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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/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.
 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.
 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.
 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.
 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;
 
   static void dump(uint64_t D, int16_t E, int Width);
   static raw_ostream &print(raw_ostream &OS, uint64_t D, int16_t E, int Width,
                             unsigned Precision);
   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

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