Single.Epsilon Field

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

Namespace:  System
Assembly:  mscorlib (in mscorlib.dll)

public const float Epsilon

The value of the Epsilon property reflects the smallest positive Single value that is significant in numeric operations or comparisons when the value of the Single 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()
   {
      float[] values = { 0f, Single.Epsilon, Single.Epsilon * .5f };

      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 = 1.401298E-45: False 
//       0 = 0: True 
//        
//       1.401298E-45 = 0: False

More precisely, the single-precision floating-point format consists of a sign, a 23-bit mantissa or significand, and an 8-bit exponent. As the following example shows, zero has an exponent of -126 and a mantissa of 0. Epsilon has an exponent of -126 and a mantissa of 1. This means that Single.Epsilon is the smallest positive Single value that is greater than zero and represents the smallest possible value and the smallest possible increment for a Single whose exponent is -126.

using System;

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

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

      // Convert the single to a 4-byte array. 
      byte[] bytes = BitConverter.GetBytes(value);
      int formattedSingle = BitConverter.ToInt32(bytes, 0);

      // Get the sign bit (byte 3, bit 7).
      result += String.Format("Sign: {0}\n", 
                              (formattedSingle >> 31) != 0 ? "1 (-)" : "0 (+)");

      // Get the exponent (byte 2 bit 7 to byte 3, bits 6) 
      int exponent =  (formattedSingle >> 23) & 0x000000FF;
      int adjustment = (exponent != 0) ? 127 : 126;
      result += String.Format("{0}Exponent: 0x{1:X4} ({1})\n", new String(' ', indent), exponent - adjustment);

      // Get the significand (bits 0-22) 
      long significand = exponent != 0 ? 
                         ((formattedSingle & 0x007FFFFF) | 0x800000) : 
                         (formattedSingle & 0x007FFFFF); 
      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: 0xFFFFFF82 (-126) 
//          Mantissa: 0x0000000000000 
//        
//        
//       1.401298E-45: Sign: 0 (+) 
//                     Exponent: 0xFFFFFF82 (-126) 
//                     Mantissa: 0x0000000000001

However, the Epsilon property is not a general measure of precision of the Single type; it applies only to Single instances that have a value of zero.

NoteNote

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 1.4e-45.

Two apparently equivalent floating-point numbers might not compare equal because of differences in their least significant digits. For example, the C# expression, (float)1/3 == (float)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, you must use a value that is greater than 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.)

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 1.175494351E-38 instead.

.NET Framework

Supported in: 4.5.2, 4.5.1, 4.5, 4, 3.5, 3.0, 2.0, 1.1, 1.0

.NET Framework Client Profile

Supported in: 4, 3.5 SP1

Portable Class Library

Supported in: Portable Class Library

.NET for Windows Store apps

Supported in: Windows 8

.NET for Windows Phone apps

Supported in: Windows Phone 8.1, Windows Phone 8, Silverlight 8.1

Windows Phone 8.1, Windows Phone 8, Windows 8.1, Windows Server 2012 R2, Windows 8, Windows Server 2012, Windows 7, Windows Vista SP2, Windows Server 2008 (Server Core Role not supported), Windows Server 2008 R2 (Server Core Role supported with SP1 or later; Itanium not supported)

The .NET Framework does not support all versions of every platform. For a list of the supported versions, see .NET Framework System Requirements.

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