浮点数添加的关联属性 [英] Associative Property on floating point number additions

问题描述

让x，y和z为浮点数.那么(x + y)+ z =/= x +(y + z)是真的吗?有人可以给我一个例子吗?我希望x，y和z是浮点数(IEEE表示)，而不仅仅是任何实数.

Let x, y, and z be floating point numbers. Then is it true that (x+y)+z =/= x+(y+z)? Can someone give me an example? I want x,y, and z be floating point numbers (IEEE representation), not just any real numbers.

推荐答案

正如您所说的那样，它通常不是正确的.存在x，y和z，使得(x+y)+z = x+(y+z).但是您可能想问(x+y)+z = x+(y+z)是否总是正确的.并非总是如此.

As you phrased the question, it is not generally true. There exist x, y, and z such that (x+y)+z = x+(y+z). But you likely intended to ask whether or not it is always true that (x+y)+z = x+(y+z). It is not always true.

一个明显的反例是(2 ⁶⁴ + -2 ⁶⁴ )+ 1和2 ⁶⁴ +(-2 ⁶⁴ + 1).使用64位二进制IEEE-754算法，前者为0 +1 = 1，后者为2 ⁶⁴ + -2 ⁶⁴ = 0.因为-2 ⁶⁴ 的大小太大，所以1丢失了，总和没有足够的精度来包含1.

An obvious counterexample is (2⁶⁴ + -2⁶⁴) + 1 and 2⁶⁴ + (-2⁶⁴ + 1). With 64-bit binary IEEE-754 arithmetic, the former is 0 + 1 = 1, and the latter is 2⁶⁴ + -2⁶⁴ = 0. In the latter, the 1 was lost because the magnitude of -2⁶⁴ is so great the sum did not have enough precision to include the 1.

还有更微妙的情况，其中一个数字的一​​部分丢失了.

There are more subtle cases, where just a part of one number is lost.

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