## 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 ```
0.10000000000000001
```
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 Python’s builtin ```
>>> 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 Other surprises follow from this one. For example, after seeing ```
>>> 0.1
0.10000000000000001
``` you may be tempted to use the ```
>>> 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.
## 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.
```
>>> 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 1 / 10 ~= J / (2**N) as J ~= 2**N / 10 and recalling that ```
>>> 2**52
4503599627370496L
>>> 2**53
9007199254740992L
>>> 2**56/10
7205759403792793L
``` That is, 56 is the only value for ```
>>> q, r = divmod(2**56, 10)
>>> r
6L
``` Since the remainder is more than half of 10, the best approximation is obtained by rounding up: ```
>>> q+1
7205759403792794L
``` 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 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
100000000000000005551115123125L
``` 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!). |