Floating point comparison functions for C#

Question

Can someone point towards (or show) some good general floating point comparison functions in C# for comparing floating point values? I want to implement functions for IsEqual, IsGreater an IsLess. I also only really care about doubles not floats.

Answer 1

Writing a useful general-purpose floating point IsEqual is very, very hard, if not outright impossible. Your current code will fail badly for a==0. How the method should behave for such cases is really a matter of definition, and arguably the code would best be tailored for the specific domain use case.

For this kind of thing, you really, really need a good test suite. That's how I did it for The Floating-Point Guide, this is what I came up with in the end (Java code, should be easy enough to translate):

public static boolean nearlyEqual(float a, float b, float epsilon) {
    final float absA = Math.abs(a);
    final float absB = Math.abs(b);
    final float diff = Math.abs(a - b);

    if (a == b) { // shortcut, handles infinities
        return true;
    } else if (a == 0 || b == 0 || absA + absB < Float.MIN_NORMAL) {
        // a or b is zero or both are extremely close to it
        // relative error is less meaningful here
        return diff < (epsilon * Float.MIN_NORMAL);
    } else { // use relative error
        return diff / (absA + absB) < epsilon;
    }
}

You can also find the test suite on the site.

Appendix: Same code in c# for doubles (as asked in questions)

public static bool NearlyEqual(double a, double b, double epsilon)
{
    const double MinNormal = 2.2250738585072014E-308d;
    double absA = Math.Abs(a);
    double absB = Math.Abs(b);
    double diff = Math.Abs(a - b);

    if (a.Equals(b))
    { // shortcut, handles infinities
        return true;
    } 
    else if (a == 0 || b == 0 || absA + absB < MinNormal) 
    {
        // a or b is zero or both are extremely close to it
        // relative error is less meaningful here
        return diff < (epsilon * MinNormal);
    }
    else
    { // use relative error
        return diff / (absA + absB) < epsilon;
    }
}

Answer 2

From Bruce Dawson's paper on comparing floats, you can also compare floats as integers. Closeness is determined by least significant bits.

public static bool AlmostEqual2sComplement( float a, float b, int maxDeltaBits ) 
{
    int aInt = BitConverter.ToInt32( BitConverter.GetBytes( a ), 0 );
    if ( aInt <  0 )
        aInt = Int32.MinValue - aInt;  // Int32.MinValue = 0x80000000

    int bInt = BitConverter.ToInt32( BitConverter.GetBytes( b ), 0 );
    if ( bInt < 0 )
        bInt = Int32.MinValue - bInt;

    int intDiff = Math.Abs( aInt - bInt );
    return intDiff <= ( 1 << maxDeltaBits );
}

EDIT: BitConverter is relatively slow. If you're willing to use unsafe code, then here is a very fast version:

    public static unsafe int FloatToInt32Bits( float f )
    {
        return *( (int*)&f );
    }

    public static bool AlmostEqual2sComplement( float a, float b, int maxDeltaBits )
    {
        int aInt = FloatToInt32Bits( a );
        if ( aInt < 0 )
            aInt = Int32.MinValue - aInt;

        int bInt = FloatToInt32Bits( b );
        if ( bInt < 0 )
            bInt = Int32.MinValue - bInt;

        int intDiff = Math.Abs( aInt - bInt );
        return intDiff <= ( 1 << maxDeltaBits );
    }

Answer 3

Be careful with some answers...

UPDATE 2019-0829, I also included Microsoft decompiled code which should be far better than mine.

1 - You could easily represent any number with 15 significatives digits in memory with a double. See Wikipedia.

2 - The problem come from calculation of floating numbers where you could loose some precision. I mean that a number like .1 could become something like .1000000000000001 ==> after calculation. When you do some calculation, results could be truncated in order to be represented in a double. That truncation brings the error you could get.

3 - To prevent the problem when comparing double values, people introduce an error margin often called epsilon. If 2 floating numbers only have a contextual epsilon as difference, then they are considered equals. double.Epsilon is the smallest number between a double value and its neigbor (next or previous) value.

4 - The difference betwen 2 double values could be more than double.epsilon. The difference between the real double value and the one computed depends on how many calculation you have done and which ones. Many peoples think that it is always double.Epsilon but they are really wrong. To have a great answer please see: Hans Passant answer. The epsilon is based on your context where it depends on the biggest number you reach during your calculation and on the number of calculation you are doing (truncation error accumulate).

5 - This is the code that I use. Be careful that I use my epsilon only for few calculations. Otherwise I multiply my epsilon by 10 or 100.

6 - As noted by SvenL, it is possible that my epsilon is not big enough. I suggest to read SvenL comment. Also, perhaps "decimal" could do the job for your case?

Microsoft decompiled code:

// Decompiled with JetBrains decompiler
// Type: MS.Internal.DoubleUtil
// Assembly: WindowsBase, Version=4.0.0.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35
// MVID: 33C590FB-77D1-4FFD-B11B-3D104CA038E5
// Assembly location: C:\Windows\Microsoft.NET\assembly\GAC_MSIL\WindowsBase\v4.0_4.0.0.0__31bf3856ad364e35\WindowsBase.dll

using MS.Internal.WindowsBase;
using System;
using System.Runtime.InteropServices;
using System.Windows;

namespace MS.Internal
{
  [FriendAccessAllowed]
  internal static class DoubleUtil
  {
    internal const double DBL_EPSILON = 2.22044604925031E-16;
    internal const float FLT_MIN = 1.175494E-38f;

    public static bool AreClose(double value1, double value2)
    {
      if (value1 == value2)
        return true;
      double num1 = (Math.Abs(value1) + Math.Abs(value2) + 10.0) * 2.22044604925031E-16;
      double num2 = value1 - value2;
      if (-num1 < num2)
        return num1 > num2;
      return false;
    }

    public static bool LessThan(double value1, double value2)
    {
      if (value1 < value2)
        return !DoubleUtil.AreClose(value1, value2);
      return false;
    }

    public static bool GreaterThan(double value1, double value2)
    {
      if (value1 > value2)
        return !DoubleUtil.AreClose(value1, value2);
      return false;
    }

    public static bool LessThanOrClose(double value1, double value2)
    {
      if (value1 >= value2)
        return DoubleUtil.AreClose(value1, value2);
      return true;
    }

    public static bool GreaterThanOrClose(double value1, double value2)
    {
      if (value1 <= value2)
        return DoubleUtil.AreClose(value1, value2);
      return true;
    }

    public static bool IsOne(double value)
    {
      return Math.Abs(value - 1.0) < 2.22044604925031E-15;
    }

    public static bool IsZero(double value)
    {
      return Math.Abs(value) < 2.22044604925031E-15;
    }

    public static bool AreClose(Point point1, Point point2)
    {
      if (DoubleUtil.AreClose(point1.X, point2.X))
        return DoubleUtil.AreClose(point1.Y, point2.Y);
      return false;
    }

    public static bool AreClose(Size size1, Size size2)
    {
      if (DoubleUtil.AreClose(size1.Width, size2.Width))
        return DoubleUtil.AreClose(size1.Height, size2.Height);
      return false;
    }

    public static bool AreClose(Vector vector1, Vector vector2)
    {
      if (DoubleUtil.AreClose(vector1.X, vector2.X))
        return DoubleUtil.AreClose(vector1.Y, vector2.Y);
      return false;
    }

    public static bool AreClose(Rect rect1, Rect rect2)
    {
      if (rect1.IsEmpty)
        return rect2.IsEmpty;
      if (!rect2.IsEmpty && DoubleUtil.AreClose(rect1.X, rect2.X) && (DoubleUtil.AreClose(rect1.Y, rect2.Y) && DoubleUtil.AreClose(rect1.Height, rect2.Height)))
        return DoubleUtil.AreClose(rect1.Width, rect2.Width);
      return false;
    }

    public static bool IsBetweenZeroAndOne(double val)
    {
      if (DoubleUtil.GreaterThanOrClose(val, 0.0))
        return DoubleUtil.LessThanOrClose(val, 1.0);
      return false;
    }

    public static int DoubleToInt(double val)
    {
      if (0.0 >= val)
        return (int) (val - 0.5);
      return (int) (val + 0.5);
    }

    public static bool RectHasNaN(Rect r)
    {
      return DoubleUtil.IsNaN(r.X) || DoubleUtil.IsNaN(r.Y) || (DoubleUtil.IsNaN(r.Height) || DoubleUtil.IsNaN(r.Width));
    }

    public static bool IsNaN(double value)
    {
      DoubleUtil.NanUnion nanUnion = new DoubleUtil.NanUnion();
      nanUnion.DoubleValue = value;
      ulong num1 = nanUnion.UintValue & 18442240474082181120UL;
      ulong num2 = nanUnion.UintValue & 4503599627370495UL;
      if (num1 == 9218868437227405312UL || num1 == 18442240474082181120UL)
        return num2 > 0UL;
      return false;
    }

    [StructLayout(LayoutKind.Explicit)]
    private struct NanUnion
    {
      [FieldOffset(0)]
      internal double DoubleValue;
      [FieldOffset(0)]
      internal ulong UintValue;
    }
  }
}

My code:

public static class DoubleExtension
    {
        // ******************************************************************
        // Base on Hans Passant Answer on:
        // https://stackoverflow.com/questions/2411392/double-epsilon-for-equality-greater-than-less-than-less-than-or-equal-to-gre

        /// <summary>
        /// Compare two double taking in account the double precision potential error.
        /// Take care: truncation errors accumulate on calculation. More you do, more you should increase the epsilon.
        public static bool AboutEquals(this double value1, double value2)
        {
            double epsilon = Math.Max(Math.Abs(value1), Math.Abs(value2)) * 1E-15;
            return Math.Abs(value1 - value2) <= epsilon;
        }

        // ******************************************************************
        // Base on Hans Passant Answer on:
        // https://stackoverflow.com/questions/2411392/double-epsilon-for-equality-greater-than-less-than-less-than-or-equal-to-gre

        /// <summary>
        /// Compare two double taking in account the double precision potential error.
        /// Take care: truncation errors accumulate on calculation. More you do, more you should increase the epsilon.
        /// You get really better performance when you can determine the contextual epsilon first.
        /// </summary>
        /// <param name="value1"></param>
        /// <param name="value2"></param>
        /// <param name="precalculatedContextualEpsilon"></param>
        /// <returns></returns>
        public static bool AboutEquals(this double value1, double value2, double precalculatedContextualEpsilon)
        {
            return Math.Abs(value1 - value2) <= precalculatedContextualEpsilon;
        }

        // ******************************************************************
        public static double GetContextualEpsilon(this double biggestPossibleContextualValue)
        {
            return biggestPossibleContextualValue * 1E-15;
        }

        // ******************************************************************
        /// <summary>
        /// Mathlab equivalent
        /// </summary>
        /// <param name="dividend"></param>
        /// <param name="divisor"></param>
        /// <returns></returns>
        public static double Mod(this double dividend, double divisor)
        {
            return dividend - System.Math.Floor(dividend / divisor) * divisor;
        }

        // ******************************************************************
    }

Answer 4

Further to Andrew Wang's answer: if the BitConverter method is too slow but you cannot use unsafe code in your project, this struct is ~6x quicker than BitConverter:

[StructLayout(LayoutKind.Explicit)]
public struct FloatToIntSafeBitConverter
{
    public static int Convert(float value)
    {
        return new FloatToIntSafeBitConverter(value).IntValue;
    }

    public FloatToIntSafeBitConverter(float floatValue): this()
    {
        FloatValue = floatValue;
    }

    [FieldOffset(0)]
    public readonly int IntValue;

    [FieldOffset(0)]
    public readonly float FloatValue;
}

(Incidentally, I tried using the accepted solution but it (well my conversion at least) failed some of the unit tests also mentioned in the answer. e.g. assertTrue(nearlyEqual(Float.MIN_VALUE, -Float.MIN_VALUE)); )

Answer 5

Continuing from the answers provided by Michael and testing, an important thing to keep in mind when translating the original Java code to C# is that Java and C# define their constants differently. C#, for instance, lacks Java's MIN_NORMAL, and the definitions for MinValue differ greatly.

Java defines MIN_VALUE to be the smallest possible positive value, while C# defines it as the smallest possible representable value overall. The equivalent value in C# is Epsilon.

The lack of MIN_NORMAL is problematic for direct translation of the original algorithm - without it, things start to break down for small values near zero. Java's MIN_NORMAL follows the IEEE specification of the smallest possible number without having the leading bit of the significand as zero, and with that in mind, we can define our own normals for both singles and doubles (which dbc mentioned in the comments to the original answer).

The following C# code for singles passes all of the tests given on The Floating Point Guide, and the double edition passes all of the tests with minor modifications in the test cases to account for the increased precision.

public static bool ApproximatelyEqualEpsilon(float a, float b, float epsilon)
{
    const float floatNormal = (1 << 23) * float.Epsilon;
    float absA = Math.Abs(a);
    float absB = Math.Abs(b);
    float diff = Math.Abs(a - b);

    if (a == b)
    {
        // Shortcut, handles infinities
        return true;
    }

    if (a == 0.0f || b == 0.0f || diff < floatNormal)
    {    
        // a or b is zero, or both are extremely close to it.
        // relative error is less meaningful here
        return diff < (epsilon * floatNormal);
    }

    // use relative error
    return diff / Math.Min((absA + absB), float.MaxValue) < epsilon;
}

The version for doubles is identical save for type changes and that the normal is defined like this instead.

const double doubleNormal = (1L << 52) * double.Epsilon;

Answer 6

For people coming here for UNITY specific

There is Mathf.Approximately so writing

if(Mathf.Approximately(a, b))

basically equals writing

if(Mathf.Abs(a - b) <= Mathf.Epsilon)

where Mathf.Epsilon

The smallest value that a float can have different from zero.

under the hood this further means (from the source code)

  // A tiny floating point value (RO).
  public static readonly float Epsilon =
      UnityEngineInternal.MathfInternal.IsFlushToZeroEnabled ? UnityEngineInternal.MathfInternal.FloatMinNormal
      : UnityEngineInternal.MathfInternal.FloatMinDenormal;
 
...
 
  [Unity.IL2CPP.CompilerServices.Il2CppEagerStaticClassConstruction]
public partial struct MathfInternal
{
    public static volatile float FloatMinNormal = 1.17549435E-38f;
    public static volatile float FloatMinDenormal = Single.Epsilon;

    public static bool IsFlushToZeroEnabled = (FloatMinDenormal == 0);
}

Answer 7

Here's how I solved it, with nullable double extension method.

    public static bool NearlyEquals(this double? value1, double? value2, double unimportantDifference = 0.0001)
    {
        if (value1 != value2)
        {
            if(value1 == null || value2 == null)
                return false;

            return Math.Abs(value1.Value - value2.Value) < unimportantDifference;
        }

        return true;
    }

...

        double? value1 = 100;
        value1.NearlyEquals(100.01); // will return false
        value1.NearlyEquals(100.000001); // will return true
        value1.NearlyEquals(100.01, 0.1); // will return true

Answer 8

Here is a much-expanded version of Simon Hewitt's class:

/// <summary>
/// Safely converts a <see cref="float"/> to an <see cref="int"/> for floating-point comparisons.
/// </summary>
[StructLayout(LayoutKind.Explicit)]
public struct FloatToInt : IEquatable<FloatToInt>, IEquatable<float>, IEquatable<int>, IComparable<FloatToInt>, IComparable<float>, IComparable<int>
{
    /// <summary>
    /// Initializes a new instance of the <see cref="FloatToInt"/> class.
    /// </summary>
    /// <param name="floatValue">The <see cref="float"/> value to be converted to an <see cref="int"/>.</param>
    public FloatToInt(float floatValue)
        : this()
    {
        FloatValue = floatValue;
    }

    /// <summary>
    /// Gets the floating-point value as an integer.
    /// </summary>
    [FieldOffset(0)]
    public readonly int IntValue;

    /// <summary>
    /// Gets the floating-point value.
    /// </summary>
    [FieldOffset(0)]
    public readonly float FloatValue;

    /// <summary>
    /// Indicates whether the current object is equal to another object of the same type.
    /// </summary>
    /// <returns>
    /// true if the current object is equal to the <paramref name="other"/> parameter; otherwise, false.
    /// </returns>
    /// <param name="other">An object to compare with this object.</param>
    public bool Equals(FloatToInt other)
    {
        return other.IntValue == IntValue;
    }

    /// <summary>
    /// Indicates whether the current object is equal to another object of the same type.
    /// </summary>
    /// <returns>
    /// true if the current object is equal to the <paramref name="other"/> parameter; otherwise, false.
    /// </returns>
    /// <param name="other">An object to compare with this object.</param>
    public bool Equals(float other)
    {
        return IntValue == new FloatToInt(other).IntValue;
    }

    /// <summary>
    /// Indicates whether the current object is equal to another object of the same type.
    /// </summary>
    /// <returns>
    /// true if the current object is equal to the <paramref name="other"/> parameter; otherwise, false.
    /// </returns>
    /// <param name="other">An object to compare with this object.</param>
    public bool Equals(int other)
    {
        return IntValue == other;
    }

    /// <summary>
    /// Compares the current object with another object of the same type.
    /// </summary>
    /// <returns>
    /// A value that indicates the relative order of the objects being compared. The return value has the following meanings: Value Meaning Less than zero This object is less than the <paramref name="other"/> parameter.Zero This object is equal to <paramref name="other"/>. Greater than zero This object is greater than <paramref name="other"/>. 
    /// </returns>
    /// <param name="other">An object to compare with this object.</param>
    public int CompareTo(FloatToInt other)
    {
        return IntValue.CompareTo(other.IntValue);
    }

    /// <summary>
    /// Compares the current object with another object of the same type.
    /// </summary>
    /// <returns>
    /// A value that indicates the relative order of the objects being compared. The return value has the following meanings: Value Meaning Less than zero This object is less than the <paramref name="other"/> parameter.Zero This object is equal to <paramref name="other"/>. Greater than zero This object is greater than <paramref name="other"/>. 
    /// </returns>
    /// <param name="other">An object to compare with this object.</param>
    public int CompareTo(float other)
    {
        return IntValue.CompareTo(new FloatToInt(other).IntValue);
    }

    /// <summary>
    /// Compares the current object with another object of the same type.
    /// </summary>
    /// <returns>
    /// A value that indicates the relative order of the objects being compared. The return value has the following meanings: Value Meaning Less than zero This object is less than the <paramref name="other"/> parameter.Zero This object is equal to <paramref name="other"/>. Greater than zero This object is greater than <paramref name="other"/>. 
    /// </returns>
    /// <param name="other">An object to compare with this object.</param>
    public int CompareTo(int other)
    {
        return IntValue.CompareTo(other);
    }

    /// <summary>
    /// Indicates whether this instance and a specified object are equal.
    /// </summary>
    /// <returns>
    /// true if <paramref name="obj"/> and this instance are the same type and represent the same value; otherwise, false.
    /// </returns>
    /// <param name="obj">Another object to compare to. </param><filterpriority>2</filterpriority>
    public override bool Equals(object obj)
    {
        if (ReferenceEquals(null, obj))
        {
            return false;
        }
        if (obj.GetType() != typeof(FloatToInt))
        {
            return false;
        }
        return Equals((FloatToInt)obj);
    }

    /// <summary>
    /// Returns the hash code for this instance.
    /// </summary>
    /// <returns>
    /// A 32-bit signed integer that is the hash code for this instance.
    /// </returns>
    /// <filterpriority>2</filterpriority>
    public override int GetHashCode()
    {
        return IntValue;
    }

    /// <summary>
    /// Implicitly converts from a <see cref="FloatToInt"/> to an <see cref="int"/>.
    /// </summary>
    /// <param name="value">A <see cref="FloatToInt"/>.</param>
    /// <returns>An integer representation of the floating-point value.</returns>
    public static implicit operator int(FloatToInt value)
    {
        return value.IntValue;
    }

    /// <summary>
    /// Implicitly converts from a <see cref="FloatToInt"/> to a <see cref="float"/>.
    /// </summary>
    /// <param name="value">A <see cref="FloatToInt"/>.</param>
    /// <returns>The floating-point value.</returns>
    public static implicit operator float(FloatToInt value)
    {
        return value.FloatValue;
    }

    /// <summary>
    /// Determines if two <see cref="FloatToInt"/> instances have the same integer representation.
    /// </summary>
    /// <param name="left">A <see cref="FloatToInt"/>.</param>
    /// <param name="right">A <see cref="FloatToInt"/>.</param>
    /// <returns>true if the two <see cref="FloatToInt"/> have the same integer representation; otherwise, false.</returns>
    public static bool operator ==(FloatToInt left, FloatToInt right)
    {
        return left.IntValue == right.IntValue;
    }

    /// <summary>
    /// Determines if two <see cref="FloatToInt"/> instances have different integer representations.
    /// </summary>
    /// <param name="left">A <see cref="FloatToInt"/>.</param>
    /// <param name="right">A <see cref="FloatToInt"/>.</param>
    /// <returns>true if the two <see cref="FloatToInt"/> have different integer representations; otherwise, false.</returns>
    public static bool operator !=(FloatToInt left, FloatToInt right)
    {
        return !(left == right);
    }
}

Answer 9

I translated the sample from Michael Borgwardt. This is the result:

public static bool NearlyEqual(float a, float b, float epsilon){
    float absA = Math.Abs (a);
    float absB = Math.Abs (b);
    float diff = Math.Abs (a - b);

    if (a == b) {
        return true;
    } else if (a == 0 || b == 0 || diff < float.Epsilon) {
        // a or b is zero or both are extremely close to it
        // relative error is less meaningful here
        return diff < epsilon;
    } else { // use relative error
        return diff / (absA + absB) < epsilon;
    }
}

Feel free to improve this answer.

Answer 10

What about: b - delta < a && a < b + delta

Answer 11

I think your second option is the best bet. Generally in floating-point comparison you often only care that one value is within a certain tolerance of another value, controlled by the selection of epsilon.

Answer 12

Although the second option is more general, the first option is better when you have an absolute tolerance, and when you have to execute many of these comparisons. If this comparison is say for every pixel in an image, the multiplication in the second options might slow your execution to unacceptable levels of performance.

Answer 13

if (Math.Abs(1.0 - 1.01) < TOLERANCE) {
//true
}

where TOLERANCE is the amount you wish to achieve. e.g. TOLERANCE = 0.01 will not result in true. But if you keep it 0.011 it will result in true since the diff is within reach.

Answer 14

static class FloatUtil {

    static bool IsEqual(float a, float b, float tolerance = 0.001f) {
      return Math.Abs(a - b) < tolerance;
    }

    static bool IsGreater(float a, float b) {
      return a > b;
    }

    static bool IsLess(float a, float b) {
      return a < b;
    }
}

The value of tolerance that is passed into IsEqual is something that the client could decide.

IsEqual(1.002, 1.001);          -->   False
IsEqual(1.002, 1.001, 0.01);    -->   True

Answer 15

Refined the answer given by @EricOuelet above:

   //to future maintainers   Based on
   //to future maintainers
   //to future maintainers   https://stackoverflow.com/questions/2411392/double-epsilon-for-equality-greater-than-less-than-less-than-or-equal-to-gre
   //to future maintainers
   public static class DoubleExtensions
   {
       /// <summary>Compare two doubles taking in account the double precision potential error.</summary>
       /// <remarks>
       /// This method works fine for floats. That's because all the float values can be represented by a double value,
       /// because they're "built" in the same way, just with more possible digits.
       /// </remarks>
       public static bool AboutEquals(this float value1, double value2) => AboutEquals((double) value1, value2);
   
       /// <summary>Compare two doubles taking in account the double precision potential error.</summary>
       /// <remarks>
       /// This method works fine for floats. That's because all the float values can be represented by a double value,
       /// because they're "built" in the same way, just with more possible digits.
       ///
       /// Also bare in mind that truncation errors accumulate on calculation. The more you do, the more you should increase the epsilon.
       /// You get really better performance when you can determine the contextual epsilon first.
       /// </remarks>
       /// <param name="value1">The first value</param>
       /// <param name="value2">The second value</param>
       /// <param name="precalculatedContextualEpsilon">The precalculated-contextual-epsilon. Use <see cref="GetContextualEpsilon"/> for this.</param>
       /// <returns>true if the two values fall within the given margin, false otherwise</returns>
       public static bool AboutEquals(this double value1, double value2, double? precalculatedContextualEpsilon = null)
       {
           precalculatedContextualEpsilon ??= Math.Max(Math.Abs(value1), Math.Abs(value2)).GetContextualEpsilon();
   
           return Math.Abs(value1 - value2) <= precalculatedContextualEpsilon;
       }
   
       public static double GetContextualEpsilon(this double biggestPossibleContextualValue) => biggestPossibleContextualValue * 1E-15;
       
       /// <summary>Matlab equivalent</summary>
       /// <param name="dividend"></param>
       /// <param name="divisor"></param>
       /// <returns></returns>
       public static double Mod(this double dividend, double divisor) => dividend - System.Math.Floor(dividend / divisor) * divisor;
   }