# Redondeo Parte II

Article: Q196652
Product(s): Microsoft Visual Basic for Windows
Version(s): 5.0,6.0,6.5,7.0
Operating System(s):
Keyword(s): kbExcel kbSQLServ KbVBA kbVBp500 kbVBp600 kbGrpDSVBDB kbDSupport kbOffice97

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- Microsoft Visual Basic Control Creation Edition for Windows, version 5.0
- Microsoft Visual Basic Learning Edition for Windows, versions 5.0, 6.0
- Microsoft Visual Basic Professional Edition for Windows, versions 5.0, 6.0
- Microsoft Visual Basic Enterprise Edition for Windows, versions 5.0, 6.0
- Microsoft Visual Basic for Applications versions 5.0, 6.0
- Microsoft SQL Server versions 6.0, 6.5, 7.0
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SUMMARY
=======

There are a number of different rounding algorithms available in Microsoft
products. Rounding algorithms range from Arithmetic Rounding in Excel's
Worksheet Round() function to Banker's Rounding in the CInt(), CLng(), and
the various Visual Basic for Applications rounding functions do and provides
samples of using the functions. In addition, the article includes sample
functions that implement various rounding algorithms.

================

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 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.

precision into account when performing rounding.

REFERENCES
==========

Visual Basic Help, version 6.0; topic: Int, Fix Functions; Round Function

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

Stewart, Microsoft Corporation.