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/* -*- Mode: C++; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 2 -*- */
/* vim: set ts=8 sts=2 et sw=2 tw=80: */
/* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
* file, You can obtain one at http://mozilla.org/MPL/2.0/. */
/* Various predicates and operations on IEEE-754 floating point types. */
#ifndef mozilla_FloatingPoint_h
#define mozilla_FloatingPoint_h
#include "mozilla/Assertions.h"
#include "mozilla/Attributes.h"
#include "mozilla/Casting.h"
#include "mozilla/MathAlgorithms.h"
#include "mozilla/MemoryChecking.h"
#include "mozilla/Types.h"
#include <algorithm>
#include <climits>
#include <limits>
#include <stdint.h>
namespace mozilla {
/*
* It's reasonable to ask why we have this header at all. Don't isnan,
* copysign, the built-in comparison operators, and the like solve these
* problems? Unfortunately, they don't. We've found that various compilers
* (MSVC, MSVC when compiling with PGO, and GCC on OS X, at least) miscompile
* the standard methods in various situations, so we can't use them. Some of
* these compilers even have problems compiling seemingly reasonable bitwise
* algorithms! But with some care we've found algorithms that seem to not
* trigger those compiler bugs.
*
* For the aforementioned reasons, be very wary of making changes to any of
* these algorithms. If you must make changes, keep a careful eye out for
* compiler bustage, particularly PGO-specific bustage.
*/
namespace detail {
/*
* These implementations assume float/double are 32/64-bit single/double
* format number types compatible with the IEEE-754 standard. C++ doesn't
* require this, but we required it in implementations of these algorithms that
* preceded this header, so we shouldn't break anything to continue doing so.
*/
template <typename T>
struct FloatingPointTrait;
template <>
struct FloatingPointTrait<float> {
protected:
using Bits = uint32_t;
static constexpr unsigned kExponentWidth = 8;
static constexpr unsigned kSignificandWidth = 23;
};
template <>
struct FloatingPointTrait<double> {
protected:
using Bits = uint64_t;
static constexpr unsigned kExponentWidth = 11;
static constexpr unsigned kSignificandWidth = 52;
};
} // namespace detail
/*
* This struct contains details regarding the encoding of floating-point
* numbers that can be useful for direct bit manipulation. As of now, the
* template parameter has to be float or double.
*
* The nested typedef |Bits| is the unsigned integral type with the same size
* as T: uint32_t for float and uint64_t for double (static assertions
* double-check these assumptions).
*
* kExponentBias is the offset that is subtracted from the exponent when
* computing the value, i.e. one plus the opposite of the mininum possible
* exponent.
* kExponentShift is the shift that one needs to apply to retrieve the
* exponent component of the value.
*
* kSignBit contains a bits mask. Bit-and-ing with this mask will result in
* obtaining the sign bit.
* kExponentBits contains the mask needed for obtaining the exponent bits and
* kSignificandBits contains the mask needed for obtaining the significand
* bits.
*
* Full details of how floating point number formats are encoded are beyond
* the scope of this comment. For more information, see
*/
template <typename T>
struct FloatingPoint final : private detail::FloatingPointTrait<T> {
private:
using Base = detail::FloatingPointTrait<T>;
public:
/**
* An unsigned integral type suitable for accessing the bitwise representation
* of T.
*/
using Bits = typename Base::Bits;
static_assert(sizeof(T) == sizeof(Bits), "Bits must be same size as T");
/** The bit-width of the exponent component of T. */
using Base::kExponentWidth;
/** The bit-width of the significand component of T. */
using Base::kSignificandWidth;
static_assert(1 + kExponentWidth + kSignificandWidth == CHAR_BIT * sizeof(T),
"sign bit plus bit widths should sum to overall bit width");
/**
* The exponent field in an IEEE-754 floating point number consists of bits
* encoding an unsigned number. The *actual* represented exponent (for all
* values finite and not denormal) is that value, minus a bias |kExponentBias|
* so that a useful range of numbers is represented.
*/
static constexpr unsigned kExponentBias = (1U << (kExponentWidth - 1)) - 1;
/**
* The amount by which the bits of the exponent-field in an IEEE-754 floating
* point number are shifted from the LSB of the floating point type.
*/
static constexpr unsigned kExponentShift = kSignificandWidth;
/** The sign bit in the floating point representation. */
static constexpr Bits kSignBit = static_cast<Bits>(1)
<< (CHAR_BIT * sizeof(Bits) - 1);
/** The exponent bits in the floating point representation. */
static constexpr Bits kExponentBits =
((static_cast<Bits>(1) << kExponentWidth) - 1) << kSignificandWidth;
/** The significand bits in the floating point representation. */
static constexpr Bits kSignificandBits =
(static_cast<Bits>(1) << kSignificandWidth) - 1;
static_assert((kSignBit & kExponentBits) == 0,
"sign bit shouldn't overlap exponent bits");
static_assert((kSignBit & kSignificandBits) == 0,
"sign bit shouldn't overlap significand bits");
static_assert((kExponentBits & kSignificandBits) == 0,
"exponent bits shouldn't overlap significand bits");
static_assert((kSignBit | kExponentBits | kSignificandBits) == ~Bits(0),
"all bits accounted for");
};
/** Determines whether a float/double is NaN. */
template <typename T>
static MOZ_ALWAYS_INLINE bool IsNaN(T aValue) {
/*
* A float/double is NaN if all exponent bits are 1 and the significand
* contains at least one non-zero bit.
*/
typedef FloatingPoint<T> Traits;
typedef typename Traits::Bits Bits;
return (BitwiseCast<Bits>(aValue) & Traits::kExponentBits) ==
Traits::kExponentBits &&
(BitwiseCast<Bits>(aValue) & Traits::kSignificandBits) != 0;
}
/** Determines whether a float/double is +Infinity or -Infinity. */
template <typename T>
static MOZ_ALWAYS_INLINE bool IsInfinite(T aValue) {
/* Infinities have all exponent bits set to 1 and an all-0 significand. */
typedef FloatingPoint<T> Traits;
typedef typename Traits::Bits Bits;
Bits bits = BitwiseCast<Bits>(aValue);
return (bits & ~Traits::kSignBit) == Traits::kExponentBits;
}
/** Determines whether a float/double is not NaN or infinite. */
template <typename T>
static MOZ_ALWAYS_INLINE bool IsFinite(T aValue) {
/*
* NaN and Infinities are the only non-finite floats/doubles, and both have
* all exponent bits set to 1.
*/
typedef FloatingPoint<T> Traits;
typedef typename Traits::Bits Bits;
Bits bits = BitwiseCast<Bits>(aValue);
return (bits & Traits::kExponentBits) != Traits::kExponentBits;
}
/**
* Determines whether a float/double is negative or -0. It is an error
* to call this method on a float/double which is NaN.
*/
template <typename T>
static MOZ_ALWAYS_INLINE bool IsNegative(T aValue) {
MOZ_ASSERT(!IsNaN(aValue), "NaN does not have a sign");
/* The sign bit is set if the double is negative. */
typedef FloatingPoint<T> Traits;
typedef typename Traits::Bits Bits;
Bits bits = BitwiseCast<Bits>(aValue);
return (bits & Traits::kSignBit) != 0;
}
/** Determines whether a float/double represents -0. */
template <typename T>
static MOZ_ALWAYS_INLINE bool IsNegativeZero(T aValue) {
/* Only the sign bit is set if the value is -0. */
typedef FloatingPoint<T> Traits;
typedef typename Traits::Bits Bits;
Bits bits = BitwiseCast<Bits>(aValue);
return bits == Traits::kSignBit;
}
/** Determines wether a float/double represents +0. */
template <typename T>
static MOZ_ALWAYS_INLINE bool IsPositiveZero(T aValue) {
/* All bits are zero if the value is +0. */
typedef FloatingPoint<T> Traits;
typedef typename Traits::Bits Bits;
Bits bits = BitwiseCast<Bits>(aValue);
return bits == 0;
}
/**
* Returns 0 if a float/double is NaN or infinite;
* otherwise, the float/double is returned.
*/
template <typename T>
static MOZ_ALWAYS_INLINE T ToZeroIfNonfinite(T aValue) {
return IsFinite(aValue) ? aValue : 0;
}
/**
* Returns the exponent portion of the float/double.
*
* Zero is not special-cased, so ExponentComponent(0.0) is
* -int_fast16_t(Traits::kExponentBias).
*/
template <typename T>
static MOZ_ALWAYS_INLINE int_fast16_t ExponentComponent(T aValue) {
/*
* The exponent component of a float/double is an unsigned number, biased
* from its actual value. Subtract the bias to retrieve the actual exponent.
*/
typedef FloatingPoint<T> Traits;
typedef typename Traits::Bits Bits;
Bits bits = BitwiseCast<Bits>(aValue);
return int_fast16_t((bits & Traits::kExponentBits) >>
Traits::kExponentShift) -
int_fast16_t(Traits::kExponentBias);
}
/** Returns +Infinity. */
template <typename T>
static MOZ_ALWAYS_INLINE T PositiveInfinity() {
/*
* Positive infinity has all exponent bits set, sign bit set to 0, and no
* significand.
*/
typedef FloatingPoint<T> Traits;
return BitwiseCast<T>(Traits::kExponentBits);
}
/** Returns -Infinity. */
template <typename T>
static MOZ_ALWAYS_INLINE T NegativeInfinity() {
/*
* Negative infinity has all exponent bits set, sign bit set to 1, and no
* significand.
*/
typedef FloatingPoint<T> Traits;
return BitwiseCast<T>(Traits::kSignBit | Traits::kExponentBits);
}
/**
* Computes the bit pattern for an infinity with the specified sign bit.
*/
template <typename T, int SignBit>
struct InfinityBits {
using Traits = FloatingPoint<T>;
static_assert(SignBit == 0 || SignBit == 1, "bad sign bit");
static constexpr typename Traits::Bits value =
(SignBit * Traits::kSignBit) | Traits::kExponentBits;
};
/**
* Computes the bit pattern for a NaN with the specified sign bit and
* significand bits.
*/
template <typename T, int SignBit, typename FloatingPoint<T>::Bits Significand>
struct SpecificNaNBits {
using Traits = FloatingPoint<T>;
static_assert(SignBit == 0 || SignBit == 1, "bad sign bit");
static_assert((Significand & ~Traits::kSignificandBits) == 0,
"significand must only have significand bits set");
static_assert(Significand & Traits::kSignificandBits,
"significand must be nonzero");
static constexpr typename Traits::Bits value =
(SignBit * Traits::kSignBit) | Traits::kExponentBits | Significand;
};
/**
* Constructs a NaN value with the specified sign bit and significand bits.
*
* There is also a variant that returns the value directly. In most cases, the
* two variants should be identical. However, in the specific case of x86
* chips, the behavior differs: returning floating-point values directly is done
* through the x87 stack, and x87 loads and stores turn signaling NaNs into
* quiet NaNs... silently. Returning floating-point values via outparam,
* however, is done entirely within the SSE registers when SSE2 floating-point
* is enabled in the compiler, which has semantics-preserving behavior you would
* expect.
*
* If preserving the distinction between signaling NaNs and quiet NaNs is
* important to you, you should use the outparam version. In all other cases,
* you should use the direct return version.
*/
template <typename T>
static MOZ_ALWAYS_INLINE void SpecificNaN(
int signbit, typename FloatingPoint<T>::Bits significand, T* result) {
typedef FloatingPoint<T> Traits;
MOZ_ASSERT(signbit == 0 || signbit == 1);
MOZ_ASSERT((significand & ~Traits::kSignificandBits) == 0);
MOZ_ASSERT(significand & Traits::kSignificandBits);
BitwiseCast<T>(
(signbit ? Traits::kSignBit : 0) | Traits::kExponentBits | significand,
result);
MOZ_ASSERT(IsNaN(*result));
}
template <typename T>
static MOZ_ALWAYS_INLINE T
SpecificNaN(int signbit, typename FloatingPoint<T>::Bits significand) {
T t;
SpecificNaN(signbit, significand, &t);
return t;
}
/** Computes the smallest non-zero positive float/double value. */
template <typename T>
static MOZ_ALWAYS_INLINE T MinNumberValue() {
typedef FloatingPoint<T> Traits;
typedef typename Traits::Bits Bits;
return BitwiseCast<T>(Bits(1));
}
namespace detail {
template <typename Float, typename SignedInteger>
inline bool NumberEqualsSignedInteger(Float aValue, SignedInteger* aInteger) {
static_assert(std::is_same_v<Float, float> || std::is_same_v<Float, double>,
"Float must be an IEEE-754 floating point type");
static_assert(std::is_signed_v<SignedInteger>,
"this algorithm only works for signed types: a different one "
"will be required for unsigned types");
static_assert(sizeof(SignedInteger) >= sizeof(int),
"this function *might* require some finessing for signed types "
"subject to integral promotion before it can be used on them");
MOZ_MAKE_MEM_UNDEFINED(aInteger, sizeof(*aInteger));
// NaNs and infinities are not integers.
if (!IsFinite(aValue)) {
return false;
}
// Otherwise do direct comparisons against the minimum/maximum |SignedInteger|
// values that can be encoded in |Float|.
constexpr SignedInteger MaxIntValue =
std::numeric_limits<SignedInteger>::max(); // e.g. INT32_MAX
constexpr SignedInteger MinValue =
std::numeric_limits<SignedInteger>::min(); // e.g. INT32_MIN
static_assert(IsPowerOfTwo(Abs(MinValue)),
"MinValue should be is a small power of two, thus exactly "
"representable in float/double both");
constexpr unsigned SignedIntegerWidth = CHAR_BIT * sizeof(SignedInteger);
constexpr unsigned ExponentShift = FloatingPoint<Float>::kExponentShift;
// Careful! |MaxIntValue| may not be the maximum |SignedInteger| value that
// can be encoded in |Float|. Its |SignedIntegerWidth - 1| bits of precision
// may exceed |Float|'s |ExponentShift + 1| bits of precision. If necessary,
// compute the maximum |SignedInteger| that fits in |Float| from IEEE-754
// first principles. (|MinValue| doesn't have this problem because as a
// [relatively] small power of two it's always representable in |Float|.)
// Per C++11 [expr.const]p2, unevaluated subexpressions of logical AND/OR and
// conditional expressions *may* contain non-constant expressions, without
// making the enclosing expression not constexpr. MSVC implements this -- but
// it sometimes warns about undefined behavior in unevaluated subexpressions.
// This bites us if we initialize |MaxValue| the obvious way including an
// |uint64_t(1) << (SignedIntegerWidth - 2 - ExponentShift)| subexpression.
// Pull that shift-amount out and give it a not-too-huge value when it's in an
// unevaluated subexpression. 🙄
constexpr unsigned PrecisionExceededShiftAmount =
ExponentShift > SignedIntegerWidth - 1
? 0
: SignedIntegerWidth - 2 - ExponentShift;
constexpr SignedInteger MaxValue =
ExponentShift > SignedIntegerWidth - 1
? MaxIntValue
: SignedInteger((uint64_t(1) << (SignedIntegerWidth - 1)) -
(uint64_t(1) << PrecisionExceededShiftAmount));
if (static_cast<Float>(MinValue) <= aValue &&
aValue <= static_cast<Float>(MaxValue)) {
auto possible = static_cast<SignedInteger>(aValue);
if (static_cast<Float>(possible) == aValue) {
*aInteger = possible;
return true;
}
}
return false;
}
template <typename Float, typename SignedInteger>
inline bool NumberIsSignedInteger(Float aValue, SignedInteger* aInteger) {
static_assert(std::is_same_v<Float, float> || std::is_same_v<Float, double>,
"Float must be an IEEE-754 floating point type");
static_assert(std::is_signed_v<SignedInteger>,
"this algorithm only works for signed types: a different one "
"will be required for unsigned types");
static_assert(sizeof(SignedInteger) >= sizeof(int),
"this function *might* require some finessing for signed types "
"subject to integral promotion before it can be used on them");
MOZ_MAKE_MEM_UNDEFINED(aInteger, sizeof(*aInteger));
if (IsNegativeZero(aValue)) {
return false;
}
return NumberEqualsSignedInteger(aValue, aInteger);
}
} // namespace detail
/**
* If |aValue| is identical to some |int32_t| value, set |*aInt32| to that value
* and return true. Otherwise return false, leaving |*aInt32| in an
* indeterminate state.
*
* This method returns false for negative zero. If you want to consider -0 to
* be 0, use NumberEqualsInt32 below.
*/
template <typename T>
static MOZ_ALWAYS_INLINE bool NumberIsInt32(T aValue, int32_t* aInt32) {
return detail::NumberIsSignedInteger(aValue, aInt32);
}
/**
* If |aValue| is identical to some |int64_t| value, set |*aInt64| to that value
* and return true. Otherwise return false, leaving |*aInt64| in an
* indeterminate state.
*
* This method returns false for negative zero. If you want to consider -0 to
* be 0, use NumberEqualsInt64 below.
*/
template <typename T>
static MOZ_ALWAYS_INLINE bool NumberIsInt64(T aValue, int64_t* aInt64) {
return detail::NumberIsSignedInteger(aValue, aInt64);
}
/**
* If |aValue| is equal to some int32_t value (where -0 and +0 are considered
* equal), set |*aInt32| to that value and return true. Otherwise return false,
* leaving |*aInt32| in an indeterminate state.
*
* |NumberEqualsInt32(-0.0, ...)| will return true. To test whether a value can
* be losslessly converted to |int32_t| and back, use NumberIsInt32 above.
*/
template <typename T>
static MOZ_ALWAYS_INLINE bool NumberEqualsInt32(T aValue, int32_t* aInt32) {
return detail::NumberEqualsSignedInteger(aValue, aInt32);
}
/**
* If |aValue| is equal to some int64_t value (where -0 and +0 are considered
* equal), set |*aInt64| to that value and return true. Otherwise return false,
* leaving |*aInt64| in an indeterminate state.
*
* |NumberEqualsInt64(-0.0, ...)| will return true. To test whether a value can
* be losslessly converted to |int64_t| and back, use NumberIsInt64 above.
*/
template <typename T>
static MOZ_ALWAYS_INLINE bool NumberEqualsInt64(T aValue, int64_t* aInt64) {
return detail::NumberEqualsSignedInteger(aValue, aInt64);
}
/**
* Computes a NaN value. Do not use this method if you depend upon a particular
* NaN value being returned.
*/
template <typename T>
static MOZ_ALWAYS_INLINE T UnspecifiedNaN() {
/*
* If we can use any quiet NaN, we might as well use the all-ones NaN,
* since it's cheap to materialize on common platforms (such as x64, where
* this value can be represented in a 32-bit signed immediate field, allowing
* it to be stored to memory in a single instruction).
*/
typedef FloatingPoint<T> Traits;
return SpecificNaN<T>(1, Traits::kSignificandBits);
}
/**
* Compare two doubles for equality, *without* equating -0 to +0, and equating
* any NaN value to any other NaN value. (The normal equality operators equate
* -0 with +0, and they equate NaN to no other value.)
*/
template <typename T>
static inline bool NumbersAreIdentical(T aValue1, T aValue2) {
using Bits = typename FloatingPoint<T>::Bits;
if (IsNaN(aValue1)) {
return IsNaN(aValue2);
}
return BitwiseCast<Bits>(aValue1) == BitwiseCast<Bits>(aValue2);
}
/**
* Compare two floating point values for bit-wise equality.
*/
template <typename T>
static inline bool NumbersAreBitwiseIdentical(T aValue1, T aValue2) {
using Bits = typename FloatingPoint<T>::Bits;
return BitwiseCast<Bits>(aValue1) == BitwiseCast<Bits>(aValue2);
}
/**
* Return true iff |aValue| and |aValue2| are equal (ignoring sign if both are
* zero) or both NaN.
*/
template <typename T>
static inline bool EqualOrBothNaN(T aValue1, T aValue2) {
if (IsNaN(aValue1)) {
return IsNaN(aValue2);
}
return aValue1 == aValue2;
}
/**
* Return NaN if either |aValue1| or |aValue2| is NaN, or the minimum of
* |aValue1| and |aValue2| otherwise.
*/
template <typename T>
static inline T NaNSafeMin(T aValue1, T aValue2) {
if (IsNaN(aValue1) || IsNaN(aValue2)) {
return UnspecifiedNaN<T>();
}
return std::min(aValue1, aValue2);
}
/**
* Return NaN if either |aValue1| or |aValue2| is NaN, or the maximum of
* |aValue1| and |aValue2| otherwise.
*/
template <typename T>
static inline T NaNSafeMax(T aValue1, T aValue2) {
if (IsNaN(aValue1) || IsNaN(aValue2)) {
return UnspecifiedNaN<T>();
}
return std::max(aValue1, aValue2);
}
namespace detail {
template <typename T>
struct FuzzyEqualsEpsilon;
template <>
struct FuzzyEqualsEpsilon<float> {
// A number near 1e-5 that is exactly representable in a float.
static float value() { return 1.0f / (1 << 17); }
};
template <>
struct FuzzyEqualsEpsilon<double> {
// A number near 1e-12 that is exactly representable in a double.
static double value() { return 1.0 / (1LL << 40); }
};
} // namespace detail
/**
* Compare two floating point values for equality, modulo rounding error. That
* is, the two values are considered equal if they are both not NaN and if they
* are less than or equal to aEpsilon apart. The default value of aEpsilon is
* near 1e-5.
*
* For most scenarios you will want to use FuzzyEqualsMultiplicative instead,
* as it is more reasonable over the entire range of floating point numbers.
* This additive version should only be used if you know the range of the
* numbers you are dealing with is bounded and stays around the same order of
* magnitude.
*/
template <typename T>
static MOZ_ALWAYS_INLINE bool FuzzyEqualsAdditive(
T aValue1, T aValue2, T aEpsilon = detail::FuzzyEqualsEpsilon<T>::value()) {
static_assert(std::is_floating_point_v<T>, "floating point type required");
return Abs(aValue1 - aValue2) <= aEpsilon;
}
/**
* Compare two floating point values for equality, allowing for rounding error
* relative to the magnitude of the values. That is, the two values are
* considered equal if they are both not NaN and they are less than or equal to
* some aEpsilon apart, where the aEpsilon is scaled by the smaller of the two
* argument values.
*
* In most cases you will want to use this rather than FuzzyEqualsAdditive, as
* this function effectively masks out differences in the bottom few bits of
* the floating point numbers being compared, regardless of what order of
* magnitude those numbers are at.
*/
template <typename T>
static MOZ_ALWAYS_INLINE bool FuzzyEqualsMultiplicative(
T aValue1, T aValue2, T aEpsilon = detail::FuzzyEqualsEpsilon<T>::value()) {
static_assert(std::is_floating_point_v<T>, "floating point type required");
// can't use std::min because of bug 965340
T smaller = Abs(aValue1) < Abs(aValue2) ? Abs(aValue1) : Abs(aValue2);
return Abs(aValue1 - aValue2) <= aEpsilon * smaller;
}
/**
* Returns true if |aValue| can be losslessly represented as an IEEE-754 single
* precision number, false otherwise. All NaN values are considered
* representable (even though the bit patterns of double precision NaNs can't
* all be exactly represented in single precision).
*/
[[nodiscard]] extern MFBT_API bool IsFloat32Representable(double aValue);
} /* namespace mozilla */
#endif /* mozilla_FloatingPoint_h */