# 4.8 — Floating point numbers

Integers are great for counting whole numbers, but sometimes we need to store very large numbers, or numbers with a fractional component. A floating point type variable is a variable that can hold a real number, such as 4320.0, -3.33, or 0.01226. The floating part of the name floating point refers to the fact that the decimal point can “float”; that is, it can support a variable number of digits before and after the decimal point.

There are three different floating point data types: float, double, and long double. As with integers, C++ does not define the actual size of these types (but it does guarantee minimum sizes). On modern architectures, floating point representation almost always follows IEEE 754 binary format. In this format, a float is 4 bytes, a double is 8, and a long double can be equivalent to a double (8 bytes), 80-bits (often padded to 12 bytes), or 16 bytes.

Floating point data types are always signed (can hold positive and negative values).

Category Type Minimum Size Typical Size
floating point float 4 bytes 4 bytes
double 8 bytes 8 bytes
long double 8 bytes 8, 12, or 16 bytes

Here are some definitions of floating point numbers:

When using floating point literals, always include at least one decimal place (even if the decimal is 0). This helps the compiler understand that the number is a floating point number and not an integer.

Note that by default, floating point literals default to type double. An f suffix is used to denote a literal of type float.

Best practice

Always make sure the type of your literals match the type of the variables they’re being assigned to or used to initialize. Otherwise an unnecessary conversion will result, possibly with a loss of precision.

Warning

Make sure you don’t use integer literals where floating point literals should be used. This includes when initializing or assigning values to floating point objects, doing floating point arithmetic, and calling functions that expect floating point values.

Printing floating point numbers

Now consider this simple program:

The results of this seemingly simple program may surprise you:

```5
6.7
9.87654e+06
```

In the first case, the std::cout printed 5, even though we typed in 5.0. By default, std::cout will not print the fractional part of a number if the fractional part is 0.

In the second case, the number prints as we expect.

In the third case, it printed the number in scientific notation (if you need a refresher on scientific notation, see lesson 4.7 -- Introduction to scientific notation).

Floating point range

Assuming IEEE 754 representation:

Size Range Precision
4 bytes ±1.18 x 10-38 to ±3.4 x 1038 6-9 significant digits, typically 7
8 bytes ±2.23 x 10-308 to ±1.80 x 10308 15-18 significant digits, typically 16
80-bits (typically uses 12 or 16 bytes) ±3.36 x 10-4932 to ±1.18 x 104932 18-21 significant digits
16 bytes ±3.36 x 10-4932 to ±1.18 x 104932 33-36 significant digits

The 80-bit floating point type is a bit of a historical anomaly. On modern processors, it is typically implemented using 12 or 16 bytes (which is a more natural size for processors to handle).

It may seem a little odd that the 80-bit floating point type has the same range as the 16-byte floating point type. This is because they have the same number of bits dedicated to the exponent -- however, the 16-byte number can store more significant digits.

Floating point precision

Consider the fraction 1/3. The decimal representation of this number is 0.33333333333333… with 3’s going out to infinity. If you were writing this number on a piece of paper, your arm would get tired at some point, and you’d eventually stop writing. And the number you were left with would be close to 0.3333333333…. (with 3’s going out to infinity) but not exactly.

On a computer, an infinite length number would require infinite memory to store, and typically we only have 4 or 8 bytes. This limited memory means floating point numbers can only store a certain number of significant digits -- and that any additional significant digits are lost. The number that is actually stored will be close to the desired number, but not exact.

The precision of a floating point number defines how many significant digits it can represent without information loss.

When outputting floating point numbers, std::cout has a default precision of 6 -- that is, it assumes all floating point variables are only significant to 6 digits (the minimum precision of a float), and hence it will truncate anything after that.

The following program shows std::cout truncating to 6 digits:

This program outputs:

```9.87654
987.654
987654
9.87654e+006
9.87654e-005
```

Note that each of these only have 6 significant digits.

Also note that std::cout will switch to outputting numbers in scientific notation in some cases. Depending on the compiler, the exponent will typically be padded to a minimum number of digits. Fear not, 9.87654e+006 is the same as 9.87654e6, just with some padding 0’s. The minimum number of exponent digits displayed is compiler-specific (Visual Studio uses 3, some others use 2 as per the C99 standard).

The number of digits of precision a floating point variable has depends on both the size (floats have less precision than doubles) and the particular value being stored (some values have more precision than others). Float values have between 6 and 9 digits of precision, with most float values having at least 7 significant digits. Double values have between 15 and 18 digits of precision, with most double values having at least 16 significant digits. Long double has a minimum precision of 15, 18, or 33 significant digits depending on how many bytes it occupies.

We can override the default precision that std::cout shows by using an `output manipulator` function named `std::setprecision()`. Output manipulators alter how data is output, and are defined in the iomanip header.

Outputs:

```3.333333253860474
3.333333333333334
```

Because we set the precision to 16 digits using `std::setprecision()`, each of the above numbers is printed with 16 digits. But, as you can see, the numbers certainly aren’t precise to 16 digits! And because floats are less precise than doubles, the float has more error.

Precision issues don’t just impact fractional numbers, they impact any number with too many significant digits. Let’s consider a big number:

Output:

```123456792
```

123456792 is greater than 123456789. The value 123456789.0 has 10 significant digits, but float values typically have 7 digits of precision (and the result of 123456792 is precise only to 7 significant digits). We lost some precision! When precision is lost because a number can’t be stored precisely, this is called a rounding error.

Consequently, one has to be careful when using floating point numbers that require more precision than the variables can hold.

Best practice

Favor double over float unless space is at a premium, as the lack of precision in a float will often lead to inaccuracies.

Rounding errors make floating point comparisons tricky

Floating point numbers are tricky to work with due to non-obvious differences between binary (how data is stored) and decimal (how we think) numbers. Consider the fraction 1/10. In decimal, this is easily represented as 0.1, and we are used to thinking of 0.1 as an easily representable number with 1 significant digit. However, in binary, 0.1 is represented by the infinite sequence: 0.00011001100110011… Because of this, when we assign 0.1 to a floating point number, we’ll run into precision problems.

You can see the effects of this in the following program:

This outputs:

```0.1
0.10000000000000001
```

On the top line, std::cout prints 0.1, as we expect.

On the bottom line, where we have std::cout show us 17 digits of precision, we see that d is actually not quite 0.1! This is because the double had to truncate the approximation due to its limited memory. The result is a number that is precise to 16 significant digits (which type double guarantees), but the number is not exactly 0.1. Rounding errors may make a number either slightly smaller or slightly larger, depending on where the truncation happens.

Rounding errors can have unexpected consequences:

```1
0.99999999999999989
```

Although we might expect that d1 and d2 should be equal, we see that they are not. If we were to compare d1 and d2 in a program, the program would probably not perform as expected. Because floating point numbers tend to be inexact, comparing floating point numbers is generally problematic -- we discuss the subject more (and solutions) in lesson 5.6 -- Relational operators and floating point comparisons.

One last note on rounding errors: mathematical operations (such as addition and multiplication) tend to make rounding errors grow. So even though 0.1 has a rounding error in the 17th significant digit, when we add 0.1 ten times, the rounding error has crept into the 16th significant digit. Continued operations would cause this error to become increasingly significant.

Key insight

Rounding errors occur when a number can’t be stored precisely. This can happen even with simple numbers, like 0.1. Therefore, rounding errors can, and do, happen all the time. Rounding errors aren’t the exception -- they’re the rule. Never assume your floating point numbers are exact.

A corollary of this rule is: be wary of using floating point numbers for financial or currency data.

NaN and Inf

There are two special categories of floating point numbers. The first is Inf, which represents infinity. Inf can be positive or negative. The second is NaN, which stands for “Not a Number”. There are several different kinds of NaN (which we won’t discuss here). NaN and Inf are only available if the compiler uses a specific format (IEEE 754) for floating point numbers. If another format is used, the following code produces undefined behavior.

Here’s a program showing all three:

And the results using Visual Studio 2008 on Windows:

```1.#INF
-1.#INF
1.#IND
```

INF stands for infinity, and IND stands for indeterminate. Note that the results of printing Inf and NaN are platform specific, so your results may vary.

Best practice

Avoid division by 0 altogether, even if your compiler supports it.

Conclusion

To summarize, the two things you should remember about floating point numbers:

1) Floating point numbers are useful for storing very large or very small numbers, including those with fractional components.

2) Floating point numbers often have small rounding errors, even when the number has fewer significant digits than the precision. Many times these go unnoticed because they are so small, and because the numbers are truncated for output. However, comparisons of floating point numbers may not give the expected results. Performing mathematical operations on these values will cause the rounding errors to grow larger. 4.9 -- Boolean values Index 4.7 -- Introduction to scientific notation

### 431 comments to 4.8 — Floating point numbers

• Bouncy Disaster

In Visual Studio (maybe other IDE have this function, I don't know) if you hover over a floating point literal, it will give you a small box that will tell you both the type and value of it, BUT it will show you its value WITH rounding error. For example:

In float f{ 123456789.0f }; hovering over literal will show (float)(123456792.0F) with rounding error after 1234567.
Or in double d{ 0.1 }; hovering over double literal 0.1 will show (double)(0.1000000000000000056)

This can be used as a helpful way to 'preview' floating point literals and find rounding errors while writting code, before even compiling it.

• Rajin100000

I have a question. When I use

3.33333333333333333333333333333333333333f it only prints 25 significant digits. But when I use 3.33333333333333333333333333333333333333, it prints 32 significant digits. Why is

not effecting float number 3.33333333333333333333333333333333333333f? Here is my code:=

setprecision() function is not working for some cases. When I use setprecision(17) and 1.04 to print, it doesn't print the extra significant digits, but when I use 1.044444 to print, it prints 17 significant digits. Why is this happening to me?

• KY

The statement here :
```
However, in binary, 0.1 is represented by the infinite sequence: 0.00011001100110011…
```

I think it should be 0.100000000000000001... something?

• KY

Also, what about 0.2 or 2/10? Would it result in an infinite sequence too? Thanks

• Alex

You can use a calculator to convert decimal to binary for the answer to both questions: https://www.binaryhexconverter.com/decimal-to-binary-converter

• Armando IG

Hello, I'm learning here and trying to do exercises on another page, they were testing the use of data types and they have this sample input: 3 12345678912345 a 334.23 14049.30493

The expected output is:
3
12345678912345
a
334.230
14049.304930000

But with my code my output is:
3
12345678912345
a
334.23
14049.30493

How do I make my code so it prints the additional zeros?

This is my code:

• nascardriver

You can use `std::fixed` to output trailing zeroes

• Calum

Hi there all, I am trying to practice using the set precision. however when I try using the <iomainip> pre-processor directive it says the the source file cannot be found. I have tirelessly looked online for a solution and it seems most answers are directed at people who understand directories and pathways. the answers are always "go to settings and make sure that the program knows where to find it." but how do you actually have to write it in??? is the library file just completely missing?? or is it that the program just cant find it?? I feel like if I want to progress in C++ I need to better understand how to solve these issues, can someone help me solve this or is there a part in this tutorial that can help me better understand how these library files work and what to do in this situation. I have gone back to making a new project followed the steps exactly, I have uninstalled and reinstalled Visual studio, I have tried a couple of troubleshooting solutions but I'm still stuck.

• nascardriver

iomainip
iomanip

• Calum

your joking.... man I feel a little embarrassed. sometimes the answer is right there! thankyou for your help I was getting so frustrated.

• J34NP3T3R

i still cant get pass this lesson. its confusing the heck out of me.
its almost as if we shouldn't be using floating point numbers.

but what if we needed to store the values of like money. like 545.15 and have that used in further operations.
how are we to safely use pointing point numbers.
why would C++ have these downsides like making 0.1 added 10 times be 0.9.
that to me looks Mathematically wrong, so why does it exist in C++ ? is it a bug or a glitch ? will it be fixed ?
if i dont tinker with the precision will adding 0.1 10x finally result to 1.0 ?

• Morningfrost

I think there are a lot of doctorate thesis about floating point rounding errors. It's something tied to how computers are built. Even a scientific oriented language like Fortran has to deal with round errors.

When dealing with financial and currency values, it's much better to use fixed point aritmethic. As fas as I know, COBOL is the only language that has a fundamental type of this sort. In C and other languages, the idea is to use integers for both the integer and the decimal part.

• J34NP3T3R

what data type should we use for money ? one INT for the integer and another INT for the decimal part ?
is that like two separate objects now instead of one ?

• Morningfrost

I hear a common way is to scale up (multiply by 100), do computations and scale down.

• J34NP3T3R

thanks

• Lucky Abby

Hello Alex and Nascardriver,

I am having a strange output from the following code. I want to split the integer part and decimal part from a number and I couldn't understand why the result is got rounded and the decimal part become zero.

Here is the code:

Thanks for explaining the situation and possible solution.

• nascardriver

Print out `number`. Floating point inaccuracy has come to visit you. Your code works (with inaccurate output) if you sacrifice the leading 9.

• Dami

I don't understand how to use the header file "iomanip"

• nascardriver

There are examples in the lesson. Is there anything in specific you're trying to do?

• DMH

My machine has a long double size of 8 bytes, but I'm required to read in 16 byte long doubles from binary file, edit them and the replace them in the file. Is there a simple way of extending the effective length so I can access, manipulate and save long doubles as if they are 16 bytes in length?

PS thanks for all the tutorials! Phenomenally helpful.

• nascardriver

You didn't say it, but it sounds like you're trying to `reinterpret_cast` the read/write your `double`s. You can't do that, it causes undefined behavior. You need to manually convert the binary representation to a C++ type and do the same the other way around when you write.
You might be able to force your compiler to use a specific width, but that's not a portable solution. You need to write a type (Or use a library) for specific-width floating point types. To do this, you need classes and bitwise operations. Both are covered on learncpp. This is not a simple task.

• Waldo Lemmer

> Visual Studio uses 3

I'm using VS Community version 16.7.3 and the exponent only contains 2 digits (e.g. 9.87654e+06)

• yeokaiwei

inf
-inf
-nan(ind)

on Visual Studio 2019.

• Robert H

I noticed that Microsoft Visual Studio recognizes floating point literals without a number after the decimal point.  For example, if we start with the expression "5 / 2", that will evaluate to 2 (due to truncation because of the integral types of the literals).  Simply adding a decimal point (to either or both literals) invokes floating point math, "5 / 2." will evaluate to 2.5, at least in Microsoft Visual Studio.  Does the C++ standard require a number after the decimal point, or is Visual Studio deviating from the standard?  Specifically, does the standard require my example to be "5 / 2.0"?  If not, is it considered bad style to not put any numbers after the decimal point of a floating point literal (if that number would be 0)?  I personally think it is reasonably readable to omit an implied 0 after the decimal point.

• nascardriver

`2.` is valid. Whether or not you use it is up to personal preference.

• Amir

I tried to find way to deal with large numbers which represented as strings, and each digit is really important and there no option to ignore last digits.
how then I can do mathematical operations (div/multi)
on this very long numbers which represented as strings .. and if I made an algorithm doing like school division or sum , there is any tips to do it more efficiency as efficient as dealing with integers ? . thank you :)

• mike

you said we should match the type of literal with the type of variable alright !

what will happen in the following program :
int main()
{
float x {};
cin >> x;
return 0;
}
if a user enter : 0.5 (no f suffic)
Will it be converted to float first or what ?

// sorry for my bad english

• nascardriver

`std::cin` doesn't care about suffixes. Suffixes are only relevant in your code.
`std::cin` will extract the 0.5 into `x`, then stop at 'f', because 'f' can't be converted to a number.

• Karl

Don't you see then they are written alphabetically

iomanip
iostream

The First 2 chars are the same and the 'm' comes before 's', don't you ever heard about sorting.

• sexy

vs allow to divide by 0?
unbelieable...

• Andreas Krug

Hello Alex & nascardriver,

in chapter 2.11 - Header files (The #include order of header files) you wrote to sort the headers alphabetically, but the code examples 5 - 8 in this chapter have iostream before iomanip.

Keep up the great work,
Andreas

• BeanSprugget

Something I found interesting is that instead of "overflowing" if the number is to large it becomes (plus or minus) infinity.

I wonder then: is it actually attempting to calculate 5.0/zero, and then becoming "Inf" after overflowing, or do compilers know that it won't work ahead of time and will automatically show "Inf"?

• Sheepy69

This is a curse image.