# Floating Point Arithmetic: Issues and Limitations¶

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 you’re 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 you’re 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. This is why you see things like:

```>>> 0.1
0.10000000000000001
```

On most machines today, that is what you’ll see if you enter 0.1 at a Python prompt. You may not, though, because the number of bits used by the hardware to store floating-point values can vary across machines, and 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
```

instead! The Python prompt uses the builtin repr() function to obtain a string version of everything it displays. For floats, repr(float) rounds the true decimal value to 17 significant digits, giving

```0.10000000000000001
```

repr(float) produces 17 significant digits because it turns out that’s enough (on most machines) so that eval(repr(x)) == x exactly for all finite floats x, but rounding to 16 digits is not enough to make that true.

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. You’ll see the same kind of thing in all languages that support your hardware’s floating-point arithmetic (although some languages may not display the difference by default, or in all output modes).

Python’s builtin str() function produces only 12 significant digits, and you may wish to use that instead. It’s unusual for eval(str(x)) to reproduce x, but the output may be more pleasant to look at:

```>>> print(str(0.1))
0.1
```

It’s important to realize that this is, in a real sense, an illusion: the value in the machine is not exactly 1/10, you’re simply rounding the display of the true machine value.

Other surprises follow from this one. For example, after seeing

```>>> 0.1
0.10000000000000001
```

you may be tempted to use the round() function to chop it back to the single digit you expect. But that makes no difference:

```>>> round(0.1, 1)
0.10000000000000001
```

The problem is that the binary floating-point value stored for “0.1” was already the best possible binary approximation to 1/10, so trying to round it again can’t make it better: it was already as good as it gets.

Another consequence is that since 0.1 is not exactly 1/10, summing ten values of 0.1 may not yield exactly 1.0, either:

```>>> sum = 0.0
>>> for i in range(10):
...     sum += 0.1
...
>>> sum
0.99999999999999989
```

Binary floating-point arithmetic holds many surprises like this. The problem with “0.1” is explained in precise detail below, in the “Representation Error” section. See The Perils of Floating Point for a more complete account of other common surprises.

As that says near the end, “there are no easy answers.” Still, don’t 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. That’s more than adequate for most tasks, but you do need to keep in mind that it’s 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 you’ll 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() method’s 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 Numerical Python package and many other packages for mathematical and statistical operations supplied by the SciPy project. See <http://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()
(3537115888337719L, 1125899906842624L)
```

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).

## Representation Error¶

This section explains the “0.1” 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 won’t display the exact decimal number you expect:

```>>> 0.1
0.10000000000000001
```

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 “double precision”. 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
4503599627370496
>>> 2**53
9007199254740992
>>> 2**56/10
7205759403792794.0
```

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 that over 2**56, or

```7205759403792794 / 72057594037927936
```

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 “sees” 1/10: what it sees is the exact fraction given above, the best 754 double approximation it can get:

```>>> .1 * 2**56
7205759403792794.0
```

If we multiply that fraction by 10**30, we can see the (truncated) value of its 30 most significant decimal digits:

```>>> 7205759403792794 * 10**30 / 2**56
100000000000000005551115123125
```

meaning that the exact number stored in the computer is approximately equal to the decimal value 0.100000000000000005551115123125. Rounding that to 17 significant digits gives the 0.10000000000000001 that Python displays (well, will display on any 754-conforming platform that does best-possible input and output conversions in its C library — yours may not!).