Float Point Numbers as Approximates & Using Errors

• Download source code - 1.4 KB

CFloat.vb is a class to handle and explain float point numbers in VB.

What are Floating-point Numbers

In computing, floating point is approximates. Floating point numbers are stored in binary format and converting from decimal number will return not the exact decimal number, but the nearest number to it that float point number can expose, as example: calculating 0.43 + 0.000001 - 0.430001, in C#, VB, VBA, Python, PHP, Java will not return 0!!

Storage Diagram

Single is stored in 4 bytes = 32 bits as the following binary storage diagram. S is signal bit, E is Exponents bit & F is fraction bit.
Fraction is called mantissa too.

Binary format of Single precision number is as follows:

Please note that the patterns may have some little differ from one compiler to other

S EEEE EEEE FFF FFFF FFFF FFFF FFFF FFFF

Binary format of Double precision number is as given below:

S EEEE EEEE EEE FFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFF FFFF

Unexpected Code Run Caused by Floating Point

Since floating point is approximates, then adding and mathematical operation does not always lead to the exact result.

Addition test:

//C#
public void TestAdd()
{
    double V = 0;
    V = 0.43 + 1E-06 - 0.430001;
    if (V != 0) {
	    Interaction.MsgBox("It will go here!");
    }
    float f = 0;
    f = 0.43f + 1E-06f - 0.430001f;
    if (f != 0) {
	    Interaction.MsgBox("It will go here!");
    }
}

'VB
Sub TestAdd()
    Dim V As Double
    V = 0.43# + 0.000001# - 0.430001#
    If V <> 0 Then
        MsgBox("It will go here!")
    End If
    Dim f As Single
    f = 0.43! + 0.000001! - 0.430001!
    If f <> 0 Then
        MsgBox("It will go here!")
    End If
End Sub

Special values:

'VB
Sub Test(x As Double)
    If x > 0 Then
        MsgBox("X > 0")
    ElseIf x = 0 Then
        MsgBox("X = 0")
    ElseIf x < 0 Then
        MsgBox("X < 0")
    Else
        MsgBox("This is a possible case! What is the value of X hear?")
        Dim R =
        (Double.NaN = 0) = False AndAlso
        (Double.NaN < 0) = False AndAlso
        (Double.NaN > 0) = False
    End If
End Sub

Sub Test(x As Single)
    If x > 0 Then
        MsgBox("X > 0")
    ElseIf x = 0 Then
        MsgBox("X = 0")
    ElseIf x < 0 Then
        MsgBox("X < 0")
    Else
        MsgBox("This is a possible case! What is the value of X hear?")
        Dim R =
        (Single.NaN = 0) = False AndAlso
        (Single.NaN < 0) = False AndAlso
        (Single.NaN > 0) = False
    End If
End Sub

//C#
public void Test(double x)
{
	if (x > 0) {
		Interaction.MsgBox("X > 0");
	} else if (x == 0) {
		Interaction.MsgBox("X = 0");
	} else if (x < 0) {
		Interaction.MsgBox("X < 0");
	} else {
		Interaction.MsgBox("This is a possible case! What is the value of X hear?");
		dynamic R = (double.NaN == 0) == false && (double.NaN < 0) == false && 
        (double.NaN > 0) == false;
	}
}

public void Test(float x)
{
	if (x > 0) {
		Interaction.MsgBox("X > 0");
	} else if (x == 0) {
		Interaction.MsgBox("X = 0");
	} else if (x < 0) {
		Interaction.MsgBox("X < 0");
	} else {
		Interaction.MsgBox("This is a possible case! What is the value of X hear?");
		dynamic R = (float.NaN == 0) == false && 
		(float.NaN < 0) == false && (float.NaN > 0) == false;
	}
}    

How to Get Float Point Value from its Exponent and Fraction

The following function will get the double value of a float point number. The Exponent Bias and Fraction Base of the float point number is dependant on the type of floating point. See the table.

The value of number is calculated from the formula:

Exponent <> 0:

Value = ±2^(Exponent - ExponentBias) * (1 + Fraction / FractionBase)

Exponent = 0:

Value = ±2^(1- ExponentBias) * ( Fraction / FractionBase)

'VB
Function GetDoubleValue(IsNegative As Boolean, Exponent As UInt16,
        Fraction As UInt64, ExponentBias As UInt16, FractionBase As UInt64) As Double
    If Exponent = 0 Then
        If Fraction = 0 Then
            Return If(IsNegative, -0#, 0#)
        End If
        Dim FractionRatio = Fraction / FractionBase
        Return If(IsNegative, -1#, 1#) * (2 ^ (1 - ExponentBias)) * FractionRatio
    Else
        If Exponent = 2 * ExponentBias + 1 Then
            If Fraction = 0 Then
                Return If(IsNegative, Double.NegativeInfinity, Double.PositiveInfinity)
            Else
                Return Double.NaN
            End If
        End If
        Dim FractionRatio = Fraction / FractionBase
        Return If(IsNegative, -1#, 1#) * (2 ^ (CInt(Exponent) - ExponentBias)) * (1# + FractionRatio)
    End If
End Function

//C#
public double GetDoubleValue(bool IsNegative, UInt16 Exponent, UInt64 Fraction, 
UInt16 ExponentBias, UInt64 FractionBase)
{
	if (Exponent == 0) {
		if (Fraction == 0) {
			return IsNegative ? -0.0 : 0.0;
		}
		dynamic FractionRatio = Fraction / FractionBase;
		return IsNegative ? -1.0 : 1.0 * (Math.Pow(2, (1 - ExponentBias))) * FractionRatio;
	} else {
		if (Exponent == 2 * ExponentBias + 1) {
			if (Fraction == 0) {
				return IsNegative ? double.NegativeInfinity : double.PositiveInfinity;
			} else {
				return double.NaN;
			}
		}
		dynamic FractionRatio = Fraction / FractionBase;
		return IsNegative ? -1.0 : 1.0 * 
           (Math.Pow(2, (Convert.ToInt32(Exponent) - ExponentBias))) * (1 + FractionRatio);
	}
}

Database primary key and float point

Since float point is approximate number; it is not a good idea to use it as primary key in a database
For example
If we have a table named Customers with a field id with the data type single then
Referring to a one row may result no record even if record is in the database
When we insert 0.4301 as id the real id will be different a lite and may this result unexpected result in database update or select
The flowing sql may result no record and this is depend on the database drive and how it convert numbers form decimal to float point type

Select Customer From Customers where id = 0.4301

Float Numbers and Programming Languages

• Floating Points in Python
• Floating Point in PHP
• PHP float numbers not adding correctly
• Precision error with floats in Java

References

• Floating point - Wikipedia
• Double-precision floating-point format - Wikipedia
• Single-precision floating-point format - Wikipedia
• IEEE Floating-Point Representation - MSDN
• Type float - MSDN
• Fun with floating point – Presentation Source - MSDN blogs
• Understanding Floating Point - Microsoft Office blogs
• Double data type numbers error
• What Every Computer Scientist Should Know About Floating-Point Arithmetic[^]