# Floating Point Math

Your language isn't broken, it's doing floating point math. Computers can only natively store integers, so they need some way of representing decimal numbers. This representation comes with some degree of inaccuracy. That's why, more often than not, `.1 + .2 != .3`.

### Why does this happen?

It's actually pretty simple. When you have a base 10 system (like ours), it can only express fractions that use a prime factor of the base. The prime factors of 10 are 2 and 5. So 1/2, 1/4, 1/5, 1/8, and 1/10 can all be expressed cleanly because the denominators all use prime factors of 10. In contrast, 1/3, 1/6, and 1/7 are all repeating decimals because their denominators use a prime factor of 3 or 7. In binary (or base 2), the only prime factor is 2. So you can only express fractions cleanly which only contain 2 as a prime factor. In binary, 1/2, 1/4, 1/8 would all be expressed cleanly as decimals. While, 1/5 or 1/10 would be repeating decimals. So 0.1 and 0.2 (1/10 and 1/5) while clean decimals in a base 10 system, are repeating decimals in the base 2 system the computer is operating in. When you do math on these repeating decimals, you end up with leftovers which carry over when you convert the computer's base 2 (binary) number into a more human readable base 10 number.

Below are some examples of sending `.1 + .2` to standard output in a variety of languages.

Language Code Result
ABAP
``WRITE / CONV f( '.1' + '.2' ).``
And
``WRITE / CONV decfloat16( '.1' + '.2' ).``

0.30000000000000004

And

0.3

Ada
``````with Ada.Text_IO; use Ada.Text_IO;
procedure Sum is
A : Float := 0.1;
B : Float := 0.2;
C : Float := A + B;
begin
Put_Line(Float'Image(C));
Put_Line(Float'Image(0.1 + 0.2));
end Sum;``````

3.00000E-01
3.00000E-01

APL
``0.1 + 0.2``

0.30000000000000004

AutoHotkey
``MsgBox, % 0.1 + 0.2``

0.300000

awk
``echo | awk '{ print 0.1 + 0.2 }'``

0.3

bc
``0.1 + 0.2``

0.3

C
``````#include<stdio.h>
int main(int argc, char** argv) {
printf("%.17f\n", .1+.2);
return 0;
}``````

0.30000000000000004

Clojure
``(+ 0.1 0.2)``

0.30000000000000004

Clojure supports arbitrary precision and ratios. `(+ 0.1M 0.2M)` returns `0.3M`, while `(+ 1/10 2/10)` returns `3/10`.

ColdFusion
``````<cfset foo = .1 + .2>
<cfoutput>#foo#</cfoutput>``````

0.3

Common Lisp
``(+ .1 .2)``
And
``(+ 1/10 2/10)``
And
``(+ 0.1d0 0.2d0)``
And
``(- 1.2 1.0)``

0.3

And

3/10

And

0.30000000000000004d0

And

0.20000005

CL’s spec doesn’t actually even require radix 2 floats (let alone specifically 32-bit singles and 64-bit doubles), but the high-performance implementations all seem to use IEEE floats with the usual sizes. This was tested on SBCL and ECL in particular.

C++
``````#include <iomanip>
std::cout << std::setprecision(17) << 0.1 + 0.2``````

0.30000000000000004

Crystal
``puts 0.1 + 0.2``
And
``puts 0.1_f32 + 0.2_f32``

0.30000000000000004

And

0.3

C#
``Console.WriteLine("{0:R}", .1 + .2);``
And
``Console.WriteLine("{0:R}", .1m + .2m);``

0.30000000000000004

And

0.3

C# has support for 128-bit decimal numbers, with 28-29 significant digits of precision. Their range, however, is smaller than that of both the single and double precision floating point types. Decimal literals are denoted with the `m` suffix.

D
``````import std.stdio;

void main(string[] args) {
writefln("%.17f", .1+.2);
writefln("%.17f", .1f+.2f);
writefln("%.17f", .1L+.2L);
}``````

0.29999999999999999
0.30000001192092896
0.30000000000000000

Dart
``print(.1 + .2);``

0.30000000000000004

dc
``0.1 0.2 + p``

.3

Delphi XE5
``writeln(0.1 + 0.2);``

3.00000000000000E-0001

Elixir
``IO.puts(0.1 + 0.2)``

0.30000000000000004

Elm
``0.1 + 0.2``

0.30000000000000004

elvish
``+ .1 .2``

0.30000000000000004

elvish uses Go’s `double` for numerical operations.

Emacs Lisp
``(+ .1 .2)``

0.30000000000000004

Erlang
``io:format("~w~n", [0.1 + 0.2]).``

0.30000000000000004

FORTRAN
``````program FLOATMATHTEST
real(kind=4) :: x4, y4
real(kind=8) :: x8, y8
real(kind=16) :: x16, y16
! REAL literals are single precision, use _8 or _16
! if the literal should be wider.
x4 = .1; x8 = .1_8; x16 = .1_16
y4 = .2; y8 = .2_8; y16 = .2_16
write (*,*) x4 + y4, x8 + y8, x16 + y16
end``````

0.300000012
0.30000000000000004
0.300000000000000000000000000000000039

Gforth
``0.1e 0.2e f+ f.``

0.3

GHC (Haskell)
``* 0.1 + 0.2 :: Double``
And
``* 0.1 + 0.2 :: Float``

`* 0.30000000000000004`

And

`* 0.3`

Haskell supports rational numbers. To get the math right, `0.1 + 0.2 :: Rational` returns `3 % 10`, which is exactly `0.3`.

Go
``````package main
import "fmt"
func main() {
fmt.Println(.1 + .2)
var a float64 = .1
var b float64 = .2
fmt.Println(a + b)
fmt.Printf("%.54f\n", .1 + .2)
}``````

0.3
0.30000000000000004
0.299999999999999988897769753748434595763683319091796875

Groovy
``println 0.1 + 0.2``

0.3

Literal decimal values in Groovy are instances of java.math.BigDecimal

Hugs (Haskell)
``0.1 + 0.2``

0.3

Io
``(0.1 + 0.2) print``

0.3

Java
``System.out.println(.1 + .2);``
And
``System.out.println(.1F + .2F);``

0.30000000000000004

And

0.3

Java has built-in support for arbitrary precision numbers using the BigDecimal class.

JavaScript
``console.log(.1 + .2);``

0.30000000000000004

The decimal.js library provides an arbitrary-precision Decimal type for JavaScript.

Julia
``.1 + .2``

0.30000000000000004

Julia has built-in rational numbers support and also a built-in arbitrary-precision BigFloat data type. To get the math right, `1//10 + 2//10` returns `3//10`.

K (Kona)
``0.1 + 0.2``

0.3

Lua
``print(.1 + .2)``
And
``print(string.format("%0.17f", 0.1 + 0.2))``

0.3

And

0.30000000000000004

Mathematica
``0.1 + 0.2``

0.3

Mathematica has a fairly thorough internal mechanism for dealing with numerical precision and supports arbitrary precision.

Matlab
``0.1 + 0.2``
And
``sprintf('%.17f',0.1+0.2)``

0.3

And

0.30000000000000004

MySQL
``SELECT .1 + .2;``

0.3

Nim
``echo(0.1 + 0.2)``

0.3

Objective-C
``````#import <Foundation/Foundation.h>
int main(int argc, const char * argv[]) {
@autoreleasepool {
NSLog(@"%.17f\n", .1+.2);
}
return 0;
}``````

0.30000000000000004

OCaml
``0.1 +. 0.2;;``

float = 0.300000000000000044

Perl 5
``perl -E 'say 0.1+0.2'``
And
``perl -e 'printf q{%.17f}, 0.1+0.2'``

0.3

And

0.30000000000000004

Perl 6
``perl6 -e 'say 0.1+0.2'``
And
``perl6 -e 'say (0.1+0.2).base(10, 17)'``
And
``perl6 -e 'say 1/10+2/10'``
And
``perl6 -e 'say (0.1.Num + 0.2.Num).base(10, 17)'``

0.3

And

0.3

And

0.3

And

0.30000000000000004

Perl 6, unlike Perl 5, uses rationals by default, so .1 is stored something like { numerator => 1, denominator => 10 }. To actually trigger the behavior, you must force the numbers to be of type Num (double in C terms) and use the base function instead of the sprintf or fmt functions (since those functions have a bug that limits the precision of the output).

PHP
``````echo .1 + .2;
var_dump(.1 + .2);``````

0.3 float(0.30000000000000004441)

PHP `echo` converts 0.30000000000000004441 to a string and shortens it to “0.3”. To achieve the desired floating point result, adjust the precision ini setting: ini_set(“precision”, 17).

PicoLisp
``[load "frac.min.l"]  # https://gist.github.com/6016d743c4c124a1c04fc12accf7ef17``
And
``[println (+ (/ 1 10) (/ 2 10))]``

(/ 3 10)

You must load file “frac.min.l”.

Postgres
``SELECT select 0.1::float + 0.2::float;``

0.3

Powershell
``PS C:\>0.1 + 0.2``

0.3

Prolog (SWI-Prolog)
``?- X is 0.1 + 0.2.``

X = 0.30000000000000004.

Pyret
``0.1 + 0.2``
And
``~0.1 + ~0.2``

0.3

And

~0.30000000000000004

Pyret has built-in support for both rational numbers and floating points. Numbers written normally are assumed to be exact. In contrast, RoughNums are represented by floating points, and are written with a `~` in front, to indicate that they are not precise answers. (The `~` is meant to visually evoke hand-waving.) Therefore, a user who sees a computation produce `~0.30000000000000004` knows to treat the value with skepticism. RoughNums also cannot be compared directly for equality; they can only be compared up to a given tolerance.

Python 2
``print(.1 + .2)``
And
``.1 + .2``
And
``float(decimal.Decimal(".1") + decimal.Decimal(".2"))``
And
``float(fractions.Fraction('0.1') + fractions.Fraction('0.2'))``

0.3

And

0.30000000000000004

And

0.3

And

0.3

Python 2’s “print” statement converts 0.30000000000000004 to a string and shortens it to “0.3”. To achieve the desired floating point result, use print(repr(.1 + .2)). This was fixed in Python 3 (see below).

Python 3
``print(.1 + .2)``
And
``.1 + .2``
And
``float(decimal.Decimal('.1') + decimal.Decimal('.2'))``
And
``float(fractions.Fraction('0.1') + fractions.Fraction('0.2'))``

0.30000000000000004

And

0.30000000000000004

And

0.3

And

0.3

Python (both 2 and 3) supports decimal arithmetic with the decimal module, and true rational numbers with the fractions module.

R
``print(.1+.2)``
And
``print(.1+.2, digits=18)``

0.3

And

0.30000000000000004

Racket (PLT Scheme)
``(+ .1 .2)``
And
``(+ 1/10 2/10)``

0.30000000000000004

And

3/10

Ruby
``puts 0.1 + 0.2``
And
``puts 1/10r + 2/10r``

0.30000000000000004

And

3/10

Ruby supports rational numbers in syntax with version 2.1 and newer directly. For older versions use Rational.
Ruby also has a library specifically for decimals: BigDecimal.

Rust
``````extern crate num;
use num::rational::Ratio;
fn main() {
println!("{}", 0.1 + 0.2);
println!("1/10 + 2/10 = {}", Ratio::new(1, 10) + Ratio::new(2, 10));
}``````

0.30000000000000004

And

1/10 + 2/10 = 3/10

Rust has rational number support from the num crate.

SageMath
``.1 + .2``
And
``RDF(.1) + RDF(.2)``
And
``RBF('.1') + RBF('.2')``
And
``QQ('1/10') + QQ('2/10')``

0.3

And

0.30000000000000004

And

[“0.300000000000000 +/- 1.64e-16”]

And

3/10

SageMath supports various fields for arithmetic: Arbitrary Precision Real Numbers, RealDoubleField, Ball Arithmetic, Rational Numbers, etc.

scala
``scala -e 'println(0.1 + 0.2)'``
And
``scala -e 'println(0.1F + 0.2F)'``
And
``scala -e 'println(BigDecimal("0.1") + BigDecimal("0.2"))'``

0.30000000000000004

And

0.3

And

0.3

Smalltalk
``0.1 + 0.2.``

0.30000000000000004

Swift
``0.1 + 0.2``
And
``NSString(format: "%.17f", 0.1 + 0.2)``

0.3

And

0.30000000000000004

TCL
``puts [expr .1 + .2]``

0.30000000000000004

Turbo Pascal 7.0
``writeln(0.1 + 0.2);``

3.0000000000E-01

Vala
``````static int main(string[] args) {
stdout.printf("%.17f\n", 0.1 + 0.2);
return 0;
}``````

0.30000000000000004

Visual Basic 6
``````a# = 0.1 + 0.2: b# = 0.3
Debug.Print Format(a - b, "0." & String(16, "0"))
Debug.Print a = b``````

0.0000000000000001
False

Appending the identifier type character `#` to any identifier forces it to Double.

WebAssembly (WAST)
``````(func \$add_f32 (result f32)
f32.const 0.1
f32.const 0.2
f32.add)
(export "add_f32" (func \$add_f32))``````
And
``````(func \$add_f64 (result f64)
f64.const 0.1
f64.const 0.2
f64.add)
(export "add_f64" (func \$add_f64))``````

0.30000001192092896

And

0.30000000000000004

https://webassembly.studio/?f=r739k6d6q4t

zsh
``echo "\$((.1+.2))"``

0.30000000000000004

I am Erik Wiffin. You can contact me at: erik.wiffin.com or erik.wiffin@gmail.com.

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