michael@0: /* -*- Mode: C++; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 2 -*- */
michael@0: /* vim: set ts=8 sts=2 et sw=2 tw=80: */
michael@0: /* This Source Code Form is subject to the terms of the Mozilla Public
michael@0:  * License, v. 2.0. If a copy of the MPL was not distributed with this
michael@0:  * file, You can obtain one at http://mozilla.org/MPL/2.0/. */
michael@0: 
michael@0: /* Various predicates and operations on IEEE-754 floating point types. */
michael@0: 
michael@0: #ifndef mozilla_FloatingPoint_h
michael@0: #define mozilla_FloatingPoint_h
michael@0: 
michael@0: #include "mozilla/Assertions.h"
michael@0: #include "mozilla/Attributes.h"
michael@0: #include "mozilla/Casting.h"
michael@0: #include "mozilla/MathAlgorithms.h"
michael@0: #include "mozilla/Types.h"
michael@0: 
michael@0: #include <stdint.h>
michael@0: 
michael@0: namespace mozilla {
michael@0: 
michael@0: /*
michael@0:  * It's reasonable to ask why we have this header at all.  Don't isnan,
michael@0:  * copysign, the built-in comparison operators, and the like solve these
michael@0:  * problems?  Unfortunately, they don't.  We've found that various compilers
michael@0:  * (MSVC, MSVC when compiling with PGO, and GCC on OS X, at least) miscompile
michael@0:  * the standard methods in various situations, so we can't use them.  Some of
michael@0:  * these compilers even have problems compiling seemingly reasonable bitwise
michael@0:  * algorithms!  But with some care we've found algorithms that seem to not
michael@0:  * trigger those compiler bugs.
michael@0:  *
michael@0:  * For the aforementioned reasons, be very wary of making changes to any of
michael@0:  * these algorithms.  If you must make changes, keep a careful eye out for
michael@0:  * compiler bustage, particularly PGO-specific bustage.
michael@0:  */
michael@0: 
michael@0: struct FloatTypeTraits
michael@0: {
michael@0:     typedef uint32_t Bits;
michael@0: 
michael@0:     static const unsigned ExponentBias = 127;
michael@0:     static const unsigned ExponentShift = 23;
michael@0: 
michael@0:     static const Bits SignBit         = 0x80000000UL;
michael@0:     static const Bits ExponentBits    = 0x7F800000UL;
michael@0:     static const Bits SignificandBits = 0x007FFFFFUL;
michael@0: };
michael@0: 
michael@0: struct DoubleTypeTraits
michael@0: {
michael@0:     typedef uint64_t Bits;
michael@0: 
michael@0:     static const unsigned ExponentBias = 1023;
michael@0:     static const unsigned ExponentShift = 52;
michael@0: 
michael@0:     static const Bits SignBit         = 0x8000000000000000ULL;
michael@0:     static const Bits ExponentBits    = 0x7ff0000000000000ULL;
michael@0:     static const Bits SignificandBits = 0x000fffffffffffffULL;
michael@0: };
michael@0: 
michael@0: template<typename T> struct SelectTrait;
michael@0: template<> struct SelectTrait<float> : public FloatTypeTraits {};
michael@0: template<> struct SelectTrait<double> : public DoubleTypeTraits {};
michael@0: 
michael@0: /*
michael@0:  *  This struct contains details regarding the encoding of floating-point
michael@0:  *  numbers that can be useful for direct bit manipulation. As of now, the
michael@0:  *  template parameter has to be float or double.
michael@0:  *
michael@0:  *  The nested typedef |Bits| is the unsigned integral type with the same size
michael@0:  *  as T: uint32_t for float and uint64_t for double (static assertions
michael@0:  *  double-check these assumptions).
michael@0:  *
michael@0:  *  ExponentBias is the offset that is subtracted from the exponent when
michael@0:  *  computing the value, i.e. one plus the opposite of the mininum possible
michael@0:  *  exponent.
michael@0:  *  ExponentShift is the shift that one needs to apply to retrieve the exponent
michael@0:  *  component of the value.
michael@0:  *
michael@0:  *  SignBit contains a bits mask. Bit-and-ing with this mask will result in
michael@0:  *  obtaining the sign bit.
michael@0:  *  ExponentBits contains the mask needed for obtaining the exponent bits and
michael@0:  *  SignificandBits contains the mask needed for obtaining the significand bits.
michael@0:  *
michael@0:  *  Full details of how floating point number formats are encoded are beyond the
michael@0:  *  scope of this comment. For more information, see
michael@0:  *  http://en.wikipedia.org/wiki/IEEE_floating_point
michael@0:  *  http://en.wikipedia.org/wiki/Floating_point#IEEE_754:_floating_point_in_modern_computers
michael@0:  */
michael@0: template<typename T>
michael@0: struct FloatingPoint : public SelectTrait<T>
michael@0: {
michael@0:     typedef SelectTrait<T> Base;
michael@0:     typedef typename Base::Bits Bits;
michael@0: 
michael@0:     static_assert((Base::SignBit & Base::ExponentBits) == 0,
michael@0:                   "sign bit shouldn't overlap exponent bits");
michael@0:     static_assert((Base::SignBit & Base::SignificandBits) == 0,
michael@0:                   "sign bit shouldn't overlap significand bits");
michael@0:     static_assert((Base::ExponentBits & Base::SignificandBits) == 0,
michael@0:                   "exponent bits shouldn't overlap significand bits");
michael@0: 
michael@0:     static_assert((Base::SignBit | Base::ExponentBits | Base::SignificandBits) ==
michael@0:                   ~Bits(0),
michael@0:                   "all bits accounted for");
michael@0: 
michael@0:     /*
michael@0:      * These implementations assume float/double are 32/64-bit single/double format
michael@0:      * number types compatible with the IEEE-754 standard.  C++ don't require this
michael@0:      * to be the case.  But we required this in implementations of these algorithms
michael@0:      * that preceded this header, so we shouldn't break anything if we keep doing so.
michael@0:      */
michael@0:     static_assert(sizeof(T) == sizeof(Bits), "Bits must be same size as T");
michael@0: };
michael@0: 
michael@0: /** Determines whether a double is NaN. */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: IsNaN(T t)
michael@0: {
michael@0:   /*
michael@0:    * A float/double is NaN if all exponent bits are 1 and the significand contains at
michael@0:    * least one non-zero bit.
michael@0:    */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   typedef typename Traits::Bits Bits;
michael@0:   Bits bits = BitwiseCast<Bits>(t);
michael@0:   return (bits & Traits::ExponentBits) == Traits::ExponentBits &&
michael@0:          (bits & Traits::SignificandBits) != 0;
michael@0: }
michael@0: 
michael@0: /** Determines whether a float/double is +Infinity or -Infinity. */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: IsInfinite(T t)
michael@0: {
michael@0:   /* Infinities have all exponent bits set to 1 and an all-0 significand. */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   typedef typename Traits::Bits Bits;
michael@0:   Bits bits = BitwiseCast<Bits>(t);
michael@0:   return (bits & ~Traits::SignBit) == Traits::ExponentBits;
michael@0: }
michael@0: 
michael@0: /** Determines whether a float/double is not NaN or infinite. */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: IsFinite(T t)
michael@0: {
michael@0:   /*
michael@0:    * NaN and Infinities are the only non-finite floats/doubles, and both have all
michael@0:    * exponent bits set to 1.
michael@0:    */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   typedef typename Traits::Bits Bits;
michael@0:   Bits bits = BitwiseCast<Bits>(t);
michael@0:   return (bits & Traits::ExponentBits) != Traits::ExponentBits;
michael@0: }
michael@0: 
michael@0: /**
michael@0:  * Determines whether a float/double is negative.  It is an error to call this method
michael@0:  * on a float/double which is NaN.
michael@0:  */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: IsNegative(T t)
michael@0: {
michael@0:   MOZ_ASSERT(!IsNaN(t), "NaN does not have a sign");
michael@0: 
michael@0:   /* The sign bit is set if the double is negative. */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   typedef typename Traits::Bits Bits;
michael@0:   Bits bits = BitwiseCast<Bits>(t);
michael@0:   return (bits & Traits::SignBit) != 0;
michael@0: }
michael@0: 
michael@0: /** Determines whether a float/double represents -0. */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: IsNegativeZero(T t)
michael@0: {
michael@0:   /* Only the sign bit is set if the value is -0. */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   typedef typename Traits::Bits Bits;
michael@0:   Bits bits = BitwiseCast<Bits>(t);
michael@0:   return bits == Traits::SignBit;
michael@0: }
michael@0: 
michael@0: /**
michael@0:  * Returns the exponent portion of the float/double.
michael@0:  *
michael@0:  * Zero is not special-cased, so ExponentComponent(0.0) is
michael@0:  * -int_fast16_t(Traits::ExponentBias).
michael@0:  */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE int_fast16_t
michael@0: ExponentComponent(T t)
michael@0: {
michael@0:   /*
michael@0:    * The exponent component of a float/double is an unsigned number, biased from its
michael@0:    * actual value.  Subtract the bias to retrieve the actual exponent.
michael@0:    */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   typedef typename Traits::Bits Bits;
michael@0:   Bits bits = BitwiseCast<Bits>(t);
michael@0:   return int_fast16_t((bits & Traits::ExponentBits) >> Traits::ExponentShift) -
michael@0:          int_fast16_t(Traits::ExponentBias);
michael@0: }
michael@0: 
michael@0: /** Returns +Infinity. */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE T
michael@0: PositiveInfinity()
michael@0: {
michael@0:   /*
michael@0:    * Positive infinity has all exponent bits set, sign bit set to 0, and no
michael@0:    * significand.
michael@0:    */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   return BitwiseCast<T>(Traits::ExponentBits);
michael@0: }
michael@0: 
michael@0: /** Returns -Infinity. */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE T
michael@0: NegativeInfinity()
michael@0: {
michael@0:   /*
michael@0:    * Negative infinity has all exponent bits set, sign bit set to 1, and no
michael@0:    * significand.
michael@0:    */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   return BitwiseCast<T>(Traits::SignBit | Traits::ExponentBits);
michael@0: }
michael@0: 
michael@0: 
michael@0: /** Constructs a NaN value with the specified sign bit and significand bits. */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE T
michael@0: SpecificNaN(int signbit, typename FloatingPoint<T>::Bits significand)
michael@0: {
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   MOZ_ASSERT(signbit == 0 || signbit == 1);
michael@0:   MOZ_ASSERT((significand & ~Traits::SignificandBits) == 0);
michael@0:   MOZ_ASSERT(significand & Traits::SignificandBits);
michael@0: 
michael@0:   T t = BitwiseCast<T>((signbit ? Traits::SignBit : 0) |
michael@0:                        Traits::ExponentBits |
michael@0:                        significand);
michael@0:   MOZ_ASSERT(IsNaN(t));
michael@0:   return t;
michael@0: }
michael@0: 
michael@0: /** Computes the smallest non-zero positive float/double value. */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE T
michael@0: MinNumberValue()
michael@0: {
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   typedef typename Traits::Bits Bits;
michael@0:   return BitwiseCast<T>(Bits(1));
michael@0: }
michael@0: 
michael@0: /**
michael@0:  * If t is equal to some int32_t value, set *i to that value and return true;
michael@0:  * otherwise return false.
michael@0:  *
michael@0:  * Note that negative zero is "equal" to zero here. To test whether a value can
michael@0:  * be losslessly converted to int32_t and back, use NumberIsInt32 instead.
michael@0:  */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: NumberEqualsInt32(T t, int32_t* i)
michael@0: {
michael@0:   /*
michael@0:    * XXX Casting a floating-point value that doesn't truncate to int32_t, to
michael@0:    *     int32_t, induces undefined behavior.  We should definitely fix this
michael@0:    *     (bug 744965), but as apparently it "works" in practice, it's not a
michael@0:    *     pressing concern now.
michael@0:    */
michael@0:   return t == (*i = int32_t(t));
michael@0: }
michael@0: 
michael@0: /**
michael@0:  * If d can be converted to int32_t and back to an identical double value,
michael@0:  * set *i to that value and return true; otherwise return false.
michael@0:  *
michael@0:  * The difference between this and NumberEqualsInt32 is that this method returns
michael@0:  * false for negative zero.
michael@0:  */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: NumberIsInt32(T t, int32_t* i)
michael@0: {
michael@0:   return !IsNegativeZero(t) && NumberEqualsInt32(t, i);
michael@0: }
michael@0: 
michael@0: /**
michael@0:  * Computes a NaN value.  Do not use this method if you depend upon a particular
michael@0:  * NaN value being returned.
michael@0:  */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE T
michael@0: UnspecifiedNaN()
michael@0: {
michael@0:   /*
michael@0:    * If we can use any quiet NaN, we might as well use the all-ones NaN,
michael@0:    * since it's cheap to materialize on common platforms (such as x64, where
michael@0:    * this value can be represented in a 32-bit signed immediate field, allowing
michael@0:    * it to be stored to memory in a single instruction).
michael@0:    */
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   return SpecificNaN<T>(1, Traits::SignificandBits);
michael@0: }
michael@0: 
michael@0: /**
michael@0:  * Compare two doubles for equality, *without* equating -0 to +0, and equating
michael@0:  * any NaN value to any other NaN value.  (The normal equality operators equate
michael@0:  * -0 with +0, and they equate NaN to no other value.)
michael@0:  */
michael@0: template<typename T>
michael@0: static inline bool
michael@0: NumbersAreIdentical(T t1, T t2)
michael@0: {
michael@0:   typedef FloatingPoint<T> Traits;
michael@0:   typedef typename Traits::Bits Bits;
michael@0:   if (IsNaN(t1))
michael@0:     return IsNaN(t2);
michael@0:   return BitwiseCast<Bits>(t1) == BitwiseCast<Bits>(t2);
michael@0: }
michael@0: 
michael@0: namespace detail {
michael@0: 
michael@0: template<typename T>
michael@0: struct FuzzyEqualsEpsilon;
michael@0: 
michael@0: template<>
michael@0: struct FuzzyEqualsEpsilon<float>
michael@0: {
michael@0:   // A number near 1e-5 that is exactly representable in
michael@0:   // floating point
michael@0:   static const float value() { return 1.0f / (1 << 17); }
michael@0: };
michael@0: 
michael@0: template<>
michael@0: struct FuzzyEqualsEpsilon<double>
michael@0: {
michael@0:   // A number near 1e-12 that is exactly representable in
michael@0:   // a double
michael@0:   static const double value() { return 1.0 / (1LL << 40); }
michael@0: };
michael@0: 
michael@0: } // namespace detail
michael@0: 
michael@0: /**
michael@0:  * Compare two floating point values for equality, modulo rounding error. That
michael@0:  * is, the two values are considered equal if they are both not NaN and if they
michael@0:  * are less than or equal to epsilon apart. The default value of epsilon is near
michael@0:  * 1e-5.
michael@0:  *
michael@0:  * For most scenarios you will want to use FuzzyEqualsMultiplicative instead,
michael@0:  * as it is more reasonable over the entire range of floating point numbers.
michael@0:  * This additive version should only be used if you know the range of the numbers
michael@0:  * you are dealing with is bounded and stays around the same order of magnitude.
michael@0:  */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: FuzzyEqualsAdditive(T val1, T val2, T epsilon = detail::FuzzyEqualsEpsilon<T>::value())
michael@0: {
michael@0:   static_assert(IsFloatingPoint<T>::value, "floating point type required");
michael@0:   return Abs(val1 - val2) <= epsilon;
michael@0: }
michael@0: 
michael@0: /**
michael@0:  * Compare two floating point values for equality, allowing for rounding error
michael@0:  * relative to the magnitude of the values. That is, the two values are
michael@0:  * considered equal if they are both not NaN and they are less than or equal to
michael@0:  * some epsilon apart, where the epsilon is scaled by the smaller of the two
michael@0:  * argument values.
michael@0:  *
michael@0:  * In most cases you will want to use this rather than FuzzyEqualsAdditive, as
michael@0:  * this function effectively masks out differences in the bottom few bits of
michael@0:  * the floating point numbers being compared, regardless of what order of magnitude
michael@0:  * those numbers are at.
michael@0:  */
michael@0: template<typename T>
michael@0: static MOZ_ALWAYS_INLINE bool
michael@0: FuzzyEqualsMultiplicative(T val1, T val2, T epsilon = detail::FuzzyEqualsEpsilon<T>::value())
michael@0: {
michael@0:   static_assert(IsFloatingPoint<T>::value, "floating point type required");
michael@0:   // can't use std::min because of bug 965340
michael@0:   T smaller = Abs(val1) < Abs(val2) ? Abs(val1) : Abs(val2);
michael@0:   return Abs(val1 - val2) <= epsilon * smaller;
michael@0: }
michael@0: 
michael@0: /**
michael@0:  * Returns true if the given value can be losslessly represented as an IEEE-754
michael@0:  * single format number, false otherwise.  All NaN values are considered
michael@0:  * representable (notwithstanding that the exact bit pattern of a double format
michael@0:  * NaN value can't be exactly represented in single format).
michael@0:  *
michael@0:  * This function isn't inlined to avoid buggy optimizations by MSVC.
michael@0:  */
michael@0: MOZ_WARN_UNUSED_RESULT
michael@0: extern MFBT_API bool
michael@0: IsFloat32Representable(double x);
michael@0: 
michael@0: } /* namespace mozilla */
michael@0: 
michael@0: #endif /* mozilla_FloatingPoint_h */