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diff --git a/Doc/faq/design.rst b/Doc/faq/design.rst index 87cc942..7c5116d 100644 --- a/Doc/faq/design.rst +++ b/Doc/faq/design.rst @@ -43,56 +43,45 @@ Why am I getting strange results with simple arithmetic operations? 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? |