# How To Implement Custom Rounding Procedures

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##### SUMMARY

##### MORE INFORMATION

### Rounding Explained

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

The simplest form of rounding is truncation. Any digits after the desiredprecision are simply ignored. The VBA Fix() function is an example oftruncation. 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 affectsthe magnitude (absolute value) of positive and negative numbers in the sameway. The Int() function is an example of asymmetric rounding because itaffects 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 wayas VBA Int() function.

### Rounding Up

SQL Server and Excel both have a function called Ceiling(), which alwaysrounds fraction values up (more positive) to the next value.Visual Basic for Applications does not have a corresponding round-upfunction. However, for negative numbers, both Fix() and Int() can be usedto round upward, in different ways.

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

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

### Arithmetic Rounding

When rounding always down or up, the resulting number is not necessarilythe closest to the original number. For example, if you round 1.9 down to1, the difference is a lot larger than if you round it up to 2. It is easyto 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? Byconvention, the half-way number is rounded up.

You can implement rounding half-way numbers in a symmetric fashion, suchthat -.5 is rounded down to -1, or in an asymmetric fashion, where -.5 isrounded 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 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 doesarithmetic rounding.

### Banker's Rounding

When you add rounded values together, always rounding .5 in the samedirection results in a bias that grows with the more numbers you addtogether. One way to minimize the bias is with banker's rounding.Banker's rounding rounds .5 up sometimes and down sometimes. The conventionis to round to the nearest even number, so that both 1.5 and 2.5 round to2, and 3.5 and 4.5 both round to 4. Banker's rounding is symmetric.

In Visual Basic for Applications, the following numeric functions performbanker'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 toremove 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 alarger bias than banker's rounding. Random rounding could result in twodifferent 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 successivecalls.No Microsoft products implement an alternate rounding procedure.

### The Round() Function is Inconsistently Implemented

The Round() function is not implemented in a consistent fashion amongdifferent 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 Applications6.0 performs banker's rounding. It has an optional second argument thatspecifies 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 variousrounding 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. Forrandomly distributed positive and negative numbers, Fix(), symmetricarithmetic rounding, banker's rounding, and alternating rounding providethe least difference from actual totals, with random rounding not farbehind.

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

### Sample User-Defined Rounding Functions

The sample code in the following Function Listing section provides sampleimplementations 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 anoptional factor. If the factor is omitted, then the functions return aninteger 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 to1/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 templatefunction that allows you to specify the number of decimal digits instead ofa 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, wherethe 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 datatype, which can represent approximately 15 decimal digits.Since not all fractional values can be expressed exactly, you might getunexpected results because the display value does not match the storedvalue.

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 mightexpect. Also, the more calculations a number is put through, the greaterpossibility that the stored binary value will deviate from the idealdecimal value.

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

You might also consider making the data types Variant and use CDec() toconvert everything to the Decimal data type, which can be exact to 28decimal 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 performsbanker's rounding to 2 decimal digits, but does not multiply the originalnumber. This avoids a possible overflow condition if the monetary amount isapproaching 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 theDecimal 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 positivenumbers. With this type of rounding, you only need to know the number to 1digit past where you are rounding to. You ignore digits past the firstdecimal place. In other words, precision is dropped as a shortcut torounding the value.For example, both 2.5 and 2.51 round up to 3, while both 2.4 and 2.49 rounddown to 2.

When you use banker's rounding (or other methods that round .5 either up ordown) or when you round negative numbers using asymmetric arithmeticrounding, dropping precision can lead to incorrect results where you mightnot round to the nearest number.

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

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

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

##### REFERENCES

Microsoft Transact SQL Help; topic: Round Function; Floor Function; CeilingFunction

(c) Microsoft Corporation 1998, All Rights Reserved. Contributions byMalcolm Stewart, Microsoft Corporation.

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Article ID: 196652 - Last Review: 12/05/2015 09:52:16 - Revision: 3.2

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