What is a simple example of floating point/rounding error?

Picture is worth a thousand words - try to draw equation f(k) :

and you will get such XY graph (X and Y are in logarithmic scale).

If computer could represent 32-bit floats without rounding error then for every k we should get zero. But instead error increases with bigger values of k because of floating point error accumulation.

hth!

 for(double d = 0; d != 0.3; d += 0.1); // never terminates 

Generally, floating point error refers to when a number that cannot be stored in the IEEE floating point representation.

Integers are stored with the right-most bit being 1, and each bit to the left being double that (2,4,8,...). It's easy to see that this can store any integer up to 2^n, where n is the number of bits.

The mantissa (decimal part) of a floating point number is stored in a similar way, but moving left to right, and each successive bit being half of the value of the previous one. (It's actually a little more complicated than this, but it will do for now).

Thus, numbers like 0.5 (1/2) are easy to store, but not every number <1 can be created by adding a fixed number of fractions of the form 1/2, 1/4, 1/8, ...

A really simple example is 0.1, or 1/10. This can be done with an infinite series (which I can't really be bothered working out), but whenever a computer stores 0.1, it's not exactly this number that is stored.

If you have access to a Unix machine, it's easy to see this:

Python 2.5.1 (r251:54863, Apr 15 2008, 22:57:26) 
[GCC 4.0.1 (Apple Inc. build 5465)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> 0.1
0.10000000000000001
>>> 

You'll want to be really careful with equality tests with floats and doubles, in whatever language you are in.

(As for your example, 0.2 is another one of those pesky numbers that cannot be stored in IEEE binary, but as long as you are testing inequalities, rather than equalities, like p <= 0.2, then you'll be okay.)

This is the simplest that comes to my mind, that should work with many languages is simply:

0.2 + 0.1

Here are some examples with the REPLs that come into my mind, but should return this result on any IEEE754-compliant language.

Python

>>> 0.2 + 0.1
0.30000000000000004

Kotlin

0.2 + 0.1
res0: kotlin.Double = 0.30000000000000004

Scala

scala> 0.2 + 0.1
val res0: Double = 0.30000000000000004

Java

jshell> 0.2 + 0.1
$1 ==> 0.30000000000000004

Ruby

irb(main):001:0> 0.2 + 0.1
=> 0.30000000000000004

Here's one that caught me :

 round(256.49999) == 256
roundf(256.49999) == 257

doubles and floats have different precision, so the first will be represented as 256.49999000000003, and the second one as 256.5, and will thus be rounded differently

A simple example in C that caught me a while back :

double d = 0;
sscanf("90.1000", "%lf", &d);
printf("%0.4f", d);

This prints 90.0999

This was in a function that converted angles in DMS to radians.

Why does it not work in the above case?

I like this one from a Python interpreter:

Python 2.7.10 (default, Oct  6 2017, 22:29:07) 
[GCC 4.2.1 Compatible Apple LLVM 9.0.0 (clang-900.0.31)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> 0.1+0.2
0.30000000000000004
>>>

super simple (Python):

a = 10000000.1
b = 1/10
print(a - b == 10000000)
print ('a:{0:.20f}\nb:{1:.20f}'.format(a, b))

prints (depending on the platform) something like:

False                                                                                                                                 
a:10000000.09999999962747097015                                                                                                       
b:0.10000000000000000555 

I think Ruby has a good example in its documentation:

sum = 0
10_000.times do
  sum = sum + 0.0001
end
print sum #=> 0.9999999999999062