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3.5 — Relational operators (comparisons)

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 all 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.

Comparison of floating point values

Directly comparing floating point values using any of these operators is dangerous. This is because small rounding errors in the floating point operands may cause unexpected results. We discuss rounding errors in detail in section 2.5 -- floating point numbers.

Here’s an example of rounding errors causing unexpected results:

This program prints an unexpected result:

d1 > d2

In the above program, d1 = 0.0100000000000005116 and d2 = 0.0099999999999997868. Both numbers are close to 0.01, but d1 is greater than, and d2 is less than. And neither are equal.

Sometimes the need to do floating point comparisons is unavoidable. In this case, the less than and greater than operators (>, >=, <, and <=) are often used with floating point values as normal. The operators will produce the correct result most of the time, only potentially failing when the two operands are close. Due to the way these operators tend to be used, a wrong result often only has slight consequences. The equality operator is much more troublesome since even the smallest of rounding errors makes it completely inaccurate. Consequently, using operator== or operator!= on floating point numbers is not advised. The most common method of doing floating point equality involves using a function that calculates how close the two values are to each other. If the two numbers 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 number (e.g. 0.0000001).

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

fabs() is a function in the <cmath> library that returns the absolute value of its parameter. fabs(a - b) returns the distance between a and b as a positive number. This function 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.

While this works, it’s not great. An epsilon of 0.00001 is good for inputs around 1.0, too big for numbers around 0.0000001, and too small for numbers 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 using epsilon as an absolute number, we’re using epsilon as a multiplier, so its effect is relative to our inputs.

Let’s examine in more detail how the approximatelyEqual() function works. On the left side of the <= operator, the absolute value of 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 1% (1% = 1/100 = 0.01). The value for epsilon can be adjusted to whatever is most appropriate for the circumstances (e.g. 0.01% = an epsilon of 0.0001). 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 many 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’ve added a new parameter: absEpsilon. First, we check to see if the distance between a and b is less than our absEpsilon, which should be set at something very small (e.g. 1e-12). This handles the case where a and b are both close to zero. If that fails, then we fall back to Knuth’s algorithm.

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.

3.6 -- Logical operators
Index
3.4 -- Sizeof, comma, and conditional operators

75 comments to 3.5 — Relational operators (comparisons)

  • Luhan

    Here:

    Where i pointed out wouldn’t have an ‘else’?

    • Alex

      It could, but in this case, it’s not necessary. If the if statement’s conditional is true, then the function returns true immediately. That means the bottom code will only execute when the conditional was false, which is what we want anyway.

  • Angel

    "In this algorithm, we’ve added a new parameter: absEpsilon. First, we check to see if the a and b are less than our absEpsilon, which should be set at something very small (e.g. 1e-12). This handles the case where a and b are both close to zero. If that fails, then we fall back to Knuth’s algorithm."

    Hello, Alex, I’m a bit confused with the line "First, we check to see if the a and b are less than our absEpsilon", do you mean to first check if the "difference" between a and b are less than our absEpsilon instead?

  • Stephane

    Hi Alex,
    1) I read this whole part, but I don’t understood the Knuth’s method, can you please explain me in details.

    2) I searched in the web about best way for comparing floating point numbers and I saw on StackOverflow Relative error, absolute, and percentage error, can you explain me what do they mean?

    • Alex

      I’m not sure I can explain it in any more detail than what’s already in the article. In short:
      1) In order to determine if two floating point numbers are equal, we take the approach that they’re equal if they’re “close enough” (to account for precision issues).
      2) We get to determine what “close enough” means. We do this by defining a value called epsilon.
      3) If the two numbers within epsilon of each other, we consider them equal.

      Then the question becomes, how do we pick epsilon?
      1) An absolute epsilon just uses a number, like 0.01, which means the numbers are considered the same if they’re within 0.01 of each other. This is simple, but doesn’t work well for both small and large numbers (0.01 is huge if you’re trying to compare two very small numbers).
      2) A relative epsilon scales your epsilon by one of the input numbers, so instead of comparing against some absolute value, you’re comparing against a number that is scaled appropriately for your inputs. In this context, the epsilon functions as a percentage of your input rather than an absolute number. This is typically done by multiplying your epsilon by one of the input numbers. Knuth’s method multiplies it by the larger of the two numbers.

  • Matt

    why don’t we use an approxEqual function which checks whether or not the two numbers are within a certain percentage of each other?

    for example something like:

    this is how we calculate percent difference in intro physics classes and will produce a reasonable result regardless of the relative closeness to zero of either a or b, and requires fewer lines of code than the suggested isAlmostEqual functions. I suppose I’m not sure why I would use the longer code you suggest rather than the simpler (to me) percent difference function used in the sciences.

  • Justinas

    There is a small misprint:
    "Both numbers are close to 0.1 …" -> should be "close to 0.01"

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