Floating point precision problem

Guess the answer

(0.125 + 0.125) * 10;

Guess the answer

(0.125 + 0.125) * 10;

mkotsur@n-racoon:~$ python -iPython 2.6.6 (r266:84292, Sep 15 2010, 15:52:39) [GCC 4.4.5] on linux2Type "help", "copyright", "credits" or "license" for more information.

>>> (0.125 + 0.125) * 10;2.5

Guess the answer

(0.1 + 0.7) * 10;

Guess the answer

(0.1 + 0.7) * 10;

mkotsur@n-racoon:~$ python -iPython 2.6.6 (r266:84292, Sep 15 2010, 15:52:39) [GCC 4.4.5] on linux2Type "help", "copyright", "credits" or "license" for more information.

>>> (0.1 + 0.7) * 10;7.9999999999999991

>>> int((0.1 + 0.7) * 10);7

Guess the answer

(0.1 + 0.1) * 10

Guess the answer

(0.1 + 0.1) * 10;

mkotsur@n-racoon:~$ python -iPython 2.6.6 (r266:84292, Sep 15 2010, 15:52:39) [GCC 4.4.5] on linux2Type "help", "copyright", "credits" or "license" for more information.

>>> (0.1 + 0.1) * 10;2.0

WTF??!11

Python, PHP, Java… => Same problems...

“The IEEE Standard for Floating-Point Arithmetic (IEEE 754-1985)

set the standard for floating-point computation for 23 years. It became the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. Its binary floating-point formats and arithmetic are preserved in the new IEEE 754-2008 standard which replaced it.”

Single float

0.125=0×200×2−1

0×2−21×2−3

0.12510=0.0012

0.0012=1.02×10−11

Single float

M=1.000exp2=−11Mantissa

Sign: 0 for “+”. 1 for “-”. 0 in our case

Exponent bias: + 127 (01111111) – half of a byte.01111100 in our case

Mantissa (fraction): integer part always 1, 23 bits of fraction00000000000000000000000 (23 zeros) in our case

Single float

What about other numbers?

0.125 = 0 01111100 00000000000000000000000

Sign Exponent Mantissa

Single float

0.7 = 0 01111110 01100110011001100110011

0.1 = 0 01111000 10011001100110011001100

0.125 = 0 01111100 00000000000000000000000

Be careful when:

1. You compare results from different sources;2. You do output floats;3. You convert float to another type;4. You use cycles or other ways to accumulate error;

How do people live with this?

1. Don't use float numbers :-)

2. Use 'near' instead of 'equals'

3.Don't trust computers.

More here:

http://en.wikipedia.org/wiki/IEEE_754-2008

http://php.net/manual/en/language.types.float.php

http://docs.python.org/tutorial/floatingpoint.html

http://en.wikipedia.org/wiki/Single_precision_floating-point_format

http://www.lahey.com/float.htm