Refactor complex

Sources/ComplexModule/Complex+AdditiveArithmetic.swift

@@ -0,0 +1,42 @@
+//===--- Complex+AdditiveArithmetic.swift ---------------------*- swift -*-===//
+//
+// This source file is part of the Swift Numerics open source project
+//
+// Copyright (c) 2019-2021 Apple Inc. and the Swift Numerics project authors
+// Licensed under Apache License v2.0 with Runtime Library Exception
+//
+// See https://swift.org/LICENSE.txt for license information
+//
+//===----------------------------------------------------------------------===//
+
+import RealModule
+
+extension Complex: AdditiveArithmetic {
+  /// The additive identity, with real and imaginary parts both zero.
+  ///
+  /// See also: `one`, `i`, `infinity`
+  @_transparent
+  public static var zero: Complex {
+    Complex(0, 0)
+  }
+
+  @_transparent
+  public static func +(z: Complex, w: Complex) -> Complex {
+    return Complex(z.x + w.x, z.y + w.y)
+  }
+
+  @_transparent
+  public static func -(z: Complex, w: Complex) -> Complex {
+    return Complex(z.x - w.x, z.y - w.y)
+  }
+
+  @_transparent
+  public static func +=(z: inout Complex, w: Complex) {
+    z = z + w
+  }
+
+  @_transparent
+  public static func -=(z: inout Complex, w: Complex) {
+    z = z - w
+  }
+}

Sources/ComplexModule/Arithmetic.swift → Sources/ComplexModule/Complex+AlgebraicField.swift

@@ -1,8 +1,8 @@
-//===--- Arithmetic.swift -------------------------------------*- swift -*-===//
+//===--- Complex+AlgebraicField.swift -------------------------*- swift -*-===//
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019-2021 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information
@@ -11,81 +11,19 @@
 
 import RealModule
 
-// MARK: - Additive structure
-extension Complex: AdditiveArithmetic {
-  @_transparent
-  public static func +(z: Complex, w: Complex) -> Complex {
-    return Complex(z.x + w.x, z.y + w.y)
-  }
-
-  @_transparent
-  public static func -(z: Complex, w: Complex) -> Complex {
-    return Complex(z.x - w.x, z.y - w.y)
-  }
-
-  @_transparent
-  public static func +=(z: inout Complex, w: Complex) {
-    z = z + w
-  }
-
+extension Complex: AlgebraicField {
+  /// The multiplicative identity `1 + 0i`.
   @_transparent
-  public static func -=(z: inout Complex, w: Complex) {
-    z = z - w
-  }
-}
-
-// MARK: - Vector space structure
-//
-// Policy: deliberately not using the * and / operators for these at the
-// moment, because then there's an ambiguity in expressions like 2*z; is
-// that Complex(2) * z or is it RealType(2) * z? This is especially
-// problematic in type inference: suppose we have:
-//
-//   let a: RealType = 1
-//   let b = 2*a
-//
-// what is the type of b? If we don't have a type context, it's ambiguous.
-// If we have a Complex type context, then b will be inferred to have type
-// Complex! Obviously, that doesn't help anyone.
-//
-// TODO: figure out if there's some way to avoid these surprising results
-// and turn these into operators if/when we have it.
-// (https://github.com/apple/swift-numerics/issues/12)
-extension Complex {
-  /// `self` scaled by `a`.
-  @usableFromInline @_transparent
-  internal func multiplied(by a: RealType) -> Complex {
-    // This can be viewed in two different ways, which are mathematically
-    // equivalent: either we are computing `self * Complex(a)` (i.e.
-    // converting `a` to be a complex value, and then using the complex
-    // multiplication) or we are using the scalar product of the vector
-    // space structure: `Complex(a*real, a*imaginary)`.
-    //
-    // Although these two interpretations are _mathematically_ equivalent,
-    // they will generate different representations of the point at
-    // infinity in general. For example, suppose `self` is represented by
-    // `(infinity, 0)`. Then `self * Complex(1)` would evaluate as
-    // `(1*infinity - 0*0, 0*infinity + 1*0) = (infinity, nan)`, but
-    // the vector space interpretation produces `(infinity, 0)`. This does
-    // not matter much, because these are two representations of the same
-    // semantic value, but note that one requires four multiplies and two
-    // additions, while the one we use requires only two real multiplications.
-    Complex(x*a, y*a)
+  public static var one: Complex {
+    Complex(1, 0)
   }
 
-  /// `self` unscaled by `a`.
-  @usableFromInline @_transparent
-  internal func divided(by a: RealType) -> Complex {
-    // See implementation notes for `multiplied` above.
-    Complex(x/a, y/a)
-  }
-}
-
-// MARK: - Multiplicative structure
-extension Complex: AlgebraicField {
+  /// The [complex conjugate][conj] of this value.
+  ///
+  /// [conj]: https://en.wikipedia.org/wiki/Complex_conjugate
   @_transparent
-  public static func *(z: Complex, w: Complex) -> Complex {
-    return Complex(z.x*w.x - z.y*w.y, z.x*w.y + z.y*w.x)
+  public var conjugate: Complex {
+    Complex(x, -y)
   }
 
   @_transparent
@@ -98,11 +36,6 @@ extension Complex: AlgebraicField {
     return z * (w.conjugate.divided(by: lenSq))
   }
 
-  @_transparent
-  public static func *=(z: inout Complex, w: Complex) {
-    z = z * w
-  }
-
   @_transparent
   public static func /=(z: inout Complex, w: Complex) {
     z = z / w

Sources/ComplexModule/Complex+Codable.swift

@@ -0,0 +1,30 @@
+//===--- Complex+Codable.swift --------------------------------*- swift -*-===//
+//
+// This source file is part of the Swift Numerics open source project
+//
+// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Licensed under Apache License v2.0 with Runtime Library Exception
+//
+// See https://swift.org/LICENSE.txt for license information
+//
+//===----------------------------------------------------------------------===//
+
+import RealModule
+
+// FloatingPoint does not refine Codable, so this is a conditional conformance.
+extension Complex: Decodable where RealType: Decodable {
+  public init(from decoder: Decoder) throws {
+    var unkeyedContainer = try decoder.unkeyedContainer()
+    let x = try unkeyedContainer.decode(RealType.self)
+    let y = try unkeyedContainer.decode(RealType.self)
+    self.init(x, y)
+  }
+}
+
+extension Complex: Encodable where RealType: Encodable {
+  public func encode(to encoder: Encoder) throws {
+    var unkeyedContainer = encoder.unkeyedContainer()
+    try unkeyedContainer.encode(x)
+    try unkeyedContainer.encode(y)
+  }
+}

Sources/ComplexModule/Differentiable.swift → Sources/ComplexModule/Complex+Differentiable.swift

@@ -2,7 +2,7 @@
 //
 // This source file is part of the Swift Numerics open source project
 //
-// Copyright (c) 2019 - 2020 Apple Inc. and the Swift Numerics project authors
+// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
 // Licensed under Apache License v2.0 with Runtime Library Exception
 //
 // See https://swift.org/LICENSE.txt for license information

Sources/ComplexModule/ElementaryFunctions.swift → Sources/ComplexModule/Complex+ElementaryFunctions.swift

@@ -1,4 +1,4 @@
-//===--- ElementaryFunctions.swift ----------------------------*- swift -*-===//
+//===--- Complex+ElementaryFunctions.swift --------------------*- swift -*-===//
 //
 // This source file is part of the Swift.org open source project
 //

Sources/ComplexModule/Complex+Hashable.swift

@@ -0,0 +1,42 @@
+//===--- Complex+Hashable.swift -------------------------------*- swift -*-===//
+//
+// This source file is part of the Swift Numerics open source project
+//
+// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Licensed under Apache License v2.0 with Runtime Library Exception
+//
+// See https://swift.org/LICENSE.txt for license information
+//
+//===----------------------------------------------------------------------===//
+
+import RealModule
+
+extension Complex: Hashable {
+  @_transparent
+  public static func ==(a: Complex, b: Complex) -> Bool {
+    // Identify all numbers with either component non-finite as a single
+    // "point at infinity".
+    guard a.isFinite || b.isFinite else { return true }
+    // For finite numbers, equality is defined componentwise. Cases where
+    // only one of a or b is infinite fall through to here as well, but this
+    // expression correctly returns false for them so we don't need to handle
+    // them explicitly.
+    return a.x == b.x && a.y == b.y
+  }
+
+  @_transparent
+  public func hash(into hasher: inout Hasher) {
+    // There are two equivalence classes to which we owe special attention:
+    // All zeros should hash to the same value, regardless of sign, and all
+    // non-finite numbers should hash to the same value, regardless of
+    // representation. The correct behavior for zero falls out for free from
+    // the hash behavior of floating-point, but we need to use a
+    // representative member for any non-finite values.
+    if isFinite {
+      hasher.combine(x)
+      hasher.combine(y)
+    } else {
+      hasher.combine(RealType.infinity)
+    }
+  }
+}

Sources/ComplexModule/Complex+IntegerLiteral.swift

@@ -0,0 +1,19 @@
+//===--- Complex+IntegerLiteral.swift -------------------------*- swift -*-===//
+//
+// This source file is part of the Swift Numerics open source project
+//
+// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Licensed under Apache License v2.0 with Runtime Library Exception
+//
+// See https://swift.org/LICENSE.txt for license information
+//
+//===----------------------------------------------------------------------===//
+
+extension Complex: ExpressibleByIntegerLiteral {
+  public typealias IntegerLiteralType = RealType.IntegerLiteralType
+
+  @inlinable
+  public init(integerLiteral value: IntegerLiteralType) {
+    self.init(RealType(integerLiteral: value))
+  }
+}

Sources/ComplexModule/Complex+Numeric.swift

@@ -0,0 +1,58 @@
+//===--- Complex+Numeric.swift --------------------------------*- swift -*-===//
+//
+// This source file is part of the Swift Numerics open source project
+//
+// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Licensed under Apache License v2.0 with Runtime Library Exception
+//
+// See https://swift.org/LICENSE.txt for license information
+//
+//===----------------------------------------------------------------------===//
+
+extension Complex: Numeric {
+
+  @_transparent
+  public static func *(z: Complex, w: Complex) -> Complex {
+    return Complex(z.x*w.x - z.y*w.y, z.x*w.y + z.y*w.x)
+  }
+
+  @_transparent
+  public static func *=(z: inout Complex, w: Complex) {
+    z = z * w
+  }
+
+  /// The complex number with specified real part and zero imaginary part.
+  ///
+  /// Equivalent to `Complex(RealType(real), 0)`.
+  @inlinable
+  public init<Other: BinaryInteger>(_ real: Other) {
+    self.init(RealType(real), 0)
+  }
+
+  /// The complex number with specified real part and zero imaginary part,
+  /// if it can be constructed without rounding.
+  @inlinable
+  public init?<Other: BinaryInteger>(exactly real: Other) {
+    guard let real = RealType(exactly: real) else { return nil }
+    self.init(real, 0)
+  }
+
+  /// The ∞-norm of the value (`max(abs(real), abs(imaginary))`).
+  ///
+  /// If you need the Euclidean norm (a.k.a. 2-norm) use the `length` or
+  /// `lengthSquared` properties instead.
+  ///
+  /// Edge cases:
+  /// - If `z` is not finite, `z.magnitude` is `.infinity`.
+  /// - If `z` is zero, `z.magnitude` is `0`.
+  /// - Otherwise, `z.magnitude` is finite and non-zero.
+  ///
+  /// See also:
+  /// - `.length`
+  /// - `.lengthSquared`
+  @_transparent
+  public var magnitude: RealType {
+    guard isFinite else { return .infinity }
+    return max(abs(x), abs(y))
+  }
+}

Sources/ComplexModule/Complex+StringConvertible.swift

@@ -0,0 +1,23 @@
+//===--- Complex+StringConvertible.swift ----------------------*- swift -*-===//
+//
+// This source file is part of the Swift Numerics open source project
+//
+// Copyright (c) 2019 - 2021 Apple Inc. and the Swift Numerics project authors
+// Licensed under Apache License v2.0 with Runtime Library Exception
+//
+// See https://swift.org/LICENSE.txt for license information
+//
+//===----------------------------------------------------------------------===//
+
+extension Complex: CustomStringConvertible {
+  public var description: String {
+    guard isFinite else { return "inf" }
+    return "(\(x), \(y))"
+  }
+}
+
+extension Complex: CustomDebugStringConvertible {
+  public var debugDescription: String {
+    "Complex<\(RealType.self)>(\(String(reflecting: x)), \(String(reflecting: y)))"
+  }
+}