Is Double.Epsilon the Right Tool for Comparing Floating-Point Numbers?

Double.Epsilon: A Reliable Precision Measure for Numerical Equality and Inequality

Introduction

The concept of "epsilon" plays a crucial role in numerical analysis when comparing floating-point numbers. Double.Epsilon, a constant defined in the System.Double class, represents the smallest representable positive floating-point value that is greater than zero.

Can Double.Epsilon be Used for Direct Comparisons?

Contrary to the statement in the MSDN documentation, Double.Epsilon cannot be directly used to determine the equality of two floating-point numbers. While it can serve as a threshold value for acceptable differences in custom algorithms, it does not provide an accurate measure of numerical equivalence.

Determining Floating-Point Equality

For practical purposes, an alternative approach is recommended. To compare two double values, x and y, for equality, an epsilon value of approximately constant * 1E-15 is typically used. This epsilon accounts for the inherent accuracy limitations of the Double data type, allowing for up to 15 significant digits of precision.

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;
}

Addressing Truncation Errors

However, it is important to consider the potential for truncation errors when working with computed values. In such cases, the epsilon value may need to be increased to accommodate the possibility of accumulated error.

Implementing Inequality Comparisons

For inequality comparisons (greater than, less than, less than or equal to, greater than or equal to), similar principles apply. The same epsilon value can be used with caution, ensuring that it is sufficiently large to account for potential error.

// Greater than
bool gt = x > (y + epsilon);

// Less than
bool lt = x < (y - epsilon);

// Less than or equal to
bool lte = x <= (y + epsilon);

// Greater than or equal to
bool gte = x >= (y - epsilon);

Conclusion

While Double.Epsilon provides a point of reference for floating-point precision, it should not be relied upon for direct equality comparisons. By adopting an epsilon-based approach with appropriate adjustment for truncation errors, developers can ensure reliable and accurate numerical comparisons when utilizing doubles.

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