See the next question.
-Why are floating point calculations so inaccurate?
+Why are floating-point calculations so inaccurate?
--------------------------------------------------
-People are often very surprised by results like this::
+Users are often surprised by results like this::
- >>> 1.2 - 1.0
- 0.199999999999999996
+ >>> 1.2 - 1.0
+ 0.199999999999999996
-and think it is a bug in Python. It's not. This has nothing to do with Python,
-but with how the underlying C platform handles floating point numbers, and
-ultimately with the inaccuracies introduced when writing down numbers as a
-string of a fixed number of digits.
-
-The internal representation of floating point numbers uses a fixed number of
-binary digits to represent a decimal number. Some decimal numbers can't be
-represented exactly in binary, resulting in small roundoff errors.
+and think it is a bug in Python. It's not. This has little to do with Python,
+and much more to do with how the underlying platform handles floating-point
+numbers.
-In decimal math, there are many numbers that can't be represented with a fixed
-number of decimal digits, e.g. 1/3 = 0.3333333333.......
+The :class:`float` type in CPython uses a C ``double`` for storage. A
+:class:`float` object's value is stored in binary floating-point with a fixed
+precision (typically 53 bits) and Python uses C operations, which in turn rely
+on the hardware implementation in the processor, to perform floating-point
+operations. This means that as far as floating-point operations are concerned,
+Python behaves like many popular languages including C and Java.
-In base 2, 1/2 = 0.1, 1/4 = 0.01, 1/8 = 0.001, etc. .2 equals 2/10 equals 1/5,
-resulting in the binary fractional number 0.001100110011001...
+Many numbers that can be written easily in decimal notation cannot be expressed
+exactly in binary floating-point. For example, after::
-Floating point numbers only have 32 or 64 bits of precision, so the digits are
-cut off at some point, and the resulting number is 0.199999999999999996 in
-decimal, not 0.2.
+ >>> x = 1.2
-A floating point number's ``repr()`` function prints as many digits are
-necessary to make ``eval(repr(f)) == f`` true for any float f. The ``str()``
-function prints fewer digits and this often results in the more sensible number
-that was probably intended::
+the value stored for ``x`` is a (very good) approximation to the decimal value
+``1.2``, but is not exactly equal to it. On a typical machine, the actual
+stored value is::
- >>> 1.1 - 0.9
- 0.20000000000000007
- >>> print(1.1 - 0.9)
- 0.2
+ 1.0011001100110011001100110011001100110011001100110011 (binary)
-One of the consequences of this is that it is error-prone to compare the result
-of some computation to a float with ``==``. Tiny inaccuracies may mean that
-``==`` fails. Instead, you have to check that the difference between the two
-numbers is less than a certain threshold::
+which is exactly::
- epsilon = 0.0000000000001 # Tiny allowed error
- expected_result = 0.4
+ 1.1999999999999999555910790149937383830547332763671875 (decimal)
- if expected_result-epsilon <= computation() <= expected_result+epsilon:
- ...
+The typical precision of 53 bits provides Python floats with 15-16
+decimal digits of accuracy.
-Please see the chapter on :ref:`floating point arithmetic <tut-fp-issues>` in
-the Python tutorial for more information.
+For a fuller explanation, please see the :ref:`floating point arithmetic
+<tut-fp-issues>` chapter in the Python tutorial.
Why are Python strings immutable?
projects / thirdparty / Python / cpython.git / commitdiff
