithmetic'}}

test/AutoDiff/differentiable_func_debuginfo.swift

@@ -23,7 +23,7 @@
 // Conclusion: mangling coverage is important.
 
 // Minimal dummy compiler-known `Differentiable` protocol.
-public protocol _Differentiable {
+public protocol Differentiable {
   associatedtype TangentVector
 }
 

test/AutoDiff/differentiable_func_type_type_checking.swift

@@ -102,7 +102,7 @@ extension Vector: Differentiable where T: Differentiable {}
 func inferredConformancesGeneric<T, U>(_: @differentiable (Vector<T>) -> Vector<U>) {}
 
 func nondiffVectorFunc(x: Vector<Int>) -> Vector<Int> {}
-// expected-error @+1 {{global function 'inferredConformancesGeneric' requires that 'Int' conform to '_Differentiable}}
+// expected-error @+1 {{global function 'inferredConformancesGeneric' requires that 'Int' conform to 'Differentiable}}
 inferredConformancesGeneric(nondiffVectorFunc)
 
 func diffVectorFunc(x: Vector<Float>) -> Vector<Float> {}

test/AutoDiff/differentiable_function_inst_lowered.sil

@@ -9,14 +9,14 @@ sil_stage lowered
 import Swift
 import Builtin
 
-struct Large : _Differentiable {
+struct Large : Differentiable {
   @_hasStorage @noDerivative let a: Float { get }
   @_hasStorage @noDerivative let b: Float { get }
   @_hasStorage @noDerivative let c: Float { get }
   @_hasStorage @noDerivative let d: Float { get }
   @_hasStorage @noDerivative let e: Float { get }
   init(a: Float, b: Float, c: Float, d: Float, e: Float)
-  struct TangentVector : _Differentiable, AdditiveArithmetic {
+  struct TangentVector : Differentiable, AdditiveArithmetic {
     init()
     typealias TangentVector = Large.TangentVector
     static var zero: Large.TangentVector { get }

test/AutoDiff/differentiable_function_silgen.swift

@@ -111,5 +111,5 @@ func appliesReabstraction(_ f: @escaping @differentiable (Float) -> Float) {
 // CHECK-SILGEN:   [[REABS_VJP:%.*]] = function_ref @$sS4fIegyd_Iegydo_S4fIegnr_Iegnro_TR : $@convention(thin) (@in_guaranteed Float, @guaranteed @callee_guaranteed (Float) -> (Float, @owned @callee_guaranteed (Float) -> Float)) -> (@out Float, @owned @callee_guaranteed (@in_guaranteed Float) -> @out Float)
 // CHECK-SILGEN:   [[NEW_VJP:%.*]] = partial_apply [callee_guaranteed] [[REABS_VJP]]([[VJP_COPY]]) : $@convention(thin) (@in_guaranteed Float, @guaranteed @callee_guaranteed (Float) -> (Float, @owned @callee_guaranteed (Float) -> Float)) -> (@out Float, @owned @callee_guaranteed (@in_guaranteed Float) -> @out Float)
 // CHECK-SILGEN:   [[NEW_DIFF_FUNC:%.*]] = differentiable_function [parameters 0] [[NEW_ORIG]] : $@callee_guaranteed (@in_guaranteed Float) -> @out Float with_derivative {[[NEW_JVP]] : $@callee_guaranteed (@in_guaranteed Float) -> (@out Float, @owned @callee_guaranteed (@in_guaranteed Float) -> @out Float), [[NEW_VJP]] : $@callee_guaranteed (@in_guaranteed Float) -> (@out Float, @owned @callee_guaranteed (@in_guaranteed Float) -> @out Float)}
-// CHECK-SILGEN:   [[DIFF_API:%.*]] = function_ref @${{.*}}pullback{{.*}}at{{.*}} : $@convention(thin) <τ_0_0, τ_0_1 where τ_0_0 : _Differentiable, τ_0_1 : _Differentiable> (@in_guaranteed τ_0_0, @guaranteed @differentiable @callee_guaranteed (@in_guaranteed τ_0_0) -> @out τ_0_1) -> @owned @callee_guaranteed (@in_guaranteed τ_0_1.TangentVector) -> @out τ_0_0.TangentVector
-// CHECK-SILGEN:   apply [[DIFF_API]]<Float, Float>({{.*}}, [[NEW_DIFF_FUNC]]) : $@convention(thin) <τ_0_0, τ_0_1 where τ_0_0 : _Differentiable, τ_0_1 : _Differentiable> (@in_guaranteed τ_0_0, @guaranteed @differentiable @callee_guaranteed (@in_guaranteed τ_0_0) -> @out τ_0_1) -> @owned @callee_guaranteed (@in_guaranteed τ_0_1.TangentVector) -> @out τ_0_0.TangentVector
+// CHECK-SILGEN:   [[DIFF_API:%.*]] = function_ref @${{.*}}pullback{{.*}}at{{.*}} : $@convention(thin) <τ_0_0, τ_0_1 where τ_0_0 : Differentiable, τ_0_1 : Differentiable> (@in_guaranteed τ_0_0, @guaranteed @differentiable @callee_guaranteed (@in_guaranteed τ_0_0) -> @out τ_0_1) -> @owned @callee_guaranteed (@in_guaranteed τ_0_1.TangentVector) -> @out τ_0_0.TangentVector
+// CHECK-SILGEN:   apply [[DIFF_API]]<Float, Float>({{.*}}, [[NEW_DIFF_FUNC]]) : $@convention(thin) <τ_0_0, τ_0_1 where τ_0_0 : Differentiable, τ_0_1 : Differentiable> (@in_guaranteed τ_0_0, @guaranteed @differentiable @callee_guaranteed (@in_guaranteed τ_0_0) -> @out τ_0_1) -> @owned @callee_guaranteed (@in_guaranteed τ_0_1.TangentVector) -> @out τ_0_0.TangentVector

test/AutoDiff/differentiable_protocol.swift

@@ -0,0 +1,46 @@
+// RUN: %target-typecheck-verify-swift
+// REQUIRES: differentiable_programming
+
+import _Differentiation
+
+// Test conformances.
+
+struct FloatWrapper {
+  var value: Float
+}
+extension FloatWrapper: AdditiveArithmetic {
+  static var zero: Self {
+    FloatWrapper(value: Float.zero)
+  }
+  static func + (lhs: Self, rhs: Self) -> Self {
+    return FloatWrapper(value: lhs.value + rhs.value)
+  }
+  static func - (lhs: Self, rhs: Self) -> Self {
+    return FloatWrapper(value: lhs.value + rhs.value)
+  }
+}
+extension FloatWrapper: Differentiable {
+  public typealias TangentVector = Self
+}
+
+struct Wrapper<T> {
+  var value: T
+}
+extension Wrapper: Equatable where T: Equatable {}
+extension Wrapper: AdditiveArithmetic where T: AdditiveArithmetic {
+  static var zero: Self {
+    Wrapper(value: T.zero)
+  }
+  static func + (lhs: Self, rhs: Self) -> Self {
+    return Wrapper(value: lhs.value + rhs.value)
+  }
+  static func - (lhs: Self, rhs: Self) -> Self {
+    return Wrapper(value: lhs.value + rhs.value)
+  }
+}
+extension Wrapper: Differentiable where T: Differentiable {
+  typealias TangentVector = Wrapper<T.TangentVector>
+  mutating func move(along direction: TangentVector) {
+    value.move(along: direction.value)
+  }
+}