ScaledNumber.h

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//===- llvm/Support/ScaledNumber.h - Support for scaled numbers -*- C++ -*-===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
//
// This file contains functions (and a class) useful for working with scaled
// numbers -- in particular, pairs of integers where one represents digits and
// another represents a scale.  The functions are helpers and live in the
// namespace ScaledNumbers.  The class ScaledNumber is useful for modelling
// certain cost metrics that need simple, integer-like semantics that are easy
// to reason about.
//
// These might remind you of soft-floats.  If you want one of those, you're in
// the wrong place.  Look at include/llvm/ADT/APFloat.h instead.
//
//===----------------------------------------------------------------------===//
 
#ifndef LLVM_SUPPORT_SCALEDNUMBER_H
#define LLVM_SUPPORT_SCALEDNUMBER_H
 
#include "llvm/Support/Compiler.h"
#include "llvm/Support/MathExtras.h"
#include <algorithm>
#include <cstdint>
#include <limits>
#include <string>
#include <tuple>
#include <utility>
 
namespace llvm {
namespace ScaledNumbers {
 
/// Maximum scale; same as APFloat for easy debug printing.
const int32_t MaxScale = 16383;
 
/// Maximum scale; same as APFloat for easy debug printing.
const int32_t MinScale = -16382;
 
/// Get the width of a number.
template <class DigitsT> inline int getWidth() { return sizeof(DigitsT) * 8; }
 
/// Conditionally round up a scaled number.
///
/// Given \c Digits and \c Scale, round up iff \c ShouldRound is \c true.
/// Always returns \c Scale unless there's an overflow, in which case it
/// returns \c 1+Scale.
///
/// \pre adding 1 to \c Scale will not overflow INT16_MAX.
template <class DigitsT>
inline std::pair<DigitsT, int16_t> getRounded(DigitsT Digits, int16_t Scale,
                                              bool ShouldRound) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
 
  if (ShouldRound)
    if (!++Digits)
      // Overflow.
      return std::make_pair(DigitsT(1) << (getWidth<DigitsT>() - 1), Scale + 1);
  return std::make_pair(Digits, Scale);
}
 
/// Convenience helper for 32-bit rounding.
inline std::pair<uint32_t, int16_t> getRounded32(uint32_t Digits, int16_t Scale,
                                                 bool ShouldRound) {
  return getRounded(Digits, Scale, ShouldRound);
}
 
/// Convenience helper for 64-bit rounding.
inline std::pair<uint64_t, int16_t> getRounded64(uint64_t Digits, int16_t Scale,
                                                 bool ShouldRound) {
  return getRounded(Digits, Scale, ShouldRound);
}
 
/// Adjust a 64-bit scaled number down to the appropriate width.
///
/// \pre Adding 64 to \c Scale will not overflow INT16_MAX.
template <class DigitsT>
inline std::pair<DigitsT, int16_t> getAdjusted(uint64_t Digits,
                                               int16_t Scale = 0) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
 
  const int Width = getWidth<DigitsT>();
  if (Width == 64 || Digits <= std::numeric_limits<DigitsT>::max())
    return std::make_pair(Digits, Scale);
 
  // Shift right and round.
  int Shift = llvm::bit_width(Digits) - Width;
  return getRounded<DigitsT>(Digits >> Shift, Scale + Shift,
                             Digits & (UINT64_C(1) << (Shift - 1)));
}
 
/// Convenience helper for adjusting to 32 bits.
inline std::pair<uint32_t, int16_t> getAdjusted32(uint64_t Digits,
                                                  int16_t Scale = 0) {
  return getAdjusted<uint32_t>(Digits, Scale);
}
 
/// Convenience helper for adjusting to 64 bits.
inline std::pair<uint64_t, int16_t> getAdjusted64(uint64_t Digits,
                                                  int16_t Scale = 0) {
  return getAdjusted<uint64_t>(Digits, Scale);
}
 
/// Multiply two 64-bit integers to create a 64-bit scaled number.
///
/// Implemented with four 64-bit integer multiplies.
LLVM_ABI std::pair<uint64_t, int16_t> multiply64(uint64_t LHS, uint64_t RHS);
 
/// Multiply two 32-bit integers to create a 32-bit scaled number.
///
/// Implemented with one 64-bit integer multiply.
template <class DigitsT>
inline std::pair<DigitsT, int16_t> getProduct(DigitsT LHS, DigitsT RHS) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
 
  if (getWidth<DigitsT>() <= 32 || (LHS <= UINT32_MAX && RHS <= UINT32_MAX))
    return getAdjusted<DigitsT>(uint64_t(LHS) * RHS);
 
  return multiply64(LHS, RHS);
}
 
/// Convenience helper for 32-bit product.
inline std::pair<uint32_t, int16_t> getProduct32(uint32_t LHS, uint32_t RHS) {
  return getProduct(LHS, RHS);
}
 
/// Convenience helper for 64-bit product.
inline std::pair<uint64_t, int16_t> getProduct64(uint64_t LHS, uint64_t RHS) {
  return getProduct(LHS, RHS);
}
 
/// Divide two 64-bit integers to create a 64-bit scaled number.
///
/// Implemented with long division.
///
/// \pre \c Dividend and \c Divisor are non-zero.
LLVM_ABI std::pair<uint64_t, int16_t> divide64(uint64_t Dividend,
                                               uint64_t Divisor);
 
/// Divide two 32-bit integers to create a 32-bit scaled number.
///
/// Implemented with one 64-bit integer divide/remainder pair.
///
/// \pre \c Dividend and \c Divisor are non-zero.
LLVM_ABI std::pair<uint32_t, int16_t> divide32(uint32_t Dividend,
                                               uint32_t Divisor);
 
/// Divide two 32-bit numbers to create a 32-bit scaled number.
///
/// Implemented with one 64-bit integer divide/remainder pair.
///
/// Returns \c (DigitsT_MAX, MaxScale) for divide-by-zero (0 for 0/0).
template <class DigitsT>
std::pair<DigitsT, int16_t> getQuotient(DigitsT Dividend, DigitsT Divisor) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
  static_assert(sizeof(DigitsT) == 4 || sizeof(DigitsT) == 8,
                "expected 32-bit or 64-bit digits");
 
  // Check for zero.
  if (!Dividend)
    return std::make_pair(0, 0);
  if (!Divisor)
    return std::make_pair(std::numeric_limits<DigitsT>::max(), MaxScale);
 
  if (getWidth<DigitsT>() == 64)
    return divide64(Dividend, Divisor);
  return divide32(Dividend, Divisor);
}
 
/// Convenience helper for 32-bit quotient.
inline std::pair<uint32_t, int16_t> getQuotient32(uint32_t Dividend,
                                                  uint32_t Divisor) {
  return getQuotient(Dividend, Divisor);
}
 
/// Convenience helper for 64-bit quotient.
inline std::pair<uint64_t, int16_t> getQuotient64(uint64_t Dividend,
                                                  uint64_t Divisor) {
  return getQuotient(Dividend, Divisor);
}
 
/// Implementation of getLg() and friends.
///
/// Returns the rounded lg of \c Digits*2^Scale and an int specifying whether
/// this was rounded up (1), down (-1), or exact (0).
///
/// Returns \c INT32_MIN when \c Digits is zero.
template <class DigitsT>
inline std::pair<int32_t, int> getLgImpl(DigitsT Digits, int16_t Scale) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
 
  if (!Digits)
    return std::make_pair(INT32_MIN, 0);
 
  // Get the floor of the lg of Digits.
  static_assert(sizeof(Digits) <= sizeof(uint64_t));
  int32_t LocalFloor = llvm::Log2_64(Digits);
 
  // Get the actual floor.
  int32_t Floor = Scale + LocalFloor;
  if (Digits == UINT64_C(1) << LocalFloor)
    return std::make_pair(Floor, 0);
 
  // Round based on the next digit.
  assert(LocalFloor >= 1);
  bool Round = Digits & UINT64_C(1) << (LocalFloor - 1);
  return std::make_pair(Floor + Round, Round ? 1 : -1);
}
 
/// Get the lg (rounded) of a scaled number.
///
/// Get the lg of \c Digits*2^Scale.
///
/// Returns \c INT32_MIN when \c Digits is zero.
template <class DigitsT> int32_t getLg(DigitsT Digits, int16_t Scale) {
  return getLgImpl(Digits, Scale).first;
}
 
/// Get the lg floor of a scaled number.
///
/// Get the floor of the lg of \c Digits*2^Scale.
///
/// Returns \c INT32_MIN when \c Digits is zero.
template <class DigitsT> int32_t getLgFloor(DigitsT Digits, int16_t Scale) {
  auto Lg = getLgImpl(Digits, Scale);
  return Lg.first - (Lg.second > 0);
}
 
/// Get the lg ceiling of a scaled number.
///
/// Get the ceiling of the lg of \c Digits*2^Scale.
///
/// Returns \c INT32_MIN when \c Digits is zero.
template <class DigitsT> int32_t getLgCeiling(DigitsT Digits, int16_t Scale) {
  auto Lg = getLgImpl(Digits, Scale);
  return Lg.first + (Lg.second < 0);
}
 
/// Implementation for comparing scaled numbers.
///
/// Compare two 64-bit numbers with different scales.  Given that the scale of
/// \c L is higher than that of \c R by \c ScaleDiff, compare them.  Return -1,
/// 1, and 0 for less than, greater than, and equal, respectively.
///
/// \pre 0 <= ScaleDiff < 64.
LLVM_ABI int compareImpl(uint64_t L, uint64_t R, int ScaleDiff);
 
/// Compare two scaled numbers.
///
/// Compare two scaled numbers.  Returns 0 for equal, -1 for less than, and 1
/// for greater than.
template <class DigitsT>
int compare(DigitsT LDigits, int16_t LScale, DigitsT RDigits, int16_t RScale) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
 
  // Check for zero.
  if (!LDigits)
    return RDigits ? -1 : 0;
  if (!RDigits)
    return 1;
 
  // Check for the scale.  Use getLgFloor to be sure that the scale difference
  // is always lower than 64.
  int32_t lgL = getLgFloor(LDigits, LScale), lgR = getLgFloor(RDigits, RScale);
  if (lgL != lgR)
    return lgL < lgR ? -1 : 1;
 
  // Compare digits.
  if (LScale < RScale)
    return compareImpl(LDigits, RDigits, RScale - LScale);
 
  return -compareImpl(RDigits, LDigits, LScale - RScale);
}
 
/// Match scales of two numbers.
///
/// Given two scaled numbers, match up their scales.  Change the digits and
/// scales in place.  Shift the digits as necessary to form equivalent numbers,
/// losing precision only when necessary.
///
/// If the output value of \c LDigits (\c RDigits) is \c 0, the output value of
/// \c LScale (\c RScale) is unspecified.
///
/// As a convenience, returns the matching scale.  If the output value of one
/// number is zero, returns the scale of the other.  If both are zero, which
/// scale is returned is unspecified.
template <class DigitsT>
int16_t matchScales(DigitsT &LDigits, int16_t &LScale, DigitsT &RDigits,
                    int16_t &RScale) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
 
  if (LScale < RScale)
    // Swap arguments.
    return matchScales(RDigits, RScale, LDigits, LScale);
  if (!LDigits)
    return RScale;
  if (!RDigits || LScale == RScale)
    return LScale;
 
  // Now LScale > RScale.  Get the difference.
  int32_t ScaleDiff = int32_t(LScale) - RScale;
  if (ScaleDiff >= 2 * getWidth<DigitsT>()) {
    // Don't bother shifting.  RDigits will get zero-ed out anyway.
    RDigits = 0;
    return LScale;
  }
 
  // Shift LDigits left as much as possible, then shift RDigits right.
  int32_t ShiftL = std::min<int32_t>(llvm::countl_zero(LDigits), ScaleDiff);
  assert(ShiftL < getWidth<DigitsT>() && "can't shift more than width");
 
  int32_t ShiftR = ScaleDiff - ShiftL;
  if (ShiftR >= getWidth<DigitsT>()) {
    // Don't bother shifting.  RDigits will get zero-ed out anyway.
    RDigits = 0;
    return LScale;
  }
 
  LDigits <<= ShiftL;
  RDigits >>= ShiftR;
 
  LScale -= ShiftL;
  RScale += ShiftR;
  assert(LScale == RScale && "scales should match");
  return LScale;
}
 
/// Get the sum of two scaled numbers.
///
/// Get the sum of two scaled numbers with as much precision as possible.
///
/// \pre Adding 1 to \c LScale (or \c RScale) will not overflow INT16_MAX.
template <class DigitsT>
std::pair<DigitsT, int16_t> getSum(DigitsT LDigits, int16_t LScale,
                                   DigitsT RDigits, int16_t RScale) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
 
  // Check inputs up front.  This is only relevant if addition overflows, but
  // testing here should catch more bugs.
  assert(LScale < INT16_MAX && "scale too large");
  assert(RScale < INT16_MAX && "scale too large");
 
  // Normalize digits to match scales.
  int16_t Scale = matchScales(LDigits, LScale, RDigits, RScale);
 
  // Compute sum.
  DigitsT Sum = LDigits + RDigits;
  if (Sum >= RDigits)
    return std::make_pair(Sum, Scale);
 
  // Adjust sum after arithmetic overflow.
  DigitsT HighBit = DigitsT(1) << (getWidth<DigitsT>() - 1);
  return std::make_pair(HighBit | Sum >> 1, Scale + 1);
}
 
/// Convenience helper for 32-bit sum.
inline std::pair<uint32_t, int16_t> getSum32(uint32_t LDigits, int16_t LScale,
                                             uint32_t RDigits, int16_t RScale) {
  return getSum(LDigits, LScale, RDigits, RScale);
}
 
/// Convenience helper for 64-bit sum.
inline std::pair<uint64_t, int16_t> getSum64(uint64_t LDigits, int16_t LScale,
                                             uint64_t RDigits, int16_t RScale) {
  return getSum(LDigits, LScale, RDigits, RScale);
}
 
/// Get the difference of two scaled numbers.
///
/// Get LHS minus RHS with as much precision as possible.
///
/// Returns \c (0, 0) if the RHS is larger than the LHS.
template <class DigitsT>
std::pair<DigitsT, int16_t> getDifference(DigitsT LDigits, int16_t LScale,
                                          DigitsT RDigits, int16_t RScale) {
  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
 
  // Normalize digits to match scales.
  const DigitsT SavedRDigits = RDigits;
  const int16_t SavedRScale = RScale;
  matchScales(LDigits, LScale, RDigits, RScale);
 
  // Compute difference.
  if (LDigits <= RDigits)
    return std::make_pair(0, 0);
  if (RDigits || !SavedRDigits)
    return std::make_pair(LDigits - RDigits, LScale);
 
  // Check if RDigits just barely lost its last bit.  E.g., for 32-bit:
  //
  //   1*2^32 - 1*2^0 == 0xffffffff != 1*2^32
  const auto RLgFloor = getLgFloor(SavedRDigits, SavedRScale);
  if (!compare(LDigits, LScale, DigitsT(1), RLgFloor + getWidth<DigitsT>()))
    return std::make_pair(std::numeric_limits<DigitsT>::max(), RLgFloor);
 
  return std::make_pair(LDigits, LScale);
}
 
/// Convenience helper for 32-bit difference.
inline std::pair<uint32_t, int16_t> getDifference32(uint32_t LDigits,
                                                    int16_t LScale,
                                                    uint32_t RDigits,
                                                    int16_t RScale) {
  return getDifference(LDigits, LScale, RDigits, RScale);
}
 
/// Convenience helper for 64-bit difference.
inline std::pair<uint64_t, int16_t> getDifference64(uint64_t LDigits,
                                                    int16_t LScale,
                                                    uint64_t RDigits,
                                                    int16_t RScale) {
  return getDifference(LDigits, LScale, RDigits, RScale);
}
 
} // end namespace ScaledNumbers
} // end namespace llvm
 
namespace llvm {
 
class raw_ostream;
class ScaledNumberBase {
public:
  static constexpr int DefaultPrecision = 10;
 
  LLVM_ABI static void dump(uint64_t D, int16_t E, int Width);
  LLVM_ABI static raw_ostream &print(raw_ostream &OS, uint64_t D, int16_t E,
                                     int Width, unsigned Precision);
  LLVM_ABI static std::string toString(uint64_t D, int16_t E, int Width,
                                       unsigned Precision);
  static int countLeadingZeros32(uint32_t N) { return llvm::countl_zero(N); }
  static int countLeadingZeros64(uint64_t N) { return llvm::countl_zero(N); }
  static uint64_t getHalf(uint64_t N) { return (N >> 1) + (N & 1); }
 
  static std::pair<uint64_t, bool> splitSigned(int64_t N) {
    if (N >= 0)
      return std::make_pair(N, false);
    uint64_t Unsigned = N == INT64_MIN ? UINT64_C(1) << 63 : uint64_t(-N);
    return std::make_pair(Unsigned, true);
  }
  static int64_t joinSigned(uint64_t U, bool IsNeg) {
    if (U > uint64_t(INT64_MAX))
      return IsNeg ? INT64_MIN : INT64_MAX;
    return IsNeg ? -int64_t(U) : int64_t(U);
  }
};
 
/// Simple representation of a scaled number.
///
/// ScaledNumber is a number represented by digits and a scale.  It uses simple
/// saturation arithmetic and every operation is well-defined for every value.
/// It's somewhat similar in behaviour to a soft-float, but is *not* a
/// replacement for one.  If you're doing numerics, look at \a APFloat instead.
/// Nevertheless, we've found these semantics useful for modelling certain cost
/// metrics.
///
/// The number is split into a signed scale and unsigned digits.  The number
/// represented is \c getDigits()*2^getScale().  In this way, the digits are
/// much like the mantissa in the x87 long double, but there is no canonical
/// form so the same number can be represented by many bit representations.
///
/// ScaledNumber is templated on the underlying integer type for digits, which
/// is expected to be unsigned.
///
/// Unlike APFloat, ScaledNumber does not model architecture floating point
/// behaviour -- while this might make it a little faster and easier to reason
/// about, it certainly makes it more dangerous for general numerics.
///
/// ScaledNumber is totally ordered.  However, there is no canonical form, so
/// there are multiple representations of most scalars.  E.g.:
///
///     ScaledNumber(8u, 0) == ScaledNumber(4u, 1)
///     ScaledNumber(4u, 1) == ScaledNumber(2u, 2)
///     ScaledNumber(2u, 2) == ScaledNumber(1u, 3)
///
/// ScaledNumber implements most arithmetic operations.  Precision is kept
/// where possible.  Uses simple saturation arithmetic, so that operations
/// saturate to 0.0 or getLargest() rather than under or overflowing.  It has
/// some extra arithmetic for unit inversion.  0.0/0.0 is defined to be 0.0.
/// Any other division by 0.0 is defined to be getLargest().
///
/// As a convenience for modifying the exponent, left and right shifting are
/// both implemented, and both interpret negative shifts as positive shifts in
/// the opposite direction.
///
/// Scales are limited to the range accepted by x87 long double.  This makes
/// it trivial to add functionality to convert to APFloat (this is already
/// relied on for the implementation of printing).
///
/// Possible (and conflicting) future directions:
///
///  1. Turn this into a wrapper around \a APFloat.
///  2. Share the algorithm implementations with \a APFloat.
///  3. Allow \a ScaledNumber to represent a signed number.
template <class DigitsT> class ScaledNumber : ScaledNumberBase {
public:
  static_assert(!std::numeric_limits<DigitsT>::is_signed,
                "only unsigned floats supported");
 
  typedef DigitsT DigitsType;
 
private:
  typedef std::numeric_limits<DigitsType> DigitsLimits;
 
  static constexpr int Width = sizeof(DigitsType) * 8;
  static_assert(Width <= 64, "invalid integer width for digits");
 
private:
  DigitsType Digits = 0;
  int16_t Scale = 0;
 
public:
  ScaledNumber() = default;
 
  constexpr ScaledNumber(DigitsType Digits, int16_t Scale)
      : Digits(Digits), Scale(Scale) {}
 
private:
  ScaledNumber(const std::pair<DigitsT, int16_t> &X)
      : Digits(X.first), Scale(X.second) {}
 
public:
  static ScaledNumber getZero() { return ScaledNumber(0, 0); }
  static ScaledNumber getOne() { return ScaledNumber(1, 0); }
  static ScaledNumber getLargest() {
    return ScaledNumber(DigitsLimits::max(), ScaledNumbers::MaxScale);
  }
  static ScaledNumber get(uint64_t N) { return adjustToWidth(N, 0); }
  static ScaledNumber getInverse(uint64_t N) {
    return get(N).invert();
  }
  static ScaledNumber getFraction(DigitsType N, DigitsType D) {
    return getQuotient(N, D);
  }
 
  int16_t getScale() const { return Scale; }
  DigitsType getDigits() const { return Digits; }
 
  /// Convert to the given integer type.
  ///
  /// Convert to \c IntT using simple saturating arithmetic, truncating if
  /// necessary.
  template <class IntT> IntT toInt() const;
 
  bool isZero() const { return !Digits; }
  bool isLargest() const { return *this == getLargest(); }
  bool isOne() const {
    if (Scale > 0 || Scale <= -Width)
      return false;
    return Digits == DigitsType(1) << -Scale;
  }
 
  /// The log base 2, rounded.
  ///
  /// Get the lg of the scalar.  lg 0 is defined to be INT32_MIN.
  int32_t lg() const { return ScaledNumbers::getLg(Digits, Scale); }
 
  /// The log base 2, rounded towards INT32_MIN.
  ///
  /// Get the lg floor.  lg 0 is defined to be INT32_MIN.
  int32_t lgFloor() const { return ScaledNumbers::getLgFloor(Digits, Scale); }
 
  /// The log base 2, rounded towards INT32_MAX.
  ///
  /// Get the lg ceiling.  lg 0 is defined to be INT32_MIN.
  int32_t lgCeiling() const {
    return ScaledNumbers::getLgCeiling(Digits, Scale);
  }
 
  bool operator==(const ScaledNumber &X) const { return compare(X) == 0; }
  bool operator<(const ScaledNumber &X) const { return compare(X) < 0; }
  bool operator!=(const ScaledNumber &X) const { return compare(X) != 0; }
  bool operator>(const ScaledNumber &X) const { return compare(X) > 0; }
  bool operator<=(const ScaledNumber &X) const { return compare(X) <= 0; }
  bool operator>=(const ScaledNumber &X) const { return compare(X) >= 0; }
 
  bool operator!() const { return isZero(); }
 
  /// Convert to a decimal representation in a string.
  ///
  /// Convert to a string.  Uses scientific notation for very large/small
  /// numbers.  Scientific notation is used roughly for numbers outside of the
  /// range 2^-64 through 2^64.
  ///
  /// \c Precision indicates the number of decimal digits of precision to use;
  /// 0 requests the maximum available.
  ///
  /// As a special case to make debugging easier, if the number is small enough
  /// to convert without scientific notation and has more than \c Precision
  /// digits before the decimal place, it's printed accurately to the first
  /// digit past zero.  E.g., assuming 10 digits of precision:
  ///
  ///     98765432198.7654... => 98765432198.8
  ///      8765432198.7654... =>  8765432198.8
  ///       765432198.7654... =>   765432198.8
  ///        65432198.7654... =>    65432198.77
  ///         5432198.7654... =>     5432198.765
  std::string toString(unsigned Precision = DefaultPrecision) {
    return ScaledNumberBase::toString(Digits, Scale, Width, Precision);
  }
 
  /// Print a decimal representation.
  ///
  /// Print a string.  See toString for documentation.
  raw_ostream &print(raw_ostream &OS,
                     unsigned Precision = DefaultPrecision) const {
    return ScaledNumberBase::print(OS, Digits, Scale, Width, Precision);
  }
  void dump() const { return ScaledNumberBase::dump(Digits, Scale, Width); }
 
  ScaledNumber &operator+=(const ScaledNumber &X) {
    std::tie(Digits, Scale) =
        ScaledNumbers::getSum(Digits, Scale, X.Digits, X.Scale);
    // Check for exponent past MaxScale.
    if (Scale > ScaledNumbers::MaxScale)
      *this = getLargest();
    return *this;
  }
  ScaledNumber &operator-=(const ScaledNumber &X) {
    std::tie(Digits, Scale) =
        ScaledNumbers::getDifference(Digits, Scale, X.Digits, X.Scale);
    return *this;
  }
  ScaledNumber &operator*=(const ScaledNumber &X);
  ScaledNumber &operator/=(const ScaledNumber &X);
  ScaledNumber &operator<<=(int16_t Shift) {
    shiftLeft(Shift);
    return *this;
  }
  ScaledNumber &operator>>=(int16_t Shift) {
    shiftRight(Shift);
    return *this;
  }
 
private:
  void shiftLeft(int32_t Shift);
  void shiftRight(int32_t Shift);
 
  /// Adjust two floats to have matching exponents.
  ///
  /// Adjust \c this and \c X to have matching exponents.  Returns the new \c X
  /// by value.  Does nothing if \a isZero() for either.
  ///
  /// The value that compares smaller will lose precision, and possibly become
  /// \a isZero().
  ScaledNumber matchScales(ScaledNumber X) {
    ScaledNumbers::matchScales(Digits, Scale, X.Digits, X.Scale);
    return X;
  }
 
public:
  /// Scale a large number accurately.
  ///
  /// Scale N (multiply it by this).  Uses full precision multiplication, even
  /// if Width is smaller than 64, so information is not lost.
  uint64_t scale(uint64_t N) const;
  uint64_t scaleByInverse(uint64_t N) const {
    // TODO: implement directly, rather than relying on inverse.  Inverse is
    // expensive.
    return inverse().scale(N);
  }
  int64_t scale(int64_t N) const {
    std::pair<uint64_t, bool> Unsigned = splitSigned(N);
    return joinSigned(scale(Unsigned.first), Unsigned.second);
  }
  int64_t scaleByInverse(int64_t N) const {
    std::pair<uint64_t, bool> Unsigned = splitSigned(N);
    return joinSigned(scaleByInverse(Unsigned.first), Unsigned.second);
  }
 
  int compare(const ScaledNumber &X) const {
    return ScaledNumbers::compare(Digits, Scale, X.Digits, X.Scale);
  }
  int compareTo(uint64_t N) const {
    return ScaledNumbers::compare<uint64_t>(Digits, Scale, N, 0);
  }
  int compareTo(int64_t N) const { return N < 0 ? 1 : compareTo(uint64_t(N)); }
 
  ScaledNumber &invert() { return *this = ScaledNumber::get(1) / *this; }
  ScaledNumber inverse() const { return ScaledNumber(*this).invert(); }
 
private:
  static ScaledNumber getProduct(DigitsType LHS, DigitsType RHS) {
    return ScaledNumbers::getProduct(LHS, RHS);
  }
  static ScaledNumber getQuotient(DigitsType Dividend, DigitsType Divisor) {
    return ScaledNumbers::getQuotient(Dividend, Divisor);
  }
 
  static int countLeadingZerosWidth(DigitsType Digits) {
    if (Width == 64)
      return countLeadingZeros64(Digits);
    if (Width == 32)
      return countLeadingZeros32(Digits);
    return countLeadingZeros32(Digits) + Width - 32;
  }
 
  /// Adjust a number to width, rounding up if necessary.
  ///
  /// Should only be called for \c Shift close to zero.
  ///
  /// \pre Shift >= MinScale && Shift + 64 <= MaxScale.
  static ScaledNumber adjustToWidth(uint64_t N, int32_t Shift) {
    assert(Shift >= ScaledNumbers::MinScale && "Shift should be close to 0");
    assert(Shift <= ScaledNumbers::MaxScale - 64 &&
           "Shift should be close to 0");
    auto Adjusted = ScaledNumbers::getAdjusted<DigitsT>(N, Shift);
    return Adjusted;
  }
 
  static ScaledNumber getRounded(ScaledNumber P, bool Round) {
    // Saturate.
    if (P.isLargest())
      return P;
 
    return ScaledNumbers::getRounded(P.Digits, P.Scale, Round);
  }
};
 
#define SCALED_NUMBER_BOP(op, base)                                            \
  template <class DigitsT>                                                     \
  ScaledNumber<DigitsT> operator op(const ScaledNumber<DigitsT> &L,            \
                                    const ScaledNumber<DigitsT> &R) {          \
    return ScaledNumber<DigitsT>(L) base R;                                    \
  }
SCALED_NUMBER_BOP(+, += )
SCALED_NUMBER_BOP(-, -= )
SCALED_NUMBER_BOP(*, *= )
SCALED_NUMBER_BOP(/, /= )
#undef SCALED_NUMBER_BOP
 
template <class DigitsT>
ScaledNumber<DigitsT> operator<<(const ScaledNumber<DigitsT> &L,
                                 int16_t Shift) {
  return ScaledNumber<DigitsT>(L) <<= Shift;
}
 
template <class DigitsT>
ScaledNumber<DigitsT> operator>>(const ScaledNumber<DigitsT> &L,
                                 int16_t Shift) {
  return ScaledNumber<DigitsT>(L) >>= Shift;
}
 
template <class DigitsT>
raw_ostream &operator<<(raw_ostream &OS, const ScaledNumber<DigitsT> &X) {
  return X.print(OS, 10);
}
 
#define SCALED_NUMBER_COMPARE_TO_TYPE(op, T1, T2)                              \
  template <class DigitsT>                                                     \
  bool operator op(const ScaledNumber<DigitsT> &L, T1 R) {                     \
    return L.compareTo(T2(R)) op 0;                                            \
  }                                                                            \
  template <class DigitsT>                                                     \
  bool operator op(T1 L, const ScaledNumber<DigitsT> &R) {                     \
    return 0 op R.compareTo(T2(L));                                            \
  }
#define SCALED_NUMBER_COMPARE_TO(op)                                           \
  SCALED_NUMBER_COMPARE_TO_TYPE(op, uint64_t, uint64_t)                        \
  SCALED_NUMBER_COMPARE_TO_TYPE(op, uint32_t, uint64_t)                        \
  SCALED_NUMBER_COMPARE_TO_TYPE(op, int64_t, int64_t)                          \
  SCALED_NUMBER_COMPARE_TO_TYPE(op, int32_t, int64_t)
SCALED_NUMBER_COMPARE_TO(< )
SCALED_NUMBER_COMPARE_TO(> )
SCALED_NUMBER_COMPARE_TO(== )
SCALED_NUMBER_COMPARE_TO(!= )
SCALED_NUMBER_COMPARE_TO(<= )
SCALED_NUMBER_COMPARE_TO(>= )
#undef SCALED_NUMBER_COMPARE_TO
#undef SCALED_NUMBER_COMPARE_TO_TYPE
 
template <class DigitsT>
uint64_t ScaledNumber<DigitsT>::scale(uint64_t N) const {
  if (Width == 64 || N <= DigitsLimits::max())
    return (get(N) * *this).template toInt<uint64_t>();
 
  // Defer to the 64-bit version.
  return ScaledNumber<uint64_t>(Digits, Scale).scale(N);
}
 
template <class DigitsT>
template <class IntT>
IntT ScaledNumber<DigitsT>::toInt() const {
  typedef std::numeric_limits<IntT> Limits;
  if (*this < 1)
    return 0;
  if (*this >= Limits::max())
    return Limits::max();
 
  IntT N = Digits;
  if (Scale > 0) {
    assert(size_t(Scale) < sizeof(IntT) * 8);
    return N << Scale;
  }
  if (Scale < 0) {
    assert(size_t(-Scale) < sizeof(IntT) * 8);
    return N >> -Scale;
  }
  return N;
}
 
template <class DigitsT>
ScaledNumber<DigitsT> &ScaledNumber<DigitsT>::
operator*=(const ScaledNumber &X) {
  if (isZero())
    return *this;
  if (X.isZero())
    return *this = X;
 
  // Save the exponents.
  int32_t Scales = int32_t(Scale) + int32_t(X.Scale);
 
  // Get the raw product.
  *this = getProduct(Digits, X.Digits);
 
  // Combine with exponents.
  return *this <<= Scales;
}
template <class DigitsT>
ScaledNumber<DigitsT> &ScaledNumber<DigitsT>::
operator/=(const ScaledNumber &X) {
  if (isZero())
    return *this;
  if (X.isZero())
    return *this = getLargest();
 
  // Save the exponents.
  int32_t Scales = int32_t(Scale) - int32_t(X.Scale);
 
  // Get the raw quotient.
  *this = getQuotient(Digits, X.Digits);
 
  // Combine with exponents.
  return *this <<= Scales;
}
template <class DigitsT> void ScaledNumber<DigitsT>::shiftLeft(int32_t Shift) {
  if (!Shift || isZero())
    return;
  assert(Shift != INT32_MIN);
  if (Shift < 0) {
    shiftRight(-Shift);
    return;
  }
 
  // Shift as much as we can in the exponent.
  int32_t ScaleShift = std::min(Shift, ScaledNumbers::MaxScale - Scale);
  Scale += ScaleShift;
  if (ScaleShift == Shift)
    return;
 
  // Check this late, since it's rare.
  if (isLargest())
    return;
 
  // Shift the digits themselves.
  Shift -= ScaleShift;
  if (Shift > countLeadingZerosWidth(Digits)) {
    // Saturate.
    *this = getLargest();
    return;
  }
 
  Digits <<= Shift;
}
 
template <class DigitsT> void ScaledNumber<DigitsT>::shiftRight(int32_t Shift) {
  if (!Shift || isZero())
    return;
  assert(Shift != INT32_MIN);
  if (Shift < 0) {
    shiftLeft(-Shift);
    return;
  }
 
  // Shift as much as we can in the exponent.
  int32_t ScaleShift = std::min(Shift, Scale - ScaledNumbers::MinScale);
  Scale -= ScaleShift;
  if (ScaleShift == Shift)
    return;
 
  // Shift the digits themselves.
  Shift -= ScaleShift;
  if (Shift >= Width) {
    // Saturate.
    *this = getZero();
    return;
  }
 
  Digits >>= Shift;
}
 
 
} // end namespace llvm
 
#endif // LLVM_SUPPORT_SCALEDNUMBER_H