cpython: a79b07e05d0d (original) (raw)

Mercurial > cpython

changeset 76909:a79b07e05d0d

Issue #14245: Merge changes from 3.2. [#14245]

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--- a/Doc/faq/design.rst +++ b/Doc/faq/design.rst @@ -43,56 +43,45 @@ Why am I getting strange results with si 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:: -

• 1.2 - 1.0

• 0.199999999999999996 +Users are often surprised by results like this::

-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.

• 1.2 - 1.0

• 0.199999999999999996

-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. - -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....... +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 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... - -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. +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. -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:: +Many numbers that can be written easily in decimal notation cannot be expressed +exactly in binary floating-point. For example, after::

• 1.1 - 0.9

• 0.20000000000000007

• print(1.1 - 0.9)

• 0.2

• x = 1.2

+ +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:: -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::

• 1.0011001100110011001100110011001100110011001100110011 (binary)

+ +which is exactly:: +

• 1.1999999999999999555910790149937383830547332763671875 (decimal)

• epsilon = 0.0000000000001 # Tiny allowed error
• expected_result = 0.4 +The typical precision of 53 bits provides Python floats with 15-16 +decimal digits of accuracy.
• if expected_result-epsilon <= computation() <= expected_result+epsilon:

•   ...[](#l1.80)

- -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[](#l1.84) +<tut-fp-issues> chapter in the Python tutorial. Why are Python strings immutable?