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

Question

I've heard of "error" when using floating point variables. Now I'm trying to solve this puzzle and I think I'm getting some rounding/floating point error. So I'm finally going to figure out the basics of floating point error.

What is a simple example of floating point/rounding error (preferably in C++) ?

Edit: For example say I have an event that has probability p of succeeding. I do this event 10 times (p does not change and all trials are independent). What is the probability of exactly 2 successful trials? I have this coded as:

double p_2x_success = pow(1-p, (double)8) * pow(p, (double)2) * (double)choose(8, 2);

Is this an opportunity for floating point error?

Answer 1

I think what you really need is this: What Every Computer Scientist Should Know About Floating-Point Arithmetic.

Answer 2

Questions 29.16 and 29.17 in C++ FAQ Lite might help.

Answer 3

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.)

Answer 4

A simple C one that caught me a while back,

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

>> 90.0999

This was in a function that converted angles in DMS to radians, which didn't in the above case.

Answer 5

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