# 5.6 — Relational operators and floating point comparisons

Relational operators are operators that let you compare two values. There are 6 relational operators:

Operator Symbol Form Operation
Greater than > x > y true if x is greater than y, false otherwise
Less than < x < y true if x is less than y, false otherwise
Greater than or equals >= x >= y true if x is greater than or equal to y, false otherwise
Less than or equals <= x <= y true if x is less than or equal to y, false otherwise
Equality == x == y true if x equals y, false otherwise
Inequality != x != y true if x does not equal y, false otherwise

You have already seen how most of these work, and they are pretty intuitive. Each of these operators evaluates to the boolean value true (1), or false (0).

Here’s some sample code using these operators with integers:

And the results from a sample run:

```Enter an integer: 4
Enter another integer: 5
4 does not equal 5
4 is less than 5
4 is less than or equal to 5
```

These operators are extremely straightforward to use when comparing integers.

Boolean conditional values

By default, conditions in an if statement or conditional operator (and a few other places) evaluate as Boolean values.

Many new programmers will write statements like this one:

This is redundant, as the `== true` doesn’t actually add any value to the condition. Instead, we should write:

Similarly, the following:

is better written as:

Best practice

Don’t add unnecessary == or != to conditions. It makes them harder to read without offering any additional value.

Comparison of floating point values

Consider the following program:

Variables d1 and d2 should both have value 0.01. But this program prints an unexpected result:

```d1 > d2
```

If you inspect the value of d1 and d2 in a debugger, you’d likely see that d1 = 0.0100000000000005116 and d2 = 0.0099999999999997868. Both numbers are close to 0.01, but d1 is greater than, and d2 is less than.

If a high level of precision is required, comparing floating point values using any of the relational operators can be dangerous. This is because floating point values are not precise, and small rounding errors in the floating point operands may cause unexpected results. We discussed rounding errors in lesson 4.8 -- Floating point numbers if you need a refresher.

When the less than and greater than operators (<, <=, >, and >=) are used with floating point values, they will usually produce the correct answer (only potentially failing when the operands are almost identical). Because of this, use of these operators with floating point operands can be acceptable, so long as the consequence of getting a wrong answer when the operands are similar is slight.

For example, consider a game (such as Space Invaders) where you want to determine whether two moving objects (such as a missile and an alien) intersect. If the objects are still far apart, these operators will return the correct answer. If the two objects are extremely close together, you might get an answer either way. In such cases, the wrong answer probably wouldn’t even be noticed (it would just look like a near miss, or near hit) and the game would continue.

Floating point equality

The equality operators (== and !=) are much more troublesome. Consider operator==, which returns true only if its operands are exactly equal. Because even the smallest rounding error will cause two floating point numbers to not be equal, operator== is at high risk for returning false when a true might be expected. Operator!= has the same kind of problem.

For this reason, use of these operators with floating point operands should be avoided.

Warning

Avoid using operator== and operator!= with floating point operands.

So how can we reasonably compare two floating point operands to see if they are equal?

The most common method of doing floating point equality involves using a function that looks to see if two numbers are almost the same. If they are “close enough”, then we call them equal. The value used to represent “close enough” is traditionally called epsilon. Epsilon is generally defined as a small positive number (e.g. 0.00000001, sometimes written 1e-8).

New developers often try to write their own “close enough” function like this:

std::abs() is a function in the <cmath> header that returns the absolute value of its argument. So `std::abs(a - b) <= epsilon` checks if the distance between a and b is less than whatever epsilon value representing "close enough" was passed in. If a and b are close enough, the function returns true to indicate they're equal. Otherwise, it returns false.

While this function can work, it's not great. An epsilon of 0.00001 is good for inputs around 1.0, too big for inputs around 0.0000001, and too small for inputs like 10,000. This means every time we call this function, we have to pick an epsilon that's appropriate for our inputs. If we know we're going to have to scale epsilon in proportion to our inputs, we might as well modify the function to do that for us.

Donald Knuth, a famous computer scientist, suggested the following method in his book “The Art of Computer Programming, Volume II: Seminumerical Algorithms (Addison-Wesley, 1969)”:

In this case, instead of epsilon being an absolute number, epsilon is now relative to the magnitude of a or b.

Let's examine in more detail how this crazy looking function works. On the left side of the <= operator, `std::abs(a - b)` tells us the distance between a and b as a positive number.

On the right side of the <= operator, we need to calculate the largest value of "close enough" we're willing to accept. To do this, the algorithm chooses the larger of a and b (as a rough indicator of the overall magnitude of the numbers), and then multiplies it by epsilon. In this function, epsilon represents a percentage. For example, if we want to say "close enough" means a and b are within 1% of the larger of a and b, we pass in an epsilon of 0.01 (1% = 1/100 = 0.01). The value for epsilon can be adjusted to whatever is most appropriate for the circumstances (e.g. an epsilon of 0.002 means within 0.2%).

To do inequality (!=) instead of equality, simply call this function and use the logical NOT operator (!) to flip the result:

Note that while the approximatelyEqual() function will work for most cases, it is not perfect, especially as the numbers approach zero:

Perhaps surprisingly, this returns:

```1
0
```

The second call didn't perform as expected. The math simply breaks down close to zero.

One way to avoid this is to use both an absolute epsilon (as we did in the first approach) and a relative epsilon (as we did in Knuth's approach):

In this algorithm, we first check if a and b are close together in absolute terms, which handles the case where a and b are both close to zero. The absEpsilon parameter should be set to something very small (e.g. 1e-12). If that fails, then we fall back to Knuth's algorithm, using the relative epsilon.

Here's our previous code testing both algorithms:

```1
0
1
```

You can see that with an appropriately picked absEpsilon, approximatelyEqualAbsRel() handles the small inputs correctly.

Comparison of floating point numbers is a difficult topic, and there's no "one size fits all" algorithm that works for every case. However, the approximatelyEqualAbsRel() should be good enough to handle most cases you'll encounter. 5.7 -- Logical operators Index 5.5 -- Comma and conditional operators

### 154 comments to 5.6 — Relational operators and floating point comparisons

• Jacob

I was quite amused to see the "close enough" scheme used here.  In my Perl package Math::Yapp (Yet Another Polynomial Package) I had to do the same thing, except I was dealing with complex numbers.  In that case I needed to see how close a complex number is to being a solution to a polynomial equation so I was almost always comparing it to 0.0.

And yes, at 9th degree polynomials, the rounding errors suddenly got so severe that I had put a disclaimer in the documentation.  And I didn't even bother with the Gram-Schmidt orthogonalization process; it was a losing proposition even at low degree.

• TavonCpp

holy shit, is comparing floating points this complicate on all programming languages?

• nascardriver

I've honestly never need to do this. If you're using floating point numbers, you'll almost exclusively use <, <=, >, >= and not ==, so there's almost no need to the functions shown here.
If you do find yourself in a situation where you need to compare floating point numbers for equality, you should try to reformulate the problem in integers instead.

• Chayim

What does it mean that 'max' multiplies the greatest 'abs' by epsilon?

• samivagyok

First, the abs function takes the absolute value of the parameter in the parentheses (e.g. abs(-6) = 6, abs(2) = 2), and after that the max function returns the bigger value out of the parameters (e.g. max(2, 6), will return 6, because 6>2). And after that the returned value from max() is multiplied by the epsilon value, which is user given. Hope the explanation helped.

• Chayim

Why is an - used (a - b) and not a comma (a, b) and how does - work?

• Chayim

Can you complete this code you mentioned:

to show how to compile and execute it?

• nascardriver

• Chayim

Why can’t the compiler use precise double floating point? It’s weird and awkward that computing should make such an error and why? Can the compiler not calculate? A computer does exactly what it’s programmed. Maybe it’s because of how the transistors operate and calculate, I’m not familiar 100% on hardware computer science.

• bobahop

If including windows.h, then you'll need to parenthesize std::max, something like this

to prevent "Illegal token on right side of ::" compile error from confusing function call with macro.

• It's+Me

Yeah, This time my mind really got f****

• swaraj

in the last example wouldn't it be a better choice to define real-epsilon as a const variable rather than a function parameter

• notaduck448

Would rounding (to the 4th or 5th decimal place) be an option for comparing floating-point numbers? The float comparison algorithm just seems unnecessarily complicated.

• John

Hello, do I need to understand how the function works floating point equality functions works, I understand what is going on there.

• Timothy Quintaro

I'm so confused. shouldn't it print out true or false rather than 1 or 0. cuz the method return is a bool not an int.

• nascardriver

`bool` is printed as 0/1 unless you set `std::boolalpha`.

• Raffaello

did I just learn my first algorithm? :) a question for @alex and @nascardriver, have you read the whole art of computer programming?
also, I wanted to see outputs of the last code in decimal form, without going through it with a debugger, so I added the header file iomanip, and used in main std::setprecision to 16 digits, but it didn't work :\

• nascardriver

hi!

Everything you learned so far is an algorithm, at least according to Wikipedia's definition.
I don't think anyone can know everything about computer programming. I'm a passionate C++ developer, I just like the language and try to learn more about it. C++ is huge, even after years of use, you'll learn something new every now and then.

Setting the precision to 16 should be enough to see a change in the output. 16 is a magic number, avoid magic numbers. You can obtain the precision necessary to represent every double through `std::numeric_limits` from the <limits> header:

• Raffaello

hah, by "the art of computer programming" I meant the whole 4 volumes wrote by donald knuth :)

• Gabe

I have the first volume, "Fundamental Algorithms." It's a great book but quite a challenge :p