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 //===- llvm/ADT/APFloat.h - Arbitrary Precision Floating Point ---*- 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
 //
 //===----------------------------------------------------------------------===//
 ///
 /// \file
 /// This file declares a class to represent arbitrary precision floating point
 /// values and provide a variety of arithmetic operations on them.
 ///
 //===----------------------------------------------------------------------===//
 
 #ifndef LLVM_ADT_APFLOAT_H
 #define LLVM_ADT_APFLOAT_H
 
 #include "llvm/ADT/APInt.h"
 #include "llvm/ADT/ArrayRef.h"
 #include "llvm/ADT/FloatingPointMode.h"
 #include "llvm/Support/ErrorHandling.h"
 #include "llvm/Support/float128.h"
 #include <memory>
 
 #define APFLOAT_DISPATCH_ON_SEMANTICS(METHOD_CALL)                             \
   do {                                                                         \
     if (usesLayout<IEEEFloat>(getSemantics()))                                 \
       return U.IEEE.METHOD_CALL;                                               \
     if (usesLayout<DoubleAPFloat>(getSemantics()))                             \
       return U.Double.METHOD_CALL;                                             \
     llvm_unreachable("Unexpected semantics");                                  \
   } while (false)
 
 namespace llvm {
 
 struct fltSemantics;
 class APSInt;
 class StringRef;
 class APFloat;
 class raw_ostream;
 
 template <typename T> class Expected;
 template <typename T> class SmallVectorImpl;
 
 /// Enum that represents what fraction of the LSB truncated bits of an fp number
 /// represent.
 ///
 /// This essentially combines the roles of guard and sticky bits.
 enum lostFraction { // Example of truncated bits:
   lfExactlyZero,    // 000000
   lfLessThanHalf,   // 0xxxxx  x's not all zero
   lfExactlyHalf,    // 100000
   lfMoreThanHalf    // 1xxxxx  x's not all zero
 };
 
 /// A self-contained host- and target-independent arbitrary-precision
 /// floating-point software implementation.
 ///
 /// APFloat uses bignum integer arithmetic as provided by static functions in
 /// the APInt class.  The library will work with bignum integers whose parts are
 /// any unsigned type at least 16 bits wide, but 64 bits is recommended.
 ///
 /// Written for clarity rather than speed, in particular with a view to use in
 /// the front-end of a cross compiler so that target arithmetic can be correctly
 /// performed on the host.  Performance should nonetheless be reasonable,
 /// particularly for its intended use.  It may be useful as a base
 /// implementation for a run-time library during development of a faster
 /// target-specific one.
 ///
 /// All 5 rounding modes in the IEEE-754R draft are handled correctly for all
 /// implemented operations.  Currently implemented operations are add, subtract,
 /// multiply, divide, fused-multiply-add, conversion-to-float,
 /// conversion-to-integer and conversion-from-integer.  New rounding modes
 /// (e.g. away from zero) can be added with three or four lines of code.
 ///
 /// Four formats are built-in: IEEE single precision, double precision,
 /// quadruple precision, and x87 80-bit extended double (when operating with
 /// full extended precision).  Adding a new format that obeys IEEE semantics
 /// only requires adding two lines of code: a declaration and definition of the
 /// format.
 ///
 /// All operations return the status of that operation as an exception bit-mask,
 /// so multiple operations can be done consecutively with their results or-ed
 /// together.  The returned status can be useful for compiler diagnostics; e.g.,
 /// inexact, underflow and overflow can be easily diagnosed on constant folding,
 /// and compiler optimizers can determine what exceptions would be raised by
 /// folding operations and optimize, or perhaps not optimize, accordingly.
 ///
 /// At present, underflow tininess is detected after rounding; it should be
 /// straight forward to add support for the before-rounding case too.
 ///
 /// The library reads hexadecimal floating point numbers as per C99, and
 /// correctly rounds if necessary according to the specified rounding mode.
 /// Syntax is required to have been validated by the caller.  It also converts
 /// floating point numbers to hexadecimal text as per the C99 %a and %A
 /// conversions.  The output precision (or alternatively the natural minimal
 /// precision) can be specified; if the requested precision is less than the
 /// natural precision the output is correctly rounded for the specified rounding
 /// mode.
 ///
 /// It also reads decimal floating point numbers and correctly rounds according
 /// to the specified rounding mode.
 ///
 /// Conversion to decimal text is not currently implemented.
 ///
 /// Non-zero finite numbers are represented internally as a sign bit, a 16-bit
 /// signed exponent, and the significand as an array of integer parts.  After
 /// normalization of a number of precision P the exponent is within the range of
 /// the format, and if the number is not denormal the P-th bit of the
 /// significand is set as an explicit integer bit.  For denormals the most
 /// significant bit is shifted right so that the exponent is maintained at the
 /// format's minimum, so that the smallest denormal has just the least
 /// significant bit of the significand set.  The sign of zeroes and infinities
 /// is significant; the exponent and significand of such numbers is not stored,
 /// but has a known implicit (deterministic) value: 0 for the significands, 0
 /// for zero exponent, all 1 bits for infinity exponent.  For NaNs the sign and
 /// significand are deterministic, although not really meaningful, and preserved
 /// in non-conversion operations.  The exponent is implicitly all 1 bits.
 ///
 /// APFloat does not provide any exception handling beyond default exception
 /// handling. We represent Signaling NaNs via IEEE-754R 2008 6.2.1 should clause
 /// by encoding Signaling NaNs with the first bit of its trailing significand as
 /// 0.
 ///
 /// TODO
 /// ====
 ///
 /// Some features that may or may not be worth adding:
 ///
 /// Binary to decimal conversion (hard).
 ///
 /// Optional ability to detect underflow tininess before rounding.
 ///
 /// New formats: x87 in single and double precision mode (IEEE apart from
 /// extended exponent range) (hard).
 ///
 /// New operations: sqrt, IEEE remainder, C90 fmod, nexttoward.
 ///
 
 // This is the common type definitions shared by APFloat and its internal
 // implementation classes. This struct should not define any non-static data
 // members.
 struct APFloatBase {
   typedef APInt::WordType integerPart;
   static constexpr unsigned integerPartWidth = APInt::APINT_BITS_PER_WORD;
 
   /// A signed type to represent a floating point numbers unbiased exponent.
   typedef int32_t ExponentType;
 
   /// \name Floating Point Semantics.
   /// @{
   enum Semantics {
     S_IEEEhalf,
     S_BFloat,
     S_IEEEsingle,
     S_IEEEdouble,
     S_IEEEquad,
     // The IBM double-double semantics. Such a number consists of a pair of
     // IEEE 64-bit doubles (Hi, Lo), where |Hi| > |Lo|, and if normal,
     // (double)(Hi + Lo) == Hi. The numeric value it's modeling is Hi + Lo.
     // Therefore it has two 53-bit mantissa parts that aren't necessarily
     // adjacent to each other, and two 11-bit exponents.
     //
     // Note: we need to make the value different from semBogus as otherwise
     // an unsafe optimization may collapse both values to a single address,
     // and we heavily rely on them having distinct addresses.
     S_PPCDoubleDouble,
     // These are legacy semantics for the fallback, inaccurate implementation
     // of IBM double-double, if the accurate semPPCDoubleDouble doesn't handle
     // the operation. It's equivalent to having an IEEE number with consecutive
     // 106 bits of mantissa and 11 bits of exponent.
     //
     // It's not equivalent to IBM double-double. For example, a legit IBM
     // double-double, 1 + epsilon:
     //
     // 1 + epsilon = 1 + (1 >> 1076)
     //
     // is not representable by a consecutive 106 bits of mantissa.
     //
     // Currently, these semantics are used in the following way:
     //
     //   semPPCDoubleDouble -> (IEEEdouble, IEEEdouble) ->
     //   (64-bit APInt, 64-bit APInt) -> (128-bit APInt) ->
     //   semPPCDoubleDoubleLegacy -> IEEE operations
     //
     // We use bitcastToAPInt() to get the bit representation (in APInt) of the
     // underlying IEEEdouble, then use the APInt constructor to construct the
     // legacy IEEE float.
     //
     // TODO: Implement all operations in semPPCDoubleDouble, and delete these
     // semantics.
     S_PPCDoubleDoubleLegacy,
     // 8-bit floating point number following IEEE-754 conventions with bit
     // layout S1E5M2 as described in https://arxiv.org/abs/2209.05433.
     S_Float8E5M2,
     // 8-bit floating point number mostly following IEEE-754 conventions
     // and bit layout S1E5M2 described in https://arxiv.org/abs/2206.02915,
     // with expanded range and with no infinity or signed zero.
     // NaN is represented as negative zero. (FN -> Finite, UZ -> unsigned zero).
     // This format's exponent bias is 16, instead of the 15 (2 ** (5 - 1) - 1)
     // that IEEE precedent would imply.
     S_Float8E5M2FNUZ,
     // 8-bit floating point number following IEEE-754 conventions with bit
     // layout S1E4M3.
     S_Float8E4M3,
     // 8-bit floating point number mostly following IEEE-754 conventions with
     // bit layout S1E4M3 as described in https://arxiv.org/abs/2209.05433.
     // Unlike IEEE-754 types, there are no infinity values, and NaN is
     // represented with the exponent and mantissa bits set to all 1s.
     S_Float8E4M3FN,
     // 8-bit floating point number mostly following IEEE-754 conventions
     // and bit layout S1E4M3 described in https://arxiv.org/abs/2206.02915,
     // with expanded range and with no infinity or signed zero.
     // NaN is represented as negative zero. (FN -> Finite, UZ -> unsigned zero).
     // This format's exponent bias is 8, instead of the 7 (2 ** (4 - 1) - 1)
     // that IEEE precedent would imply.
     S_Float8E4M3FNUZ,
     // 8-bit floating point number mostly following IEEE-754 conventions
     // and bit layout S1E4M3 with expanded range and with no infinity or signed
     // zero.
     // NaN is represented as negative zero. (FN -> Finite, UZ -> unsigned zero).
     // This format's exponent bias is 11, instead of the 7 (2 ** (4 - 1) - 1)
     // that IEEE precedent would imply.
     S_Float8E4M3B11FNUZ,
     // 8-bit floating point number following IEEE-754 conventions with bit
     // layout S1E3M4.
     S_Float8E3M4,
     // Floating point number that occupies 32 bits or less of storage, providing
     // improved range compared to half (16-bit) formats, at (potentially)
     // greater throughput than single precision (32-bit) formats.
     S_FloatTF32,
     // 8-bit floating point number with (all the) 8 bits for the exponent
     // like in FP32. There are no zeroes, no infinities, and no denormal values.
     // This format has unsigned representation only. (U -> Unsigned only).
     // NaN is represented with all bits set to 1. Bias is 127.
     // This format represents the scale data type in the MX specification from:
     // https://www.opencompute.org/documents/ocp-microscaling-formats-mx-v1-0-spec-final-pdf
     S_Float8E8M0FNU,
     // 6-bit floating point number with bit layout S1E3M2. Unlike IEEE-754
     // types, there are no infinity or NaN values. The format is detailed in
     // https://www.opencompute.org/documents/ocp-microscaling-formats-mx-v1-0-spec-final-pdf
     S_Float6E3M2FN,
     // 6-bit floating point number with bit layout S1E2M3. Unlike IEEE-754
     // types, there are no infinity or NaN values. The format is detailed in
     // https://www.opencompute.org/documents/ocp-microscaling-formats-mx-v1-0-spec-final-pdf
     S_Float6E2M3FN,
     // 4-bit floating point number with bit layout S1E2M1. Unlike IEEE-754
     // types, there are no infinity or NaN values. The format is detailed in
     // https://www.opencompute.org/documents/ocp-microscaling-formats-mx-v1-0-spec-final-pdf
     S_Float4E2M1FN,
     // TODO: Documentation is missing.
     S_x87DoubleExtended,
     S_MaxSemantics = S_x87DoubleExtended,
   };
 
   static const llvm::fltSemantics &EnumToSemantics(Semantics S);
   static Semantics SemanticsToEnum(const llvm::fltSemantics &Sem);
 
   static const fltSemantics &IEEEhalf() LLVM_READNONE;
   static const fltSemantics &BFloat() LLVM_READNONE;
   static const fltSemantics &IEEEsingle() LLVM_READNONE;
   static const fltSemantics &IEEEdouble() LLVM_READNONE;
   static const fltSemantics &IEEEquad() LLVM_READNONE;
   static const fltSemantics &PPCDoubleDouble() LLVM_READNONE;
   static const fltSemantics &PPCDoubleDoubleLegacy() LLVM_READNONE;
   static const fltSemantics &Float8E5M2() LLVM_READNONE;
   static const fltSemantics &Float8E5M2FNUZ() LLVM_READNONE;
   static const fltSemantics &Float8E4M3() LLVM_READNONE;
   static const fltSemantics &Float8E4M3FN() LLVM_READNONE;
   static const fltSemantics &Float8E4M3FNUZ() LLVM_READNONE;
   static const fltSemantics &Float8E4M3B11FNUZ() LLVM_READNONE;
   static const fltSemantics &Float8E3M4() LLVM_READNONE;
   static const fltSemantics &FloatTF32() LLVM_READNONE;
   static const fltSemantics &Float8E8M0FNU() LLVM_READNONE;
   static const fltSemantics &Float6E3M2FN() LLVM_READNONE;
   static const fltSemantics &Float6E2M3FN() LLVM_READNONE;
   static const fltSemantics &Float4E2M1FN() LLVM_READNONE;
   static const fltSemantics &x87DoubleExtended() LLVM_READNONE;
 
   /// A Pseudo fltsemantic used to construct APFloats that cannot conflict with
   /// anything real.
   static const fltSemantics &Bogus() LLVM_READNONE;
 
   // Returns true if any number described by this semantics can be precisely
   // represented by the specified semantics. Does not take into account
   // the value of fltNonfiniteBehavior, hasZero, hasSignedRepr.
   static bool isRepresentableBy(const fltSemantics &A, const fltSemantics &B);
 
   /// @}
 
   /// IEEE-754R 5.11: Floating Point Comparison Relations.
   enum cmpResult {
     cmpLessThan,
     cmpEqual,
     cmpGreaterThan,
     cmpUnordered
   };
 
   /// IEEE-754R 4.3: Rounding-direction attributes.
   using roundingMode = llvm::RoundingMode;
 
   static constexpr roundingMode rmNearestTiesToEven =
                                                 RoundingMode::NearestTiesToEven;
   static constexpr roundingMode rmTowardPositive = RoundingMode::TowardPositive;
   static constexpr roundingMode rmTowardNegative = RoundingMode::TowardNegative;
   static constexpr roundingMode rmTowardZero     = RoundingMode::TowardZero;
   static constexpr roundingMode rmNearestTiesToAway =
                                                 RoundingMode::NearestTiesToAway;
 
   /// IEEE-754R 7: Default exception handling.
   ///
   /// opUnderflow or opOverflow are always returned or-ed with opInexact.
   ///
   /// APFloat models this behavior specified by IEEE-754:
   ///   "For operations producing results in floating-point format, the default
   ///    result of an operation that signals the invalid operation exception
   ///    shall be a quiet NaN."
   enum opStatus {
     opOK = 0x00,
     opInvalidOp = 0x01,
     opDivByZero = 0x02,
     opOverflow = 0x04,
     opUnderflow = 0x08,
     opInexact = 0x10
   };
 
   /// Category of internally-represented number.
   enum fltCategory {
     fcInfinity,
     fcNaN,
     fcNormal,
     fcZero
   };
 
   /// Convenience enum used to construct an uninitialized APFloat.
   enum uninitializedTag {
     uninitialized
   };
 
   /// Enumeration of \c ilogb error results.
   enum IlogbErrorKinds {
     IEK_Zero = INT_MIN + 1,
     IEK_NaN = INT_MIN,
     IEK_Inf = INT_MAX
   };
 
   static unsigned int semanticsPrecision(const fltSemantics &);
   static ExponentType semanticsMinExponent(const fltSemantics &);
   static ExponentType semanticsMaxExponent(const fltSemantics &);
   static unsigned int semanticsSizeInBits(const fltSemantics &);
   static unsigned int semanticsIntSizeInBits(const fltSemantics&, bool);
   static bool semanticsHasZero(const fltSemantics &);
   static bool semanticsHasSignedRepr(const fltSemantics &);
   static bool semanticsHasInf(const fltSemantics &);
   static bool semanticsHasNaN(const fltSemantics &);
   static bool isIEEELikeFP(const fltSemantics &);
 
   // Returns true if any number described by \p Src can be precisely represented
   // by a normal (not subnormal) value in \p Dst.
   static bool isRepresentableAsNormalIn(const fltSemantics &Src,
                                         const fltSemantics &Dst);
 
   /// Returns the size of the floating point number (in bits) in the given
   /// semantics.
   static unsigned getSizeInBits(const fltSemantics &Sem);
 };
 
 namespace detail {
 
 using integerPart = APFloatBase::integerPart;
 using uninitializedTag = APFloatBase::uninitializedTag;
 using roundingMode = APFloatBase::roundingMode;
 using opStatus = APFloatBase::opStatus;
 using cmpResult = APFloatBase::cmpResult;
 using fltCategory = APFloatBase::fltCategory;
 using ExponentType = APFloatBase::ExponentType;
 static constexpr uninitializedTag uninitialized = APFloatBase::uninitialized;
 static constexpr roundingMode rmNearestTiesToEven =
     APFloatBase::rmNearestTiesToEven;
 static constexpr roundingMode rmNearestTiesToAway =
     APFloatBase::rmNearestTiesToAway;
 static constexpr roundingMode rmTowardNegative = APFloatBase::rmTowardNegative;
 static constexpr roundingMode rmTowardPositive = APFloatBase::rmTowardPositive;
 static constexpr roundingMode rmTowardZero = APFloatBase::rmTowardZero;
 static constexpr unsigned integerPartWidth = APFloatBase::integerPartWidth;
 static constexpr cmpResult cmpEqual = APFloatBase::cmpEqual;
 static constexpr cmpResult cmpLessThan = APFloatBase::cmpLessThan;
 static constexpr cmpResult cmpGreaterThan = APFloatBase::cmpGreaterThan;
 static constexpr cmpResult cmpUnordered = APFloatBase::cmpUnordered;
 static constexpr opStatus opOK = APFloatBase::opOK;
 static constexpr opStatus opInvalidOp = APFloatBase::opInvalidOp;
 static constexpr opStatus opDivByZero = APFloatBase::opDivByZero;
 static constexpr opStatus opOverflow = APFloatBase::opOverflow;
 static constexpr opStatus opUnderflow = APFloatBase::opUnderflow;
 static constexpr opStatus opInexact = APFloatBase::opInexact;
 static constexpr fltCategory fcInfinity = APFloatBase::fcInfinity;
 static constexpr fltCategory fcNaN = APFloatBase::fcNaN;
 static constexpr fltCategory fcNormal = APFloatBase::fcNormal;
 static constexpr fltCategory fcZero = APFloatBase::fcZero;
 
 class IEEEFloat final {
 public:
   /// \name Constructors
   /// @{
 
   IEEEFloat(const fltSemantics &); // Default construct to +0.0
   IEEEFloat(const fltSemantics &, integerPart);
   IEEEFloat(const fltSemantics &, uninitializedTag);
   IEEEFloat(const fltSemantics &, const APInt &);
   explicit IEEEFloat(double d);
   explicit IEEEFloat(float f);
   IEEEFloat(const IEEEFloat &);
   IEEEFloat(IEEEFloat &&);
   ~IEEEFloat();
 
   /// @}
 
   /// Returns whether this instance allocated memory.
   bool needsCleanup() const { return partCount() > 1; }
 
   /// \name Convenience "constructors"
   /// @{
 
   /// @}
 
   /// \name Arithmetic
   /// @{
 
   opStatus add(const IEEEFloat &, roundingMode);
   opStatus subtract(const IEEEFloat &, roundingMode);
   opStatus multiply(const IEEEFloat &, roundingMode);
   opStatus divide(const IEEEFloat &, roundingMode);
   /// IEEE remainder.
   opStatus remainder(const IEEEFloat &);
   /// C fmod, or llvm frem.
   opStatus mod(const IEEEFloat &);
   opStatus fusedMultiplyAdd(const IEEEFloat &, const IEEEFloat &, roundingMode);
   opStatus roundToIntegral(roundingMode);
   /// IEEE-754R 5.3.1: nextUp/nextDown.
   opStatus next(bool nextDown);
 
   /// @}
 
   /// \name Sign operations.
   /// @{
 
   void changeSign();
 
   /// @}
 
   /// \name Conversions
   /// @{
 
   opStatus convert(const fltSemantics &, roundingMode, bool *);
   opStatus convertToInteger(MutableArrayRef<integerPart>, unsigned int, bool,
                             roundingMode, bool *) const;
   opStatus convertFromAPInt(const APInt &, bool, roundingMode);
   opStatus convertFromSignExtendedInteger(const integerPart *, unsigned int,
                                           bool, roundingMode);
   opStatus convertFromZeroExtendedInteger(const integerPart *, unsigned int,
                                           bool, roundingMode);
   Expected<opStatus> convertFromString(StringRef, roundingMode);
   APInt bitcastToAPInt() const;
   double convertToDouble() const;
 #ifdef HAS_IEE754_FLOAT128
   float128 convertToQuad() const;
 #endif
   float convertToFloat() const;
 
   /// @}
 
   /// The definition of equality is not straightforward for floating point, so
   /// we won't use operator==.  Use one of the following, or write whatever it
   /// is you really mean.
   bool operator==(const IEEEFloat &) const = delete;
 
   /// IEEE comparison with another floating point number (NaNs compare
   /// unordered, 0==-0).
   cmpResult compare(const IEEEFloat &) const;
 
   /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
   bool bitwiseIsEqual(const IEEEFloat &) const;
 
   /// Write out a hexadecimal representation of the floating point value to DST,
   /// which must be of sufficient size, in the C99 form [-]0xh.hhhhp[+-]d.
   /// Return the number of characters written, excluding the terminating NUL.
   unsigned int convertToHexString(char *dst, unsigned int hexDigits,
                                   bool upperCase, roundingMode) const;
 
   /// \name IEEE-754R 5.7.2 General operations.
   /// @{
 
   /// IEEE-754R isSignMinus: Returns true if and only if the current value is
   /// negative.
   ///
   /// This applies to zeros and NaNs as well.
   bool isNegative() const { return sign; }
 
   /// IEEE-754R isNormal: Returns true if and only if the current value is normal.
   ///
   /// This implies that the current value of the float is not zero, subnormal,
   /// infinite, or NaN following the definition of normality from IEEE-754R.
   bool isNormal() const { return !isDenormal() && isFiniteNonZero(); }
 
   /// Returns true if and only if the current value is zero, subnormal, or
   /// normal.
   ///
   /// This means that the value is not infinite or NaN.
   bool isFinite() const { return !isNaN() && !isInfinity(); }
 
   /// Returns true if and only if the float is plus or minus zero.
   bool isZero() const { return category == fltCategory::fcZero; }
 
   /// IEEE-754R isSubnormal(): Returns true if and only if the float is a
   /// denormal.
   bool isDenormal() const;
 
   /// IEEE-754R isInfinite(): Returns true if and only if the float is infinity.
   bool isInfinity() const { return category == fcInfinity; }
 
   /// Returns true if and only if the float is a quiet or signaling NaN.
   bool isNaN() const { return category == fcNaN; }
 
   /// Returns true if and only if the float is a signaling NaN.
   bool isSignaling() const;
 
   /// @}
 
   /// \name Simple Queries
   /// @{
 
   fltCategory getCategory() const { return category; }
   const fltSemantics &getSemantics() const { return *semantics; }
   bool isNonZero() const { return category != fltCategory::fcZero; }
   bool isFiniteNonZero() const { return isFinite() && !isZero(); }
   bool isPosZero() const { return isZero() && !isNegative(); }
   bool isNegZero() const { return isZero() && isNegative(); }
 
   /// Returns true if and only if the number has the smallest possible non-zero
   /// magnitude in the current semantics.
   bool isSmallest() const;
 
   /// Returns true if this is the smallest (by magnitude) normalized finite
   /// number in the given semantics.
   bool isSmallestNormalized() const;
 
   /// Returns true if and only if the number has the largest possible finite
   /// magnitude in the current semantics.
   bool isLargest() const;
 
   /// Returns true if and only if the number is an exact integer.
   bool isInteger() const;
 
   /// @}
 
   IEEEFloat &operator=(const IEEEFloat &);
   IEEEFloat &operator=(IEEEFloat &&);
 
   /// Overload to compute a hash code for an APFloat value.
   ///
   /// Note that the use of hash codes for floating point values is in general
   /// frought with peril. Equality is hard to define for these values. For
   /// example, should negative and positive zero hash to different codes? Are
   /// they equal or not? This hash value implementation specifically
   /// emphasizes producing different codes for different inputs in order to
   /// be used in canonicalization and memoization. As such, equality is
   /// bitwiseIsEqual, and 0 != -0.
   friend hash_code hash_value(const IEEEFloat &Arg);
 
   /// Converts this value into a decimal string.
   ///
   /// \param FormatPrecision The maximum number of digits of
   ///   precision to output.  If there are fewer digits available,
   ///   zero padding will not be used unless the value is
   ///   integral and small enough to be expressed in
   ///   FormatPrecision digits.  0 means to use the natural
   ///   precision of the number.
   /// \param FormatMaxPadding The maximum number of zeros to
   ///   consider inserting before falling back to scientific
   ///   notation.  0 means to always use scientific notation.
   ///
   /// \param TruncateZero Indicate whether to remove the trailing zero in
   ///   fraction part or not. Also setting this parameter to false forcing
   ///   producing of output more similar to default printf behavior.
   ///   Specifically the lower e is used as exponent delimiter and exponent
   ///   always contains no less than two digits.
   ///
   /// Number       Precision    MaxPadding      Result
   /// ------       ---------    ----------      ------
   /// 1.01E+4              5             2       10100
   /// 1.01E+4              4             2       1.01E+4
   /// 1.01E+4              5             1       1.01E+4
   /// 1.01E-2              5             2       0.0101
   /// 1.01E-2              4             2       0.0101
   /// 1.01E-2              4             1       1.01E-2
   void toString(SmallVectorImpl<char> &Str, unsigned FormatPrecision = 0,
                 unsigned FormatMaxPadding = 3, bool TruncateZero = true) const;
 
   /// If this value has an exact multiplicative inverse, store it in inv and
   /// return true.
   bool getExactInverse(APFloat *inv) const;
 
   // If this is an exact power of two, return the exponent while ignoring the
   // sign bit. If it's not an exact power of 2, return INT_MIN
   LLVM_READONLY
   int getExactLog2Abs() const;
 
   // If this is an exact power of two, return the exponent. If it's not an exact
   // power of 2, return INT_MIN
   LLVM_READONLY
   int getExactLog2() const {
     return isNegative() ? INT_MIN : getExactLog2Abs();
   }
 
   /// Returns the exponent of the internal representation of the APFloat.
   ///
   /// Because the radix of APFloat is 2, this is equivalent to floor(log2(x)).
   /// For special APFloat values, this returns special error codes:
   ///
   ///   NaN -> \c IEK_NaN
   ///   0   -> \c IEK_Zero
   ///   Inf -> \c IEK_Inf
   ///
   friend int ilogb(const IEEEFloat &Arg);
 
   /// Returns: X * 2^Exp for integral exponents.
   friend IEEEFloat scalbn(IEEEFloat X, int Exp, roundingMode);
 
   friend IEEEFloat frexp(const IEEEFloat &X, int &Exp, roundingMode);
 
   /// \name Special value setters.
   /// @{
 
   void makeLargest(bool Neg = false);
   void makeSmallest(bool Neg = false);
   void makeNaN(bool SNaN = false, bool Neg = false,
                const APInt *fill = nullptr);
   void makeInf(bool Neg = false);
   void makeZero(bool Neg = false);
   void makeQuiet();
 
   /// Returns the smallest (by magnitude) normalized finite number in the given
   /// semantics.
   ///
   /// \param Negative - True iff the number should be negative
   void makeSmallestNormalized(bool Negative = false);
 
   /// @}
 
   cmpResult compareAbsoluteValue(const IEEEFloat &) const;
 
 private:
   /// \name Simple Queries
   /// @{
 
   integerPart *significandParts();
   const integerPart *significandParts() const;
   unsigned int partCount() const;
 
   /// @}
 
   /// \name Significand operations.
   /// @{
 
   integerPart addSignificand(const IEEEFloat &);
   integerPart subtractSignificand(const IEEEFloat &, integerPart);
   lostFraction addOrSubtractSignificand(const IEEEFloat &, bool subtract);
   lostFraction multiplySignificand(const IEEEFloat &, IEEEFloat,
                                    bool ignoreAddend = false);
   lostFraction multiplySignificand(const IEEEFloat&);
   lostFraction divideSignificand(const IEEEFloat &);
   void incrementSignificand();
   void initialize(const fltSemantics *);
   void shiftSignificandLeft(unsigned int);
   lostFraction shiftSignificandRight(unsigned int);
   unsigned int significandLSB() const;
   unsigned int significandMSB() const;
   void zeroSignificand();
   unsigned int getNumHighBits() const;
   /// Return true if the significand excluding the integral bit is all ones.
   bool isSignificandAllOnes() const;
   bool isSignificandAllOnesExceptLSB() const;
   /// Return true if the significand excluding the integral bit is all zeros.
   bool isSignificandAllZeros() const;
   bool isSignificandAllZerosExceptMSB() const;
 
   /// @}
 
   /// \name Arithmetic on special values.
   /// @{
 
   opStatus addOrSubtractSpecials(const IEEEFloat &, bool subtract);
   opStatus divideSpecials(const IEEEFloat &);
   opStatus multiplySpecials(const IEEEFloat &);
   opStatus modSpecials(const IEEEFloat &);
   opStatus remainderSpecials(const IEEEFloat&);
 
   /// @}
 
   /// \name Miscellany
   /// @{
 
   bool convertFromStringSpecials(StringRef str);
   opStatus normalize(roundingMode, lostFraction);
   opStatus addOrSubtract(const IEEEFloat &, roundingMode, bool subtract);
   opStatus handleOverflow(roundingMode);
   bool roundAwayFromZero(roundingMode, lostFraction, unsigned int) const;
   opStatus convertToSignExtendedInteger(MutableArrayRef<integerPart>,
                                         unsigned int, bool, roundingMode,
                                         bool *) const;
   opStatus convertFromUnsignedParts(const integerPart *, unsigned int,
                                     roundingMode);
   Expected<opStatus> convertFromHexadecimalString(StringRef, roundingMode);
   Expected<opStatus> convertFromDecimalString(StringRef, roundingMode);
   char *convertNormalToHexString(char *, unsigned int, bool,
                                  roundingMode) const;
   opStatus roundSignificandWithExponent(const integerPart *, unsigned int, int,
                                         roundingMode);
   ExponentType exponentNaN() const;
   ExponentType exponentInf() const;
   ExponentType exponentZero() const;
 
   /// @}
 
   template <const fltSemantics &S> APInt convertIEEEFloatToAPInt() const;
   APInt convertHalfAPFloatToAPInt() const;
   APInt convertBFloatAPFloatToAPInt() const;
   APInt convertFloatAPFloatToAPInt() const;
   APInt convertDoubleAPFloatToAPInt() const;
   APInt convertQuadrupleAPFloatToAPInt() const;
   APInt convertF80LongDoubleAPFloatToAPInt() const;
   APInt convertPPCDoubleDoubleLegacyAPFloatToAPInt() const;
   APInt convertFloat8E5M2APFloatToAPInt() const;
   APInt convertFloat8E5M2FNUZAPFloatToAPInt() const;
   APInt convertFloat8E4M3APFloatToAPInt() const;
   APInt convertFloat8E4M3FNAPFloatToAPInt() const;
   APInt convertFloat8E4M3FNUZAPFloatToAPInt() const;
   APInt convertFloat8E4M3B11FNUZAPFloatToAPInt() const;
   APInt convertFloat8E3M4APFloatToAPInt() const;
   APInt convertFloatTF32APFloatToAPInt() const;
   APInt convertFloat8E8M0FNUAPFloatToAPInt() const;
   APInt convertFloat6E3M2FNAPFloatToAPInt() const;
   APInt convertFloat6E2M3FNAPFloatToAPInt() const;
   APInt convertFloat4E2M1FNAPFloatToAPInt() const;
   void initFromAPInt(const fltSemantics *Sem, const APInt &api);
   template <const fltSemantics &S> void initFromIEEEAPInt(const APInt &api);
   void initFromHalfAPInt(const APInt &api);
   void initFromBFloatAPInt(const APInt &api);
   void initFromFloatAPInt(const APInt &api);
   void initFromDoubleAPInt(const APInt &api);
   void initFromQuadrupleAPInt(const APInt &api);
   void initFromF80LongDoubleAPInt(const APInt &api);
   void initFromPPCDoubleDoubleLegacyAPInt(const APInt &api);
   void initFromFloat8E5M2APInt(const APInt &api);
   void initFromFloat8E5M2FNUZAPInt(const APInt &api);
   void initFromFloat8E4M3APInt(const APInt &api);
   void initFromFloat8E4M3FNAPInt(const APInt &api);
   void initFromFloat8E4M3FNUZAPInt(const APInt &api);
   void initFromFloat8E4M3B11FNUZAPInt(const APInt &api);
   void initFromFloat8E3M4APInt(const APInt &api);
   void initFromFloatTF32APInt(const APInt &api);
   void initFromFloat8E8M0FNUAPInt(const APInt &api);
   void initFromFloat6E3M2FNAPInt(const APInt &api);
   void initFromFloat6E2M3FNAPInt(const APInt &api);
   void initFromFloat4E2M1FNAPInt(const APInt &api);
 
   void assign(const IEEEFloat &);
   void copySignificand(const IEEEFloat &);
   void freeSignificand();
 
   /// Note: this must be the first data member.
   /// The semantics that this value obeys.
   const fltSemantics *semantics;
 
   /// A binary fraction with an explicit integer bit.
   ///
   /// The significand must be at least one bit wider than the target precision.
   union Significand {
     integerPart part;
     integerPart *parts;
   } significand;
 
   /// The signed unbiased exponent of the value.
   ExponentType exponent;
 
   /// What kind of floating point number this is.
   ///
   /// Only 2 bits are required, but VisualStudio incorrectly sign extends it.
   /// Using the extra bit keeps it from failing under VisualStudio.
   fltCategory category : 3;
 
   /// Sign bit of the number.
   unsigned int sign : 1;
 };
 
 hash_code hash_value(const IEEEFloat &Arg);
 int ilogb(const IEEEFloat &Arg);
 IEEEFloat scalbn(IEEEFloat X, int Exp, roundingMode);
 IEEEFloat frexp(const IEEEFloat &Val, int &Exp, roundingMode RM);
 
 // This mode implements more precise float in terms of two APFloats.
 // The interface and layout is designed for arbitrary underlying semantics,
 // though currently only PPCDoubleDouble semantics are supported, whose
 // corresponding underlying semantics are IEEEdouble.
 class DoubleAPFloat final {
   // Note: this must be the first data member.
   const fltSemantics *Semantics;
   std::unique_ptr<APFloat[]> Floats;
 
   opStatus addImpl(const APFloat &a, const APFloat &aa, const APFloat &c,
                    const APFloat &cc, roundingMode RM);
 
   opStatus addWithSpecial(const DoubleAPFloat &LHS, const DoubleAPFloat &RHS,
                           DoubleAPFloat &Out, roundingMode RM);
 
 public:
   DoubleAPFloat(const fltSemantics &S);
   DoubleAPFloat(const fltSemantics &S, uninitializedTag);
   DoubleAPFloat(const fltSemantics &S, integerPart);
   DoubleAPFloat(const fltSemantics &S, const APInt &I);
   DoubleAPFloat(const fltSemantics &S, APFloat &&First, APFloat &&Second);
   DoubleAPFloat(const DoubleAPFloat &RHS);
   DoubleAPFloat(DoubleAPFloat &&RHS);
 
   DoubleAPFloat &operator=(const DoubleAPFloat &RHS);
   inline DoubleAPFloat &operator=(DoubleAPFloat &&RHS);
 
   bool needsCleanup() const { return Floats != nullptr; }
 
   inline APFloat &getFirst();
   inline const APFloat &getFirst() const;
   inline APFloat &getSecond();
   inline const APFloat &getSecond() const;
 
   opStatus add(const DoubleAPFloat &RHS, roundingMode RM);
   opStatus subtract(const DoubleAPFloat &RHS, roundingMode RM);
   opStatus multiply(const DoubleAPFloat &RHS, roundingMode RM);
   opStatus divide(const DoubleAPFloat &RHS, roundingMode RM);
   opStatus remainder(const DoubleAPFloat &RHS);
   opStatus mod(const DoubleAPFloat &RHS);
   opStatus fusedMultiplyAdd(const DoubleAPFloat &Multiplicand,
                             const DoubleAPFloat &Addend, roundingMode RM);
   opStatus roundToIntegral(roundingMode RM);
   void changeSign();
   cmpResult compareAbsoluteValue(const DoubleAPFloat &RHS) const;
 
   fltCategory getCategory() const;
   bool isNegative() const;
 
   void makeInf(bool Neg);
   void makeZero(bool Neg);
   void makeLargest(bool Neg);
   void makeSmallest(bool Neg);
   void makeSmallestNormalized(bool Neg);
   void makeNaN(bool SNaN, bool Neg, const APInt *fill);
 
   cmpResult compare(const DoubleAPFloat &RHS) const;
   bool bitwiseIsEqual(const DoubleAPFloat &RHS) const;
   APInt bitcastToAPInt() const;
   Expected<opStatus> convertFromString(StringRef, roundingMode);
   opStatus next(bool nextDown);
 
   opStatus convertToInteger(MutableArrayRef<integerPart> Input,
                             unsigned int Width, bool IsSigned, roundingMode RM,
                             bool *IsExact) const;
   opStatus convertFromAPInt(const APInt &Input, bool IsSigned, roundingMode RM);
   opStatus convertFromSignExtendedInteger(const integerPart *Input,
                                           unsigned int InputSize, bool IsSigned,
                                           roundingMode RM);
   opStatus convertFromZeroExtendedInteger(const integerPart *Input,
                                           unsigned int InputSize, bool IsSigned,
                                           roundingMode RM);
   unsigned int convertToHexString(char *DST, unsigned int HexDigits,
                                   bool UpperCase, roundingMode RM) const;
 
   bool isDenormal() const;
   bool isSmallest() const;
   bool isSmallestNormalized() const;
   bool isLargest() const;
   bool isInteger() const;
 
   void toString(SmallVectorImpl<char> &Str, unsigned FormatPrecision,
                 unsigned FormatMaxPadding, bool TruncateZero = true) const;
 
   bool getExactInverse(APFloat *inv) const;
 
   LLVM_READONLY
   int getExactLog2() const;
   LLVM_READONLY
   int getExactLog2Abs() const;
 
   friend DoubleAPFloat scalbn(const DoubleAPFloat &X, int Exp, roundingMode);
   friend DoubleAPFloat frexp(const DoubleAPFloat &X, int &Exp, roundingMode);
   friend hash_code hash_value(const DoubleAPFloat &Arg);
 };
 
 hash_code hash_value(const DoubleAPFloat &Arg);
 DoubleAPFloat scalbn(const DoubleAPFloat &Arg, int Exp, roundingMode RM);
 DoubleAPFloat frexp(const DoubleAPFloat &X, int &Exp, roundingMode);
 
 } // End detail namespace
 
 // This is a interface class that is currently forwarding functionalities from
 // detail::IEEEFloat.
 class APFloat : public APFloatBase {
   typedef detail::IEEEFloat IEEEFloat;
   typedef detail::DoubleAPFloat DoubleAPFloat;
 
   static_assert(std::is_standard_layout<IEEEFloat>::value);
 
   union Storage {
     const fltSemantics *semantics;
     IEEEFloat IEEE;
     DoubleAPFloat Double;
 
     explicit Storage(IEEEFloat F, const fltSemantics &S);
     explicit Storage(DoubleAPFloat F, const fltSemantics &S)
         : Double(std::move(F)) {
       assert(&S == &PPCDoubleDouble());
     }
 
     template <typename... ArgTypes>
     Storage(const fltSemantics &Semantics, ArgTypes &&... Args) {
       if (usesLayout<IEEEFloat>(Semantics)) {
         new (&IEEE) IEEEFloat(Semantics, std::forward<ArgTypes>(Args)...);
         return;
       }
       if (usesLayout<DoubleAPFloat>(Semantics)) {
         new (&Double) DoubleAPFloat(Semantics, std::forward<ArgTypes>(Args)...);
         return;
       }
       llvm_unreachable("Unexpected semantics");
     }
 
     ~Storage() {
       if (usesLayout<IEEEFloat>(*semantics)) {
         IEEE.~IEEEFloat();
         return;
       }
       if (usesLayout<DoubleAPFloat>(*semantics)) {
         Double.~DoubleAPFloat();
         return;
       }
       llvm_unreachable("Unexpected semantics");
     }
 
     Storage(const Storage &RHS) {
       if (usesLayout<IEEEFloat>(*RHS.semantics)) {
         new (this) IEEEFloat(RHS.IEEE);
         return;
       }
       if (usesLayout<DoubleAPFloat>(*RHS.semantics)) {
         new (this) DoubleAPFloat(RHS.Double);
         return;
       }
       llvm_unreachable("Unexpected semantics");
     }
 
     Storage(Storage &&RHS) {
       if (usesLayout<IEEEFloat>(*RHS.semantics)) {
         new (this) IEEEFloat(std::move(RHS.IEEE));
         return;
       }
       if (usesLayout<DoubleAPFloat>(*RHS.semantics)) {
         new (this) DoubleAPFloat(std::move(RHS.Double));
         return;
       }
       llvm_unreachable("Unexpected semantics");
     }
 
     Storage &operator=(const Storage &RHS) {
       if (usesLayout<IEEEFloat>(*semantics) &&
           usesLayout<IEEEFloat>(*RHS.semantics)) {
         IEEE = RHS.IEEE;
       } else if (usesLayout<DoubleAPFloat>(*semantics) &&
                  usesLayout<DoubleAPFloat>(*RHS.semantics)) {
         Double = RHS.Double;
       } else if (this != &RHS) {
         this->~Storage();
         new (this) Storage(RHS);
       }
       return *this;
     }
 
     Storage &operator=(Storage &&RHS) {
       if (usesLayout<IEEEFloat>(*semantics) &&
           usesLayout<IEEEFloat>(*RHS.semantics)) {
         IEEE = std::move(RHS.IEEE);
       } else if (usesLayout<DoubleAPFloat>(*semantics) &&
                  usesLayout<DoubleAPFloat>(*RHS.semantics)) {
         Double = std::move(RHS.Double);
       } else if (this != &RHS) {
         this->~Storage();
         new (this) Storage(std::move(RHS));
       }
       return *this;
     }
   } U;
 
   template <typename T> static bool usesLayout(const fltSemantics &Semantics) {
     static_assert(std::is_same<T, IEEEFloat>::value ||
                   std::is_same<T, DoubleAPFloat>::value);
     if (std::is_same<T, DoubleAPFloat>::value) {
       return &Semantics == &PPCDoubleDouble();
     }
     return &Semantics != &PPCDoubleDouble();
   }
 
   IEEEFloat &getIEEE() {
     if (usesLayout<IEEEFloat>(*U.semantics))
       return U.IEEE;
     if (usesLayout<DoubleAPFloat>(*U.semantics))
       return U.Double.getFirst().U.IEEE;
     llvm_unreachable("Unexpected semantics");
   }
 
   const IEEEFloat &getIEEE() const {
     if (usesLayout<IEEEFloat>(*U.semantics))
       return U.IEEE;
     if (usesLayout<DoubleAPFloat>(*U.semantics))
       return U.Double.getFirst().U.IEEE;
     llvm_unreachable("Unexpected semantics");
   }
 
   void makeZero(bool Neg) { APFLOAT_DISPATCH_ON_SEMANTICS(makeZero(Neg)); }
 
   void makeInf(bool Neg) { APFLOAT_DISPATCH_ON_SEMANTICS(makeInf(Neg)); }
 
   void makeNaN(bool SNaN, bool Neg, const APInt *fill) {
     APFLOAT_DISPATCH_ON_SEMANTICS(makeNaN(SNaN, Neg, fill));
   }
 
   void makeLargest(bool Neg) {
     APFLOAT_DISPATCH_ON_SEMANTICS(makeLargest(Neg));
   }
 
   void makeSmallest(bool Neg) {
     APFLOAT_DISPATCH_ON_SEMANTICS(makeSmallest(Neg));
   }
 
   void makeSmallestNormalized(bool Neg) {
     APFLOAT_DISPATCH_ON_SEMANTICS(makeSmallestNormalized(Neg));
   }
 
   explicit APFloat(IEEEFloat F, const fltSemantics &S) : U(std::move(F), S) {}
   explicit APFloat(DoubleAPFloat F, const fltSemantics &S)
       : U(std::move(F), S) {}
 
   cmpResult compareAbsoluteValue(const APFloat &RHS) const {
     assert(&getSemantics() == &RHS.getSemantics() &&
            "Should only compare APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.compareAbsoluteValue(RHS.U.IEEE);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.compareAbsoluteValue(RHS.U.Double);
     llvm_unreachable("Unexpected semantics");
   }
 
 public:
   APFloat(const fltSemantics &Semantics) : U(Semantics) {}
   APFloat(const fltSemantics &Semantics, StringRef S);
   APFloat(const fltSemantics &Semantics, integerPart I) : U(Semantics, I) {}
   template <typename T,
             typename = std::enable_if_t<std::is_floating_point<T>::value>>
   APFloat(const fltSemantics &Semantics, T V) = delete;
   // TODO: Remove this constructor. This isn't faster than the first one.
   APFloat(const fltSemantics &Semantics, uninitializedTag)
       : U(Semantics, uninitialized) {}
   APFloat(const fltSemantics &Semantics, const APInt &I) : U(Semantics, I) {}
   explicit APFloat(double d) : U(IEEEFloat(d), IEEEdouble()) {}
   explicit APFloat(float f) : U(IEEEFloat(f), IEEEsingle()) {}
   APFloat(const APFloat &RHS) = default;
   APFloat(APFloat &&RHS) = default;
 
   ~APFloat() = default;
 
   bool needsCleanup() const { APFLOAT_DISPATCH_ON_SEMANTICS(needsCleanup()); }
 
   /// Factory for Positive and Negative Zero.
   ///
   /// \param Negative True iff the number should be negative.
   static APFloat getZero(const fltSemantics &Sem, bool Negative = false) {
     APFloat Val(Sem, uninitialized);
     Val.makeZero(Negative);
     return Val;
   }
 
   /// Factory for Positive and Negative One.
   ///
   /// \param Negative True iff the number should be negative.
   static APFloat getOne(const fltSemantics &Sem, bool Negative = false) {
     APFloat Val(Sem, 1U);
     if (Negative)
       Val.changeSign();
     return Val;
   }
 
   /// Factory for Positive and Negative Infinity.
   ///
   /// \param Negative True iff the number should be negative.
   static APFloat getInf(const fltSemantics &Sem, bool Negative = false) {
     APFloat Val(Sem, uninitialized);
     Val.makeInf(Negative);
     return Val;
   }
 
   /// Factory for NaN values.
   ///
   /// \param Negative - True iff the NaN generated should be negative.
   /// \param payload - The unspecified fill bits for creating the NaN, 0 by
   /// default.  The value is truncated as necessary.
   static APFloat getNaN(const fltSemantics &Sem, bool Negative = false,
                         uint64_t payload = 0) {
     if (payload) {
       APInt intPayload(64, payload);
       return getQNaN(Sem, Negative, &intPayload);
     } else {
       return getQNaN(Sem, Negative, nullptr);
     }
   }
 
   /// Factory for QNaN values.
   static APFloat getQNaN(const fltSemantics &Sem, bool Negative = false,
                          const APInt *payload = nullptr) {
     APFloat Val(Sem, uninitialized);
     Val.makeNaN(false, Negative, payload);
     return Val;
   }
 
   /// Factory for SNaN values.
   static APFloat getSNaN(const fltSemantics &Sem, bool Negative = false,
                          const APInt *payload = nullptr) {
     APFloat Val(Sem, uninitialized);
     Val.makeNaN(true, Negative, payload);
     return Val;
   }
 
   /// Returns the largest finite number in the given semantics.
   ///
   /// \param Negative - True iff the number should be negative
   static APFloat getLargest(const fltSemantics &Sem, bool Negative = false) {
     APFloat Val(Sem, uninitialized);
     Val.makeLargest(Negative);
     return Val;
   }
 
   /// Returns the smallest (by magnitude) finite number in the given semantics.
   /// Might be denormalized, which implies a relative loss of precision.
   ///
   /// \param Negative - True iff the number should be negative
   static APFloat getSmallest(const fltSemantics &Sem, bool Negative = false) {
     APFloat Val(Sem, uninitialized);
     Val.makeSmallest(Negative);
     return Val;
   }
 
   /// Returns the smallest (by magnitude) normalized finite number in the given
   /// semantics.
   ///
   /// \param Negative - True iff the number should be negative
   static APFloat getSmallestNormalized(const fltSemantics &Sem,
                                        bool Negative = false) {
     APFloat Val(Sem, uninitialized);
     Val.makeSmallestNormalized(Negative);
     return Val;
   }
 
   /// Returns a float which is bitcasted from an all one value int.
   ///
   /// \param Semantics - type float semantics
   static APFloat getAllOnesValue(const fltSemantics &Semantics);
 
   /// Returns true if the given semantics has actual significand.
   ///
   /// \param Sem - type float semantics
   static bool hasSignificand(const fltSemantics &Sem) {
     return &Sem != &Float8E8M0FNU();
   }
 
   /// Used to insert APFloat objects, or objects that contain APFloat objects,
   /// into FoldingSets.
   void Profile(FoldingSetNodeID &NID) const;
 
   opStatus add(const APFloat &RHS, roundingMode RM) {
     assert(&getSemantics() == &RHS.getSemantics() &&
            "Should only call on two APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.add(RHS.U.IEEE, RM);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.add(RHS.U.Double, RM);
     llvm_unreachable("Unexpected semantics");
   }
   opStatus subtract(const APFloat &RHS, roundingMode RM) {
     assert(&getSemantics() == &RHS.getSemantics() &&
            "Should only call on two APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.subtract(RHS.U.IEEE, RM);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.subtract(RHS.U.Double, RM);
     llvm_unreachable("Unexpected semantics");
   }
   opStatus multiply(const APFloat &RHS, roundingMode RM) {
     assert(&getSemantics() == &RHS.getSemantics() &&
            "Should only call on two APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.multiply(RHS.U.IEEE, RM);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.multiply(RHS.U.Double, RM);
     llvm_unreachable("Unexpected semantics");
   }
   opStatus divide(const APFloat &RHS, roundingMode RM) {
     assert(&getSemantics() == &RHS.getSemantics() &&
            "Should only call on two APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.divide(RHS.U.IEEE, RM);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.divide(RHS.U.Double, RM);
     llvm_unreachable("Unexpected semantics");
   }
   opStatus remainder(const APFloat &RHS) {
     assert(&getSemantics() == &RHS.getSemantics() &&
            "Should only call on two APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.remainder(RHS.U.IEEE);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.remainder(RHS.U.Double);
     llvm_unreachable("Unexpected semantics");
   }
   opStatus mod(const APFloat &RHS) {
     assert(&getSemantics() == &RHS.getSemantics() &&
            "Should only call on two APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.mod(RHS.U.IEEE);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.mod(RHS.U.Double);
     llvm_unreachable("Unexpected semantics");
   }
   opStatus fusedMultiplyAdd(const APFloat &Multiplicand, const APFloat &Addend,
                             roundingMode RM) {
     assert(&getSemantics() == &Multiplicand.getSemantics() &&
            "Should only call on APFloats with the same semantics");
     assert(&getSemantics() == &Addend.getSemantics() &&
            "Should only call on APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.fusedMultiplyAdd(Multiplicand.U.IEEE, Addend.U.IEEE, RM);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.fusedMultiplyAdd(Multiplicand.U.Double, Addend.U.Double,
                                        RM);
     llvm_unreachable("Unexpected semantics");
   }
   opStatus roundToIntegral(roundingMode RM) {
     APFLOAT_DISPATCH_ON_SEMANTICS(roundToIntegral(RM));
   }
 
   // TODO: bool parameters are not readable and a source of bugs.
   // Do something.
   opStatus next(bool nextDown) {
     APFLOAT_DISPATCH_ON_SEMANTICS(next(nextDown));
   }
 
   /// Negate an APFloat.
   APFloat operator-() const {
     APFloat Result(*this);
     Result.changeSign();
     return Result;
   }
 
   /// Add two APFloats, rounding ties to the nearest even.
   /// No error checking.
   APFloat operator+(const APFloat &RHS) const {
     APFloat Result(*this);
     (void)Result.add(RHS, rmNearestTiesToEven);
     return Result;
   }
 
   /// Subtract two APFloats, rounding ties to the nearest even.
   /// No error checking.
   APFloat operator-(const APFloat &RHS) const {
     APFloat Result(*this);
     (void)Result.subtract(RHS, rmNearestTiesToEven);
     return Result;
   }
 
   /// Multiply two APFloats, rounding ties to the nearest even.
   /// No error checking.
   APFloat operator*(const APFloat &RHS) const {
     APFloat Result(*this);
     (void)Result.multiply(RHS, rmNearestTiesToEven);
     return Result;
   }
 
   /// Divide the first APFloat by the second, rounding ties to the nearest even.
   /// No error checking.
   APFloat operator/(const APFloat &RHS) const {
     APFloat Result(*this);
     (void)Result.divide(RHS, rmNearestTiesToEven);
     return Result;
   }
 
   void changeSign() { APFLOAT_DISPATCH_ON_SEMANTICS(changeSign()); }
   void clearSign() {
     if (isNegative())
       changeSign();
   }
   void copySign(const APFloat &RHS) {
     if (isNegative() != RHS.isNegative())
       changeSign();
   }
 
   /// A static helper to produce a copy of an APFloat value with its sign
   /// copied from some other APFloat.
   static APFloat copySign(APFloat Value, const APFloat &Sign) {
     Value.copySign(Sign);
     return Value;
   }
 
   /// Assuming this is an IEEE-754 NaN value, quiet its signaling bit.
   /// This preserves the sign and payload bits.
   APFloat makeQuiet() const {
     APFloat Result(*this);
     Result.getIEEE().makeQuiet();
     return Result;
   }
 
   opStatus convert(const fltSemantics &ToSemantics, roundingMode RM,
                    bool *losesInfo);
   opStatus convertToInteger(MutableArrayRef<integerPart> Input,
                             unsigned int Width, bool IsSigned, roundingMode RM,
                             bool *IsExact) const {
     APFLOAT_DISPATCH_ON_SEMANTICS(
         convertToInteger(Input, Width, IsSigned, RM, IsExact));
   }
   opStatus convertToInteger(APSInt &Result, roundingMode RM,
                             bool *IsExact) const;
   opStatus convertFromAPInt(const APInt &Input, bool IsSigned,
                             roundingMode RM) {
     APFLOAT_DISPATCH_ON_SEMANTICS(convertFromAPInt(Input, IsSigned, RM));
   }
   opStatus convertFromSignExtendedInteger(const integerPart *Input,
                                           unsigned int InputSize, bool IsSigned,
                                           roundingMode RM) {
     APFLOAT_DISPATCH_ON_SEMANTICS(
         convertFromSignExtendedInteger(Input, InputSize, IsSigned, RM));
   }
   opStatus convertFromZeroExtendedInteger(const integerPart *Input,
                                           unsigned int InputSize, bool IsSigned,
                                           roundingMode RM) {
     APFLOAT_DISPATCH_ON_SEMANTICS(
         convertFromZeroExtendedInteger(Input, InputSize, IsSigned, RM));
   }
   Expected<opStatus> convertFromString(StringRef, roundingMode);
   APInt bitcastToAPInt() const {
     APFLOAT_DISPATCH_ON_SEMANTICS(bitcastToAPInt());
   }
 
   /// Converts this APFloat to host double value.
   ///
   /// \pre The APFloat must be built using semantics, that can be represented by
   /// the host double type without loss of precision. It can be IEEEdouble and
   /// shorter semantics, like IEEEsingle and others.
   double convertToDouble() const;
 
   /// Converts this APFloat to host float value.
   ///
   /// \pre The APFloat must be built using semantics, that can be represented by
   /// the host float type without loss of precision. It can be IEEEquad and
   /// shorter semantics, like IEEEdouble and others.
 #ifdef HAS_IEE754_FLOAT128
   float128 convertToQuad() const;
 #endif
 
   /// Converts this APFloat to host float value.
   ///
   /// \pre The APFloat must be built using semantics, that can be represented by
   /// the host float type without loss of precision. It can be IEEEsingle and
   /// shorter semantics, like IEEEhalf.
   float convertToFloat() const;
 
   bool operator==(const APFloat &RHS) const { return compare(RHS) == cmpEqual; }
 
   bool operator!=(const APFloat &RHS) const { return compare(RHS) != cmpEqual; }
 
   bool operator<(const APFloat &RHS) const {
     return compare(RHS) == cmpLessThan;
   }
 
   bool operator>(const APFloat &RHS) const {
     return compare(RHS) == cmpGreaterThan;
   }
 
   bool operator<=(const APFloat &RHS) const {
     cmpResult Res = compare(RHS);
     return Res == cmpLessThan || Res == cmpEqual;
   }
 
   bool operator>=(const APFloat &RHS) const {
     cmpResult Res = compare(RHS);
     return Res == cmpGreaterThan || Res == cmpEqual;
   }
 
   cmpResult compare(const APFloat &RHS) const {
     assert(&getSemantics() == &RHS.getSemantics() &&
            "Should only compare APFloats with the same semantics");
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.compare(RHS.U.IEEE);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.compare(RHS.U.Double);
     llvm_unreachable("Unexpected semantics");
   }
 
   bool bitwiseIsEqual(const APFloat &RHS) const {
     if (&getSemantics() != &RHS.getSemantics())
       return false;
     if (usesLayout<IEEEFloat>(getSemantics()))
       return U.IEEE.bitwiseIsEqual(RHS.U.IEEE);
     if (usesLayout<DoubleAPFloat>(getSemantics()))
       return U.Double.bitwiseIsEqual(RHS.U.Double);
     llvm_unreachable("Unexpected semantics");
   }
 
   /// We don't rely on operator== working on double values, as
   /// it returns true for things that are clearly not equal, like -0.0 and 0.0.
   /// As such, this method can be used to do an exact bit-for-bit comparison of
   /// two floating point values.
   ///
   /// We leave the version with the double argument here because it's just so
   /// convenient to write "2.0" and the like.  Without this function we'd
   /// have to duplicate its logic everywhere it's called.
   bool isExactlyValue(double V) const {
     bool ignored;
     APFloat Tmp(V);
     Tmp.convert(getSemantics(), APFloat::rmNearestTiesToEven, &ignored);
     return bitwiseIsEqual(Tmp);
   }
 
   unsigned int convertToHexString(char *DST, unsigned int HexDigits,
                                   bool UpperCase, roundingMode RM) const {
     APFLOAT_DISPATCH_ON_SEMANTICS(
         convertToHexString(DST, HexDigits, UpperCase, RM));
   }
 
   bool isZero() const { return getCategory() == fcZero; }
   bool isInfinity() const { return getCategory() == fcInfinity; }
   bool isNaN() const { return getCategory() == fcNaN; }
 
   bool isNegative() const { return getIEEE().isNegative(); }
   bool isDenormal() const { APFLOAT_DISPATCH_ON_SEMANTICS(isDenormal()); }
   bool isSignaling() const { return getIEEE().isSignaling(); }
 
   bool isNormal() const { return !isDenormal() && isFiniteNonZero(); }
   bool isFinite() const { return !isNaN() && !isInfinity(); }
 
   fltCategory getCategory() const { return getIEEE().getCategory(); }
   const fltSemantics &getSemantics() const { return *U.semantics; }
   bool isNonZero() const { return !isZero(); }
   bool isFiniteNonZero() const { return isFinite() && !isZero(); }
   bool isPosZero() const { return isZero() && !isNegative(); }
   bool isNegZero() const { return isZero() && isNegative(); }
   bool isPosInfinity() const { return isInfinity() && !isNegative(); }
   bool isNegInfinity() const { return isInfinity() && isNegative(); }
   bool isSmallest() const { APFLOAT_DISPATCH_ON_SEMANTICS(isSmallest()); }
   bool isLargest() const { APFLOAT_DISPATCH_ON_SEMANTICS(isLargest()); }
   bool isInteger() const { APFLOAT_DISPATCH_ON_SEMANTICS(isInteger()); }
   bool isIEEE() const { return usesLayout<IEEEFloat>(getSemantics()); }
 
   bool isSmallestNormalized() const {
     APFLOAT_DISPATCH_ON_SEMANTICS(isSmallestNormalized());
   }
 
   /// Return the FPClassTest which will return true for the value.
   FPClassTest classify() const;
 
   APFloat &operator=(const APFloat &RHS) = default;
   APFloat &operator=(APFloat &&RHS) = default;
 
   void toString(SmallVectorImpl<char> &Str, unsigned FormatPrecision = 0,
                 unsigned FormatMaxPadding = 3, bool TruncateZero = true) const {
     APFLOAT_DISPATCH_ON_SEMANTICS(
         toString(Str, FormatPrecision, FormatMaxPadding, TruncateZero));
   }
 
   void print(raw_ostream &) const;
   void dump() const;
 
   bool getExactInverse(APFloat *inv) const {
     APFLOAT_DISPATCH_ON_SEMANTICS(getExactInverse(inv));
   }
 
   LLVM_READONLY
   int getExactLog2Abs() const {
     APFLOAT_DISPATCH_ON_SEMANTICS(getExactLog2Abs());
   }
 
   LLVM_READONLY
   int getExactLog2() const {
     APFLOAT_DISPATCH_ON_SEMANTICS(getExactLog2());
   }
 
   friend hash_code hash_value(const APFloat &Arg);
   friend int ilogb(const APFloat &Arg) { return ilogb(Arg.getIEEE()); }
   friend APFloat scalbn(APFloat X, int Exp, roundingMode RM);
   friend APFloat frexp(const APFloat &X, int &Exp, roundingMode RM);
   friend IEEEFloat;
   friend DoubleAPFloat;
 };
 
 static_assert(sizeof(APFloat) == sizeof(detail::IEEEFloat),
               "Empty base class optimization is not performed.");
 
 /// See friend declarations above.
 ///
 /// These additional declarations are required in order to compile LLVM with IBM
 /// xlC compiler.
 hash_code hash_value(const APFloat &Arg);
 inline APFloat scalbn(APFloat X, int Exp, APFloat::roundingMode RM) {
   if (APFloat::usesLayout<detail::IEEEFloat>(X.getSemantics()))
     return APFloat(scalbn(X.U.IEEE, Exp, RM), X.getSemantics());
   if (APFloat::usesLayout<detail::DoubleAPFloat>(X.getSemantics()))
     return APFloat(scalbn(X.U.Double, Exp, RM), X.getSemantics());
   llvm_unreachable("Unexpected semantics");
 }
 
 /// Equivalent of C standard library function.
 ///
 /// While the C standard says Exp is an unspecified value for infinity and nan,
 /// this returns INT_MAX for infinities, and INT_MIN for NaNs.
 inline APFloat frexp(const APFloat &X, int &Exp, APFloat::roundingMode RM) {
   if (APFloat::usesLayout<detail::IEEEFloat>(X.getSemantics()))
     return APFloat(frexp(X.U.IEEE, Exp, RM), X.getSemantics());
   if (APFloat::usesLayout<detail::DoubleAPFloat>(X.getSemantics()))
     return APFloat(frexp(X.U.Double, Exp, RM), X.getSemantics());
   llvm_unreachable("Unexpected semantics");
 }
 /// Returns the absolute value of the argument.
 inline APFloat abs(APFloat X) {
   X.clearSign();
   return X;
 }
 
 /// Returns the negated value of the argument.
 inline APFloat neg(APFloat X) {
   X.changeSign();
   return X;
 }
 
 /// Implements IEEE-754 2019 minimumNumber semantics. Returns the smaller of the
 /// 2 arguments if both are not NaN. If either argument is a NaN, returns the
 /// other argument. -0 is treated as ordered less than +0.
 LLVM_READONLY
 inline APFloat minnum(const APFloat &A, const APFloat &B) {
   if (A.isNaN())
     return B;
   if (B.isNaN())
     return A;
   if (A.isZero() && B.isZero() && (A.isNegative() != B.isNegative()))
     return A.isNegative() ? A : B;
   return B < A ? B : A;
 }
 
 /// Implements IEEE-754 2019 maximumNumber semantics. Returns the larger of the
 /// 2 arguments if both are not NaN. If either argument is a NaN, returns the
 /// other argument. +0 is treated as ordered greater than -0.
 LLVM_READONLY
 inline APFloat maxnum(const APFloat &A, const APFloat &B) {
   if (A.isNaN())
     return B;
   if (B.isNaN())
     return A;
   if (A.isZero() && B.isZero() && (A.isNegative() != B.isNegative()))
     return A.isNegative() ? B : A;
   return A < B ? B : A;
 }
 
 /// Implements IEEE 754-2019 minimum semantics. Returns the smaller of 2
 /// arguments, returning a quiet NaN if an argument is a NaN and treating -0
 /// as less than +0.
 LLVM_READONLY
 inline APFloat minimum(const APFloat &A, const APFloat &B) {
   if (A.isNaN())
     return A.makeQuiet();
   if (B.isNaN())
     return B.makeQuiet();
   if (A.isZero() && B.isZero() && (A.isNegative() != B.isNegative()))
     return A.isNegative() ? A : B;
   return B < A ? B : A;
 }
 
 /// Implements IEEE 754-2019 minimumNumber semantics. Returns the smaller
 /// of 2 arguments, not propagating NaNs and treating -0 as less than +0.
 LLVM_READONLY
 inline APFloat minimumnum(const APFloat &A, const APFloat &B) {
   if (A.isNaN())
     return B.isNaN() ? B.makeQuiet() : B;
   if (B.isNaN())
     return A;
   if (A.isZero() && B.isZero() && (A.isNegative() != B.isNegative()))
     return A.isNegative() ? A : B;
   return B < A ? B : A;
 }
 
 /// Implements IEEE 754-2019 maximum semantics. Returns the larger of 2
 /// arguments, returning a quiet NaN if an argument is a NaN and treating -0
 /// as less than +0.
 LLVM_READONLY
 inline APFloat maximum(const APFloat &A, const APFloat &B) {
   if (A.isNaN())
     return A.makeQuiet();
   if (B.isNaN())
     return B.makeQuiet();
   if (A.isZero() && B.isZero() && (A.isNegative() != B.isNegative()))
     return A.isNegative() ? B : A;
   return A < B ? B : A;
 }
 
 /// Implements IEEE 754-2019 maximumNumber semantics. Returns the larger
 /// of 2 arguments, not propagating NaNs and treating -0 as less than +0.
 LLVM_READONLY
 inline APFloat maximumnum(const APFloat &A, const APFloat &B) {
   if (A.isNaN())
     return B.isNaN() ? B.makeQuiet() : B;
   if (B.isNaN())
     return A;
   if (A.isZero() && B.isZero() && (A.isNegative() != B.isNegative()))
     return A.isNegative() ? B : A;
   return A < B ? B : A;
 }
 
 inline raw_ostream &operator<<(raw_ostream &OS, const APFloat &V) {
   V.print(OS);
   return OS;
 }
 
 // We want the following functions to be available in the header for inlining.
 // We cannot define them inline in the class definition of `DoubleAPFloat`
 // because doing so would instantiate `std::unique_ptr<APFloat[]>` before
 // `APFloat` is defined, and that would be undefined behavior.
 namespace detail {
 
 DoubleAPFloat &DoubleAPFloat::operator=(DoubleAPFloat &&RHS) {
   if (this != &RHS) {
     this->~DoubleAPFloat();
     new (this) DoubleAPFloat(std::move(RHS));
   }
   return *this;
 }
 
 APFloat &DoubleAPFloat::getFirst() { return Floats[0]; }
 const APFloat &DoubleAPFloat::getFirst() const { return Floats[0]; }
 APFloat &DoubleAPFloat::getSecond() { return Floats[1]; }
 const APFloat &DoubleAPFloat::getSecond() const { return Floats[1]; }
 
 } // namespace detail
 
 } // namespace llvm
 
 #undef APFLOAT_DISPATCH_ON_SEMANTICS
 #endif // LLVM_ADT_APFLOAT_H

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