FE.ES-终结0.1+0.2,答到点上的那种

柔光的暖阳◎ 2022-03-06 06:54 111阅读 0赞

随便逛了一下知乎和思否的博文，大都没回答到点上。

# I.浮点数的二进制存储 #

采用 [IEEE 754][] 规范来存储浮点数：  
`1`位【正负符号】+`11`位【指数】+`52`位【有效数字】，如下图  
![clipboard.png][]

由于`0.1.toString(2)`=`0.0001100110011001100110011001100110011001100110011001101`  
所以  
0.1 = 2^-4 \* \[1\].1001100110011001100110011001100110011001100110011010  
0.2 = 2^-3 \* \[1\].1001100110011001100110011001100110011001100110011010

# II. 到底怎么相加 #

铁律是52位有效数字，也就是：

0.1 = 2^-3 *  0.1100110011001100110011001100110011001100110011001101(0)
    0.2 = 2^-3 *  1.1001100110011001100110011001100110011001100110011010
    sum = 2^-2 * 1.0011001100110011001100110011001100110011001100110011(1)

由于有效数字变成了53位，根据IEEE754 [rounding mode][] 的 Round to Nearest,若x在a和b之间，选择**最低有效位为零的值**。

a = 2^-2 * 1.0011001100110011001100110011001100110011001100110011
    x = 2^-2 * 1.0011001100110011001100110011001100110011001100110011(1)
    b = 2^-2 * 1.0011001100110011001100110011001100110011001100110100

当比较0.1+0.2 和 0.3时，实际比较的是

0.1 + 0.2 => 0:01111111101:0011001100110011001100110011001100110011001100110[100]
    0.3       => 0:01111111101:0011001100110011001100110011001100110011001100110[011]

转成10进制，看上去是：

0.1 + 0.2 => 0.300000000000000044408920985006...
    0.3       => 0.299999999999999988897769753748...

结果很明显啦

参考资料：  
[https://stackoverflow.com/que...][https_stackoverflow.com_que...]

[IEEE 754]: https://people.eecs.berkeley.edu/~wkahan/ieee754status/IEEE754.PDF
[clipboard.png]: /images/20220306/f093ec68bc7344d5bd8e0e4c0ca6e2c9.png
[rounding mode]: https://en.wikipedia.org/wiki/IEEE_754-1985#Rounding_floating-point_numbers
[https_stackoverflow.com_que...]: https://stackoverflow.com/questions/588004/is-floating-point-math-broken/28679423#28679423?newreg=8cd7c97e47ab49c89a62e6fe2e7b41bb