Demystify C# floating-point equality and relation operations

Introduction

My introduction is as often: why to bother at all? The C# double value type is known to everybody.

Do we know that type really? What is the result of this?

double a = 1.0 / 6.0;
Console.WriteLine(a + a + a + a + a + a == 1.0);

Answer: False (adding six sixth does not result in 1.0)!

Or what do we get for this?

double a 1E300 / 1E-100;
Console.WriteLine(a - a == 0.0);

Answer: False (a is +∞, and ∞ - ∞ is NaN, which is not 0.0)!

Or finally, what do we get with the following:

Random r = new Random();
foreach (var n in Enumerable.Range(0, 10))
{
    double sum = 0.0;
    foreach (int count in Enumerable.Range(0, 100))
    {
        sum += Math.Round(r.NextDouble() / 50.0, 2);
    }
    Console.WriteLine(sum);
}

Answer: you get random sums, of course, but not always to two decimal digits only (rounding is not "exact", since the results is still a double value...)!

This article details on how to deal with issues in comparing double values. The following items are discussed:

• double type at a glance
• double Equality
• Quirks of double Equality
• Make double Equality tolerant
• double Relation operations
• Quirks of double Relations
• Make double Relations tolerant
• Summary

double type at a glance

The C# double type is defined according to IEEE-754 specification. That means that double:

• is a floating point type
• has a range from about -10³⁰⁸ to 10³⁰⁸
• has a precision of about 15 decimal digits
• has a smallest number (closest to 0.0) of about +/- 10^-308
• has two zero values: +/- 0.0
• has two infinty values: +/- ∞
• has a NaN "value" (Not a Number)
• has the commonly known arithmetic operations
• has the commonly known equality and relation operations
• has Math library support for further functions like absolute value, rounding, etc.

double Equality

As we've seen in the intro and in the overview, double has some special values:

• - 0.0
• + 0.0
• - ∞
• + ∞
• NaN

Let's look at the equality table for all these values (plus 1.0 representing a "normal" double value):

Equality (x==y)	y=1.0	y=-0.0	y=+0.0	y=-∞	y=+∞	y=NaN
x=1.0	True	False	False	False	False	False
x=-0.0	False	True	True	False	False	False
x=+0.0	False	True	True	False	False	False
x=-∞	False	False	False	True	False	False
x=+∞	False	False	False	False	True	False
x=NaN	False	False	False	False	False	False

Inequality (x!=y)	y=1.0	y=-0.0	y=+0.0	y=-∞	y=+∞	y=NaN
x=1.0	False	True	True	True	True	True
x=-0.0	True	False	False	True	True	True
x=+0.0	True	False	False	True	True	True
x=-∞	True	True	True	False	True	True
x=+∞	True	True	True	True	False	True
x=NaN	True	True	True	True	True	True

Notes:

• The surprising equality is given for NaN: two NaN values are not identical!
• +0.0 and -0.0 are identical - luckily

Bonus

What to do if you want to know if a value is NaN?
The equality operator returns always False for a == double.NaN...

Answer: use the specific function double.IsNaN(a). Likewise, it's prudent to check for infinity by the specific fucntions double.IsPositiveInfinity(a) and double.IsNegativeInfinity(a) respectively.

Quirks of double Equality

The nature of floating point arithmetic is that they do implicit rounding. Not all values can be stored with unlimited precision, leaving an approximation of these values only. So, this approximation may result in a tiny "rounding error".

Adding up values with rounding errors may result in even larger errors.

double a = 1.0/6.0; // has a small rounding error
Console.WriteLine("equal = {0}", a+a+a+a+a+a == 1.0); // False --> *not* equal
Console.WriteLine("delta = {0}", a+a+a+a+a+a - 1.0);  // delta = -1.11022302462516E-16 --> *not* 0.0

This means, you cannot trust the equality operators == and != on double values. As a conclusion, we need an equality comparison that allows for "tiny" delta.

Note: As shown in the introduction, rounding does not help in this. A reliable comparison is needed.

Make double Equality tolerant

To gain trust into the equality operator again, we must provide an equality function that allows for tolerant comparison. For that, we must make sure that the equality/inequality table is met.

A possible solution:

public static class RealExtensions
{
    public struct Epsilon
    {
        public Epsilon(double value) { _value = value; }
        private double _value;
        internal bool IsEqual   (double a, double b) { return (a == b) ||  (Math.Abs(a - b) < _value); }
        internal bool IsNotEqual(double a, double b) { return (a != b) && !(Math.Abs(a - b) < _value); }
    }
    public static bool EQ(this double a, double b, Epsilon e) { return e.IsEqual   (a, b); }
    public static bool NE(this double a, double b, Epsilon e) { return e.IsNotEqual(a, b); }
}

Usage:

var epsilon = new RealExtension.Epsilon(1E-3);
...
double x = 1.0 / 6.0;
double y = x+x+x+x+x+x;
if (y.EQ(1.0, epsilon)) ...

Notes:

• These equality/inequality functions work also for the special values.
• Choose the Epsilon value carefully: e.g. angle epsilon are usually very different to length epsilons.
• Epsilon values can be expected to be in the range 1E-1 to 1E-15 (a meaningful limit is 1E-15 since the double prercision is about 15 decimal digits).
• The double.Epsilon is not useful for epsilon - it is the smallest possible only.

That's all! Well, not yet...

What about the relation operators: <, <=, >=, >. They also involve equality and inequality operations.

double Relation operations

First, let's list again the tables of the operations:

Less-than (x<y)	y=1.0	y=-0.0	y=+0.0	y=-∞	y=+∞	y=NaN
x=1.0	False	False	False	False	True	False
x=-0.0	True	False	False	False	True	False
x=+0.0	True	False	False	False	True	False
x=-∞	True	True	True	False	True	False
x=+∞	False	False	False	False	False	False
x=NaN	False	False	False	False	False	False

Less-equal (x<=y)	y=1.0	y=-0.0	y=+0.0	y=-∞	y=+∞	y=NaN
x=1.0	True	False	False	False	True	False
x=-0.0	True	True	True	False	True	False
x=+0.0	True	True	True	False	True	False
x=-∞	True	True	True	True	True	False
x=+∞	False	False	False	False	True	False
x=NaN	False	False	False	False	False	False

Greater-equal (x>=y)	y=1.0	y=-0.0	y=+0.0	y=-∞	y=+∞	y=NaN
x=1.0	True	True	True	True	False	False
x=-0.0	False	True	True	True	False	False
x=+0.0	False	True	True	True	False	False
x=-∞	False	False	False	True	False	False
x=+∞	True	True	True	True	True	False
x=NaN	False	False	False	False	False	False

Greater-than (x>y)	y=1.0	y=-0.0	y=+0.0	y=-∞	y=+∞	y=NaN
x=1.0	False	True	True	True	False	False
x=-0.0	False	False	False	True	False	False
x=+0.0	False	False	False	True	False	False
x=-∞	False	False	False	False	False	False
x=+∞	True	True	True	True	False	False
x=NaN	False	False	False	False	False	False

Notes:

• double.NaN results always in False (i.e. a < b is not the opposite of a >= b).
• +0.0 and -0.0 are identical again.

What is the issue with the equality/inequality functions is similar with the relations. Let's look into that in the following section.

Quirks of double Relations

Since one cannot trust the native equality/inequality operators, the relation operators cannot be trusted neither.

double a = 1.0 / 6.0;
double b = a + a + a + a + a + a;
...
if (1.0 < b) ... // delta = +1.11022302462516E-16 --> False (delta is close to zero but not exactly)
...
if (b < 1.0) ... // delta = -1.11022302462516E-16 --> True (delta is close to zero but not exactly)

Likewise for all relation operators.

This calls for specific epsilon based relation functions as above for equality/inequality functions.

Make double Relations tolerant

Let's just start with a possible solution, based on the EQ/NE from above.

public static class RealExtensions
{
    public struct Epsilon
    {
        public Epsilon(double value) { _value = value; }
        private double _value;
        internal bool IsEqual   (double a, double b) { return (a == b) ||  (Math.Abs(a - b) < _value); }
        internal bool IsNotEqual(double a, double b) { return (a != b) && !(Math.Abs(a - b) < _value); }
    }
    public static bool EQ(this double a, double b, Epsilon e) { return e.IsEqual   (a, b); }
    public static bool LE(this double a, double b, Epsilon e) { return e.IsEqual   (a, b) || (a < b); }
    public static bool GE(this double a, double b, Epsilon e) { return e.IsEqual   (a, b) || (a > b); }

    public static bool NE(this double a, double b, Epsilon e) { return e.IsNotEqual(a, b); }
    public static bool LT(this double a, double b, Epsilon e) { return e.IsNotEqual(a, b) && (a < b); }
    public static bool GT(this double a, double b, Epsilon e) { return e.IsNotEqual(a, b) && (a > b); }
}

Usage:

var epsilon = new RealExtension.Epsilon(1E-3);
...
double x = 1.0 / 6.0;
double y = x+x+x+x+x+x;
if (y.LT(1.0, epsilon)) ...

Notes:

• These relation functions work also for the special values.

*************** Now, that's all! Finally. *******************

Summary

Native equality and relation operations on floating point values cannot be trusted. A possible solution is to invent the six operations based on some epsilon value. Special care must be taken for the special values like Infinity and NaN.

What else?

The float value type suffers the same problem. I.e. that type needs that same solution as for double, adapted to float.

The decimal has a bit a different situation: it is not considered as a C# floating-point type but has similar issues with implicit rounding errors. I.e. the same solution (adapted to decimal) will solve the problem here too.

Some useful links:

• Wikipedia: IEEE-754[^]
• IEEE 754 Overview by Steve Hollasch[^]
• C# double type (C# reference)[^]
• Floating-Point Types Table (C# Reference)[^]
• decimal (C# Reference)[^]
• Double.Equals Method (Double)[^]
• Double.CompareTo Method (Object)[^]

Revisions

Version	Date	Description
V1.0	2012-05-13	Initial version
V1.1	2012-05-14	Fix formatting
V1.2	2012-05-14	Added double.IsNaN(x), double.IsPositiveInfinity(x), and double.IsNegativeInfinity(x)