我对任何涉及浮点减法的答案都非常谨慎(例如，fabs(a-b) < epsilon)。首先，浮点数在更大的量级上变得更稀疏，在足够大的量级上，当间隔大于时，您可能只需要做a == b。其次，减去两个非常接近的浮点数(因为您正在寻找接近相等的浮点数)正是您得到灾难性抵消的方式。

虽然不能移植，但我认为grom的答案在避免这些问题方面做得最好。

I found that the Google C++ Testing Framework contains a nice cross-platform template-based implementation of AlmostEqual2sComplement which works on both doubles and floats. Given that it is released under the BSD license, using it in your own code should be no problem, as long as you retain the license. I extracted the below code from http://code.google.com/p/googletest/source/browse/trunk/include/gtest/internal/gtest-internal.h https://github.com/google/googletest/blob/master/googletest/include/gtest/internal/gtest-internal.h and added the license on top.

一定要将GTEST_OS_WINDOWS定义为某个值(或者将使用它的代码更改为适合您的代码库的代码-毕竟它是BSD许可的)。

使用的例子:

double left  = // something
double right = // something
const FloatingPoint<double> lhs(left), rhs(right);

if (lhs.AlmostEquals(rhs)) {
  //they're equal!
}

代码如下:

// 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)


// 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:
#if GTEST_OS_WINDOWS
  typedef __int64 Int;
  typedef unsigned __int64 UInt;
#else
  typedef long long Int;  // NOLINT
  typedef unsigned long long UInt;  // NOLINT
#endif  // GTEST_OS_WINDOWS
};


// 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.  Therefore, 4 should be enough for ordinary use.
  //
  // See the following article for more details on ULP:
  // http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm.
  static const size_t kMaxUlps = 4;

  // 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;

    return DistanceBetweenSignAndMagnitudeNumbers(u_.bits_, rhs.u_.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);
    return (biased1 >= biased2) ? (biased1 - biased2) : (biased2 - biased1);
  }

  FloatingPointUnion u_;
};

编辑:这篇文章是4年前写的。它可能仍然有效，代码也很好，但有些人发现了改进。最好从谷歌Test源代码中获得最新版本的AlmostEquals，而不是我粘贴在这里的那个。

有关更深入的方法，请参阅比较浮点数。以下是该链接的代码片段:

// Usable AlmostEqual function    
bool AlmostEqual2sComplement(float A, float B, int maxUlps)    
{    
    // Make sure maxUlps is non-negative and small enough that the    
    // default NAN won't compare as equal to anything.    
    assert(maxUlps > 0 && maxUlps < 4 * 1024 * 1024);    
    int aInt = *(int*)&A;    
    // Make aInt lexicographically ordered as a twos-complement int    
    if (aInt < 0)    
        aInt = 0x80000000 - aInt;    
    // Make bInt lexicographically ordered as a twos-complement int    
    int bInt = *(int*)&B;    
    if (bInt < 0)    
        bInt = 0x80000000 - bInt;    
    int intDiff = abs(aInt - bInt);    
    if (intDiff <= maxUlps)    
        return true;    
    return false;    
}

意识到这是一个老话题，但这篇文章是我发现的关于比较浮点数的最直接的文章之一，如果你想探索更多，它也有更详细的参考资料，它的主要站点涵盖了处理浮点数的完整范围的问题《浮点指南:比较》。

我们可以在浮点公差中找到一篇更实用的文章，并指出有绝对公差测试，在c++中归结为:

bool absoluteToleranceCompare(double x, double y)
{
    return std::fabs(x - y) <= std::numeric_limits<double>::epsilon() ;
}

及相对耐量试验:

bool relativeToleranceCompare(double x, double y)
{
    double maxXY = std::max( std::fabs(x) , std::fabs(y) ) ;
    return std::fabs(x - y) <= std::numeric_limits<double>::epsilon()*maxXY ;
}

文章指出，当x和y较大时，绝对检验失败;当x和y较小时，相对检验失败。假设绝对耐受性和相对耐受性是相同的，综合测试将是这样的:

bool combinedToleranceCompare(double x, double y)
{
    double maxXYOne = std::max( { 1.0, std::fabs(x) , std::fabs(y) } ) ;

    return std::fabs(x - y) <= std::numeric_limits<double>::epsilon()*maxXYOne ;
}

/// testing whether two doubles are almost equal. We consider two doubles
/// equal if the difference is within the range [0, epsilon).
///
/// epsilon: a positive number (supposed to be small)
///
/// if either x or y is 0, then we are comparing the absolute difference to
/// epsilon.
/// if both x and y are non-zero, then we are comparing the relative difference
/// to epsilon.
bool almost_equal(double x, double y, double epsilon)
{
    double diff = x - y;
    if (x != 0 && y != 0){
        diff = diff/y; 
    }

    if (diff < epsilon && -1.0*diff < epsilon){
        return true;
    }
    return false;
}

我在我的小项目中使用了这个函数，它是有效的，但注意以下几点:

双精度误差可以为你制造惊喜。假设epsilon = 1.0e-6，那么根据上面的代码，1.0和1.000001不应该被认为是相等的，但在我的机器上，函数认为它们是相等的，这是因为1.000001不能精确地转换为二进制格式，它可能是1.0000009xxx。我用1.0和1.0000011测试了它，这次我得到了预期的结果。

你写的代码有bug:

return (diff < EPSILON) && (-diff > EPSILON);

正确的代码应该是:

return (diff < EPSILON) && (diff > -EPSILON);

(…是的，这是不同的)

我想知道晶圆厂是否会让你在某些情况下失去懒惰的评价。我会说这取决于编译器。你可能想两种都试试。如果它们在平均水平上是相等的，则采用晶圆厂实现。

如果你有一些关于两个浮点数中哪一个比另一个更大的信息，你可以根据比较的顺序来更好地利用惰性求值。

最后，通过内联这个函数可能会得到更好的结果。不过不太可能有太大改善……

编辑:OJ，谢谢你纠正你的代码。我相应地删除了我的评论