floating point inaccuracies in c - c

Question

I know floating point values are limited in the numbers the can express accurately and i have found many sites that describe why this happens. But i have not found any information of how to deal with this problem efficiently. But I'm sure NASA isn't OK with 0.2/0.1 = 0.199999. Example:
#include <stdio.h>
#include <float.h>
#include <math.h>
int main(void)
{
float number = 4.20;
float denominator = 0.25;
printf("number = %f\n", number);
printf("denominator = %f\n", denominator);
printf("quotient as a float = %f should be 16.8\n", number/denominator);
printf("the remainder of 4.20 / 0.25 = %f\n", number - ((int) number/denominator)*denominator);
printf("now if i divide 0.20000 by 0.1 i get %f not 2\n", ( number - ((int) number/denominator)*denominator)/0.1);
}
output:
number = 4.200000
denominator = 0.250000
quotient as a float = 16.799999 should be 16.8
the remainder of 4.20 / 0.25 = 0.200000
now if i divide 0.20000 by 0.1 i get 1.999998 not 2
So how do i do arithmetic with floats (or decimals or doubles) and get accurate results. Hope i haven't just missed something super obvious. Any help would be awesome! Thanks.

Answer 1

The solution is to not use floats for applications where you can't accept roundoff errors. Use an extended precision library (a.k.a. arbitrary precision library) like GNU MP Bignum. See this Wikipedia page for a nice list of arbitrary-precision libraries. See also the Wikipedia article on rational data types and this thread for more info.
If you are going to use floating point representations (float, double, etc.) then write code using accepted methods for dealing with roundoff errors (e.g., avoiding ==). There's lots of on-line literature about how to do this and the methods vary widely depending on the application and algorithms involved.

Answer 2

Floating point is pretty fine, most of the time. Here are the key things I try to keep in mind:
There's really a big difference between float and double. double gives you enough precision for most things, most of the time; float surprisingly often gives you not enough. Unless you know what you're doing and have a really good reason, just always use double.
There are some things that floating point is not good for. Although C doesn't support it natively, fixed point is often a good alternative. You're essentially using fixed point if you do your financial calculations in cents rather than dollars -- that is, if you use an int or a long int representing pennies, and remember to put a decimal point two places from the right when it's time to print out as dollars.
The algorithm you use can really matter. Naïve or "obvious" algorithms can easily end up magnifying the effects of roundoff error, while more sophisticated algorithms minimize them. One simple example is that the order you add up floating-point numbers can matter.
Never worry about 16.8 versus 16.799999. That sort of thing always happens, but it's not a problem, unless you make it a problem. If you want one place past the decimal, just print it using %.1f, and printf will round it for you. (Also don't try to compare floating-point numbers for exact equality, but I assume you've heard that by now.)
Related to the above, remember that 0.1 is not representable exactly in binary (just as 1/3 is not representable exactly in decimal). This is just one of many reasons that you'll always get what look like tiny roundoff "errors", even though they're perfectly normal and needn't cause problems.
Occasionally you need a multiple precision (MP or "bignum") library, which can represent numbers to arbitrary precision, but these are (relatively) slow and (relatively) cumbersome to use, and fortunately you usually don't need them. But it's good to know they exist, and if you're a math nurd they can be a lot of fun to use.
Occasionally a library for representing rational numbers is useful. Such a library represents, for example, the number 1/3 as the pair of numbers (1, 3), so it doesn't have the inaccuracies inherent in trying to represent that number as 0.333333333.
Others have recommended the paper What Every Computer Scientist Should Know About Floating-Point Arithmetic, which is very good, and the standard reference, although it's long and fairly technical. An easier and shorter read I can recommend is this handout from a class I used to teach: https://www.eskimo.com/~scs/cclass/handouts/sciprog.html#precision . This is a little dated by now, but it should get you started on the basics.

Answer 3

There's isn't a good answer and it's often a problem.
If data is integral, e.g. amounts of money in cents, then store it as integers, which can mean a double that is constrained to hold an integer number of cents rather than a rational number of dollars. But that only helps in a few circumstances.
As a general rule, you get inaccuracies when trying to divide by numbers that are close to zero. So you just have to write the algorithms to avoid or suppress such operations. There are lots of discussions of "numerically stable" versus "unstable" algorithms and it's too big a subject to do justice to it here. And then, usually, it's best to treat floating point numbers as though they have small random errors. If they ultimately represent measurements of analogue values in the real world, there must be a certain tolerance or inaccuracy in them anyway.
If you are doing maths rather than processing data, simply don't use C or C++. Use a symbolic algebra package such a Maple, which stores values such as sqrt(2) as an expression rather than a floating point number, so sqrt(2) * sqrt(2) will always give exactly 2, rather than a number very close to 2.