# 15. Floating Point Arithmetic: Issues and LimitationsÂ¶

Floating-point numbers are represented in computer hardware as base 2 (binary)

fractions. For example, the
**
decimal
**
fraction
```
0.125
```

has value 1/10 + 2/100 + 5/1000, and in the same way the
**
binary
**
fraction
```
0.001
```

has value 0/2 + 0/4 + 1/8. These two fractions have identical values, the only

real difference being that the first is written in base 10 fractional notation,

and the second in base 2.

Unfortunately, most decimal fractions cannot be represented exactly as binary

fractions. A consequence is that, in general, the decimal floating-point

numbers you enter are only approximated by the binary floating-point numbers

actually stored in the machine.

The problem is easier to understand at first in base 10. Consider the fraction

1/3. You can approximate that as a base 10 fraction:

`0.3`

or, better,

`0.33`

or, better,

`0.333`

and so on. No matter how many digits youâre willing to write down, the result

will never be exactly 1/3, but will be an increasingly better approximation of

1/3.

In the same way, no matter how many base 2 digits youâre willing to use, the

decimal value 0.1 cannot be represented exactly as a base 2 fraction. In base

2, 1/10 is the infinitely repeating fraction

`0.0001100110011001100110011001100110011001100110011...`

Stop at any finite number of bits, and you get an approximation. On most

machines today, floats are approximated using a binary fraction with

the numerator using the first 53 bits starting with the most significant bit and

with the denominator as a power of two. In the case of 1/10, the binary fraction

is
```
3602879701896397
/
2
**
55
```

which is close to but not exactly

equal to the true value of 1/10.

Many users are not aware of the approximation because of the way values are

displayed. Python only prints a decimal approximation to the true decimal

value of the binary approximation stored by the machine. On most machines, if

Python were to print the true decimal value of the binary approximation stored

for 0.1, it would have to display

`>>> 0.1`

0.1000000000000000055511151231257827021181583404541015625

That is more digits than most people find useful, so Python keeps the number

of digits manageable by displaying a rounded value instead

`>>> 1 / 10`

0.1

Just remember, even though the printed result looks like the exact value

of 1/10, the actual stored value is the nearest representable binary fraction.

Interestingly, there are many different decimal numbers that share the same

nearest approximate binary fraction. For example, the numbers
```
0.1
```

and

```
0.10000000000000001
```

and

```
0.1000000000000000055511151231257827021181583404541015625
```

are all

approximated by
```
3602879701896397
/
2
**
55
```

. Since all of these decimal

values share the same approximation, any one of them could be displayed

while still preserving the invariant
```
eval(repr(x))
==
x
```

.

Historically, the Python prompt and built-in
```
repr()
```

function would choose

the one with 17 significant digits,
```
0.10000000000000001
```

. Starting with

Python 3.1, Python (on most systems) is now able to choose the shortest of

these and simply display
```
0.1
```

.

Note that this is in the very nature of binary floating-point: this is not a bug

in Python, and it is not a bug in your code either. Youâll see the same kind of

thing in all languages that support your hardwareâs floating-point arithmetic

(although some languages may not
*
display
*
the difference by default, or in all

output modes).

For more pleasant output, you may wish to use string formatting to produce a limited number of significant digits:

`>>> format(math.pi, '.12g') # give 12 significant digits`

'3.14159265359'

>>> format(math.pi, '.2f') # give 2 digits after the point

'3.14'

>>> repr(math.pi)

'3.141592653589793'

Itâs important to realize that this is, in a real sense, an illusion: youâre

simply rounding the
*
display
*
of the true machine value.

One illusion may beget another. For example, since 0.1 is not exactly 1/10,

summing three values of 0.1 may not yield exactly 0.3, either:

`>>> .1 + .1 + .1 == .3`

False

Also, since the 0.1 cannot get any closer to the exact value of 1/10 and

0.3 cannot get any closer to the exact value of 3/10, then pre-rounding with

```
round()
```

function cannot help:

`>>> round(.1, 1) + round(.1, 1) + round(.1, 1) == round(.3, 1)`

False

Though the numbers cannot be made closer to their intended exact values,

the
```
round()
```

function can be useful for post-rounding so that results

with inexact values become comparable to one another:

`>>> round(.1 + .1 + .1, 10) == round(.3, 10)`

True

Binary floating-point arithmetic holds many surprises like this. The problem

with â0.1â is explained in precise detail below, in the âRepresentation Errorâ

section. See The Perils of Floating Point

for a more complete account of other common surprises.

As that says near the end, âthere are no easy answers.â Still, donât be unduly

wary of floating-point! The errors in Python float operations are inherited

from the floating-point hardware, and on most machines are on the order of no

more than 1 part in 2**53 per operation. Thatâs more than adequate for most

tasks, but you do need to keep in mind that itâs not decimal arithmetic and

that every float operation can suffer a new rounding error.

While pathological cases do exist, for most casual use of floating-point

arithmetic youâll see the result you expect in the end if you simply round the

display of your final results to the number of decimal digits you expect.

```
str()
```

usually suffices, and for finer control see the
```
str.format()
```

methodâs format specifiers in
Format String Syntax
.

For use cases which require exact decimal representation, try using the

```
decimal
```

module which implements decimal arithmetic suitable for

accounting applications and high-precision applications.

Another form of exact arithmetic is supported by the
```
fractions
```

module

which implements arithmetic based on rational numbers (so the numbers like

1/3 can be represented exactly).

If you are a heavy user of floating point operations you should take a look

at the NumPy package and many other packages for mathematical and

statistical operations supplied by the SciPy project. See <https://scipy.org>.

Python provides tools that may help on those rare occasions when you really

*
do
*
want to know the exact value of a float. The

```
float.as_integer_ratio()
```

method expresses the value of a float as a

fraction:

`>>> x = 3.14159`

>>> x.as_integer_ratio()

(3537115888337719, 1125899906842624)

Since the ratio is exact, it can be used to losslessly recreate the

original value:

`>>> x == 3537115888337719 / 1125899906842624`

True

The
```
float.hex()
```

method expresses a float in hexadecimal (base

16), again giving the exact value stored by your computer:

`>>> x.hex()`

'0x1.921f9f01b866ep+1'

This precise hexadecimal representation can be used to reconstruct

the float value exactly:

`>>> x == float.fromhex('0x1.921f9f01b866ep+1')`

True

Since the representation is exact, it is useful for reliably porting values

across different versions of Python (platform independence) and exchanging

data with other languages that support the same format (such as Java and C99).

Another helpful tool is the
```
math.fsum()
```

function which helps mitigate

loss-of-precision during summation. It tracks âlost digitsâ as values are

added onto a running total. That can make a difference in overall accuracy

so that the errors do not accumulate to the point where they affect the

final total:

`>>> sum([0.1] * 10) == 1.0`

False

>>> math.fsum([0.1] * 10) == 1.0

True

## 15.1. Representation ErrorÂ¶

This section explains the â0.1â example in detail, and shows how you can perform

an exact analysis of cases like this yourself. Basic familiarity with binary

floating-point representation is assumed.

*
Representation error
*
refers to the fact that some (most, actually)

decimal fractions cannot be represented exactly as binary (base 2) fractions.

This is the chief reason why Python (or Perl, C, C++, Java, Fortran, and many

others) often wonât display the exact decimal number you expect.

Why is that? 1/10 is not exactly representable as a binary fraction. Almost all

machines today (November 2000) use IEEE-754 floating point arithmetic, and

almost all platforms map Python floats to IEEE-754 âdouble precisionâ. 754

doubles contain 53 bits of precision, so on input the computer strives to

convert 0.1 to the closest fraction it can of the form
*
J
*
/2**
*
N
*
where
*
J
*
is

an integer containing exactly 53 bits. Rewriting

`1 / 10 ~= J / (2**N)`

as

`J ~= 2**N / 10`

and recalling that
*
J
*
has exactly 53 bits (is
```
>=
2**52
```

but
```
<
2**53
```

),

the best value for
*
N
*
is 56:

`>>> 2**52 <= 2**56 // 10 < 2**53`

True

That is, 56 is the only value for
*
N
*
that leaves
*
J
*
with exactly 53 bits. The

best possible value for
*
J
*
is then that quotient rounded:

`>>> q, r = divmod(2**56, 10)`

>>> r

6

Since the remainder is more than half of 10, the best approximation is obtained

by rounding up:

`>>> q+1`

7205759403792794

Therefore the best possible approximation to 1/10 in 754 double precision is:

`7205759403792794 / 2 ** 56`

Dividing both the numerator and denominator by two reduces the fraction to:

`3602879701896397 / 2 ** 55`

Note that since we rounded up, this is actually a little bit larger than 1/10;

if we had not rounded up, the quotient would have been a little bit smaller than

1/10. But in no case can it be
*
exactly
*
1/10!

So the computer never âseesâ 1/10: what it sees is the exact fraction given

above, the best 754 double approximation it can get:

`>>> 0.1 * 2 ** 55`

3602879701896397.0

If we multiply that fraction by 10**55, we can see the value out to

55 decimal digits:

`>>> 3602879701896397 * 10 ** 55 // 2 ** 55`

1000000000000000055511151231257827021181583404541015625

meaning that the exact number stored in the computer is equal to

the decimal value 0.1000000000000000055511151231257827021181583404541015625.

Instead of displaying the full decimal value, many languages (including

older versions of Python), round the result to 17 significant digits:

`>>> format(0.1, '.17f')`

'0.10000000000000001'

The
```
fractions
```

and
```
decimal
```

modules make these calculations

easy:

`>>> from decimal import Decimal`

>>> from fractions import Fraction

>>> Fraction.from_float(0.1)

Fraction(3602879701896397, 36028797018963968)

>>> (0.1).as_integer_ratio()

(3602879701896397, 36028797018963968)

>>> Decimal.from_float(0.1)

Decimal('0.1000000000000000055511151231257827021181583404541015625')

>>> format(Decimal.from_float(0.1), '.17')

'0.10000000000000001'