Why is floating point arithmetic in C# imprecise?

Question

Why does the following program print what it prints?

class Program
{
    static void Main(string[] args)
    {
        float f1 = 0.09f*100f;
        float f         


        

Answer 1

Floating point only has so many digits of precision. If you're seeing f1 == f2, it is because any difference requires more precision than a 32-bit float can represent.

I recommend reading What Every Computer Scientist Should Read About Floating Point

Answer 2

In this particular case, it’s because .09 and .999999 cannot be represented with exact precision in binary (similarly, 1/3 cannot be represented with exact precision in decimal). For example, 0.111111111111111111101111 base 2 is 0.999998986721038818359375 base 10. Adding 1 to the previous binary value, 0.11111111111111111111 base 2 is 0.99999904632568359375 base 10. There isn’t a binary value for exactly 0.999999. Floating point precision is also limited by the space allocated for storing the exponent and the fractional part of the mantissa. Also, like integer types, floating point can overflow its range, although its range is larger than integer ranges.

Running this bit of C++ code in the Xcode debugger,

float myFloat = 0.1;

shows that myFloat gets the value 0.100000001. It is off by 0.000000001. Not a lot, but if the computation has several arithmetic operations, the imprecision can be compounded.

imho a very good explanation of floating point is in Chapter 14 of Introduction to Computer Organization with x86-64 Assembly Language & GNU/Linux by Bob Plantz of California State University at Sonoma (retired) http://bob.cs.sonoma.edu/getting_book.html. The following is based on that chapter.

Floating point is like scientific notation, where a value is stored as a mixed number greater than or equal to 1.0 and less than 2.0 (the mantissa), times another number to some power (the exponent). Floating point uses base 2 rather than base 10, but in the simple model Plantz gives, he uses base 10 for clarity’s sake. Imagine a system where two positions of storage are used for the mantissa, one position is used for the sign of the exponent* (0 representing + and 1 representing -), and one position is used for the exponent. Now add 0.93 and 0.91. The answer is 1.8, not 1.84.

9311 represents 0.93, or 9.3 times 10 to the -1.

9111 represents 0.91, or 9.1 times 10 to the -1.

The exact answer is 1.84, or 1.84 times 10 to the 0, which would be 18400 if we had 5 positions, but, having only four positions, the answer is 1800, or 1.8 times 10 to the zero, or 1.8. Of course, floating point data types can use more than four positions of storage, but the number of positions is still limited.

Not only is precision limited by space, but “an exact representation of fractional values in binary is limited to sums of inverse powers of two.” (Plantz, op. cit.).

0.11100110 (binary) = 0.89843750 (decimal)

0.11100111 (binary) = 0.90234375 (decimal)

There is no exact representation of 0.9 decimal in binary. Even carrying the fraction out more places doesn’t work, as you get into repeating 1100 forever on the right.

Beginning programmers often see floating point arithmetic as more accurate than integer. It is true that even adding two very large integers can cause overflow. Multiplication makes it even more likely that the result will be very large and, thus, overflow. And when used with two integers, the / operator in C/C++ causes the fractional part to be lost. However, ... floating point representations have their own set of inaccuracies. (Plantz, op. cit.)

*In floating point, both the sign of the number and the sign of the exponent are represented.

Answer 3

The main thing is that this isn't just .Net: it's a limitation of the underlying system most every language will use to represent a float in memory. The precision only goes so far.

You can also have some fun with relatively simple numbers, when you take into account that it's not even base ten. 0.1, for example, is a repeating decimal when represented in binary.