Where do I find the machine epsilon in C#?

Question

The machine epsilon is canonically defined as the smallest number which added to one, gives a result different from one.

There is a Double.Epsilon but the name is very misleading: it is the smallest (denormalized) Double value representable, and thus useless for any kind of numeric programming.

I'd like to get the true epsilon for the Double type, so that not to have to hardcode tolerances into my program. How do I do this ?

Answer 1

It's(on my machine):

   1.11022302462516E-16

You can easily calculate it:

        double machEps = 1.0d;

        do {
           machEps /= 2.0d;
        }
        while ((double)(1.0 + machEps) != 1.0);

        Console.WriteLine( "Calculated machine epsilon: " + machEps );

Edited:

I calcualted 2 times epsilon, now it should be correct.

Answer 2

The Math.NET library defines a Precision class, which has a DoubleMachineEpsilon property.

You could check how they do it.

According to that it is:

    /// <summary>
    /// The base number for binary values
    /// </summary>
    private const int BinaryBaseNumber = 2;

    /// <summary>
    /// The number of binary digits used to represent the binary number for a double precision floating
    /// point value. i.e. there are this many digits used to represent the
    /// actual number, where in a number as: 0.134556 * 10^5 the digits are 0.134556 and the exponent is 5.
    /// </summary>
    private const int DoublePrecision = 53;

    private static readonly double doubleMachinePrecision = Math.Pow(BinaryBaseNumber, -DoublePrecision);

So it is 1,11022302462516E-16 according to this source.

Answer 3

Just hard-code the value:

const double e1 = 2.2204460492503131e-16;

or use the power of two:

static readonly double e2 = Math.Pow(2, -52);

or use your definition (more or less):

static readonly double e3 = BitConverter.Int64BitsToDouble(BitConverter.DoubleToInt64Bits(1.0) + 1L) - 1.0;

And see Wikipedia: machine epsilon.