Should I Use Double.Epsilon for Floating-Point Comparisons?

Double.Epsilon in Floating-Point Comparisons

Question:

The MSDN documentation for Double.Epsilon states that it should not be used to determine equality between floating-point numbers. However, some resources suggest using it for comparisons. Is it possible to use Double.Epsilon for equality and other comparisons, such as greater than and less than?

Answer:

The Epsilon value represents the smallest representable non-denormal floating point value other than 0. While it may seem like a natural candidate for epsilon in comparisons, it is not precise enough for that purpose.

Double.Epsilon reflects the smallest representable truncation error, but it varies greatly depending on the magnitude of the numbers being compared. For instance, for two numbers close to 1, Double.Epsilon might be 0.0000000000000002220446, while for two numbers close to 1,000,000,000,000,000, it could be 222,044,604,925,093,800. This means that relying on Double.Epsilon for equality or other comparisons would not provide consistent results.

Alternative Epsilon Estimation for Equality:

One recommended approach for comparing floating-point values for equality is to use an epsilon of constant * 1E-15. This approximates that the Double type can represent values accurately to 15 digits. For example:

public static bool AboutEqual(double x, double y) {
    double epsilon = Math.Max(Math.Abs(x), Math.Abs(y)) * 1E-15;
    return Math.Abs(x - y) <= epsilon;
}

Comparison Operators:

While Double.Epsilon cannot be used directly for comparisons, the AboutEqual method can be adapted for greater than, less than, less than or equal to, and greater than or equal to:

• Greater than: x > y && !AboutEqual(x, y)
• Less than: x < y && !AboutEqual(x, y)
• Less than or equal to: AboutEqual(x, y) || x < y
• Greater than or equal to: AboutEqual(x, y) || x > y

Additional Considerations:

It's important to note that truncation errors can accumulate, especially when dealing with computed values. Therefore, you may need to adjust the epsilon value accordingly to achieve appropriate precision.

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