加 0.1 + 0.1 时，浮点加法 go 究竟在哪里出错？

Question

I'm trying to find out why 0.1 + 0.1 +... 100 times equals 9.99999999999998

I started by reasoning that 0.1 is represented by a 64 bit value as 1 X 10^-1. (1 bit for negative, (0), 11 bits for exponent (10000000001), 52 bits for negative fraction (0000....0001))

But when I think about floating point addition... I can't figure how it could mess this up.

Example 1: Notice how we have to limit precision when we convert fractions to Real numbers. Put 1/3 in a calculator and it will have to cut it off at some point: 1/3 = 0.333333333333333333... (Limit its precision)

Example 2: double sum= 0;

    for(int i=0;i<100;i++){
        sum+=0.1;    // 0.1 + 0.1 = 1.0 * 10^-1
    }
    System.out.println(sum); // = 9.99999999999998 because floating point addition is limited by precision in each step.

And because of this, the best way to compare the numbers is see if they are equal at some precision:

    double epsilon = 0.0000000000001;
    
    if(Math.abs(sum-10) < epsilon){

        System.out.println("Equal within epsilon"); // This one wins here
    } else {
        System.out.println("Not equal within epsilon");
    }

That's great but can someone tell me how under-the-hood, addition of 0.1 is going wrong?

Answer 1

That's great but can someone tell me how under-the-hood, addition of 0.1 is going wrong?

The problem is that when you use double to represent 0.1 , the actual value of the double is not exactly 0.1. It is as close as possible to 0.1 as can be represented as a IEE 754 double precision value. But it is "off" by a tiny amount... a "delta".

^{(The actual double value for 0.1d is precisely 3602879701896397 / 2 55 .)}

So when you add the double representation of "almost by not exactly" 0.1 one hundred times, the sum you get is off by one hundred times the "delta". That is large enough that 100 x 0.1d is not equal to 10 (or 10.0d ).

If you want a more detailed understanding, read Is floating point math broken? . (Answer: no it isn't, but it is more complicated than you learned in high school.)

If you want even more, read"What Every Computer Scientist Should Know About Floating-Point Arithmetic" by David Goldberg.