比较两个双精度浮点数或两个浮点数最有效的方法是什么?
简单地这样做是不正确的:
bool CompareDoubles1 (double A, double B)
{
return A == B;
}
比如:
bool CompareDoubles2 (double A, double B)
{
diff = A - B;
return (diff < EPSILON) && (-diff < EPSILON);
}
似乎是浪费加工。
有人知道更聪明的浮点比较器吗?
General-purpose comparison of floating-point numbers is generally meaningless. How to compare really depends on a problem at hand. In many problems, numbers are sufficiently discretized to allow comparing them within a given tolerance. Unfortunately, there are just as many problems, where such trick doesn't really work. For one example, consider working with a Heaviside (step) function of a number in question (digital stock options come to mind) when your observations are very close to the barrier. Performing tolerance-based comparison wouldn't do much good, as it would effectively shift the issue from the original barrier to two new ones. Again, there is no general-purpose solution for such problems and the particular solution might require going as far as changing the numerical method in order to achieve stability.
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 ;
}
比较浮点数取决于上下文。因为即使改变操作的顺序也会产生不同的结果,所以知道你希望这些数字有多“相等”是很重要的。
在研究浮点数比较时,比较Bruce Dawson编写的浮点数是一个很好的开始。
以下定义来自Knuth的《The art of computer programming》:
bool approximatelyEqual(float a, float b, float epsilon)
{
return fabs(a - b) <= ( (fabs(a) < fabs(b) ? fabs(b) : fabs(a)) * epsilon);
}
bool essentiallyEqual(float a, float b, float epsilon)
{
return fabs(a - b) <= ( (fabs(a) > fabs(b) ? fabs(b) : fabs(a)) * epsilon);
}
bool definitelyGreaterThan(float a, float b, float epsilon)
{
return (a - b) > ( (fabs(a) < fabs(b) ? fabs(b) : fabs(a)) * epsilon);
}
bool definitelyLessThan(float a, float b, float epsilon)
{
return (b - a) > ( (fabs(a) < fabs(b) ? fabs(b) : fabs(a)) * epsilon);
}
当然,选择取决于上下文,并决定你想要的数字有多相等。
比较浮点数的另一种方法是查看数字的ULP(最后位置的单位)。虽然没有专门处理比较,但“每个计算机科学家都应该知道浮点数”这篇论文是了解浮点数如何工作以及陷阱是什么,包括什么是ULP的很好的资源。