KB196652 - How To Implement Custom Rounding Procedures

MORE INFORMATION

Rounding Explained

You need to round when you want to convert a number of greater precision into a number of lesser precision. The most common case is when you need to convert a floating-point number into an integer.

Rounding Down

The simplest form of rounding is truncation. Any digits after the desired precision are simply ignored. The VBA Fix() function is an example of truncation. For example, Fix(3.5) is 3, and Fix(-3.5) is -3.

The Int() function rounds down to the highest integer less than the value. Both Int() and Fix() act the same way with positive numbers - truncating - but give different results for negative numbers: Int(-3.5) gives -4.

The Fix() function is an example of symmetric rounding because it affects the magnitude (absolute value) of positive and negative numbers in the same way. The Int() function is an example of asymmetric rounding because it affects the magnitude of positive and negative numbers differently.

Excel has similar spreadsheet functions: Int(), Floor(), and RoundDown(). Int() works the same way as Int() does in Visual Basic for Applications. Floor() truncates positive values, but does not work with negative numbers. The RoundDown() function works the same way as the VBA Fix() function.

Microsoft SQL Server has a Round() function that can act like the VBA Fix() function. SQL Server also has a Floor() function, which works the same way as VBA Int() function.

Rounding Up

SQL Server and Excel both have a function called Ceiling(), which always rounds fraction values up (more positive) to the next value.

Visual Basic for Applications does not have a corresponding round-up function. However, for negative numbers, both Fix() and Int() can be used to round upward, in different ways.

Fix() rounds towards 0 (up in the absolute sense, but down in terms of absolute magnitude). Fix(-3.5) is -3.5.

Int() rounds away from 0 (up in terms of absolute magnitude, but down in the absolute sense). Int(-3.5) is -4.

Arithmetic Rounding

When rounding always down or up, the resulting number is not necessarily the closest to the original number. For example, if you round 1.9 down to 1, the difference is a lot larger than if you round it up to 2. It is easy to see that numbers from 1.6 to 2.4 should be rounded to 2.

However, what about 1.5, which is equidistant between 1 and 2? By convention, the half-way number is rounded up.

You can implement rounding half-way numbers in a symmetric fashion, such that -.5 is rounded down to -1, or in an asymmetric fashion, where -.5 is rounded up to 0.

The following functions provide symmetric arithmetic rounding:

The Excel Round() spreadsheet function.
The SQL Server Round() function can do symmetric arithmetic rounding.

The following function provide asymmetric arithmetic rounding:

The Round() method of the Java Math library.

Visual Basic for Applications does not have any function that does arithmetic rounding.

Banker's Rounding

When you add rounded values together, always rounding .5 in the same direction results in a bias that grows with the more numbers you add together. One way to minimize the bias is with banker's rounding.

Banker's rounding rounds .5 up sometimes and down sometimes. The convention is to round to the nearest even number, so that both 1.5 and 2.5 round to 2, and 3.5 and 4.5 both round to 4. Banker's rounding is symmetric.

In Visual Basic for Applications, the following numeric functions perform banker's rounding: CByte(), CInt(), CLng(), CCur(), and Round().

There are no Excel spreadsheet functions that perform banker's rounding.

Random Rounding

Even banker's rounding can bias totals. You can take an extra step to remove bias by rounding .5 up or down in a truly random fashion. This way, even if the data is deliberately biased, bias might be minimized. However, using random rounding with randomly distributed data might result in a larger bias than banker's rounding. Random rounding could result in two different totals on the same data.

No Microsoft products implement any sort of random rounding procedure.

Alternate Rounding

Alternate rounding is rounding between .5 up and .5 down on successive calls.

No Microsoft products implement an alternate rounding procedure.

The Round() Function is Inconsistently Implemented

The Round() function is not implemented in a consistent fashion among different Microsoft products for historical reasons.

The following table relates product to implementation:

   Product                             Implementation
   ----------------------------------------------------------------------
   Visual Basic for Applications 6.0   Banker's Rounding
   Excel Worksheet                     Symmetric Arithmetic Rounding
   SQL Server                          Either Symmetric Arithmetic Rounding
                                       or Symmetric Round Down (Fix)
                                       depending on arguments

   Java Math library                   Asymmetric Arithmetic Rounding

The Round() function in Visual Basic 6.0 and Visual Basic for Applications 6.0 performs banker's rounding. It has an optional second argument that specifies the number of decimal digits to round to:

   Debug.Print Round(2.45, 1) returns 2.4.

Sample Data

The following table shows some sample data and the effects of various rounding methods on the numbers and totals generated.

   Number/Int./Fix/Ceiling/Asym. Arith./Sym. Arith./Banker's/Random/Alt.
   ---------------------------------------------------------------------
   -2.6   -3   -2  -2      -3          -3           -3       -3     -3
   -2.5   -3   -2  -2      -2          -3           -2       -2     -3
   -2.4   -3   -2  -2      -2          -2           -2       -2     -2
   -1.6   -2   -1  -1      -2          -2           -2       -2     -2
   -1.5   -2   -1  -1      -1          -2           -2       -1     -1
   -1.4   -2   -1  -1      -1          -1           -1       -1     -1
   -0.6   -1    0   0      -1          -1           -1       -1     -1
   -0.5   -1    0   0       0          -1            0       -1     -1
   -0.4   -1    0   0       0           0            0        0      0
    0.4    0    0   1       0           0            0        0      0
    0.5    0    0   1       1           1            0        1      1
    0.6    0    0   1       1           1            1        1      1
    1.4    1    1   2       1           1            1        1      1
    1.5    1    1   2       2           2            2        1      1
    1.6    1    1   2       2           2            2        2      2
    2.4    2    2   3       2           2            2        2      2
    2.5    2    2   3       3           3            2        3      3
    2.6    2    2   3       3           3            3        3      3

Total of all numbers:

   Number/Int./Fix/Ceiling/Asym. Arith./Sym. Arith./Banker's/Random/Alt.
   ---------------------------------------------------------------------
   0.0    -9   0   9       3            0           0        1      0

Total of all negative numbers:

   Number/Int./Fix/Ceiling/Asym. Arith./Sym. Arith./Banker's/Random/Alt.
   ---------------------------------------------------------------------
   -13.5  -18  -9  -9      -12          -15         -13      -13    -14

Total of all positive numbers:

   Number/Int./Fix/Ceiling/Asym. Arith./Sym. Arith./Banker's/Random/Alt.
   ---------------------------------------------------------------------
   13.5   9    9   18      15           15          13       14     14

The table shows the difference between the various rounding methods. For randomly distributed positive and negative numbers, Fix(), symmetric arithmetic rounding, banker's rounding, and alternating rounding provide the least difference from actual totals, with random rounding not far behind.

However, if the numbers are either all positive or all negative, banker's rounding, alternating rounding, and random rounding provide the least difference from the actual totals.

Sample User-Defined Rounding Functions

The sample code in the following Function Listing section provides sample implementations for each of the rounding types described.

The functions provided are:

   AsymDown      Asymmetrically rounds numbers down - similar to Int().
                 Negative numbers get more negative.

   SymDown       Symmetrically rounds numbers down - similar to Fix().
                 Truncates all numbers toward 0.
                 Same as AsymDown for positive numbers.

   AsymUp        Asymmetrically rounds numbers fractions up.
                 Same as SymDown for negative numbers.
                 Similar to Ceiling.

   SymUp         Symmetrically rounds fractions up - that is, away from 0.
                 Same as AsymUp for positive numbers.
                 Same as AsymDown for negative numbers.

   AsymArith     Asymmetric arithmetic rounding - rounds .5 up always.
                 Similar to Java worksheet Round function.

   SymArith      Symmetric arithmetic rounding - rounds .5 away from 0.
                 Same as AsymArith for positive numbers.
                 Similar to Excel Worksheet Round function.

   BRound        Banker's rounding.
                 Rounds .5 up or down to achieve an even number.
                 Symmetrical by definition.

   RandRound     Random rounding.
                 Rounds .5 up or down in a random fashion.

   AltRound      Alternating rounding.
                 Alternates between rounding .5 up or down.

   ATruncDigits  Same as AsyncTrunc but takes different arguments.

All of these functions take two arguments: the number to be rounded and an optional factor. If the factor is omitted, then the functions return an integer created by one of the above methods. If the factor is specified, the number is scaled by the factor to create different rounding effects. For example AsymArith(2.55, 10) produces 2.6, that is, it rounds to 1/factor = 1/10 = 0.1.

NOTE: A factor of 0 generates a run-time error: 1/factor = 1/0.

The following table shows the effects of various factors:

   Expression       Result  Comment
   --------------------------------------------------------------------
   AsymArith(2.5)     3     Rounds up to next integer.
   BRound(2.18, 20)   2.2   Rounds to the nearest 5 cents (1/20 dollar).
   SymDown(25, .1)   20     Rounds down to an even multiple of 10.

The exception to the above description is ADownDigits, which is a template function that allows you to specify the number of decimal digits instead of a factor.

   Expression            Result Comment
   ---------------------------------------------------------------------
   ADownDigits(2.18, 1)    2.1  Rounds down to next multiple of 10 ^ -1.

Function Listing

   Function AsymDown(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
     AsymDown = Int(X * Factor) / Factor
   End Function

   Function SymDown(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
     SymDown = Fix(X * Factor) / Factor
   '  Alternately:
   '  SymDown = AsymDown(Abs(X), Factor) * Sgn(X)
   End Function

   Function AsymUp(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
   Dim Temp As Double
     Temp = Int(X * Factor)
     AsymUp = (Temp + IIf(X = Temp, 0, 1)) / Factor
   End Function

   Function SymUp(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
   Dim Temp As Double
     Temp = Fix(X * Factor)
     SymUp = (Temp + IIf(X = Temp, 0, Sgn(X))) / Factor
   End Function

   Function AsymArith(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
     AsymArith = Int(X * Factor + 0.5) / Factor
   End Function

   Function SymArith(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
     SymArith = Fix(X * Factor + 0.5 * Sgn(X)) / Factor
   '  Alternately:
   '  SymArith = Abs(AsymArith(X, Factor)) * Sgn(X)
   End Function

   Function BRound(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
   '  For smaller numbers:
   '  BRound = CLng(X * Factor) / Factor
   Dim Temp As Double, FixTemp As Double
     Temp = X * Factor
     FixTemp = Fix(Temp + 0.5 * Sgn(X))
     ' Handle rounding of .5 in a special manner
     If Temp - Int(Temp) = 0.5 Then
       If FixTemp / 2 <> Int(FixTemp / 2) Then ' Is Temp odd
         ' Reduce Magnitude by 1 to make even
         FixTemp = FixTemp - Sgn(X)
       End If
     End If
     BRound = FixTemp / Factor
   End Function

   Function RandRound(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
   ' Should Execute Randomize statement somewhere prior to calling.
   Dim Temp As Double, FixTemp As Double
     Temp = X * Factor
     FixTemp = Fix(Temp + 0.5 * Sgn(X))
     ' Handle rounding of .5 in a special manner.
     If Temp - Int(Temp) = 0.5 Then
       ' Reduce Magnitude by 1 in half the cases.
       FixTemp = FixTemp - Int(Rnd * 2) * Sgn(X)
     End If
     RandRound = FixTemp / Factor
   End Function

   Function AltRound(ByVal X As Double, _
            Optional ByVal Factor As Double = 1) As Double
   Static fReduce As Boolean
   Dim Temp As Double, FixTemp As Double
     Temp = X * Factor
     FixTemp = Fix(Temp + 0.5 * Sgn(X))
     ' Handle rounding of .5 in a special manner.
     If Temp - Int(Temp) = 0.5 Then
       ' Alternate between rounding .5 down (negative) and up (positive).
       If (fReduce And Sgn(X) = 1) Or (Not fReduce And Sgn(X) = -1) Then
       ' Or, replace the previous If statement with the following to
       ' alternate between rounding .5 to reduce magnitude and increase
       ' magnitude.
       ' If fReduce Then
         FixTemp = FixTemp - Sgn(X)
       End If
       fReduce = Not fReduce
     End If
     AltRound = FixTemp / Factor
   End Function

   Function ADownDigits(ByVal X As Double, _
            Optional ByVal Digits As Integer = 0) As Double
     ADownDigits = AsymDown(X, 10 ^ Digits)
   End Function

NOTE: With the exception of Excel's MRound() worksheet function, the built- in rounding functions take arguments in the manner of ADownDigits, where the second argument specifies the number of digits instead of a factor.

The rounding implementations presented here use a factor, like MRound(), which is more flexible because you do not have to round to a power of 10. You can write wrapper functions in the manner of ADownDigits.

Floating Point Limitations

All of the rounding implementations presented here use the double data type, which can represent approximately 15 decimal digits.

Since not all fractional values can be expressed exactly, you might get unexpected results because the display value does not match the stored value.

For example, the number 2.25 might be stored internally as 2.2499999..., which would round down with arithmetic rounding, instead of up as you might expect. Also, the more calculations a number is put through, the greater possibility that the stored binary value will deviate from the ideal decimal value.

If this is the case, you may want to choose a different data type, such as Currency, which is exact to 4 decimal places.

You might also consider making the data types Variant and use CDec() to convert everything to the Decimal data type, which can be exact to 28 decimal digits.

Rounding Currency Values

When you use the Currency data type, which is exact to 4 decimal digits, you typically want to round to 2 decimal digits for cents.

The Round2CB function below is a hard-coded variation that performs banker's rounding to 2 decimal digits, but does not multiply the original number. This avoids a possible overflow condition if the monetary amount is approaching the limits of the Currency data type.

   Function Round2CB (ByVal X As Currency) As Currency
     Round2CB = CCur(X / 100) * 100
   End Function

Rounding Decimal Values

The following is an example of asymmetric arithmetic rounding using the Decimal data type:

   Function AsymArithDec(ByVal X As Variant, _
            Optional ByVal Factor As Variant = 1) As Variant
     If Not IsNumeric(X) Then
       AsymArithDec = X
     Else
       If Not IsNumeric(Factor) Then Factor = 1
       AsymArithDec = Int(CDec(X * Factor) + .5)
     End If
   End Function

Dropping Precision as a Shortcut in Rounding

As taught in school, rounding is usually arithmetic rounding using positive numbers. With this type of rounding, you only need to know the number to 1 digit past where you are rounding to. You ignore digits past the first decimal place. In other words, precision is dropped as a shortcut to rounding the value.

For example, both 2.5 and 2.51 round up to 3, while both 2.4 and 2.49 round down to 2.

When you use banker's rounding (or other methods that round .5 either up or down) or when you round negative numbers using asymmetric arithmetic rounding, dropping precision can lead to incorrect results where you might not round to the nearest number.

For example, with banker's rounding, 2.5 rounds down to 2 and 2.51 rounds up to 3.

With asymmetric arithmetic rounding, -2.5 rounds up to -2 while -2.51 rounds down to -3.

The user-defined functions presented in this article take the number's full precision into account when performing rounding.

How To Implement Custom Rounding Procedures

SUMMARY