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//===- llvm/ADT/APFloat.h - Arbitrary Precision Floating Point ---*- C++ -*-==//
// The LLVM Compiler Infrastructure
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
/// \file
/// \brief
/// This file declares a class to represent arbitrary precision floating point
/// values and provide a variety of arithmetic operations on them.
#include "llvm/ADT/APInt.h"
namespace llvm {
struct fltSemantics;
class APSInt;
class StringRef;
/// 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
/// \brief 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.
class APFloat {
/// A signed type to represent a floating point numbers unbiased exponent.
typedef signed short ExponentType;
/// \name Floating Point Semantics.
/// @{
static const fltSemantics IEEEhalf;
static const fltSemantics IEEEsingle;
static const fltSemantics IEEEdouble;
static const fltSemantics IEEEquad;
static const fltSemantics PPCDoubleDouble;
static const fltSemantics x87DoubleExtended;
/// A Pseudo fltsemantic used to construct APFloats that cannot conflict with
/// anything real.
static const fltSemantics Bogus;
/// @}
static unsigned int semanticsPrecision(const fltSemantics &);
/// IEEE-754R 5.11: Floating Point Comparison Relations.
enum cmpResult {
/// IEEE-754R 4.3: Rounding-direction attributes.
enum roundingMode {
/// IEEE-754R 7: Default exception handling.
/// opUnderflow or opOverflow are always returned or-ed with opInexact.
enum opStatus {
opOK = 0x00,
opInvalidOp = 0x01,
opDivByZero = 0x02,
opOverflow = 0x04,
opUnderflow = 0x08,
opInexact = 0x10
/// Category of internally-represented number.
enum fltCategory {
/// Convenience enum used to construct an uninitialized APFloat.
enum uninitializedTag {
/// \name Constructors
/// @{
APFloat(const fltSemantics &); // Default construct to 0.0
APFloat(const fltSemantics &, StringRef);
APFloat(const fltSemantics &, integerPart);
APFloat(const fltSemantics &, uninitializedTag);
APFloat(const fltSemantics &, const APInt &);
explicit APFloat(double d);
explicit APFloat(float f);
APFloat(const APFloat &);
APFloat(APFloat &&);
/// @}
/// \brief Returns whether this instance allocated memory.
bool needsCleanup() const { return partCount() > 1; }
/// \name Convenience "constructors"
/// @{
/// 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);
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);
return Val;
/// Factory for QNaN values.
/// \param Negative - True iff the NaN generated should be negative.
/// \param type - 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,
unsigned type = 0) {
if (type) {
APInt fill(64, type);
return getQNaN(Sem, Negative, &fill);
} else {
return getQNaN(Sem, Negative, nullptr);
/// Factory for QNaN values.
static APFloat getQNaN(const fltSemantics &Sem, bool Negative = false,
const APInt *payload = nullptr) {
return makeNaN(Sem, false, Negative, payload);
/// Factory for SNaN values.
static APFloat getSNaN(const fltSemantics &Sem, bool Negative = false,
const APInt *payload = nullptr) {
return makeNaN(Sem, true, Negative, payload);
/// 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);
/// 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);
/// 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);
/// Returns a float which is bitcasted from an all one value int.
/// \param BitWidth - Select float type
/// \param isIEEE - If 128 bit number, select between PPC and IEEE
static APFloat getAllOnesValue(unsigned BitWidth, bool isIEEE = false);
/// @}
/// Used to insert APFloat objects, or objects that contain APFloat objects,
/// into FoldingSets.
void Profile(FoldingSetNodeID &NID) const;
/// \brief Used by the Bitcode serializer to emit APInts to Bitcode.
void Emit(Serializer &S) const;
/// \brief Used by the Bitcode deserializer to deserialize APInts.
static APFloat ReadVal(Deserializer &D);
/// \name Arithmetic
/// @{
opStatus add(const APFloat &, roundingMode);
opStatus subtract(const APFloat &, roundingMode);
opStatus multiply(const APFloat &, roundingMode);
opStatus divide(const APFloat &, roundingMode);
/// IEEE remainder.
opStatus remainder(const APFloat &);
/// C fmod, or llvm frem.
opStatus mod(const APFloat &, roundingMode);
opStatus fusedMultiplyAdd(const APFloat &, const APFloat &, roundingMode);
opStatus roundToIntegral(roundingMode);
/// IEEE-754R 5.3.1: nextUp/nextDown.
opStatus next(bool nextDown);
/// @}
/// \name Sign operations.
/// @{
void changeSign();
void clearSign();
void copySign(const APFloat &);
/// @}
/// \name Conversions
/// @{
opStatus convert(const fltSemantics &, roundingMode, bool *);
opStatus convertToInteger(integerPart *, unsigned int, bool, roundingMode,
bool *) const;
opStatus convertToInteger(APSInt &, 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);
opStatus convertFromString(StringRef, roundingMode);
APInt bitcastToAPInt() const;
double convertToDouble() const;
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 APFloat &) const LLVM_DELETED_FUNCTION;
/// IEEE comparison with another floating point number (NaNs compare
/// unordered, 0==-0).
cmpResult compare(const APFloat &) const;
/// Bitwise comparison for equality (QNaNs compare equal, 0!=-0).
bool bitwiseIsEqual(const APFloat &) 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 == 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 != 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 and only if the number has the largest possible finite
/// magnitude in the current semantics.
bool isLargest() const;
/// @}
APFloat &operator=(const APFloat &);
APFloat &operator=(APFloat &&);
/// \brief 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 APFloat &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.
/// 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) const;
/// If this value has an exact multiplicative inverse, store it in inv and
/// return true.
bool getExactInverse(APFloat *inv) const;
/// \name Simple Queries
/// @{
integerPart *significandParts();
const integerPart *significandParts() const;
unsigned int partCount() const;
/// @}
/// \name Significand operations.
/// @{
integerPart addSignificand(const APFloat &);
integerPart subtractSignificand(const APFloat &, integerPart);
lostFraction addOrSubtractSignificand(const APFloat &, bool subtract);
lostFraction multiplySignificand(const APFloat &, const APFloat *);
lostFraction divideSignificand(const APFloat &);
void incrementSignificand();
void initialize(const fltSemantics *);
void shiftSignificandLeft(unsigned int);
lostFraction shiftSignificandRight(unsigned int);
unsigned int significandLSB() const;
unsigned int significandMSB() const;
void zeroSignificand();
/// Return true if the significand excluding the integral bit is all ones.
bool isSignificandAllOnes() const;
/// Return true if the significand excluding the integral bit is all zeros.
bool isSignificandAllZeros() const;
/// @}
/// \name Arithmetic on special values.
/// @{
opStatus addOrSubtractSpecials(const APFloat &, bool subtract);
opStatus divideSpecials(const APFloat &);
opStatus multiplySpecials(const APFloat &);
opStatus modSpecials(const APFloat &);
/// @}
/// \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);
static APFloat makeNaN(const fltSemantics &Sem, bool SNaN, bool Negative,
const APInt *fill);
void makeInf(bool Neg = false);
void makeZero(bool Neg = false);
/// @}
/// \name Miscellany
/// @{
bool convertFromStringSpecials(StringRef str);
opStatus normalize(roundingMode, lostFraction);
opStatus addOrSubtract(const APFloat &, roundingMode, bool subtract);
cmpResult compareAbsoluteValue(const APFloat &) const;
opStatus handleOverflow(roundingMode);
bool roundAwayFromZero(roundingMode, lostFraction, unsigned int) const;
opStatus convertToSignExtendedInteger(integerPart *, unsigned int, bool,
roundingMode, bool *) const;
opStatus convertFromUnsignedParts(const integerPart *, unsigned int,
opStatus convertFromHexadecimalString(StringRef, roundingMode);
opStatus convertFromDecimalString(StringRef, roundingMode);
char *convertNormalToHexString(char *, unsigned int, bool,
roundingMode) const;
opStatus roundSignificandWithExponent(const integerPart *, unsigned int, int,
/// @}
APInt convertHalfAPFloatToAPInt() const;
APInt convertFloatAPFloatToAPInt() const;
APInt convertDoubleAPFloatToAPInt() const;
APInt convertQuadrupleAPFloatToAPInt() const;
APInt convertF80LongDoubleAPFloatToAPInt() const;
APInt convertPPCDoubleDoubleAPFloatToAPInt() const;
void initFromAPInt(const fltSemantics *Sem, const APInt &api);
void initFromHalfAPInt(const APInt &api);
void initFromFloatAPInt(const APInt &api);
void initFromDoubleAPInt(const APInt &api);
void initFromQuadrupleAPInt(const APInt &api);
void initFromF80LongDoubleAPInt(const APInt &api);
void initFromPPCDoubleDoubleAPInt(const APInt &api);
void assign(const APFloat &);
void copySignificand(const APFloat &);
void freeSignificand();
/// 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;
/// See friend declaration above.
/// This additional declaration is required in order to compile LLVM with IBM
/// xlC compiler.
hash_code hash_value(const APFloat &Arg);
} // namespace llvm