1. Floating point precision
problem

2. Guess the answer
(0.125 + 0.125) * 10;

3. Guess the answer
(0.125 + 0.125) * 10;
mkotsur@n-racoon:~$ python -i
Python 2.6.6 (r266:84292, Sep 15 2010, 15:52:39)
[GCC 4.4.5] on linux2
Type "help", "copyright", "credits" or "license" for more
information.
>>> (0.125 + 0.125) * 10;
2.5

4. Guess the answer
(0.1 + 0.7) * 10;

5. Guess the answer
(0.1 + 0.7) * 10;
mkotsur@n-racoon:~$ python -i
Python 2.6.6 (r266:84292, Sep 15 2010, 15:52:39)
[GCC 4.4.5] on linux2
Type "help", "copyright", "credits" or "license" for more
information.
>>> (0.1 + 0.7) * 10;
7.9999999999999991
>>> int((0.1 + 0.7) * 10);
7

6. Guess the answer
(0.1 + 0.1) * 10

7. Guess the answer
(0.1 + 0.1) * 10;
mkotsur@n-racoon:~$ python -i
Python 2.6.6 (r266:84292, Sep 15 2010, 15:52:39)
[GCC 4.4.5] on linux2
Type "help", "copyright", "credits" or "license" for more
information.
>>> (0.1 + 0.1) * 10;
2.0

8. 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.”

9. Single float
0.125=0×20
0×2−1
0×2−2
1×2−3
0.12510=0.0012
0.0012=1.02×10−11

10. Single float
M =1.000exp2=−11
Mantissa
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 fraction
00000000000000000000000 (23 zeros) in our case

11. Single float
What about other numbers?
0.125 = 0 01111100 00000000000000000000000
Sign Exponent Mantissa

12. Single float
0.7 = 0 01111110 01100110011001100110011
0.1 = 0 01111000 10011001100110011001100
0.125 = 0 01111100 00000000000000000000000

13. 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;

14. How do people live with this?
1. Don't use float numbers :-)
2. Use 'near' instead of 'equals'
3.Don't trust computers.

15. 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-p
http://www.lahey.com/float.htm