APFloat.h

Go to the documentation of this file.

//===- 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/Compiler.h"
#include "llvm/Support/ErrorHandling.h"
#include "llvm/Support/float128.h"
#include <memory>
#include <optional>
 
#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.
///
 
namespace detail {
class IEEEFloat;
class DoubleAPFloat;
} // namespace detail
 
// This is the common type definitions shared by APFloat and its internal
// implementation classes. This struct should not define any non-static data
// members.
class APFloatBase {
public:
  typedef APInt::WordType integerPart;
  static constexpr unsigned integerPartWidth = APInt::APINT_BITS_PER_WORD;
 
  /// A signed type to represent a floating point numbers unbiased exponent.
  using ExponentType = int32_t;
 
  /// \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,
    // 8-bit floating point number mostly following IEEE-754 conventions with
    // bit layout S0E5M3 as described in PTX ISA page.
    // https://docs.nvidia.com/cuda/developer-preview/13.4/parallel-thread-execution/index.html#alternate-floating-point-data-formats
    // Unlike IEEE-754 types, there are no infinity values, and NaN is
    // represented with the exponent and mantissa bits set to all 1s.
    S_Float8E5M3FNU,
    // TODO: Documentation is missing.
    S_x87DoubleExtended,
    S_MaxSemantics = S_x87DoubleExtended,
  };
 
  LLVM_ABI static const llvm::fltSemantics &EnumToSemantics(Semantics S);
  LLVM_ABI static Semantics SemanticsToEnum(const llvm::fltSemantics &Sem);
 
private:
  LLVM_ABI static const fltSemantics semIEEEhalf;
  LLVM_ABI static const fltSemantics semBFloat;
  LLVM_ABI static const fltSemantics semIEEEsingle;
  LLVM_ABI static const fltSemantics semIEEEdouble;
  LLVM_ABI static const fltSemantics semIEEEquad;
  LLVM_ABI static const fltSemantics semFloat8E5M2;
  LLVM_ABI static const fltSemantics semFloat8E5M2FNUZ;
  LLVM_ABI static const fltSemantics semFloat8E4M3;
  LLVM_ABI static const fltSemantics semFloat8E4M3FN;
  LLVM_ABI static const fltSemantics semFloat8E4M3FNUZ;
  LLVM_ABI static const fltSemantics semFloat8E4M3B11FNUZ;
  LLVM_ABI static const fltSemantics semFloat8E3M4;
  LLVM_ABI static const fltSemantics semFloatTF32;
  LLVM_ABI static const fltSemantics semFloat8E8M0FNU;
  LLVM_ABI static const fltSemantics semFloat8E5M3FNU;
  LLVM_ABI static const fltSemantics semFloat6E3M2FN;
  LLVM_ABI static const fltSemantics semFloat6E2M3FN;
  LLVM_ABI static const fltSemantics semFloat4E2M1FN;
  LLVM_ABI static const fltSemantics semX87DoubleExtended;
  LLVM_ABI static const fltSemantics semBogus;
  LLVM_ABI static const fltSemantics semPPCDoubleDouble;
  LLVM_ABI static const fltSemantics semPPCDoubleDoubleLegacy;
 
  friend class detail::IEEEFloat;
  friend class detail::DoubleAPFloat;
  friend class APFloat;
 
public:
  static const fltSemantics &IEEEhalf() { return semIEEEhalf; }
  static const fltSemantics &BFloat() { return semBFloat; }
  static const fltSemantics &IEEEsingle() { return semIEEEsingle; }
  static const fltSemantics &IEEEdouble() { return semIEEEdouble; }
  static const fltSemantics &IEEEquad() { return semIEEEquad; }
  static const fltSemantics &PPCDoubleDouble() { return semPPCDoubleDouble; }
  static const fltSemantics &PPCDoubleDoubleLegacy() {
    return semPPCDoubleDoubleLegacy;
  }
  static const fltSemantics &Float8E5M2() { return semFloat8E5M2; }
  static const fltSemantics &Float8E5M2FNUZ() { return semFloat8E5M2FNUZ; }
  static const fltSemantics &Float8E4M3() { return semFloat8E4M3; }
  static const fltSemantics &Float8E4M3FN() { return semFloat8E4M3FN; }
  static const fltSemantics &Float8E4M3FNUZ() { return semFloat8E4M3FNUZ; }
  static const fltSemantics &Float8E4M3B11FNUZ() {
    return semFloat8E4M3B11FNUZ;
  }
  static const fltSemantics &Float8E3M4() { return semFloat8E3M4; }
  static const fltSemantics &FloatTF32() { return semFloatTF32; }
  static const fltSemantics &Float8E8M0FNU() { return semFloat8E8M0FNU; }
  static const fltSemantics &Float8E5M3FNU() { return semFloat8E5M3FNU; }
  static const fltSemantics &Float6E3M2FN() { return semFloat6E3M2FN; }
  static const fltSemantics &Float6E2M3FN() { return semFloat6E2M3FN; }
  static const fltSemantics &Float4E2M1FN() { return semFloat4E2M1FN; }
  static const fltSemantics &x87DoubleExtended() {
    return semX87DoubleExtended;
  }
 
  /// A Pseudo fltsemantic used to construct APFloats that cannot conflict with
  /// anything real.
  static const fltSemantics &Bogus() { return semBogus; }
 
  // 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.
  LLVM_ABI 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
  };
 
  LLVM_ABI static unsigned int semanticsPrecision(const fltSemantics &);
  LLVM_ABI static ExponentType semanticsMinExponent(const fltSemantics &);
  LLVM_ABI static ExponentType semanticsMaxExponent(const fltSemantics &);
  LLVM_ABI static unsigned int semanticsSizeInBits(const fltSemantics &);
  LLVM_ABI static unsigned int semanticsIntSizeInBits(const fltSemantics &,
                                                      bool);
  LLVM_ABI static bool semanticsHasZero(const fltSemantics &);
  LLVM_ABI static bool semanticsHasSignedRepr(const fltSemantics &);
  LLVM_ABI static bool semanticsHasInf(const fltSemantics &);
  LLVM_ABI static bool semanticsHasNaN(const fltSemantics &);
  LLVM_ABI static bool isIEEELikeFP(const fltSemantics &);
  LLVM_ABI static bool hasSignBitInMSB(const fltSemantics &);
 
  // Returns true if any number described by \p Src can be precisely represented
  // by a normal (not subnormal) value in \p Dst.
  LLVM_ABI static bool isRepresentableAsNormalIn(const fltSemantics &Src,
                                                 const fltSemantics &Dst);
 
  /// Returns the size of the floating point number (in bits) in the given
  /// semantics.
  LLVM_ABI static unsigned getSizeInBits(const fltSemantics &Sem);
 
  /// Returns true if the given string is a valid arbitrary floating-point
  /// format interpretation for llvm.convert.to.arbitrary.fp and
  /// llvm.convert.from.arbitrary.fp intrinsics.
  LLVM_ABI static bool isValidArbitraryFPFormat(StringRef Format);
 
  /// Returns the size in bits of a valid arbitrary floating-point format
  /// string, or 0 if the string is not a valid format. Covers every format
  /// accepted by isValidArbitraryFPFormat, not only those
  /// getArbitraryFPSemantics can currently lower.
  LLVM_ABI static unsigned getArbitraryFPFormatSizeInBits(StringRef Format);
 
  /// Returns the fltSemantics for a given arbitrary FP format string,
  /// or nullptr if invalid.
  LLVM_ABI static const fltSemantics *getArbitraryFPSemantics(StringRef Format);
};
 
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
  /// @{
 
  LLVM_ABI IEEEFloat(const fltSemantics &); // Default construct to +0.0
  LLVM_ABI IEEEFloat(const fltSemantics &, integerPart);
  LLVM_ABI IEEEFloat(const fltSemantics &, uninitializedTag);
  LLVM_ABI IEEEFloat(const fltSemantics &, const APInt &);
  LLVM_ABI explicit IEEEFloat(double d);
  LLVM_ABI explicit IEEEFloat(float f);
  LLVM_ABI IEEEFloat(const IEEEFloat &);
  LLVM_ABI IEEEFloat(IEEEFloat &&);
  LLVM_ABI ~IEEEFloat();
 
  /// @}
 
  /// Returns whether this instance allocated memory.
  bool needsCleanup() const { return partCount() > 1; }
 
  /// \name Convenience "constructors"
  /// @{
 
  /// @}
 
  /// \name Arithmetic
  /// @{
 
  LLVM_ABI opStatus add(const IEEEFloat &, roundingMode);
  LLVM_ABI opStatus subtract(const IEEEFloat &, roundingMode);
  LLVM_ABI opStatus multiply(const IEEEFloat &, roundingMode);
  LLVM_ABI opStatus divide(const IEEEFloat &, roundingMode);
  /// IEEE remainder.
  LLVM_ABI opStatus remainder(const IEEEFloat &);
  /// C fmod, or llvm frem.
  LLVM_ABI opStatus mod(const IEEEFloat &);
  LLVM_ABI opStatus fusedMultiplyAdd(const IEEEFloat &, const IEEEFloat &,
                                     roundingMode);
  LLVM_ABI opStatus roundToIntegral(roundingMode);
  /// IEEE-754R 5.3.1: nextUp/nextDown.
  LLVM_ABI opStatus next(bool nextDown);
 
  /// @}
 
  /// \name Sign operations.
  /// @{
 
  LLVM_ABI void changeSign();
 
  /// @}
 
  /// \name Conversions
  /// @{
 
  LLVM_ABI opStatus convert(const fltSemantics &, roundingMode, bool *);
  LLVM_ABI opStatus convertToInteger(MutableArrayRef<integerPart>, unsigned int,
                                     bool, roundingMode, bool *) const;
  LLVM_ABI opStatus convertFromAPInt(const APInt &, bool, roundingMode);
  LLVM_ABI Expected<opStatus> convertFromString(StringRef, roundingMode);
  LLVM_ABI APInt bitcastToAPInt() const;
  LLVM_ABI double convertToDouble() const;
#ifdef HAS_IEE754_FLOAT128
  LLVM_ABI float128 convertToQuad() const;
#endif
  LLVM_ABI 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).
  LLVM_ABI cmpResult compare(const IEEEFloat &) const;
 
  /// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
  LLVM_ABI 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.
  LLVM_ABI 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.
  LLVM_ABI 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.
  LLVM_ABI 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.
  LLVM_ABI bool isSmallest() const;
 
  /// Returns true if this is the smallest (by magnitude) normalized finite
  /// number in the given semantics.
  LLVM_ABI bool isSmallestNormalized() const;
 
  /// Returns true if and only if the number has the largest possible finite
  /// magnitude in the current semantics.
  LLVM_ABI bool isLargest() const;
 
  /// Returns true if and only if the number is an exact integer.
  LLVM_ABI bool isInteger() const;
 
  /// @}
 
  LLVM_ABI IEEEFloat &operator=(const IEEEFloat &);
  LLVM_ABI 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.
  LLVM_ABI 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
  LLVM_ABI void toString(SmallVectorImpl<char> &Str,
                         unsigned FormatPrecision = 0,
                         unsigned FormatMaxPadding = 3,
                         bool TruncateZero = true) const;
 
  LLVM_ABI LLVM_READONLY int getExactLog2Abs() const;
 
  LLVM_ABI friend int ilogb(const IEEEFloat &Arg);
 
  LLVM_ABI friend IEEEFloat scalbn(IEEEFloat X, int Exp, roundingMode);
 
  LLVM_ABI friend IEEEFloat frexp(const IEEEFloat &X, int &Exp, roundingMode);
 
  /// \name Special value setters.
  /// @{
 
  LLVM_ABI void makeLargest(bool Neg = false);
  LLVM_ABI void makeSmallest(bool Neg = false);
  LLVM_ABI void makeNaN(bool SNaN = false, bool Neg = false,
                        const APInt *fill = nullptr);
  LLVM_ABI void makeInf(bool Neg = false);
  LLVM_ABI void makeZero(bool Neg = false);
  LLVM_ABI void makeQuiet();
 
  /// Returns the smallest (by magnitude) normalized finite number in the given
  /// semantics.
  ///
  /// \param Negative - True iff the number should be negative
  LLVM_ABI void makeSmallestNormalized(bool Negative = false);
 
  /// @}
 
  LLVM_ABI cmpResult compareAbsoluteValue(const IEEEFloat &) const;
 
  LLVM_ABI APInt getNaNPayload() const;
 
private:
  /// \name Simple Queries
  /// @{
 
  integerPart *significandParts();
  const integerPart *significandParts() const;
  LLVM_ABI unsigned int partCount() const;
 
  /// @}
 
  /// \name Significand operations.
  /// @{
 
  integerPart addSignificand(const IEEEFloat &);
  integerPart subtractSignificand(const IEEEFloat &, integerPart);
  // Exported for IEEEFloatUnitTestHelper.
  LLVM_ABI 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 convertFloat8E5M3FNUAPFloatToAPInt() 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 initFromFloat8E5M3FNUAPInt(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;
 
  friend class IEEEFloatUnitTestHelper;
};
 
LLVM_ABI hash_code hash_value(const IEEEFloat &Arg);
LLVM_ABI int ilogb(const IEEEFloat &Arg);
LLVM_ABI IEEEFloat scalbn(IEEEFloat X, int Exp, roundingMode);
LLVM_ABI 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;
  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);
  opStatus convertToSignExtendedInteger(MutableArrayRef<integerPart> Input,
                                        unsigned int Width, bool IsSigned,
                                        roundingMode RM, bool *IsExact) const;
 
  // Convert an unsigned integer Src to a floating point number,
  // rounding according to RM.  The sign of the floating point number is not
  // modified.
  opStatus convertFromUnsignedParts(const integerPart *Src,
                                    unsigned int SrcCount, roundingMode RM);
 
  // Handle overflow.  Sign is preserved.  We either become infinity or
  // the largest finite number.
  opStatus handleOverflow(roundingMode RM);
 
public:
  LLVM_ABI DoubleAPFloat(const fltSemantics &S);
  LLVM_ABI DoubleAPFloat(const fltSemantics &S, uninitializedTag);
  LLVM_ABI DoubleAPFloat(const fltSemantics &S, integerPart);
  LLVM_ABI DoubleAPFloat(const fltSemantics &S, const APInt &I);
  LLVM_ABI DoubleAPFloat(const fltSemantics &S, APFloat &&First,
                         APFloat &&Second);
  LLVM_ABI DoubleAPFloat(const DoubleAPFloat &RHS);
  LLVM_ABI DoubleAPFloat(DoubleAPFloat &&RHS);
  ~DoubleAPFloat();
 
  LLVM_ABI 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;
 
  LLVM_ABI opStatus add(const DoubleAPFloat &RHS, roundingMode RM);
  LLVM_ABI opStatus subtract(const DoubleAPFloat &RHS, roundingMode RM);
  LLVM_ABI opStatus multiply(const DoubleAPFloat &RHS, roundingMode RM);
  LLVM_ABI opStatus divide(const DoubleAPFloat &RHS, roundingMode RM);
  LLVM_ABI opStatus remainder(const DoubleAPFloat &RHS);
  LLVM_ABI opStatus mod(const DoubleAPFloat &RHS);
  LLVM_ABI opStatus fusedMultiplyAdd(const DoubleAPFloat &Multiplicand,
                                     const DoubleAPFloat &Addend,
                                     roundingMode RM);
  LLVM_ABI opStatus roundToIntegral(roundingMode RM);
  LLVM_ABI void changeSign();
  LLVM_ABI cmpResult compareAbsoluteValue(const DoubleAPFloat &RHS) const;
 
  LLVM_ABI fltCategory getCategory() const;
  LLVM_ABI bool isNegative() const;
 
  LLVM_ABI void makeInf(bool Neg);
  LLVM_ABI void makeZero(bool Neg);
  LLVM_ABI void makeLargest(bool Neg);
  LLVM_ABI void makeSmallest(bool Neg);
  LLVM_ABI void makeSmallestNormalized(bool Neg);
  LLVM_ABI void makeNaN(bool SNaN, bool Neg, const APInt *fill);
 
  LLVM_ABI cmpResult compare(const DoubleAPFloat &RHS) const;
  LLVM_ABI bool bitwiseIsEqual(const DoubleAPFloat &RHS) const;
  LLVM_ABI APInt bitcastToAPInt() const;
  LLVM_ABI Expected<opStatus> convertFromString(StringRef, roundingMode);
  LLVM_ABI opStatus next(bool nextDown);
 
  LLVM_ABI opStatus convertToInteger(MutableArrayRef<integerPart> Input,
                                     unsigned int Width, bool IsSigned,
                                     roundingMode RM, bool *IsExact) const;
  LLVM_ABI opStatus convertFromAPInt(const APInt &Input, bool IsSigned,
                                     roundingMode RM);
  LLVM_ABI unsigned int convertToHexString(char *DST, unsigned int HexDigits,
                                           bool UpperCase,
                                           roundingMode RM) const;
 
  LLVM_ABI bool isDenormal() const;
  LLVM_ABI bool isSmallest() const;
  LLVM_ABI bool isSmallestNormalized() const;
  LLVM_ABI bool isLargest() const;
  LLVM_ABI bool isInteger() const;
 
  LLVM_ABI APInt getNaNPayload() const;
 
  LLVM_ABI void toString(SmallVectorImpl<char> &Str, unsigned FormatPrecision,
                         unsigned FormatMaxPadding,
                         bool TruncateZero = true) const;
 
  LLVM_ABI LLVM_READONLY int getExactLog2Abs() const;
 
  LLVM_ABI friend int ilogb(const DoubleAPFloat &X);
  LLVM_ABI friend DoubleAPFloat scalbn(const DoubleAPFloat &X, int Exp,
                                       roundingMode);
  LLVM_ABI friend DoubleAPFloat frexp(const DoubleAPFloat &X, int &Exp,
                                      roundingMode);
  LLVM_ABI friend hash_code hash_value(const DoubleAPFloat &Arg);
};
 
LLVM_ABI hash_code hash_value(const DoubleAPFloat &Arg);
LLVM_ABI DoubleAPFloat scalbn(const DoubleAPFloat &Arg, int Exp,
                              roundingMode RM);
LLVM_ABI DoubleAPFloat frexp(const DoubleAPFloat &X, int &Exp, roundingMode);
 
} // End detail namespace
 
// How the nonfinite values Inf and NaN are represented.
enum class fltNonfiniteBehavior {
  // Represents standard IEEE 754 behavior. A value is nonfinite if the
  // exponent field is all 1s. In such cases, a value is Inf if the
  // significand bits are all zero, and NaN otherwise
  IEEE754,
 
  // This behavior is present in the Float8ExMyFN* types (Float8E4M3FN,
  // Float8E5M2FNUZ, Float8E4M3FNUZ, and Float8E4M3B11FNUZ). There is no
  // representation for Inf, and operations that would ordinarily produce Inf
  // produce NaN instead.
  // The details of the NaN representation(s) in this form are determined by the
  // `fltNanEncoding` enum. We treat all NaNs as quiet, as the available
  // encodings do not distinguish between signalling and quiet NaN.
  NanOnly,
 
  // This behavior is present in Float6E3M2FN, Float6E2M3FN, and
  // Float4E2M1FN types, which do not support Inf or NaN values.
  FiniteOnly,
};
 
// How NaN values are represented. This is curently only used in combination
// with fltNonfiniteBehavior::NanOnly, and using a variant other than IEEE
// while having IEEE non-finite behavior is liable to lead to unexpected
// results.
enum class fltNanEncoding {
  // Represents the standard IEEE behavior where a value is NaN if its
  // exponent is all 1s and the significand is non-zero.
  IEEE,
 
  // Represents the behavior in the Float8E4M3FN floating point type where NaN
  // is represented by having the exponent and mantissa set to all 1s.
  // This behavior matches the FP8 E4M3 type described in
  // https://arxiv.org/abs/2209.05433. We treat both signed and unsigned NaNs
  // as non-signalling, although the paper does not state whether the NaN
  // values are signalling or not.
  AllOnes,
 
  // Represents the behavior in Float8E{5,4}E{2,3}FNUZ floating point types
  // where NaN is represented by a sign bit of 1 and all 0s in the exponent
  // and mantissa (i.e. the negative zero encoding in a IEEE float). Since
  // there is only one NaN value, it is treated as quiet NaN. This matches the
  // behavior described in https://arxiv.org/abs/2206.02915 .
  NegativeZero,
};
 
/* Represents floating point arithmetic semantics.  */
struct fltSemantics {
  /* The largest E such that 2^E is representable; this matches the
     definition of IEEE 754.  */
  APFloatBase::ExponentType maxExponent;
 
  /* The smallest E such that 2^E is a normalized number; this
     matches the definition of IEEE 754.  */
  APFloatBase::ExponentType minExponent;
 
  /* Number of bits in the significand.  This includes the integer
     bit.  */
  unsigned int precision;
 
  /* Number of bits actually used in the semantics. */
  unsigned int sizeInBits;
 
  fltNonfiniteBehavior nonFiniteBehavior = fltNonfiniteBehavior::IEEE754;
 
  fltNanEncoding nanEncoding = fltNanEncoding::IEEE;
 
  /* Whether this semantics has an encoding for Zero */
  bool hasZero = true;
 
  /* Whether this semantics can represent signed values */
  bool hasSignedRepr = true;
 
  /* Whether the sign bit of this semantics is the most significant bit */
  bool hasSignBitInMSB = true;
 
  /* Whether the format supports IEEE754 denormal representation.
     If both hasDenormals and hasZero are false exponent 0 is assumed to be a
     regular exponent instead of being reserved. This changes the bias by +1. */
  bool hasDenormals = true;
 
  /* Whether the integer bit is explicitly represented between significant and
     exponent, for example as specified by the x86 double extended precision
     format.
 
     For bit patterns designated as undefined under the standard the following
     conversions will happen when converting from bits. These follow x87
     behaviour:
     - exponent = all 1's, integer bit 0, significand 0 ("pseudoinfinity")
     - exponent = all 1's, integer bit 0, significand nonzero ("pseudoNaN")
     - exponent!=0 nor all 1's, integer bit 0 ("unnormal")
     - exponent = 0, integer bit 1 ("pseudodenormal")
     The first three are treated as NaNs, the last one as Normal */
  bool hasExplicitIntegerBit = false;
};
 
// This is a interface class that is currently forwarding functionalities from
// detail::IEEEFloat.
class APFloat : public APFloatBase {
  using IEEEFloat = detail::IEEEFloat;
  using DoubleAPFloat = detail::DoubleAPFloat;
 
  static_assert(std::is_standard_layout<IEEEFloat>::value);
 
  union Storage {
    const fltSemantics *semantics;
    IEEEFloat IEEE;
    DoubleAPFloat Double;
 
    LLVM_ABI 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");
    }
 
    LLVM_ABI ~Storage();
    LLVM_ABI Storage(const Storage &RHS);
    LLVM_ABI Storage(Storage &&RHS);
    LLVM_ABI Storage &operator=(const Storage &RHS);
    LLVM_ABI Storage &operator=(Storage &&RHS);
  } 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) {}
 
public:
  APFloat(const fltSemantics &Semantics) : U(Semantics) {}
  LLVM_ABI 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
  LLVM_ABI 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.
  LLVM_ABI 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.
  [[nodiscard]] APFloat makeQuiet() const {
    APFloat Result(*this);
    Result.getIEEE().makeQuiet();
    return Result;
  }
 
  LLVM_ABI opStatus convert(const fltSemantics &ToSemantics, roundingMode RM,
                            bool *losesInfo);
  // Convert a floating point number to an integer according to the
  // rounding mode.  We provide deterministic values in case of an invalid
  // operation exception, namely zero for NaNs and the minimal or maximal value
  // respectively for underflow or overflow.
  // The *IsExact output tells whether the result is exact, in the sense that
  // converting it back to the original floating point type produces the
  // original value.  This is almost equivalent to result==opOK, except for
  // negative zeroes.
  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));
  }
  // Same as convertToInteger(integerPart*, ...), except the result is returned
  // in an APSInt, whose initial bit-width and signed-ness are used to determine
  // the precision of the conversion.
  LLVM_ABI opStatus convertToInteger(APSInt &Result, roundingMode RM,
                                     bool *IsExact) const;
 
  // Convert a two's complement integer Input to a floating point number,
  // rounding according to RM.  IsSigned is true if the integer is signed,
  // in which case it must be sign-extended.
  opStatus convertFromAPInt(const APInt &Input, bool IsSigned,
                            roundingMode RM) {
    APFLOAT_DISPATCH_ON_SEMANTICS(convertFromAPInt(Input, IsSigned, RM));
  }
 
  /// Fill this APFloat with the result of a string conversion.
  ///
  /// The following strings are accepted for conversion purposes:
  /// * Decimal floating-point literals (e.g., `0.1e-5`)
  /// * Hexadecimal floating-point literals (e.g., `0x1.0p-5`)
  /// * Positive infinity via "inf", "INFINITY", "Inf", "+Inf", or "+inf".
  /// * Negative infinity via "-inf", "-INFINITY", or "-Inf".
  /// * Quiet NaNs via "nan", "NaN", "nan(...)", or "NaN(...)", where the
  ///   "..." is either a decimal or hexadecimal integer representing the
  ///   payload. A negative sign may be optionally provided.
  /// * Signaling NaNs via "snan", "sNaN", "snan(...)", or "sNaN(...)", where
  ///   the "..." is either a decimal or hexadecimal integer representing the
  ///   payload. A negative sign may be optionally provided.
  ///
  /// If the input string is none of these forms, then an error is returned.
  ///
  /// If a floating-point exception occurs during conversion, then no error is
  /// returned, and the exception is indicated via opStatus.
  LLVM_ABI 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.
  LLVM_ABI 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
  LLVM_ABI 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.
  LLVM_ABI 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;
  }
 
  // IEEE comparison with another floating point number (NaNs compare unordered,
  // 0==-0).
  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");
  }
 
  // Compares the absolute value of this APFloat with another.  Both operands
  // must be finite non-zero.
  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");
  }
 
  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 isSmallestNormalized() const {
    APFLOAT_DISPATCH_ON_SEMANTICS(isSmallestNormalized());
  }
 
  /// If the value is a NaN value, return an integer containing the payload of
  /// this value. This payload will include the quiet bit as part of the
  /// returned integer.
  APInt getNaNPayload() const {
    assert(isNaN() && "Can only call this on a NaN value");
    APFLOAT_DISPATCH_ON_SEMANTICS(getNaNPayload());
  }
 
  /// Return the FPClassTest which will return true for the value.
  LLVM_ABI 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));
  }
 
  LLVM_ABI void print(raw_ostream &) const;
 
#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
  LLVM_DUMP_METHOD void dump() const;
#endif
 
  /// If this value is normal and has an exact, normal, multiplicative inverse,
  /// store it in inv and return true.
  LLVM_ABI 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 {
    APFLOAT_DISPATCH_ON_SEMANTICS(getExactLog2Abs());
  }
 
  // 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 true if this value is exactly 2^N.
  LLVM_READONLY
  bool isPowerOf2(int N) const { return N != INT_MIN && getExactLog2() == N; }
 
  // Returns true if this value is exactly -(2^N).
  LLVM_READONLY
  bool isNegPowerOf2(int N) const {
    return N != INT_MIN && isNegative() && getExactLog2Abs() == N;
  }
 
  // Returns true if this value is exactly +1.0.
  LLVM_READONLY bool isOne() const { return isPowerOf2(0); }
 
  // Returns true if this value is exactly -1.0.
  LLVM_READONLY bool isMinusOne() const { return isNegPowerOf2(0); }
 
  LLVM_ABI friend hash_code hash_value(const APFloat &Arg);
  friend int ilogb(const APFloat &Arg);
  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.
LLVM_ABI hash_code hash_value(const APFloat &Arg);
 
/// 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
///
inline int ilogb(const APFloat &Arg) {
  if (APFloat::usesLayout<detail::IEEEFloat>(Arg.getSemantics()))
    return ilogb(Arg.U.IEEE);
  if (APFloat::usesLayout<detail::DoubleAPFloat>(Arg.getSemantics()))
    return ilogb(Arg.U.Double);
  llvm_unreachable("Unexpected semantics");
}
 
/// Returns: X * 2^Exp for integral exponents.
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 2008 minNum semantics. Returns the smaller of the
/// 2 arguments if both are not NaN. If either argument is a qNaN, returns the
/// other argument. If either argument is sNaN, return a qNaN.
/// -0 is treated as ordered less than +0.
LLVM_READONLY
inline APFloat minnum(const APFloat &A, const APFloat &B) {
  if (A.isSignaling())
    return A.makeQuiet();
  if (B.isSignaling())
    return B.makeQuiet();
  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 2008 maxNum semantics. Returns the larger of the
/// 2 arguments if both are not NaN. If either argument is a qNaN, returns the
/// other argument. If either argument is sNaN, return a qNaN.
/// +0 is treated as ordered greater than -0.
LLVM_READONLY
inline APFloat maxnum(const APFloat &A, const APFloat &B) {
  if (A.isSignaling())
    return A.makeQuiet();
  if (B.isSignaling())
    return B.makeQuiet();
  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;
}
 
/// Implement IEEE 754-2019 exp functions
LLVM_READONLY
LLVM_ABI std::optional<APFloat>
exp(const APFloat &X, RoundingMode RM = APFloat::rmNearestTiesToEven,
    APFloat::opStatus *Status = nullptr);
 
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]; }
 
inline DoubleAPFloat::~DoubleAPFloat() { delete[] Floats; }
 
} // namespace detail
 
} // namespace llvm
 
#undef APFLOAT_DISPATCH_ON_SEMANTICS
#endif // LLVM_ADT_APFLOAT_H