test/Prototypes/DoubleWidth.swift.gyb

// RUN: %target-run-simple-swiftgyb
// REQUIRES: executable_test
// REQUIRES: CPU=x86_64

import StdlibUnittest

/// A fixed-width integer that has twice the bit width of its base type.
///
/// You can use the `DoubleWidth` type to continue calculations with the result
/// of a full width arithmetic operation. Normally, when you perform a full
/// width operation, the result is a tuple of the high and low parts of the
/// result.
///
///     let a = 2241543570477705381
///     let b = 186319822866995413
///     let c = a.multipliedFullWidth(by: b)
///     // c == (high: 22640526660490081, low: 7959093232766896457)
///
/// The tuple `c` can't be used in any further comparisons or calculations. To
/// use this value, create a `DoubleWidth` instance from the result. You can
/// use the `DoubleWidth` instance in the same way that you would use any other
/// integer type.
///
///     let d = DoubleWidth(a.multipliedFullWidth(by: b))
///     // d == 417644001000058515200174966092417353
///
///     // Check the calculation:
///     print(d / DoubleWidth(a) == b)
///     // Prints "true"
///
///     if d > Int.max {
///         print("Too big to be an 'Int'!")
///     } else {
///         print("Small enough to fit in an 'Int'")
///     }
///     // Prints "Too big to be an 'Int'!"
///
/// The `DoubleWidth` type is not intended as a replacement for a variable-width
/// integer type. Nesting `DoubleWidth` instances, in particular, may result in
/// undesirable performance.
@_fixed_layout // FIXME(sil-serialize-all)
public struct DoubleWidth<Base : FixedWidthInteger>
  where Base.Magnitude : UnsignedInteger,
  Base.Words : Collection, Base.Magnitude.Words : Collection {

  public typealias High = Base
  public typealias Low = Base.Magnitude

#if _endian(big)
  @usableFromInline // FIXME(sil-serialize-all)
  internal var _storage: (high: High, low: Low)
#else
  @usableFromInline // FIXME(sil-serialize-all)
  internal var _storage: (low: Low, high: High)
#endif

  /// The high part of the value.
  @_transparent
  public var high: High {
    return _storage.high
  }

  /// The low part of the value.
  @_transparent
  public var low: Low {
    return _storage.low
  }

  /// Creates a new instance from the given tuple of high and low parts.
  ///
  /// - Parameter value: The tuple to use as the source of the new instance's
  ///   high and low parts.
  @_transparent
  public init(_ value: (high: High, low: Low)) {
#if _endian(big)
    self._storage = (high: value.0, low: value.1)
#else
    self._storage = (low: value.1, high: value.0)
#endif
  }

  // We expect users to invoke the public initializer above as demonstrated in
  // the documentation (that is, by passing in the result of a full width
  // operation).
  //
  // Internally, we'll need to create new instances by supplying high and low
  // parts directly; ((double parentheses)) greatly impair readability,
  // especially when nested:
  //
  //   DoubleWidth<DoubleWidth>((DoubleWidth((0, 0)), DoubleWidth((0, 0))))
  //
  // For that reason, we'll include an internal initializer that takes two
  // separate arguments.
  @inlinable // FIXME(sil-serialize-all)
  @usableFromInline  @_transparent
  internal init(_ _high: High, _ low: Low) {
    self.init((_high, low))
  }

  @_transparent
  public init() {
    self.init(0, 0)
  }
}

extension DoubleWidth : CustomStringConvertible {
  @inlinable // FIXME(sil-serialize-all)
  public var description: String {
    return String(self, radix: 10)
  }
}

extension DoubleWidth : CustomDebugStringConvertible {
  @inlinable // FIXME(sil-serialize-all)
  public var debugDescription: String {
    return "(\(_storage.high), \(_storage.low))"
  }
}

extension DoubleWidth : Comparable {
  @inlinable // FIXME(sil-serialize-all)
  public static func ==(lhs: DoubleWidth, rhs: DoubleWidth) -> Bool {
    return lhs._storage.low == rhs._storage.low &&
      lhs._storage.high == rhs._storage.high
  }

  @inlinable // FIXME(sil-serialize-all)
  public static func <(lhs: DoubleWidth, rhs: DoubleWidth) -> Bool {
    if lhs._storage.high < rhs._storage.high {
      return true
    }
    if lhs._storage.high > rhs._storage.high {
      return false
    }
    return lhs._storage.low < rhs._storage.low
  }
}

extension DoubleWidth : Hashable {
  @inlinable // FIXME(sil-serialize-all)
  public var hashValue: Int {
    return _hashValue(for: self)
  }

  @inlinable // FIXME(sil-serialize-all)
  public func hash(into hasher: inout Hasher) {
    hasher.combine(low)
    hasher.combine(high)
  }
}

extension DoubleWidth : Numeric {
  public typealias Magnitude = DoubleWidth<Low>

  @inlinable // FIXME(sil-serialize-all)
  public var magnitude: Magnitude {
    let result = Magnitude(Low(truncatingIfNeeded: _storage.high), _storage.low)
    if Base.isSigned && _storage.high < (0 as High) {
      return ~result &+ 1
    } else {
      return result
    }
  }

  @inlinable // FIXME(sil-serialize-all)
  @usableFromInline // FIXME(sil-serialize-all)
  internal init(_ _magnitude: Magnitude) {
    self.init(High(_magnitude._storage.high), _magnitude._storage.low)
  }

  @inlinable // FIXME(sil-serialize-all)
  public init<T : BinaryInteger>(_ source: T) {
    guard let result = DoubleWidth<Base>(exactly: source) else {
      _preconditionFailure("Value is outside the representable range")
    }
    self = result
  }

  @inlinable // FIXME(sil-serialize-all)
  public init?<T : BinaryInteger>(exactly source: T) {
    // Can't represent a negative 'source' if Base is unsigned.
    guard DoubleWidth.isSigned || source >= 0 else { return nil }
    
    // Is 'source' entirely representable in Low?
    if let low = Low(exactly: source.magnitude) {
      self.init(source < (0 as T) ? (~0, ~low &+ 1) : (0, low))
    } else {
      // At this point we know source.bitWidth > Base.bitWidth, or else we
      // would've taken the first branch.
      let lowInT = source & T(~0 as Low)
      let highInT = source >> Low.bitWidth
      
      let low = Low(lowInT)
      guard let high = High(exactly: highInT) else { return nil }
      self.init(high, low)
    }
  }
}

#if false

// This conformance is only possible once the type is in the stdlib, as it uses
// Builtin
extension DoubleWidth: _ExpressibleByBuiltinIntegerLiteral {
  @inlinable // FIXME(sil-serialize-all)
  public init(_builtinIntegerLiteral _x: _MaxBuiltinIntegerType) {
    var _x = _x
    self = DoubleWidth()

    // If we can capture the entire literal in a single Int64, stop there.
    // This avoids some potential deep recursion due to literal expressions in
    // other DoubleWidth methods.
    let (_value, _overflow) = Builtin.s_to_s_checked_trunc_Int2048_Int64(_x)
    if !Bool(_overflow) {
      self = DoubleWidth(Int64(_value))
      return
    }

    // Convert all but the most significant 64 bits as unsigned integers.
    let _shift = Builtin.sext_Int64_Int2048((64 as Int64)._value)
    let lowWordCount = (bitWidth - 1) / 64
    for i in 0..<lowWordCount {
      let value =
        DoubleWidth(UInt64(Builtin.s_to_u_checked_trunc_Int2048_Int64(_x).0))
          &<< DoubleWidth(i * 64)
      self |= value
      _x = Builtin.ashr_Int2048(_x, _shift)
    }

    // Finally, convert the most significant 64 bits and check for overflow.
    let overflow: Bool
    if Base.isSigned {
      let (_value, _overflow) = Builtin.s_to_s_checked_trunc_Int2048_Int64(_x)
      self |= DoubleWidth(Int64(_value)) &<< DoubleWidth(lowWordCount * 64)
      overflow = Bool(_overflow)
    } else {
      let (_value, _overflow) = Builtin.s_to_u_checked_trunc_Int2048_Int64(_x)
      self |= DoubleWidth(UInt64(_value)) &<< DoubleWidth(lowWordCount * 64)
      overflow = Bool(_overflow)
    }
    _precondition(!overflow, "Literal integer out of range for this type")
  }
}

#endif


extension DoubleWidth : FixedWidthInteger {
  @_fixed_layout // FIXME(sil-serialize-all)
  public struct Words : Collection {
    @_frozen // FIXME(sil-serialize-all)
    public enum _IndexValue {
      case low(Low.Words.Index)
      case high(High.Words.Index)
    }
    
    @_fixed_layout // FIXME(sil-serialize-all)
    public struct Index : Comparable {
      public var _value: _IndexValue

      @inlinable // FIXME(sil-serialize-all)
      public init(_ _value: _IndexValue) { self._value = _value }

      @inlinable // FIXME(sil-serialize-all)
      public static func ==(lhs: Index, rhs: Index) -> Bool {
        switch (lhs._value, rhs._value) {
        case let (.low(l), .low(r)): return l == r
        case let (.high(l), .high(r)): return l == r
        default: return false
        }
      }

      @inlinable // FIXME(sil-serialize-all)
      public static func <(lhs: Index, rhs: Index) -> Bool {
        switch (lhs._value, rhs._value) {
        case let (.low(l), .low(r)): return l < r
        case (.low(_), .high(_)): return true
        case (.high(_), .low(_)): return false
        case let (.high(l), .high(r)): return l < r
        }
      }
    }

    public var _high: High.Words
    public var _low: Low.Words

    @inlinable // FIXME(sil-serialize-all)
    public init(_ value: DoubleWidth<Base>) {
      // Multiples of word size only.
      guard Base.bitWidth == Base.Magnitude.bitWidth &&
        (UInt.bitWidth % Base.bitWidth == 0 ||
        Base.bitWidth % UInt.bitWidth == 0) else {
        fatalError("Access to words is not supported on this type")
      }
      self._high = value._storage.high.words
      self._low = value._storage.low.words
      _sanityCheck(!_low.isEmpty)
    }

    @inlinable // FIXME(sil-serialize-all)
    public var startIndex: Index {
      return Index(.low(_low.startIndex))
    }

    @inlinable // FIXME(sil-serialize-all)
    public var endIndex: Index {
      return Index(.high(_high.endIndex))
    }
    
    @inlinable // FIXME(sil-serialize-all)
    public var count: Int {
      if Base.bitWidth < UInt.bitWidth { return 1 }
      return Int(_low.count) + Int(_high.count)
    }
  
    @inlinable // FIXME(sil-serialize-all)
    public func index(after i: Index) -> Index {
      switch i._value {
      case let .low(li):
        if Base.bitWidth < UInt.bitWidth {
          return Index(.high(_high.endIndex)) 
        }
        let next = _low.index(after: li)
        if next == _low.endIndex { 
          return Index(.high(_high.startIndex)) 
        }
        return Index(.low(next))
      case let .high(hi):
        return Index(.high(_high.index(after: hi)))
      }
    }
  
    @inlinable // FIXME(sil-serialize-all)
    public subscript(_ i: Index) -> UInt {
      if Base.bitWidth < UInt.bitWidth {
        _precondition(i == Index(.low(_low.startIndex)), "Invalid index")
        _sanityCheck(2 * Base.bitWidth <= UInt.bitWidth)
        return _low.first! | (_high.first! &<< Base.bitWidth._lowWord) 
      }
      switch i._value {
      case let .low(li): return _low[li]
      case let .high(hi): return _high[hi]
      }
    }
  }

  @inlinable // FIXME(sil-serialize-all)
  public var words: Words {
    return Words(self)
  }

  @inlinable // FIXME(sil-serialize-all)
  public static var isSigned: Bool {
    return Base.isSigned
  }

  @inlinable // FIXME(sil-serialize-all)
  public static var max: DoubleWidth {
    return self.init(High.max, Low.max)
  }

  @inlinable // FIXME(sil-serialize-all)
  public static var min: DoubleWidth {
    return self.init(High.min, Low.min)
  }

  @inlinable // FIXME(sil-serialize-all)
  public static var bitWidth: Int {
    return High.bitWidth + Low.bitWidth
  }

% for (operator, name) in [('+', 'adding'), ('-', 'subtracting')]:
%   highAffectedByLowOverflow = 'Base.max' if operator == '+' else 'Base.min'
  @inlinable // FIXME(sil-serialize-all)
  public func ${name}ReportingOverflow(_ rhs: DoubleWidth)
    -> (partialValue: DoubleWidth, overflow: Bool) {
    let (low, lowOverflow) =
      _storage.low.${name}ReportingOverflow(rhs._storage.low)
    let (high, highOverflow) =
      _storage.high.${name}ReportingOverflow(rhs._storage.high)
    let result = (high &${operator} (lowOverflow ? 1 : 0), low)
    let overflow = highOverflow ||
      high == ${highAffectedByLowOverflow} && lowOverflow
    return (partialValue: DoubleWidth(result), overflow: overflow)
  }
% end

  @inlinable // FIXME(sil-serialize-all)
  public func multipliedReportingOverflow(
    by rhs: DoubleWidth
  ) -> (partialValue: DoubleWidth, overflow: Bool) {
    let (carry, product) = multipliedFullWidth(by: rhs)
    let result = DoubleWidth(truncatingIfNeeded: product)
    
    let isNegative = DoubleWidth.isSigned &&
      (self < (0 as DoubleWidth)) != (rhs < (0 as DoubleWidth))
    let didCarry = isNegative
      ? carry != ~(0 as DoubleWidth)
      : carry != (0 as DoubleWidth)
    let hadPositiveOverflow =
      DoubleWidth.isSigned && !isNegative && product.leadingZeroBitCount == 0

    return (result, didCarry || hadPositiveOverflow)
  }

  @inlinable // FIXME(sil-serialize-all)
  public func quotientAndRemainder(
    dividingBy other: DoubleWidth
  ) -> (quotient: DoubleWidth, remainder: DoubleWidth) {
    let (quotient, remainder) =
      Magnitude._divide(self.magnitude, by: other.magnitude)
    guard DoubleWidth.isSigned else {
      return (DoubleWidth(quotient), DoubleWidth(remainder))
    }
    let isNegative = (self.high < (0 as High)) != (other.high < (0 as High))
    let quotient_ = isNegative
      ? quotient == DoubleWidth.min.magnitude
        ? DoubleWidth.min
        : 0 - DoubleWidth(quotient)
      : DoubleWidth(quotient)
    let remainder_ = self.high < (0 as High)
      ? 0 - DoubleWidth(remainder)
      : DoubleWidth(remainder)
    return (quotient_, remainder_)
  }

  @inlinable // FIXME(sil-serialize-all)
  public func dividedReportingOverflow(
    by other: DoubleWidth
  ) -> (partialValue: DoubleWidth, overflow: Bool) {
    if other == (0 as DoubleWidth) { return (self, true) }
    if DoubleWidth.isSigned && other == -1 && self == .min {
      return (self, true)
    }
    return (quotientAndRemainder(dividingBy: other).quotient, false)
  }

  @inlinable // FIXME(sil-serialize-all)
  public func remainderReportingOverflow(
    dividingBy other: DoubleWidth
  ) -> (partialValue: DoubleWidth, overflow: Bool) {
    if other == (0 as DoubleWidth) { return (self, true) }
    if DoubleWidth.isSigned && other == -1 && self == .min { return (0, true) }
    return (quotientAndRemainder(dividingBy: other).remainder, false)
  }

  @inlinable // FIXME(sil-serialize-all)
  public func multipliedFullWidth(
    by other: DoubleWidth
  ) -> (high: DoubleWidth, low: DoubleWidth.Magnitude) {
    let isNegative = DoubleWidth.isSigned &&
      (self < (0 as DoubleWidth)) != (other < (0 as DoubleWidth))

    func mul(_ x: Low, _ y: Low) -> (partial: Low, carry: Low) {
      let (high, low) = x.multipliedFullWidth(by: y)
      return (low, high)
    }
        
    func sum(_ x: Low, _ y: Low, _ z: Low) -> (partial: Low, carry: Low) {
      let (sum1, overflow1) = x.addingReportingOverflow(y)
      let (sum2, overflow2) = sum1.addingReportingOverflow(z)
      let carry: Low = (overflow1 ? 1 : 0) + (overflow2 ? 1 : 0)
      return (sum2, carry)
    }
        
    let lhs = self.magnitude
    let rhs = other.magnitude
        
    let a = mul(rhs._storage.low, lhs._storage.low)
    let b = mul(rhs._storage.low, lhs._storage.high)
    let c = mul(rhs._storage.high, lhs._storage.low)
    let d = mul(rhs._storage.high, lhs._storage.high)
        
    let mid1 = sum(a.carry, b.partial, c.partial)
    let mid2 = sum(b.carry, c.carry, d.partial)
        
    let low =
      DoubleWidth<Low>(mid1.partial, a.partial)
    let high =
      DoubleWidth(High(mid2.carry + d.carry), mid1.carry + mid2.partial)
        
    if isNegative {
      let (lowComplement, overflow) = (~low).addingReportingOverflow(1)
      return (~high + (overflow ? 1 : 0 as DoubleWidth), lowComplement)
    } else {
      return (high, low)
    }
  }

  @inlinable // FIXME(sil-serialize-all)
  public func dividingFullWidth(
    _ dividend: (high: DoubleWidth, low: DoubleWidth.Magnitude)
  ) -> (quotient: DoubleWidth, remainder: DoubleWidth) {
    let other = DoubleWidth<DoubleWidth>(dividend)
    let (quotient, remainder) =
      Magnitude._divide(other.magnitude, by: self.magnitude)
    guard DoubleWidth.isSigned else {
      return (DoubleWidth(quotient), DoubleWidth(remainder))
    }
    let isNegative =
      (self.high < (0 as High)) != (other.high.high < (0 as High))
    let quotient_ = isNegative
      ? quotient == DoubleWidth.min.magnitude
        ? DoubleWidth.min
        : 0 - DoubleWidth(quotient)
      : DoubleWidth(quotient)
    let remainder_ = other.high.high < (0 as High)
      ? 0 - DoubleWidth(remainder)
      : DoubleWidth(remainder)
    return (quotient_, remainder_)
  }

% for operator in ['&', '|', '^']:
  @inlinable // FIXME(sil-serialize-all)
  public static func ${operator}=(
    lhs: inout DoubleWidth, rhs: DoubleWidth
  ) {
    lhs._storage.low ${operator}= rhs._storage.low
    lhs._storage.high ${operator}= rhs._storage.high
  }
% end

  @inlinable // FIXME(sil-serialize-all)
  public static func <<=(lhs: inout DoubleWidth, rhs: DoubleWidth) {
    if DoubleWidth.isSigned && rhs < (0 as DoubleWidth) {
      lhs >>= 0 - rhs
      return
    }
    
    // Shift is larger than this type's bit width.
    if rhs._storage.high != (0 as High) ||
      rhs._storage.low >= DoubleWidth.bitWidth
    {
      lhs = 0
      return
    }

    lhs &<<= rhs
  }
  
  @inlinable // FIXME(sil-serialize-all)
  public static func >>=(lhs: inout DoubleWidth, rhs: DoubleWidth) {
    if DoubleWidth.isSigned && rhs < (0 as DoubleWidth) {
      lhs <<= 0 - rhs
      return
    }

    // Shift is larger than this type's bit width.
    if rhs._storage.high != (0 as High) ||
      rhs._storage.low >= DoubleWidth.bitWidth
    {
      lhs = lhs < (0 as DoubleWidth) ? ~0 : 0
      return
    }

    lhs &>>= rhs
  }

  /// Returns this value "masked" by its bit width.
  ///
  /// "Masking" notionally involves repeatedly incrementing or decrementing this
  /// value by `self.bitWidth` until the result is contained in the range
  /// `0..<self.bitWidth`.
  @usableFromInline  @_transparent
  internal func _masked() -> DoubleWidth {
    // FIXME(integers): test types with bit widths that aren't powers of 2
    if DoubleWidth.bitWidth.nonzeroBitCount == 1 {
      return self & DoubleWidth(DoubleWidth.bitWidth &- 1)
    }
    if DoubleWidth.isSigned && self._storage.high < (0 as High) {
      return self % DoubleWidth(DoubleWidth.bitWidth) + self
    }
    return self % DoubleWidth(DoubleWidth.bitWidth)
  }

  @inlinable // FIXME(sil-serialize-all)
  public static func &<<=(lhs: inout DoubleWidth, rhs: DoubleWidth) {
    let rhs = rhs._masked()

    guard rhs._storage.low < Base.bitWidth else {
      lhs._storage.high = High(
        truncatingIfNeeded: lhs._storage.low &<<
          (rhs._storage.low &- Low(Base.bitWidth)))
      lhs._storage.low = 0
      return
    }

    guard rhs._storage.low != (0 as Low) else { return }
    lhs._storage.high &<<= High(rhs._storage.low)
    lhs._storage.high |= High(
      truncatingIfNeeded: lhs._storage.low &>>
        (Low(Base.bitWidth) &- rhs._storage.low))
    lhs._storage.low &<<= rhs._storage.low
  }
  
  @inlinable // FIXME(sil-serialize-all)
  public static func &>>=(lhs: inout DoubleWidth, rhs: DoubleWidth) {
    let rhs = rhs._masked()

    guard rhs._storage.low < Base.bitWidth else {
      lhs._storage.low = Low(
        truncatingIfNeeded: lhs._storage.high &>>
          High(rhs._storage.low &- Low(Base.bitWidth)))
      lhs._storage.high = lhs._storage.high < (0 as High) ? ~0 : 0
      return
    }

    guard rhs._storage.low != (0 as Low) else { return }
    lhs._storage.low &>>= rhs._storage.low
    lhs._storage.low |= Low(
      truncatingIfNeeded: lhs._storage.high &<<
        High(Low(Base.bitWidth) &- rhs._storage.low))
    lhs._storage.high &>>= High(rhs._storage.low)
  }
  
%{
binaryOperators = [
  ('+', 'adding', '_', '+'),
  ('-', 'subtracting', '_', '-'),
  ('*', 'multiplied', 'by', '*'),
  ('/', 'divided', 'by', '/'),
  ('%', 'remainder', 'dividingBy', '/'),
]
}%
% for (operator, name, firstArg, kind) in binaryOperators:

  // FIXME(integers): remove this once the operators are back to Numeric
  @inlinable // FIXME(sil-serialize-all)
  public static func ${operator} (
    lhs: DoubleWidth, rhs: DoubleWidth
  ) -> DoubleWidth {
    var lhs = lhs
    lhs ${operator}= rhs
    return lhs
  }

%   argumentLabel = (firstArg + ':') if firstArg != '_' else ''
  @inlinable // FIXME(sil-serialize-all)
  public static func ${operator}=(
    lhs: inout DoubleWidth, rhs: DoubleWidth
  ) {
    let (result, overflow) = lhs.${name}ReportingOverflow(${argumentLabel}rhs)
    _precondition(!overflow, "Overflow in ${operator}=")
    lhs = result
  }
% end

  @inlinable // FIXME(sil-serialize-all)
  public init(_truncatingBits bits: UInt) {
    _storage.low = Low(_truncatingBits: bits)
    _storage.high = High(_truncatingBits: bits >> UInt(Low.bitWidth))
  }

  @inlinable // FIXME(sil-serialize-all)
  public init(integerLiteral x: Int) {
    self.init(x)
  }
  @inlinable // FIXME(sil-serialize-all)
  public var leadingZeroBitCount: Int {
    return high == (0 as High)
      ? High.bitWidth + low.leadingZeroBitCount
      : high.leadingZeroBitCount
  }

  @inlinable // FIXME(sil-serialize-all)
  public var trailingZeroBitCount: Int {
    return low == (0 as Low)
      ? Low.bitWidth + high.trailingZeroBitCount
      : low.trailingZeroBitCount
  }

  @inlinable // FIXME(sil-serialize-all)
  public var nonzeroBitCount: Int {
    return high.nonzeroBitCount + low.nonzeroBitCount
  }

  @_transparent
  public var byteSwapped: DoubleWidth {
    return DoubleWidth(
      High(truncatingIfNeeded: low.byteSwapped),
      Low(truncatingIfNeeded: high.byteSwapped))
  }
}

extension DoubleWidth : UnsignedInteger where Base : UnsignedInteger {
  /// Returns the quotient and remainder after dividing a triple-width magnitude
  /// `lhs` by a double-width magnitude `rhs`.
  ///
  /// This operation is conceptually that described by Burnikel and Ziegler
  /// (1998).
  @inlinable // FIXME(sil-serialize-all)
  @usableFromInline
  internal static func _divide(
    _ lhs: (high: Low, mid: Low, low: Low), by rhs: Magnitude
  ) -> (quotient: Low, remainder: Magnitude) {
    // The following invariants are guaranteed to hold by dividingFullWidth or
    // quotientAndRemainder before this method is invoked:
    _sanityCheck(lhs.high != (0 as Low))
    _sanityCheck(rhs.leadingZeroBitCount == 0)
    _sanityCheck(Magnitude(lhs.high, lhs.mid) < rhs)

    // Estimate the quotient.
    var quotient = lhs.high == rhs.high
      ? Low.max
      : rhs.high.dividingFullWidth((lhs.high, lhs.mid)).quotient
    // Compute quotient * rhs.
    // TODO: This could be performed more efficiently.
    var product =
      DoubleWidth<Magnitude>(
        0, Magnitude(quotient.multipliedFullWidth(by: rhs.low)))
    let (x, y) = quotient.multipliedFullWidth(by: rhs.high)
    product += DoubleWidth<Magnitude>(Magnitude(0, x), Magnitude(y, 0))
    // Compute the remainder after decrementing quotient as necessary.
    var remainder =
      DoubleWidth<Magnitude>(
        Magnitude(0, lhs.high), Magnitude(lhs.mid, lhs.low))
    while remainder < product {
      quotient = quotient &- 1
      remainder += DoubleWidth<Magnitude>(0, rhs)
    }
    remainder -= product

    return (quotient, remainder.low)
  }

  /// Returns the quotient and remainder after dividing a quadruple-width
  /// magnitude `lhs` by a double-width magnitude `rhs`.
  @inlinable // FIXME(sil-serialize-all)
  @usableFromInline
  internal static func _divide(
    _ lhs: DoubleWidth<Magnitude>, by rhs: Magnitude
  ) -> (quotient: Magnitude, remainder: Magnitude) {
    guard _fastPath(rhs > (0 as Magnitude)) else {
      fatalError("Division by zero")
    }
    guard _fastPath(rhs >= lhs.high) else {
      fatalError("Division results in an overflow")
    }

    if lhs.high == (0 as Magnitude) {
      return lhs.low.quotientAndRemainder(dividingBy: rhs)
    }

    if rhs.high == (0 as Low) {
      let a = lhs.high.high % rhs.low
      let b = a == (0 as Low)
        ? lhs.high.low % rhs.low
        : rhs.low.dividingFullWidth((a, lhs.high.low)).remainder
      let (x, c) = b == (0 as Low)
        ? lhs.low.high.quotientAndRemainder(dividingBy: rhs.low)
        : rhs.low.dividingFullWidth((b, lhs.low.high))
      let (y, d) = c == (0 as Low)
        ? lhs.low.low.quotientAndRemainder(dividingBy: rhs.low)
        : rhs.low.dividingFullWidth((c, lhs.low.low))
      return (Magnitude(x, y), Magnitude(0, d))
    }

    // Left shift both rhs and lhs, then divide and right shift the remainder.
    let shift = rhs.leadingZeroBitCount
    let rhs = rhs &<< shift
    let lhs = lhs &<< shift
    if lhs.high.high == (0 as Low)
      && Magnitude(lhs.high.low, lhs.low.high) < rhs {
      let (quotient, remainder) =
        Magnitude._divide((lhs.high.low, lhs.low.high, lhs.low.low), by: rhs)
      return (Magnitude(0, quotient), remainder &>> shift)
    }
    let (x, a) =
      Magnitude._divide((lhs.high.high, lhs.high.low, lhs.low.high), by: rhs)
    let (y, b) =
      Magnitude._divide((a.high, a.low, lhs.low.low), by: rhs)
    return (Magnitude(x, y), b &>> shift)
  }

  /// Returns the quotient and remainder after dividing a double-width
  /// magnitude `lhs` by a double-width magnitude `rhs`.
  @inlinable // FIXME(sil-serialize-all)
  @usableFromInline
  internal static func _divide(
    _ lhs: Magnitude, by rhs: Magnitude
  ) -> (quotient: Magnitude, remainder: Magnitude) {
    guard _fastPath(rhs > (0 as Magnitude)) else {
      fatalError("Division by zero")
    }
    guard rhs < lhs else {
      if _fastPath(rhs > lhs) { return (0, lhs) }
      return (1, 0)
    }

    if lhs.high == (0 as Low) {
      let (quotient, remainder) =
        lhs.low.quotientAndRemainder(dividingBy: rhs.low)
      return (Magnitude(quotient), Magnitude(remainder))
    }

    if rhs.high == (0 as Low) {
      let (x, a) = lhs.high.quotientAndRemainder(dividingBy: rhs.low)
      let (y, b) = a == (0 as Low)
        ? lhs.low.quotientAndRemainder(dividingBy: rhs.low)
        : rhs.low.dividingFullWidth((a, lhs.low))
      return (Magnitude(x, y), Magnitude(0, b))
    }

    // Left shift both rhs and lhs, then divide and right shift the remainder.
    let shift = rhs.leadingZeroBitCount
    let rhs = rhs &<< shift
    let high = (lhs &>> (Magnitude.bitWidth &- shift)).low
    let lhs = lhs &<< shift
    let (quotient, remainder) =
      Magnitude._divide((high, lhs.high, lhs.low), by: rhs)
    return (Magnitude(0, quotient), remainder &>> shift)
  }
}

extension DoubleWidth : SignedNumeric, SignedInteger
  where Base : SignedInteger {}


//===----------------------------------------------------------------------===//
// Tests
//===----------------------------------------------------------------------===//

var dwTests = TestSuite("DoubleWidth")

typealias UInt128 = DoubleWidth<UInt64>
typealias UInt256 = DoubleWidth<UInt128>
typealias UInt512 = DoubleWidth<UInt256>
typealias UInt1024 = DoubleWidth<UInt512>

typealias Int128 = DoubleWidth<Int64>
typealias Int256 = DoubleWidth<Int128>
typealias Int512 = DoubleWidth<Int256>
typealias Int1024 = DoubleWidth<Int512>

func checkSignedIntegerConformance<T: SignedInteger>(_ x: T) {}
func checkUnsignedIntegerConformance<T: UnsignedInteger>(_ x: T) {}

dwTests.test("Literals") {
  let w: DoubleWidth<UInt8> = 100
  expectTrue(w == 100 as Int)
  
  let x: DoubleWidth<UInt8> = 1000
  expectTrue(x == 1000 as Int)
  
  let y: DoubleWidth<Int8> = 1000
  expectTrue(y == 1000 as Int)
  
  let z: DoubleWidth<Int8> = -1000
  expectTrue(z == -1000 as Int)
  
  expectCrashLater()
  _ = -1 as DoubleWidth<UInt8>
}

#if false
// Uncomment these tests once _ExpressibleByBuiltinIntegerLiteral
// conformance is available
dwTests.test("Literals/Large/Signed") {
  let a: Int256 =
    0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
  let b: Int256 =
    -0x8000000000000000000000000000000000000000000000000000000000000000
  expectEqual(a, Int256.max)
  expectEqual(b, Int256.min)
  expectCrashLater()
  _ = -0x8000000000000000000000000000000000000000000000000000000000000001
    as Int256
}

dwTests.test("Literals/Large/SignedOverflow") {
  expectCrashLater()
  _ = 0x8000000000000000000000000000000000000000000000000000000000000000
    as Int256
}

dwTests.test("Literals/Large/Unsigned") {
  let a: UInt256 =
    0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
  let b: UInt256 = 0
  expectEqual(a, UInt256.max)
  expectEqual(b, UInt256.min)
  expectCrashLater()
  _ = -1 as UInt256
}

dwTests.test("Literals/Large/UnsignedOverflow") {
  expectCrashLater()
  _ = 0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff0
    as UInt256
}
#endif

dwTests.test("Arithmetic/unsigned") {
  let x: DoubleWidth<UInt8> = 1000
  let y: DoubleWidth<UInt8> = 1111
  expectEqual(x + 1, 1001)
  expectEqual(x + x, 2000)
  expectEqual(x - (1 as DoubleWidth<UInt8>), 999)
  expectEqual(x - x, 0)
  expectEqual(y - x, 111)

  expectEqual(x * 7, 7000)
  expectEqual(y * 7, 7777)

  expectEqual(x / 3, 333)
  expectEqual(x / x, 1)
  expectEqual(x / y, 0)
  expectEqual(y / x, 1)

  expectEqual(x % 3, 1)
  expectEqual(x % y, x)
}

dwTests.test("Arithmetic/signed") {
  let x: DoubleWidth<Int8> = 1000
  let y: DoubleWidth<Int8> = -1111
  expectEqual(x + 1, 1001)
  expectEqual(x + x, 2000)
  expectEqual(x - (1 as DoubleWidth<Int8>), 999)
  expectEqual(x - x, 0)
  expectEqual(0 - x, -1000)
  expectEqual(x + y, -111)
  expectEqual(x - y, 2111)

  expectEqual(x * 7, 7000)
  expectEqual(y * 7, -7777)
  expectEqual(x * -7, -7000)
  expectEqual(y * -7, 7777)

  expectEqual(x / 3, 333)
  expectEqual(x / -3, -333)
  expectEqual(x / x, 1)
  expectEqual(x / y, 0)
  expectEqual(y / x, -1)
  expectEqual(y / y, 1)

  expectEqual(x % 3, 1)
  expectEqual(x % -3, 1)
  expectEqual(y % 3, -1)
  expectEqual(y % -3, -1)

  expectEqual(-y, 1111)
  expectEqual(-x, -1000)
}

dwTests.test("Nested") {
  do {
    let x = UInt1024.max
    let (y, o) = x.addingReportingOverflow(1)
    expectEqual(y, 0)
    expectTrue(y == (0 as Int))
    expectTrue(o)
  }

  do {
    let x = Int1024.max
    let (y, o) = x.addingReportingOverflow(1)
    expectEqual(y, Int1024.min)
    expectLT(y, 0)
    expectTrue(y < (0 as Int))
    expectTrue(y < (0 as UInt))
    expectTrue(o)
  }

  expectFalse(UInt1024.isSigned)
  expectEqual(UInt1024.bitWidth, 1024)
  expectTrue(Int1024.isSigned)
  expectEqual(Int1024.bitWidth, 1024)

  expectEqualSequence(
    UInt1024.max.words, repeatElement(UInt.max, count: 1024 / UInt.bitWidth))
}

dwTests.test("inits") {
  typealias DWU16 = DoubleWidth<UInt8>

  expectTrue(DWU16(UInt16.max) == UInt16.max)
  expectNil(DWU16(exactly: UInt32.max))
  expectEqual(DWU16(truncatingIfNeeded: UInt64.max), DWU16.max)

  expectCrashLater()
  _ = DWU16(UInt32.max)
}

dwTests.test("Magnitude") {
  typealias DWU16 = DoubleWidth<UInt8>
  typealias DWI16 = DoubleWidth<Int8>

  expectTrue(DWU16.min.magnitude == UInt16.min.magnitude)
  expectTrue((42 as DWU16).magnitude == (42 as UInt16).magnitude)
  expectTrue(DWU16.max.magnitude == UInt16.max.magnitude)

  expectTrue(DWI16.min.magnitude == Int16.min.magnitude)
  expectTrue((-42 as DWI16).magnitude == (-42 as Int16).magnitude)
  expectTrue(DWI16().magnitude == Int16(0).magnitude) // See SR-6602.
  expectTrue((42 as DWI16).magnitude == (42 as Int16).magnitude)
  expectTrue(DWI16.max.magnitude == Int16.max.magnitude)
}

dwTests.test("TwoWords") {
  typealias DW = DoubleWidth<Int>

  expectEqual(-1 as DW, DW(truncatingIfNeeded: -1 as Int8))

  expectNil(Int(exactly: DW(Int.min) - 1))
  expectNil(Int(exactly: DW(Int.max) + 1))

  expectTrue(DW(Int.min) - 1 < Int.min)
  expectTrue(DW(Int.max) + 1 > Int.max)
}

dwTests.test("Bitshifts") {
  typealias DWU64 = DoubleWidth<DoubleWidth<DoubleWidth<UInt8>>>
  typealias DWI64 = DoubleWidth<DoubleWidth<DoubleWidth<Int8>>>

  func f<T: FixedWidthInteger, U: FixedWidthInteger>(_ x: T, type: U.Type) {
    let y = U(x)
    expectEqual(T.bitWidth, U.bitWidth)
    for i in -(T.bitWidth + 1)...(T.bitWidth + 1) {
      expectTrue(x << i == y << i)
      expectTrue(x >> i == y >> i)

      expectTrue(x &<< i == y &<< i)
      expectTrue(x &>> i == y &>> i)
    }
  }

  f(1 as UInt64, type: DWU64.self)
  f(~(~0 as UInt64 >> 1), type: DWU64.self)
  f(UInt64.max, type: DWU64.self)
  // 0b01010101_10100101_11110000_10100101_11110000_10100101_11110000_10100101
  f(17340530535757639845 as UInt64, type: DWU64.self)

  f(1 as Int64, type: DWI64.self)
  f(Int64.min, type: DWI64.self)
  f(Int64.max, type: DWI64.self)
  // 0b01010101_10100101_11110000_10100101_11110000_10100101_11110000_10100101
  f(6171603459878809765 as Int64, type: DWI64.self)
}

dwTests.test("Remainder/DividingBy0") {
  func f(_ x: Int1024, _ y: Int1024) -> Int1024 {
    return x % y
  }
  expectCrashLater()
  _ = f(42, 0)
}

dwTests.test("RemainderReportingOverflow/DividingByMinusOne") {
  func f(_ x: Int256, _ y: Int256) -> Int256 {
    return x.remainderReportingOverflow(dividingBy: y).partialValue
  }
  expectEqual(f(.max, -1), 0)
  expectEqual(f(.min, -1), 0)
}

dwTests.test("Division/By0") {
  func f(_ x: Int1024, _ y: Int1024) -> Int1024 {
    return x / y
  }
  expectCrashLater()
  _ = f(42, 0)
}

dwTests.test("DivideMinByMinusOne") {
  func f(_ x: Int1024) -> Int1024 {
    return x / -1
  }
  expectCrashLater()
  _ = f(Int1024.min)
}

dwTests.test("MultiplyMinByMinusOne") {
  func f(_ x: Int1024) -> Int1024 {
    return x * -1
  }
  expectCrashLater()
  _ = f(Int1024.min)
}

typealias DWI16 = DoubleWidth<Int8>
typealias DWU16 = DoubleWidth<UInt8>

dwTests.test("Conversions") {
  expectTrue(DWI16(1 << 15 - 1) == Int(1 << 15 - 1))
  expectTrue(DWI16(-1 << 15) == Int(-1 << 15))
  expectTrue(DWU16(1 << 16 - 1) == Int(1 << 16 - 1))
  expectTrue(DWU16(0) == Int(0))

  expectTrue(DWI16(Double(1 << 15 - 1)) == Int(1 << 15 - 1))
  expectTrue(DWI16(Double(-1 << 15)) == Int(-1 << 15))
  expectTrue(DWU16(Double(1 << 16 - 1)) == Int(1 << 16 - 1))
  expectTrue(DWU16(Double(0)) == Int(0))

  expectTrue(DWI16(Double(1 << 15 - 1) + 0.9) == Int(1 << 15 - 1))
  expectTrue(DWI16(Double(-1 << 15) - 0.9) == Int(-1 << 15))
  expectTrue(DWU16(Double(1 << 16 - 1) + 0.9) == Int(1 << 16 - 1))
  expectTrue(DWU16(Double(0) - 0.9) == Int(0))

  expectEqual(DWI16(0.00001), 0)
  expectEqual(DWU16(0.00001), 0)
}

dwTests.test("Exact Conversions") {
  expectEqual(DWI16(Double(1 << 15 - 1)), DWI16(exactly: Double(1 << 15 - 1))!)
  expectEqual(DWI16(Double(-1 << 15)), DWI16(exactly: Double(-1 << 15))!)
  expectEqual(DWU16(Double(1 << 16 - 1)), DWU16(exactly: Double(1 << 16 - 1))!)
  expectEqual(DWU16(Double(0)), DWU16(exactly: Double(0))!)

  expectNil(DWI16(exactly: Double(1 << 15 - 1) + 0.9))
  expectNil(DWI16(exactly: Double(-1 << 15) - 0.9))
  expectNil(DWU16(exactly: Double(1 << 16 - 1) + 0.9))
  expectNil(DWU16(exactly: Double(0) - 0.9))

  expectNil(DWI16(exactly: Double(1 << 15)))
  expectNil(DWI16(exactly: Double(-1 << 15) - 1))
  expectNil(DWU16(exactly: Double(1 << 16)))
  expectNil(DWU16(exactly: Double(-1)))

  expectNil(DWI16(exactly: 0.00001))
  expectNil(DWU16(exactly: 0.00001))

  expectNil(DWU16(exactly: Double.nan))
  expectNil(DWU16(exactly: Float.nan))
  expectNil(DWU16(exactly: Double.infinity))
  expectNil(DWU16(exactly: Float.infinity))
}

dwTests.test("Conversions/SignedMax+1") {
  expectCrashLater()
  _ = DWI16(1 << 15)
}

dwTests.test("Conversions/SignedMin-1") {
  expectCrashLater()
  _ = DWI16(-1 << 15 - 1)
}

dwTests.test("Conversions/UnsignedMax+1") {
  expectCrashLater()
  _ = DWU16(1 << 16)
}

dwTests.test("Conversions/Unsigned-1") {
  expectCrashLater()
  _ = DWU16(-1)
}

dwTests.test("Conversions/String") {
  expectEqual(String(Int256.max, radix: 16),
    "7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff")
  expectEqual(String(Int256.min, radix: 16),
    "-8000000000000000000000000000000000000000000000000000000000000000")
  
  expectEqual(String(Int256.max, radix: 2), """
    1111111111111111111111111111111111111111111111111111111111111111\
    1111111111111111111111111111111111111111111111111111111111111111\
    1111111111111111111111111111111111111111111111111111111111111111\
    111111111111111111111111111111111111111111111111111111111111111
    """)
  expectEqual(String(Int256.min, radix: 2), """
    -100000000000000000000000000000000000000000000000000000000000000\
    0000000000000000000000000000000000000000000000000000000000000000\
    0000000000000000000000000000000000000000000000000000000000000000\
    00000000000000000000000000000000000000000000000000000000000000000
    """)
  
  expectEqual(String(Int128.max, radix: 10),
    "170141183460469231731687303715884105727")
  expectEqual(String(Int128.min, radix: 10),
    "-170141183460469231731687303715884105728")
}

dwTests.test("Words") {
  expectEqualSequence((0 as DoubleWidth<Int8>).words, [0])
  expectEqualSequence((1 as DoubleWidth<Int8>).words, [1])
  expectEqualSequence((-1 as DoubleWidth<Int8>).words, [UInt.max])
  expectEqualSequence((256 as DoubleWidth<Int8>).words, [256])
  expectEqualSequence((-256 as DoubleWidth<Int8>).words, [UInt.max - 255])
  expectEqualSequence(DoubleWidth<Int8>.max.words, [32767])
  expectEqualSequence(DoubleWidth<Int8>.min.words, [UInt.max - 32767])

  expectEqualSequence((0 as Int1024).words,
    repeatElement(0 as UInt, count: 1024 / UInt.bitWidth))
  expectEqualSequence((-1 as Int1024).words,
    repeatElement(UInt.max, count: 1024 / UInt.bitWidth))
  expectEqualSequence((1 as Int1024).words,
    [1] + Array(repeating: 0, count: 1024 / UInt.bitWidth - 1))
}

dwTests.test("Conditional Conformance") {
  checkSignedIntegerConformance(0 as Int128)
  checkSignedIntegerConformance(0 as Int1024)

  checkUnsignedIntegerConformance(0 as UInt128)
  checkUnsignedIntegerConformance(0 as UInt1024)
}

runAllTests()