FPCompare.h

// ==========================================================================
// floating-point AlmostEquals checker
// Lior Kogan (koganlior1@gmail.com), 2011
// based on code extracted from Google Test (http://code.google.com/p/googletest/)
//
// the following code can be used to compare two floating-point values.
// example:
// double f= // something
// double g= // something
// const FloatingPoint<double> lhs(f), rhs(g);
// if (lhs.AlmostEquals(rhs)) { ... }
// ==========================================================================

#pragma once

#define GTEST_OS_WINDOWS 1

// ==========================================================================
// Copyright 2005, Google Inc.
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
//     * Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
//     * Redistributions in binary form must reproduce the above
// copyright notice, this list of conditions and the following disclaimer
// in the documentation and/or other materials provided with the
// distribution.
//     * Neither the name of Google Inc. nor the names of its
// contributors may be used to endorse or promote products derived from
// this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Authors: wan@google.com (Zhanyong Wan), eefacm@gmail.com (Sean Mcafee)
//
// The Google C++ Testing Framework (Google Test)

// ==========================================================================
// from gtest-1.5.0\include\gtest\internal\gtest-port.h:

// This template class serves as a compile-time function from size to
// type.  It maps a size in bytes to a primitive type with that
// size. e.g.
//
//   TypeWithSize<4>::UInt
//
// is typedef-ed to be unsigned int (unsigned integer made up of 4 bytes).
//
// Such functionality should belong to STL, but I cannot find it there.
//
// Google Test uses this class in the implementation of floating-point comparison.
//
// For now it only handles UInt (unsigned int) as that's all Google Test
// needs.  Other types can be easily added in the future if need arises.
template <size_t size>
class TypeWithSize
{
public:
    // This prevents the user from using TypeWithSize<N> with incorrect values of N.
    typedef void UInt;
};

// The specialization for size 4.
template <>
class TypeWithSize<4>
{
public:
    // unsigned int has size 4 in both gcc and MSVC.
    // As base/basictypes.h doesn't compile on Windows, we cannot use uint32, uint64, and etc here.
    typedef int          Int ;
    typedef unsigned int UInt;
};

// The specialization for size 8.
template <>
class TypeWithSize<8>
{
public:
    typedef          __int64 Int ;
    typedef unsigned __int64 UInt;
};

// ==========================================================================
// from gtest-1.5.0.zip\gtest-1.5.0\include\gtest\internal\gtest-internal.h:

// This template class represents an IEEE floating-point number
// (either single-precision or double-precision, depending on the
// template parameters).
//
// The purpose of this class is to do more sophisticated number
// comparison.  (Due to round-off error, etc, it's very unlikely that
// two floating-points will be equal exactly.  Hence a naive
// comparison by the == operation often doesn't work.)
//
// Format of IEEE floating-point:
//
//   The most-significant bit being the leftmost, an IEEE
//   floating-point looks like
//
//     sign_bit exponent_bits fraction_bits
//
//   Here, sign_bit is a single bit that designates the sign of the
//   number.
//
//   For float, there are 8 exponent bits and 23 fraction bits.
//
//   For double, there are 11 exponent bits and 52 fraction bits.
//
//   More details can be found at
//   http://en.wikipedia.org/wiki/IEEE_floating-point_standard.
//
// Template parameter:
//
//   RawType: the raw floating-point type (either float or double)
template <typename RawType>
class FloatingPoint
{
public:
    // Defines the unsigned integer type that has the same size as the floating point number
    typedef typename TypeWithSize<sizeof(RawType)>::UInt Bits;

    // Constants.

    // # of bits in a number.
    static const size_t kBitCount = 8*sizeof(RawType);

    // # of fraction bits in a number.
    static const size_t kFractionBitCount =
    std::numeric_limits<RawType>::digits - 1;

    // # of exponent bits in a number.
    static const size_t kExponentBitCount = kBitCount - 1 - kFractionBitCount;

    // The mask for the sign bit.
    static const Bits kSignBitMask = static_cast<Bits>(1) << (kBitCount - 1);

    // The mask for the fraction bits.
    static const Bits kFractionBitMask =
    ~static_cast<Bits>(0) >> (kExponentBitCount + 1);

    // The mask for the exponent bits.
    static const Bits kExponentBitMask = ~(kSignBitMask | kFractionBitMask);

    // How many ULP's (Units in the Last Place) we want to tolerate when
    // comparing two numbers.  The larger the value, the more error we
    // allow.  A 0 value means that two numbers must be exactly the same
    // to be considered equal.
    //
    // The maximum error of a single floating-point operation is 0.5
    // units in the last place.  On Intel CPU's, all floating-point
    // calculations are done with 80-bit precision, while double has 64
    // bits.
    //
    // See the following article for more details on ULP:
    // http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm.
    static const size_t kMaxUlps= 5000000;

    // Constructs a FloatingPoint from a raw floating-point number.
    //
    // On an Intel CPU, passing a non-normalized NAN (Not a Number)
    // around may change its bits, although the new value is guaranteed
    // to be also a NAN.  Therefore, don't expect this constructor to
    // preserve the bits in x when x is a NAN.
    explicit FloatingPoint(const RawType& x) { u_.value_ = x; }

    // Static methods

    // Reinterprets a bit pattern as a floating-point number.
    //
    // This function is needed to test the AlmostEquals() method.
    static RawType ReinterpretBits(const Bits bits)
    {
        FloatingPoint fp(0);
        fp.u_.bits_ = bits;
        return fp.u_.value_;
    }

    // Returns the floating-point number that represent positive infinity.
    static RawType Infinity()
    {
        return ReinterpretBits(kExponentBitMask);
    }

    // Non-static methods

    // Returns the bits that represents this number.
    const Bits &bits() const { return u_.bits_; }

    // Returns the exponent bits of this number.
    Bits exponent_bits() const { return kExponentBitMask & u_.bits_; }

    // Returns the fraction bits of this number.
    Bits fraction_bits() const { return kFractionBitMask & u_.bits_; }

    // Returns the sign bit of this number.
    Bits sign_bit() const { return kSignBitMask & u_.bits_; }

    // Returns true iff this is NAN (not a number).
    bool is_nan() const
    {
        // It's a NAN if the exponent bits are all ones and the fraction
        // bits are not entirely zeros.
        return (exponent_bits() == kExponentBitMask) && (fraction_bits() != 0);
    }

    // Returns true iff this number is at most kMaxUlps ULP's away from
    // rhs.  In particular, this function:
    //
    //   - returns false if either number is (or both are) NAN.
    //   - treats really large numbers as almost equal to infinity.
    //   - thinks +0.0 and -0.0 are 0 DLP's apart.
    bool AlmostEquals(const FloatingPoint& rhs) const
    {
        // The IEEE standard says that any comparison operation involving
        // a NAN must return false.
        if (is_nan() || rhs.is_nan())
            return false;

        // Lior Kogan, 25/9/2010: e.g. for comparing 1e-13 with exact 0
        if (fabs(u_.value_ - rhs.u_.value_) < 1e-12)
            return true;

        Bits bits= DistanceBetweenSignAndMagnitudeNumbers(u_.bits_, rhs.u_.bits_);

        if (bits > kMaxUlps && bits<100000000)
            __debugbreak();

        return bits <= kMaxUlps;
  }

 private:
    // The data type used to store the actual floating-point number.
    union FloatingPointUnion
    {
        RawType value_;  // The raw floating-point number.
        Bits bits_;      // The bits that represent the number.
    };

    // Converts an integer from the sign-and-magnitude representation to
    // the biased representation.  More precisely, let N be 2 to the
    // power of (kBitCount - 1), an integer x is represented by the
    // unsigned number x + N.
    //
    // For instance,
    //
    //   -N + 1 (the most negative number representable using
    //          sign-and-magnitude) is represented by 1;
    //   0      is represented by N; and
    //   N - 1  (the biggest number representable using
    //          sign-and-magnitude) is represented by 2N - 1.
    //
    // Read http://en.wikipedia.org/wiki/Signed_number_representations
    // for more details on signed number representations.
    static Bits SignAndMagnitudeToBiased(const Bits &sam)
    {
        if (kSignBitMask & sam) // sam represents a negative number.
            return ~sam + 1;
        else                    // sam represents a positive number.
            return kSignBitMask | sam;
    }

    // Given two numbers in the sign-and-magnitude representation,
    // returns the distance between them as an unsigned number.
    static Bits DistanceBetweenSignAndMagnitudeNumbers(const Bits &sam1,
                                                       const Bits &sam2)
    {
        const Bits biased1 = SignAndMagnitudeToBiased(sam1);
        const Bits biased2 = SignAndMagnitudeToBiased(sam2);
        Bits bits= (biased1 >= biased2) ? (biased1 - biased2) : (biased2 - biased1);
        return bits;
    }

    FloatingPointUnion u_;
};

// ==========================================================================
// Lior Kogan, 25/9/2010
// ==========================================================================

// check if two floating-points are almost equal
template<typename T>
static bool IsAlmostEq(T x, T y)
{
    static_assert(!std::numeric_limits<T>::is_exact , "IsAlmostEq: floating-point type expected");

    FloatingPoint<T> f(x);
    FloatingPoint<T> g(y);

    return f.AlmostEquals(g);
}

// assert that 2 floating-points are almost equal
static void AssertAlmostEq(const double f, const double g)
{
    assert(IsAlmostEq(f, g));
}