Opeartions with floating numbers

Question

Few days before, in our class we talked about floating numbers. So there was an expression to be checked in some programming languages :

0.1 + 0.2 == 0.3

Returned false. But trying

100*0.1 + 100*0.2 == 100*0.3

Returned true.

So, what actually I know is, 0.1 cannot be represented exactly (0.00011001100110011....) and here appears some errors. But why second expression is true ? Why multiplying with 100 it works ?

And another one good question why this expression is not 0 ? is there a trick with operations?

>>> math.exp(6)/10 - math.exp(6)*0.1
-7.105427357601002e-15

Read http://stackoverflow.com/tags/floating-point/info for a summary that doesn't need to be repeated again and links to further documentation — Pascal Cuoq, Sep 19 '15 at 22:06

score 1 · Accepted Answer · answered Sep 19 '15 at 21:16

As an example, the set of double precision floating point numbers used in most programming languages is quite roughly the set of numbers +/- m * 2^k where m is an integer from 2^52 to 2^53, and k is an integer from roughly -1024 to +1024. Numbers like 0.1 are not in that set. Any integer multiplied or divided by a power of two is never equal to 0.1.

Any floating-point calculation calculates the exact result, then rounds it to the nearest floating-point number.

When you compare 0.1 + 0.2 to 0.3, you actually take the floating point number closest to 0.1, and the floating point number closest to 0.2, add the numbers, round the result to the nearest floating-point number, and compare to the floating-point number closest to 0.3.

Both results will be very close together. It is more or less coincidence whether they are equal, or whether one result is smaller or the other is smaller. If you do the same with 100 * 0.1, 100 * 0.2, and 100 * 0.3, the same thing happens. You will end up with two numbers that are very close together, and it is more or less coincidence whether the are equal, or one or the other is smaller.

In your last question, the first expression divides by 10. The second multiplies by the floating point number closest to 0.1. There is no floating point number equal to 0.1, the nearest floating point number is slightly larger or smaller than 0.1. So the results will be very close together (they are), but there is no guarantee that they would be the same.

Opeartions with floating numbers

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