14. 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 built-in 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 built-in 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
0.9999999999999999

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.

>>> 0.1
0.10000000000000001

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

J ~= 2**N / 10

>>> 2**52
4503599627370496L
>>> 2**53
9007199254740992L
>>> 2**56/10
7205759403792793L

>>> q, r = divmod(2**56, 10)
>>> r
6L

>>> q+1
7205759403792794L

7205759403792794 / 72057594037927936

>>> .1 * 2**56
7205759403792794.0

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