Double.Epsilon Double.Epsilon Double.Epsilon Double.Epsilon Field

Definition

Represents the smallest positive Double value that is greater than zero. This field is constant.

public: double Epsilon = 4.94065645841247E-324;
public const double Epsilon = 4.94065645841247E-324;
val mutable Epsilon : double
Public Const Epsilon As Double  = 4.94065645841247E-324

Field Value

Remarks

The value of the Epsilon property reflects the smallest positive Double value that is significant in numeric operations or comparisons when the value of the Double instance is zero. For example, the following code shows that zero and Epsilon are considered to be unequal values, whereas zero and half the value of Epsilon are considered to be equal.

using System;

public class Example
{
   public static void Main()
   {
      double[] values = { 0, Double.Epsilon, Double.Epsilon * .5 };
      
      for (int ctr = 0; ctr <= values.Length - 2; ctr++)
      {
         for (int ctr2 = ctr + 1; ctr2 <= values.Length - 1; ctr2++)
         {
            Console.WriteLine("{0:r} = {1:r}: {2}", 
                              values[ctr], values[ctr2],  
                              values[ctr].Equals(values[ctr2]));
         }
         Console.WriteLine();
      }      
   }
}
// The example displays the following output:
//       0 = 4.94065645841247E-324: False
//       0 = 0: True
//       
//       4.94065645841247E-324 = 0: False
Module Example
   Public Sub Main()
      Dim values() As Double = { 0, Double.Epsilon, Double.Epsilon * .5 }
      
      For ctr As Integer = 0 To values.Length - 2
         For ctr2 As Integer = ctr + 1 To values.Length - 1
            Console.WriteLine("{0:r} = {1:r}: {2}", _
                              values(ctr), values(ctr2), _ 
                              values(ctr).Equals(values(ctr2)))
         Next
         Console.WriteLine()
      Next      
   End Sub
End Module
' The example displays the following output:
'       0 = 4.94065645841247E-324: False
'       0 = 0: True
'       
'       4.94065645841247E-324 = 0: False

More precisely, the floating point format consists of a sign, a 52-bit mantissa or significand, and an 11-bit exponent. As the following example shows, zero has an exponent of -1022 and a mantissa of 0. Epsilon has an exponent of -1022 and a mantissa of 1. This means that Epsilon is the smallest positive Double value greater than zero and represents the smallest possible value and the smallest possible increment for a Double whose exponent is -1022.

using System;

public class Example
{
   public static void Main()
   {
      double[] values = { 0.0, Double.Epsilon };
      foreach (var value in values) {
         Console.WriteLine(GetComponentParts(value));
         Console.WriteLine();
      }   
   }

   private static string GetComponentParts(double value)
   {
      string result = String.Format("{0:R}: ", value);
      int indent = result.Length;

      // Convert the double to an 8-byte array.
      byte[] bytes = BitConverter.GetBytes(value);
      // Get the sign bit (byte 7, bit 7).
      result += String.Format("Sign: {0}\n", 
                              (bytes[7] & 0x80) == 0x80 ? "1 (-)" : "0 (+)");

      // Get the exponent (byte 6 bits 4-7 to byte 7, bits 0-6)
      int exponent = (bytes[7] & 0x07F) << 4;
      exponent = exponent | ((bytes[6] & 0xF0) >> 4);  
      int adjustment = exponent != 0 ? 1023 : 1022;
      result += String.Format("{0}Exponent: 0x{1:X4} ({1})\n", new String(' ', indent), exponent - adjustment);

      // Get the significand (bits 0-51)
      long significand = ((bytes[6] & 0x0F) << 48); 
      significand = significand | ((long) bytes[5] << 40);
      significand = significand | ((long) bytes[4] << 32);
      significand = significand | ((long) bytes[3] << 24);
      significand = significand | ((long) bytes[2] << 16);
      significand = significand | ((long) bytes[1] << 8);
      significand = significand | bytes[0];    
      result += String.Format("{0}Mantissa: 0x{1:X13}\n", new String(' ', indent), significand);    

      return result;   
   }
}
//       // The example displays the following output:
//       0: Sign: 0 (+)
//          Exponent: 0xFFFFFC02 (-1022)
//          Mantissa: 0x0000000000000
//       
//       
//       4.94065645841247E-324: Sign: 0 (+)
//                              Exponent: 0xFFFFFC02 (-1022)
//                              Mantissa: 0x0000000000001
Module Example
   Public Sub Main()
      Dim values() As Double = { 0.0, Double.Epsilon }
      For Each value In values
         Console.WriteLine(GetComponentParts(value))
         Console.WriteLine()
      Next   
   End Sub

   Private Function GetComponentParts(value As Double) As String
      Dim result As String =  String.Format("{0:R}: ", value)
      Dim indent As Integer =  result.Length

      ' Convert the double to an 8-byte array.
      Dim bytes() As Byte = BitConverter.GetBytes(value)
      ' Get the sign bit (byte 7, bit 7).
      result += String.Format("Sign: {0}{1}", 
                              If((bytes(7) And &H80) = &H80, "1 (-)", "0 (+)"),
                              vbCrLf)

      ' Get the exponent (byte 6 bits 4-7 to byte 7, bits 0-6)
      Dim exponent As Integer =  (bytes(7) And &H07F) << 4
      exponent = exponent Or ((bytes(6) And &HF0) >> 4)  
      Dim adjustment As Integer = If(exponent <> 0, 1023, 1022)
      result += String.Format("{0}Exponent: 0x{1:X4} ({1}){2}", 
                              New String(" "c, indent), exponent - adjustment,
                              vbCrLf)

      ' Get the significand (bits 0-51)
      Dim significand As Long =  ((bytes(6) And &H0F) << 48) 
      significand = significand Or (bytes(5) << 40)
      significand = significand Or (bytes(4) << 32)
      significand = significand Or (bytes(3) << 24)
      significand = significand Or (bytes(2) << 16)
      significand = significand Or (bytes(1) << 8)
      significand = significand Or bytes(0)    
      result += String.Format("{0}Mantissa: 0x{1:X13}{2}", 
                              New String(" "c, indent), significand, vbCrLf)    

      Return result   
   End Function
End Module
' The example displays the following output:
'       0: Sign: 0 (+)
'          Exponent: 0xFFFFFC02 (-1022)
'          Mantissa: 0x0000000000000
'       
'       
'       4.94065645841247E-324: Sign: 0 (+)
'                              Exponent: 0xFFFFFC02 (-1022)
'                              Mantissa: 0x0000000000001

However, the Epsilon property is not a general measure of precision of the Double type; it applies only to Double instances that have a value of zero or an exponent of -1022.

Note

The value of the Epsilon property is not equivalent to machine epsilon, which represents the upper bound of the relative error due to rounding in floating-point arithmetic.

The value of this constant is 4.94065645841247e-324.

Two apparently equivalent floating-point numbers might not compare equal because of differences in their least significant digits. For example, the C# expression, (double)1/3 == (double)0.33333, does not compare equal because the division operation on the left side has maximum precision while the constant on the right side is precise only to the specified digits. If you create a custom algorithm that determines whether two floating-point numbers can be considered equal, we do not recommend that you base your algorithm on the value of the Epsilon constant to establish the acceptable absolute margin of difference for the two values to be considered equal. (Typically, that margin of difference is many times greater than Epsilon.) For information about comparing two double-precision floating-point values, see Double and Equals(Double).

Platform Notes

On ARM systems, the value of the Epsilon constant is too small to be detected, so it equates to zero. You can define an alternative epsilon value that equals 2.2250738585072014E-308 instead.

Applies to