我为java编写这篇文章，但是您可能会发现它很有用。它使用长变量而不是双变量，但会处理nan、亚法线等。

public static boolean equal(double a, double b) {
    final long fm = 0xFFFFFFFFFFFFFL;       // fraction mask
    final long sm = 0x8000000000000000L;    // sign mask
    final long cm = 0x8000000000000L;       // most significant decimal bit mask
    long c = Double.doubleToLongBits(a), d = Double.doubleToLongBits(b);        
    int ea = (int) (c >> 52 & 2047), eb = (int) (d >> 52 & 2047);
    if (ea == 2047 && (c & fm) != 0 || eb == 2047 && (d & fm) != 0) return false;   // NaN 
    if (c == d) return true;                            // identical - fast check
    if (ea == 0 && eb == 0) return true;                // ±0 or subnormals
    if ((c & sm) != (d & sm)) return false;             // different signs
    if (abs(ea - eb) > 1) return false;                 // b > 2*a or a > 2*b
    d <<= 12; c <<= 12;
    if (ea < eb) c = c >> 1 | sm;
    else if (ea > eb) d = d >> 1 | sm;
    c -= d;
    return c < 65536 && c > -65536;     // don't use abs(), because:
    // There is a posibility c=0x8000000000000000 which cannot be converted to positive
}
public static boolean zero(double a) { return (Double.doubleToLongBits(a) >> 52 & 2047) < 3; }

请记住，在一些浮点运算之后，number可能与我们期望的非常不同。没有代码可以解决这个问题。

/// 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。

Qt实现了两个函数，也许你可以从中学到一些东西:

static inline bool qFuzzyCompare(double p1, double p2)
{
    return (qAbs(p1 - p2) <= 0.000000000001 * qMin(qAbs(p1), qAbs(p2)));
}

static inline bool qFuzzyCompare(float p1, float p2)
{
    return (qAbs(p1 - p2) <= 0.00001f * qMin(qAbs(p1), qAbs(p2)));
}

您可能需要以下函数，因为

请注意，比较p1或p2为0.0的值是无效的， 也不会比较其中一个值为NaN或无穷大的值。 如果其中一个值总是0.0，则使用qFuzzyIsNull代替。如果一个人 其中的值很可能是0.0，一种解决方案是将两者都加上1.0 值。

static inline bool qFuzzyIsNull(double d)
{
    return qAbs(d) <= 0.000000000001;
}

static inline bool qFuzzyIsNull(float f)
{
    return qAbs(f) <= 0.00001f;
}

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 ;
}

在这个版本中，你可以检查，这些数字之间的差异并不比某些分数(比如，0.0001%)更大:

bool floatApproximatelyEquals(const float a, const float b) {
    if (b == 0.) return a == 0.; // preventing division by zero
    return abs(1. - a / b) < 1e-6;
}

请注意Sneftel关于浮动可能的分数限制的评论。

还要注意的是，它不同于使用绝对的epsilon的方法——这里你不需要担心“数量级”——数字可能是，比如说1e100，或者1e-100，它们总是会被一致地比较，而且你不必为每一种情况更新epsilon。

你必须为浮点数比较做这个处理，因为浮点数不能像整数类型那样完美地比较。下面是各种比较运算符的函数。

浮点数等于(==)

我也更喜欢减法技术，而不是依赖于fabs()或abs()，但我必须在从64位PC到ATMega328微控制器(Arduino)的各种架构上快速配置它，才能真正看到它是否会产生很大的性能差异。

所以，让我们忘记这些绝对值的东西，只做一些减法和比较!

从微软的例子修改如下:

/// @brief      See if two floating point numbers are approximately equal.
/// @param[in]  a        number 1
/// @param[in]  b        number 2
/// @param[in]  epsilon  A small value such that if the difference between the two numbers is
///                      smaller than this they can safely be considered to be equal.
/// @return     true if the two numbers are approximately equal, and false otherwise
bool is_float_eq(float a, float b, float epsilon) {
    return ((a - b) < epsilon) && ((b - a) < epsilon);
}
bool is_double_eq(double a, double b, double epsilon) {
    return ((a - b) < epsilon) && ((b - a) < epsilon);
}

使用示例:

constexpr float EPSILON = 0.0001; // 1e-4
is_float_eq(1.0001, 0.99998, EPSILON);

我不完全确定，但在我看来，对基于epsilon的方法的一些批评，正如这个高度好评的答案下面的评论所描述的那样，可以通过使用变量epsilon来解决，根据比较的浮点值缩放，像这样:

float a = 1.0001;
float b = 0.99998;
float epsilon = std::max(std::fabs(a), std::fabs(b)) * 1e-4;

is_float_eq(a, b, epsilon);

通过这种方式，epsilon值随浮点值伸缩，因此它的值不会小到不重要。

为了完整起见，让我们添加剩下的:

大于(>)小于(<):

/// @brief      See if floating point number `a` is > `b`
/// @param[in]  a        number 1
/// @param[in]  b        number 2
/// @param[in]  epsilon  a small value such that if `a` is > `b` by this amount, `a` is considered
///             to be definitively > `b`
/// @return     true if `a` is definitively > `b`, and false otherwise
bool is_float_gt(float a, float b, float epsilon) {
    return a > b + epsilon;
}
bool is_double_gt(double a, double b, double epsilon) {
    return a > b + epsilon;
}

/// @brief      See if floating point number `a` is < `b`
/// @param[in]  a        number 1
/// @param[in]  b        number 2
/// @param[in]  epsilon  a small value such that if `a` is < `b` by this amount, `a` is considered
///             to be definitively < `b`
/// @return     true if `a` is definitively < `b`, and false otherwise
bool is_float_lt(float a, float b, float epsilon) {
    return a < b - epsilon;
}
bool is_double_lt(double a, double b, double epsilon) {
    return a < b - epsilon;
}

大于或等于(>=)，小于或等于(<=)

/// @brief      Returns true if `a` is definitively >= `b`, and false otherwise
bool is_float_ge(float a, float b, float epsilon) {
    return a > b - epsilon;
}
bool is_double_ge(double a, double b, double epsilon) {
    return a > b - epsilon;
}

/// @brief      Returns true if `a` is definitively <= `b`, and false otherwise
bool is_float_le(float a, float b, float epsilon) {
    return a < b + epsilon;
}
bool is_double_le(double a, double b, double epsilon) {
    return a < b + epsilon;
}

额外的改进:

A good default value for epsilon in C++ is std::numeric_limits<T>::epsilon(), which evaluates to either 0 or FLT_EPSILON, DBL_EPSILON, or LDBL_EPSILON. See here: https://en.cppreference.com/w/cpp/types/numeric_limits/epsilon. You can also see the float.h header for FLT_EPSILON, DBL_EPSILON, and LDBL_EPSILON. See https://en.cppreference.com/w/cpp/header/cfloat and https://www.cplusplus.com/reference/cfloat/ You could template the functions instead, to handle all floating point types: float, double, and long double, with type checks for these types via a static_assert() inside the template. Scaling the epsilon value is a good idea to ensure it works for really large and really small a and b values. This article recommends and explains it: http://realtimecollisiondetection.net/blog/?p=89. So, you should scale epsilon by a scaling value equal to max(1.0, abs(a), abs(b)), as that article explains. Otherwise, as a and/or b increase in magnitude, the epsilon would eventually become so small relative to those values that it becomes lost in the floating point error. So, we scale it to become larger in magnitude like they are. However, using 1.0 as the smallest allowed scaling factor for epsilon also ensures that for really small-magnitude a and b values, epsilon itself doesn't get scaled so small that it also becomes lost in the floating point error. So, we limit the minimum scaling factor to 1.0. If you want to "encapsulate" the above functions into a class, don't. Instead, wrap them up in a namespace if you like in order to namespace them. Ex: if you put all of the stand-alone functions into a namespace called float_comparison, then you could access the is_eq() function like this, for instance: float_comparison::is_eq(1.0, 1.5);. It might also be nice to add comparisons against zero, not just comparisons between two values. So, here is a better type of solution with the above improvements in place: namespace float_comparison { /// Scale the epsilon value to become large for large-magnitude a or b, /// but no smaller than 1.0, per the explanation above, to ensure that /// epsilon doesn't ever fall out in floating point error as a and/or b /// increase in magnitude. template<typename T> static constexpr T scale_epsilon(T a, T b, T epsilon = std::numeric_limits<T>::epsilon()) noexcept { static_assert(std::is_floating_point_v<T>, "Floating point comparisons " "require type float, double, or long double."); T scaling_factor; // Special case for when a or b is infinity if (std::isinf(a) || std::isinf(b)) { scaling_factor = 0; } else { scaling_factor = std::max({(T)1.0, std::abs(a), std::abs(b)}); } T epsilon_scaled = scaling_factor * std::abs(epsilon); return epsilon_scaled; } // Compare two values /// Equal: returns true if a is approximately == b, and false otherwise template<typename T> static constexpr bool is_eq(T a, T b, T epsilon = std::numeric_limits<T>::epsilon()) noexcept { static_assert(std::is_floating_point_v<T>, "Floating point comparisons " "require type float, double, or long double."); // test `a == b` first to see if both a and b are either infinity // or -infinity return a == b || std::abs(a - b) <= scale_epsilon(a, b, epsilon); } /* etc. etc.: is_eq() is_ne() is_lt() is_le() is_gt() is_ge() */ // Compare against zero /// Equal: returns true if a is approximately == 0, and false otherwise template<typename T> static constexpr bool is_eq_zero(T a, T epsilon = std::numeric_limits<T>::epsilon()) noexcept { static_assert(std::is_floating_point_v<T>, "Floating point comparisons " "require type float, double, or long double."); return is_eq(a, (T)0.0, epsilon); } /* etc. etc.: is_eq_zero() is_ne_zero() is_lt_zero() is_le_zero() is_gt_zero() is_ge_zero() */ } // namespace float_comparison

参见:

The macro forms of some of the functions above in my repo here: utilities.h. UPDATE 29 NOV 2020: it's a work-in-progress, and I'm going to make it a separate answer when ready, but I've produced a better, scaled-epsilon version of all of the functions in C in this file here: utilities.c. Take a look. ADDITIONAL READING I need to do now have done: Floating-point tolerances revisited, by Christer Ericson. VERY USEFUL ARTICLE! It talks about scaling epsilon in order to ensure it never falls out in floating point error, even for really large-magnitude a and/or b values!