Casting a result to float in method returning float changes result

Question

Why does this code print False in .NET 4? It seems some unexpected behavior is being caused by the explicit cast.

I'd like an answer beyond "floating point is inaccurate" or "don't do that".

float a(float x, float y)
{
  return ( x * y );
}

float b(float x, float y)
{
  return (float)( x * y );
}

void Main()
{
  Console.WriteLine( a( 10f, 1f/10f ) == b( 10f, 1f/10f ) );
}

PS: This code came from a unit test, not release code. The code was written this way deliberately. I suspected it would fail eventually but I wanted to know exactly when and exactly why. The answer proves the validity of this technique because it provides an understanding that goes beyond the usual understanding of floating point determinism. And that was the point of writing this code this way; deliberate exploration.

PPS: The unit test was passing in .NET 3.5, but now fails after the upgrade to .NET 4.

Answer 1

David's comment is correct but insufficiently strong. There is no guarantee that doing that calculation twice in the same program will produce the same results.

The C# specification is extremely clear on this point:

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Floating-point operations may be performed with higher precision than the result type of the operation. For example, some hardware architectures support an “extended” or “long double” floating-point type with greater range and precision than the double type, and implicitly perform all floating-point operations using this higher precision type. Only at excessive cost in performance can such hardware architectures be made to perform floating-point operations with less precision, and rather than require an implementation to forfeit both performance and precision, C# allows a higher precision type to be used for all floating-point operations. Other than delivering more precise results, this rarely has any measurable effects. However, in expressions of the form x * y / z, where the multiplication produces a result that is outside the double range, but the subsequent division brings the temporary result back into the double range, the fact that the expression is evaluated in a higher range format may cause a finite result to be produced instead of an infinity.

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The C# compiler, the jitter and the runtime all have broad lattitude to give you more accurate results than are required by the specification, at any time, at a whim -- they are not required to choose to do so consistently and in fact they do not.

If you don't like that then do not use binary floating point numbers; either use decimals or arbitrary precision rationals.

I don't understand why casting to float in a method that returns float makes the difference it does

Excellent point.

Your sample program demonstrates how small changes can cause large effects. You note that in some version of the runtime, casting to float explicitly gives a different result than not doing so. When you explicitly cast to float, the C# compiler gives a hint to the runtime to say "take this thing out of extra high precision mode if you happen to be using this optimization". As the specification notes, this has a potential performance cost.

That doing so happens to round to the "right answer" is merely a happy accident; the right answer is obtained because in this case losing precision happened to lose it in the correct direction.

How is .net 4 different?

You ask what the difference is between 3.5 and 4.0 runtimes; the difference is clearly that in 4.0, the jitter chooses to go to higher precision in your particular case, and the 3.5 jitter chooses not to. That does not mean that this situation was impossible in 3.5; it has been possible in every version of the runtime and every version of the C# compiler. You've just happened to run across a case where, on your machine, they differ in their details. But the jitter has always been allowed to make this optimization, and always has done so at its whim.

The C# compiler is also completely within its rights to choose to make similar optimizations when computing constant floats at compile time. Two seemingly-identical calculations in constants may have different results depending upon details of the compiler's runtime state.

More generally, your expectation that floating point numbers should have the algebraic properties of real numbers is completely out of line with reality; they do not have those algebraic properties. Floating point operations are not even associative; they certainly do not obey the laws of multiplicative inverses as you seem to expect them to. Floating point numbers are only an approximation of real arithmetic; an approximation that is close enough for, say, simulating a physical system, or computing summary statistics, or some such thing.