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Problem in comparing Floating point numbers and how to compare them correctly?

In this article, we will see what is the problem in comparing floating-point numbers and we will discuss the correct way to compare two floating-point numbers.
What is the problem in comparing Floating-Point Numbers usually?
Let us first compare two floating-point numbers with the help of relational operator (==).
Example: Using “==” for comparison

CPP

 `// C++ program to compare``// floating point numbers` `#include ``using` `namespace` `std;` `void` `compareFloatNum(``double` `a, ``double` `b)``{``    ``if` `(a == b) {``        ``cout << ``"The numbers are equal"``             ``<< endl;``    ``}``    ``else` `{``        ``cout << ``"The numbers are not equal"``             ``<< endl;``    ``}``}` `// Driver code``int` `main()``{``    ``double` `a = (0.3 * 3) + 0.1;``    ``double` `b = 1;``    ``compareFloatNum(a, b);``}`

Java

 `// Java program to compare``// floating point numbers``class` `GFG``{` `    ``static` `void` `compareFloatNum(``double` `a, ``double` `b)``    ``{``        ``if` `(a == b)``        ``{``            ``System.out.print(``"The numbers are equal"` `+ ``"\n"``);``        ``}``        ``else``        ``{``            ``System.out.print(``"The numbers are not equal"` `+ ``"\n"``);``        ``}``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``double` `a = (``0.3` `* ``3``) + ``0.1``;``        ``double` `b = ``1``;``        ``compareFloatNum(a, b);``    ``}``}` `// This code is contributed by 29AjayKumar`

Python

 `# Python program to compare``# floating point numbers``def` `compareFloatNum(a, b):``    ``if` `(a ``=``=` `b):``        ``print``(``"The numbers are equal"``)` `    ``else``:``        ``print``(``"The numbers are not equal"``)` `# Driver code` `a ``=` `(``0.3` `*` `3``) ``+` `0.1``b ``=` `1``compareFloatNum(a, b)` `# This code is contributed by mohit kumar 29`

C#

 `// C# program to compare``// floating point numbers``using` `System;` `class` `GFG``{` `    ``static` `void` `comparefloatNum(``double` `a, ``double` `b)``    ``{``        ``if` `(a == b)``        ``{``            ``Console.Write(``"The numbers are equal"` `+ ``"\n"``);``        ``}``        ``else``        ``{``            ``Console.Write(``"The numbers are not equal"` `+ ``"\n"``);``        ``}``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``double` `a = (0.3 * 3) + 0.1;``        ``double` `b = 1;``        ``comparefloatNum(a, b);``    ``}``}` `// This code is contributed by PrinciRaj1992`

Javascript

 ``

Output:

`The numbers are not equal`

Time complexity of this program is O(1), as it only performs a comparison between two floating point numbers.

The space complexity is also O(1), as the program uses only a constant amount of memory for storing the two floating point numbers and a few local variables used for the comparison.

Why does this problem occur?
In the case of floating-point numbers, the relational operator (==) does not produce correct output, this is due to the internal precision errors in rounding up floating-point numbers.
In the above example, we can see the inaccuracy in comparing two floating-point numbers using “==” operator. The two numbers ‘a’ and ‘b’ are equal ( as (0.3 * 3) + 0.1 = 1 ) but the program results in an incorrect output.
Let’s take a closer look at the numbers in the next snippet.

CPP

 `// C++ program to compare``// floating point numbers` `#include ``using` `namespace` `std;` `void` `printFloatNum(``double` `a, ``double` `b)``{``    ``// To print decimal numbers up to 20 digits``    ``cout << setprecision(20);` `    ``cout << ``"a is : "` `<< a << endl;``    ``cout << ``"b is : "` `<< b << endl;``}` `// Driver code``int` `main()``{``    ``double` `a = (0.3 * 3) + 0.1;``    ``double` `b = 1;``    ``printFloatNum(a, b);``}`

Python3

 `# Python 3 program to compare``# floating point numbers``def` `printFloatNum(a,  b):` `    ``# To print decimal numbers up to 20 digits``    ``print``(``"a is : %.20f"` `%``a)``    ``print``(``"b is : %.20f"` `%``b)` `# Driver code``if` `__name__ ``=``=` `"__main__"``:` `    ``a ``=` `(``0.3` `*` `3``) ``+` `0.1``    ``b ``=` `1``    ``printFloatNum(a, b)` `    ``# This code is contributed by ukasp.`

Javascript

 `// JavaScript program to compare floating point numbers` `function` `printFloatNum(a, b) {``    ``// To print decimal numbers up to 20 digits``    ``console.log(`a is : \${a.toFixed(20)}`);``    ``console.log(`b is : \${b.toFixed(20)}`);``}` `// Driver code``    ``let a = (0.3 * 3) + 0.1;``    ``let b = 1;``    ``printFloatNum(a, b);`

Java

 `import` `java.util.*;` `public` `class` `CompareFloatingPointNumbers {``    ``public` `static` `void` `printFloatNum(``double` `a, ``double` `b) {``        ``// To print decimal numbers up to 20 digits``        ``System.out.println(String.format(``"a is : %.20f"``, a));``        ``System.out.println(String.format(``"b is : %.20f"``, b));``    ``}` `    ``public` `static` `void` `main(String[] args) {``        ``double` `a = (``0.3` `* ``3``) + ``0.1``;``        ``double` `b = ``1``;``        ``printFloatNum(a, b);``    ``}``}`

Output:

```a is : 0.99999999999999988898
b is : 1```

Time complexity:
The program has a constant time complexity, as it only performs a fixed set of operations and does not depend on the input size. Therefore, the time complexity is O(1).

Space complexity:
The program uses a fixed amount of memory for the double variables a and b, as well as for the output printed to the console. Therefore, the space complexity is also O(1).

Now we can see the internal rounding error in floating-point numbers. Number ‘a’ is not correctly rounded up to 1,
there is an internal error in rounding up, a very small error but makes a huge difference when we are comparing the numbers.
How to compare floating-point numbers correctly?
If we do have to compare two floating-point numbers then rather than using “==” operator we will find the absolute difference between the numbers (which if were correctly represented, the difference would have been 0) and compare it with a very small number 1e-9 (i.e 10^-9, this number is very small) and if the difference is less than this number, we can safely say that the two floating-point numbers are equal.
Example:

C++

 `// C++ program to compare``// floating point numbers correctly` `#include ``using` `namespace` `std;` `void` `compareFloatNum(``double` `a, ``double` `b)``{` `    ``// Correct method to compare``    ``// floating-point numbers``    ``if` `(``abs``(a - b) < 1e-9) {``        ``cout << ``"The numbers are equal "``             ``<< endl;``    ``}``    ``else` `{``        ``cout << ``"The numbers are not equal "``             ``<< endl;``    ``}``}` `// Driver code``int` `main()``{``    ``double` `a = (0.3 * 3) + 0.1;``    ``double` `b = 1;``    ``compareFloatNum(a, b);``}`

Java

 `// Java program to compare``// floating point numbers correctly``class` `GFG``{` `static` `void` `compareFloatNum(``double` `a, ``double` `b)``{` `    ``// Correct method to compare``    ``// floating-point numbers``    ``if` `(Math.abs(a - b) < 1e-``9``)``    ``{``        ``System.out.print(``"The numbers are equal "``            ``+``"\n"``);``    ``}``    ``else``    ``{``        ``System.out.print(``"The numbers are not equal "``            ``+``"\n"``);``    ``}``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``double` `a = (``0.3` `* ``3``) + ``0.1``;``    ``double` `b = ``1``;``    ``compareFloatNum(a, b);``}``}` `// This code is contributed by Rajput-Ji`

Python3

 `# Python program to compare``# floating point numbers correctly` `def` `compareFloatNum(a, b):``    ` `    ``# Correct method to compare``    ``# floating-point numbers``    ``if` `(``abs``(a ``-` `b) < ``1e``-``9``):``        ``print``(``"The numbers are equal "``);``    ``else``:``        ``print``(``"The numbers are not equal "``);``    ` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``a ``=` `(``0.3` `*` `3``) ``+` `0.1``;``    ``b ``=` `1``;``    ``compareFloatNum(a, b);` `# This code is contributed by PrinciRaj1992`

C#

 `    ` `// C# program to compare``// floating point numbers correctly``using` `System;` `class` `GFG``{` `static` `void` `comparefloatNum(``double` `a, ``double` `b)``{` `    ``// Correct method to compare``    ``// floating-point numbers``    ``if` `(Math.Abs(a - b) < 1e-9)``    ``{``        ``Console.Write(``"The numbers are equal "``            ``+``"\n"``);``    ``}``    ``else``    ``{``        ``Console.Write(``"The numbers are not equal "``            ``+``"\n"``);``    ``}``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``double` `a = (0.3 * 3) + 0.1;``    ``double` `b = 1;``    ``comparefloatNum(a, b);``}``}` `// This code is contributed by 29AjayKumar`

Javascript

 ``

Output:

`The numbers are equal`

This code results in the correct output, so whenever two floating point numbers are two be compared then rather than using “==” operator, we will use the above technique.

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