Rounding error in C++

Question

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Why can't decimal numbers be represented exactly in binary?

Language c++ I'm declaring an array and i save numbers (type double) in it. Then i start comparing the difference between each two elements in the array. for example

a[1] = 0.05
a[2] = 0.1
a[3] = 0.15

so when i do the following

if(a[3] - a[2] == a[2] - a[1] )

the condition becomes false!!

After debugging, i found out that 0.05 is saved in the array as 0.0499......993 and the case is similar with 0.10 and 0.15

How can i overcome this problem?

A useful read: [What Every Computer Scientist Should Know About Floating-Point Arithmetic](http://download.oracle.com/docs/cd/E19957-01/806-3568/ncg_goldberg.html) — eldarerathis, May 09 '11 at 14:51
@Rob: I disagree. The question here is "How can I overcome this problem?", not "What causes this problem?". However, I bet this is a duplicate of some other question :) — Magnus Hoff, May 09 '11 at 15:07

score 8 · Answer 1 · answered May 09 '11 at 14:52

8

You should never compare floating point numbers for exact equality.

You should read What Every Computer Scientist Should Know About Floating-Point Arithmetic to figure out why.

answered May 09 '11 at 14:52

unwind

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score 3 · Answer 2 · answered May 09 '11 at 14:54

If you want to do exact calculations, you might want to look into using rational numbers. It is possible to implement a class for rational numbers in C++. Boost.Rational is an example of this.

If you want to use floating point, you probably want to do comparisons for "closeness", rather than equality, like this:

const float EPSILON = 0.0001; //< Some acceptable limit for equivalence
float d1 = a[3] - a[2];
float d2 = a[2] - a[1];

if (fabs(d1 - d2) < EPSILON) {
    // Consider d1 and d2 eqivalent
}

Sujoy · Answer 3 · 2011-05-09T14:58:23.410

2

Floating point mathematics is one area where computers will not give results as you would normally expect. See this for reference.

What you can do is consider something like delta = 0.00001
and check if fabs((a[3]-a[2]) - (a[2]-a[1])) < delta

Note: If this has anything to do with currency or monetary data then just use integers/long etc. Using float/double to represent money is a bad bad thing, for reasons mentioned above and explained further in the link above.

edited May 09 '11 at 14:58

answered May 09 '11 at 14:52

Sujoy

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Being "up to [the] par" has nothing to do with it; floating-point inaccuracy is a mathematical fact. If you don't want the fallout, use fixed-point instead. Of course, most mainstream languages don't provide native fixed-point types, which _is_ a flaw. – Lightness Races in Orbit May 09 '11 at 14:54
1

It should probably be `abs((a[3]-a[2]) - (a[2]-a[1])) < delta`, because the error can go either way. – Mihai May 09 '11 at 14:54
you need to check the **absolute** value of the difference against delta – Alnitak May 09 '11 at 14:54
@Tomalak: yes i understand, i have changed the wording accordingly. @Mihai: thanks for pointing that out – Sujoy May 09 '11 at 14:57
@Sujoy: You added `abs` wrong. – Lightness Races in Orbit May 09 '11 at 14:58

score 0 · Answer 4 · answered May 09 '11 at 14:56

You've got several options:

(1) Use a type that stores these numbers precisely. For these numbers, it's easiest probably to use int and store them as 100x the original number.

(2) figure out that comparing doubles for equality is a bad strategy. See, when you store numbers as doubles, there are numbers that can't be represented exactly in binary in a fixed number of bits. 0.05 (as well as 0.5 5 50 and so on) is one of those numbers.

Rounding error in C++

4 Answers4