# JS - Why does 0.1 + 0.2 ≠ 0.3

July 03, 2021

This is a JS issue that most frontend devs come across at some point. If you haven’t done so already then go to the console and just type 0.1 + 0.2 === 0.3.

0.1 + 0.2 === 0.3 // false

If you haven’t done this before then the answer might surprise you, so what is going on?

## Numbers in JS

JS uses a floating point format to store numbers, specifically IEEE 754 - binary64. Whereas in many other languages, such as C# or Java, you get multiple integer and floating point types, in JS you get just the one - Number (technically with the introduction of ES2020 you now have BigInt but for now I’m going to pretend that doesn’t exist). This is great as it means you don’t have to think about data types and for the majority of use cases everything will be fine, until it isn’t…

To start to see what is happening there are a couple of things you need to understand.

• JS cannot precisely store some decimal numbers
• When you log a JS Number the value shown may not be entirely accurate

To see both of these things happening we can take the simple value (in decimal anyway) of 0.1. If you take this value, store it against the variable x and return it then you get 0.1. Brilliant!

const x = 0.1
x // 0.1

Except that the precise Number value that has been stored against x is not in fact 0.1. A way to find out exactly what has been stored is to force a greater precision to be returned.

const x = 0.1
x // 0.1
x.toPrecision(55) // 0.1000000000000000055511151231257827021181583404541015625

So when we asked JS to store 0.1, it made it look like it had done exactly this but actually had stored a much more specific value.

### A Repeating Problem

How JS stores Numbers are stored is shown in more detail later, but the important thing to be aware of is that JS numbers are rational numbers (can be represented as a fraction) and using the binary format it is impossible to exactly represent some numbers.

The fundamental issue is that in this format some numbers end up with a repeating binary pattern. So 0.1 ($1/10$) would end up being represented with the repeated pattern $1.1\.{0}01\.{1}$ Whenever we have this repeating pattern it means it is not possible for the format to precisely store the value - some precision will be lost as it has to cut the repeating pattern at some point (in IEEE 754 - binary64 it will actually store up to 52 bits as you will see later). We have exactly the same problem with base 10 - if you try to represent $1/3$ as a decimal you end up with a repeating pattern $0.3333333\.{3}$.

Again the takeaway is that it is impossible for 0.1 to be accurately stored using the JS Number format!

### JS Number::toString

So that covers why JS struggles to precisely store some numbers but it doesn’t explain why, on the face of it, it looks like a value has been accurately stored.

The reason the console gives you 0.1 when you print x above is due to the fact that toString is being applied to the Number before it is displayed. You can read the ECMAScript Number::toString specification for more detail but the key thing to be aware of is that the output will not be the exact Number but the lowest precision decimal representation that the Number value is closer to than any other Number representation. The reason the console logs 0.1 is because the value stored is the closest possible value to 0.1 that can be stored.

It does this to try and get the best decimal representation of the Number being stored. The success of this is evidenced by the fact that you enter 0.1 and you get 0.1 out!

## Back to 0.1 + 0.2 ≠ 0.3

We now know that some numbers can not be precisely represented in the format that JS uses to store Numbers and we also know that when we console.log numbers we aren’t necessarily getting the precise Number that JS has stored.

So what is really happening is that an imperfect representation of 0.1 is being added to an imperfect representation of 0.2 - these are being added together to get a value that is close to 0.3. However this value is not the closest possible value to 0.3 that can stored by a JS Number so instead of 0.3 you get 0.30000000000000004 when toString is applied.

const x = 0.1
const y = 0.2
x // 0.1
y // 0.2
x.toPrecision(55) // 0.1000000000000000055511151231257827021181583404541015625
y.toPrecision(55) // 0.2000000000000000111022302462515654042363166809082031250
const z = x + y
z // 0.30000000000000004
z.toPrecision(55) // 0.3000000000000000444089209850062616169452667236328125000

If you just directly assign 0.3 it is still not possible to store precisely 0.3, but the value stored is the closest possible so when you log it you do see 0.3.

const p = 0.3 // 0.3
p // 0.3
p.toPrecision(55) // 0.2999999999999999888977697537484345957636833190917968750

So as can be seen in these snippets what is really happening is not:

0.1 + 0.2 === 0.3

But instead (after the values for 0.1 and 0.2 have been added):

0.3000000000000000444089209850062616169452667236328125000 === 0.2999999999999999888977697537484345957636833190917968750

It seems much more reasonable for false to be the return value when viewed like this!

## Some Maths

So hopefully this gives a much better understanding of the original snippet:

0.1 + 0.2 === 0.3 // false

Personally I would stop reading if that is all you are interested in. I wanted to understand the maths behind exactly why 0.1 + 0.2 = 0.30000000000000004 though, and that is what the rest of this post is about.

So, going back to a previous snippet:

const x = 0.1
const y = 0.2
x // 0.1
y // 0.2
x.toPrecision(55) // 0.1000000000000000055511151231257827021181583404541015625
y.toPrecision(55) // 0.2000000000000000111022302462515654042363166809082031250
const z = x + y
z // 0.30000000000000004
z.toPrecision(55) // 0.3000000000000000444089209850062616169452667236328125000

If you take the Number value that is actually stored for 0.1, that we show above:

0.1000000000000000055511151231257827021181583404541015625

And the value for 0.2:

0.2000000000000000111022302462515654042363166809082031250

If we were working with decimals then the expectation would be that our value for 0.3 would be equal to the sum of these values. However:

    0.1000000000000000055511151231257827021181583404541015625
+   0.2000000000000000111022302462515654042363166809082031250
-------------------------------------------------------------
0.3000000000000000166533453693773481063544750213623046875

Which is different to the 0.3000000000000000444089209850062616169452667236328125000 we actually get!

The reason this happens is that the arithmetic is done on the binary representation. To really be able to follow through the example of 0.1 + 0.2 you therefore need to understand how Numbers are stored.

### Storing Numbers

JS numbers are stored using a double-precision 64-bit binary format (IEEE 754 - binary64). This means that out of the 64 bits available, the first bit represents the sign, the next 11 bits represent the exponent and the final 52 bits represent the mantissa/significand/fraction. The formula to calculate the number being represented using these 3 parts is:

$sign * 2^{exponent} * (1 + mantissa)$

Each of the three parts - sign, exponent and mantissa are encoded within the bits available to them.

#### Sign bit

Encoded as 0 = positive and 1 = negative.

So if the first bit is 0 then the sign in the formula will be +1 and if the first bit is 1 then the sign in the formulat will be -1.

#### Exponent

This can be slightly more complex to understand as we need to be able to store a range of positive and negative numbers for the exponent. The exponent in this formula is used to essentially tell us how many places to move the binary point and using the 11 bits available we can store from -1022 to +1023 (there are some special cases that mean we can’t quite use the full range).

The way the encoding allows for this range is by using a 0 offset of 1023 (01111111111) - meaning that 01111111111 is equal to an exponent of 0. So all exponents are stored as:

encoding - 1023 = exponent

Exponent Value Encoding Binary
0 1023 (1023 - 1023 = 0) $01111111111$
10 1033 (1033 - 1023 = 10) $10000001001$
500 1523 (1523 - 1023 = 500) $10111110011$
-300 723 (723 - 1023 = -300) $01011010011$
-1023 1023 (1023 - 1023 = 0) $00000000001$

#### Mantissa

This is the fractional part of the number and is the value that is going to be shifted by the exponent. Essentially, to calculate the correct value you need to imagine that there is a binary point just before the start of the value - so if the mantissa is 10000000 (pretending we are only using 8 bits) then the actual value would be 0.10000000. The below shows some example mantissa encodings but only to 8 bits (in reality there would be 52 bits):

Mantissa encoding = $10000000$
With implicit binary point = $0.10000000$
Mantissa value = $2^{-1}$
= $1/2$
= $0.5$

Mantissa encoding = $11110000$
With implicit binary point = $0.11110000$
Mantissa value = $2^{-1} + 2^{-2} + 2^{-3} + 2^{-4}$
= $1/2 + 1/4 + 1/8 + 1/16$
= $0.5 + 0.25 + 0.125 + 0.0625$
= $0.9375$

#### An Example

So taking the example of the number 10, using IEEE 754 - binary64, it is actually stored as:

Part Value Binary (only showing 8 bit mantissa)
Sign bit +1 0
Exponent 3 (encoded as 1026) 10000000010
Mantissa 0.25 01000000

Remembering out formula from above:

$sign * 2^{exponent} * (1 + mantissa)$

We end up with:

x = 1 * $2^{3}$ * (1 + 0.25)
x = 8 * 1.25
x = 10

For an integer, like 10, this format can perfectly represent the number with no potential errors. In fact all integers within the format range can be perfectly represented, it is when trying to represent decimals that we get imprecision.

If you’d like to play around with the IEEE 754 format there are a number of different sites available, I personally found the following calculator very useful - http://weitz.de/ieee/

### Back to our original sum

Now we understand how Numbers are stored, it becomes much easier to understand how operators work and why 0.1 + 0.2 = 0.30000000000000004.

There is a great video that walks through adding IEEE-754 floating point numbers - HOW TO: Adding IEEE-754 Floating Point Numbers. The below follows this, just with the 64 bit format as opposed to the 32 bit format used in the video.

Going back to our original example sum:

0.1 + 0.2

Before we can do the calculation, we need to understand exactly how these numbers are stored.

#### 0.1

Sign bit = +1
Exponent = -4
Binary Mantissa = 0.1001100110011001100110011001100110011001100110011010
Decimal Mantissa = 0.60000000000000008882

$sign * 2^{exponent} * (1 + mantissa)$

x = +1 * $2^{-4}$ * (1 + 0.60000000000000008882)
x = 0.0625 * 1.60000000000000008882
x = 0.1000000000000000055511151231257827021181583404541015625

Binary notation:

1.1001100110011001100110011001100110011001100110011010 * $2^{-4}$

#### 0.2

Sign bit = +1
Exponent = -3
Binary Mantissa = 1.1001100110011001100110011001100110011001100110011010
Decimal Mantissa = 1.60000000000000008882

$sign * 2^{exponent} * (1 + mantissa)$

x = +1 * $2^{-3}$ * (1 + 0.60000000000000008882)
x = 0.125 * 1.60000000000000008882
x = 0.2000000000000000111022302462515654042363166809082031250

Binary notation:

1.1001100110011001100110011001100110011001100110011010 * $2^{-3}$

#### 0.1 + 0.2

Now we have the binary notation for 0.1 and 0.2.

0.1 = 1.1001100110011001100110011001100110011001100110011010 * $2^{-4}$
0.2 = 1.1001100110011001100110011001100110011001100110011010 * $2^{-3}$

We need to standardise the exponents between these calculations, we do this by converting 0.1 to have an exponent of -3:

0.1 = 0.11001100110011001100110011001100110011001100110011010 * $2^{-3}$
0.2 = 1.1001100110011001100110011001100110011001100110011010 * $2^{-3}$

Now it is a matter of binary addition:

    0.1100110011001100110011001100110011001100110011001101
+   1.1001100110011001100110011001100110011001100110011010
-----------------------------------------------------------
10.0110011001100110011001100110011001100110011001100111

And we end up with:

0.3 = 10.0110011001100110011001100110011001100110011001100111 * $2^{-3}$

We can then convert this back into the standard notation by changing the exponent, so:

0.3 = 1.00110011001100110011001100110011001100110011001100111 * $2^{-2}$

One important thing to be aware of here is that the mantissa now has 53 bits but we know that the format can only store 52 bits. We need to eliminate the additional bit by rounding it back to 52 bits:

0.3 = 1.0011001100110011001100110011001100110011001100110100 * $2^{-2}$

Finally we can convert this into decimal format (the sign bit will still be 1):

0.3 = +1 * $2^{-2}$ * 1.20000000000000017764
0.3 = 0.25 * 1.20000000000000017764
0.3 = 0.3000000000000000444089209850062616169452667236328125000

And finally we have managed to get to the exact number that JS gives us:

const x = 0.1 + 0.2
z // 0.30000000000000004
z.toPrecision(55) // 0.3000000000000000444089209850062616169452667236328125000

Written by Ciaran Grimes - Questions or corrections? Please let me know