B Floating Point Arithmetic: Issues and Limitations
This manual section was written by Tim Peters tim.one at home.com.
Floatingpoint 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
floatingpoint numbers you enter are only approximated by the binary
floatingpoint 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 to 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 floatingpoint 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 (implicitly) 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 floatingpoint: this is
not a bug in Python, it is not a bug in your code either, and you'll
see the same kind of thing in all languages that support your
hardware's floatingpoint 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 floatingpoint 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, adding 0.1
to itself 10 times may not yield exactly 1.0, either:
>>> sum = 0.0
>>> for i in range(10):
... sum += 0.1
...
>>> sum
0.99999999999999989
Binary floatingpoint arithmetic holds many surprises like this. The
problem with "0.1" is explained in precise detail below, in the
"Representation Error" section.
Still,
don't be unduly wary of floatingpoint! The errors in Python float
operations are inherited from the floatingpoint 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
floatingpoint 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 discussion of Pythons's % format
operator: the %g , %f and %e format codes
supply flexible and easy ways to round float results for display.
