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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
+ * http://en.wikipedia.org/wiki/IEEE_floating_point
+ * http://en.wikipedia.org/wiki/Floating_point#IEEE_754:_floating_point_in_modern_computers
+ */
+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 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(!std::isnan(aValue), "NaN does not have a sign");
+ return std::signbit(aValue);
+}
+
+/** 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 std::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(std::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 (!std::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 (std::isnan(aValue1)) {
+ return std::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 (std::isnan(aValue1)) {
+ return std::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 (std::isnan(aValue1) || std::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 (std::isnan(aValue1) || std::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 */