是浮点加法还是乘法关联？ [英] Is Floating point addition and multiplication associative?

问题描述

我在添加三个浮点值并将其与1比较时遇到了问题。

I had a problem when I was adding three floating point values and comparing them to 1.

cout << ((0.7 + 0.2 + 0.1)==1)<<endl;     //output is 0
cout << ((0.7 + 0.1 + 0.2)==1)<<endl;     //output is 1

为什么这些值会不同？

推荐答案

浮点加法不一定是关联的。

Floating point addition is not necessarily associative. If you change the order in which you add things up, this can change the result.

主题的标准文件是每个计算机科学家应该知道的浮点算术。它给出了以下示例：

The standard paper on the subject is What Every Computer Scientist Should Know about Floating Point Arithmetic. It gives the following example:

另一个灰色区域涉及括号的解释。由于四舍五入误差，代数的关联定律不一定适用于浮点数。例如，当x = 1e30，y = -1e30和z = 1时（在前一种情况下为1，在后一种情况下为0），表达式（x + y）+ z具有完全不同于x +（y + z） ）。

Another grey area concerns the interpretation of parentheses. Due to roundoff errors, the associative laws of algebra do not necessarily hold for floating-point numbers. For example, the expression (x+y)+z has a totally different answer than x+(y+z) when x = 1e30, y = -1e30 and z = 1 (it is 1 in the former case, 0 in the latter).

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