Is Excel VBA's Rnd() really this bad? - excel

Question

I need a pseudo random number generator for 2D Monte Carlo simulation that doesn't have the characteristic hyperplanes that you get with simple LCGs. I tested the random number generator Rnd() in Excel 2013 using the following code (takes about 5 secs to run):
Sub ZoomRNG()
Randomize
For i = 1 To 1000
Found = False
Do
x = Rnd() ' 2 random numbers between 0.0 and 1.0
y = Rnd()
If ((x > 0.5) And (x < 0.51)) Then
If ((y > 0.5) And (y < 0.51)) Then
' Write if both x & y in a narrow range
Cells(i, 1) = i
Cells(i, 2) = x
Cells(i, 3) = y
Found = True
End If
End If
Loop While (Not Found)
Next i
End Sub
Here is a simple plot of x vs y from running the above code
Not only is it not very random-looking, it has more obvious hyperplanes than the infamous RANDU algorithm does in 2D. Basically, am I using the function incorrectly or is the Rnd() function in VBA actually not the least bit usable?
For comparison, here's what I get for the Mersenne Twister MT19937 in C++.

Answer 1

To yield a better random generator and to make its performance faster, I modified your code like this:
Const N = 1000 'Put this on top of your code module
Sub ZoomRNG()
Dim RandXY(1 To N, 1 To 3) As Single, i As Single, x As Single, y As Single
For i = 1 To N
Randomize 'Put this in the loop to generate a better random numbers
Do
x = Rnd
y = Rnd
If x > 0.5 And x < 0.51 Then
If y > 0.5 And y < 0.51 Then
RandXY(i, 1) = i
RandXY(i, 2) = x
RandXY(i, 3) = y
Exit Do
End If
End If
Loop
Next
Cells(1, 9).Resize(N, 3) = RandXY
End Sub
I obtain this after plotting the result
The result looks better than your code's output. Modifying the above code a little bit to something like this
Const N = 1000
Sub ZoomRNG()
Dim RandXY(1 To N, 1 To 3) As Single, i As Single, x As Single, y As Single
For i = 1 To N
Randomize
Do
x = Rnd
If x > 0.5 And x < 0.51 Then
y = Rnd
If y > 0.5 And y < 0.51 Then
RandXY(i, 1) = i
RandXY(i, 2) = x
RandXY(i, 3) = y
Exit Do
End If
End If
Loop
Next
Cells(1, 9).Resize(N, 3) = RandXY
End Sub
yields a better result than the previous one
Sure the Mersenne Twister MT19937 in C++ is still better, but the last result is quite good for conducting Monte-Carlo simulations. FWIW, you might be interested in reading this paper: On the accuracy of statistical procedures in Microsoft Excel 2010.

Answer 2

That seems like it would take on average 1000 * 100 * 100 iterations to complete and VBA is usually a bit slower than native Excel formulas. Consider this example
Sub ZoomRNG()
t = Timer
[a1:a1000] = "=ROW()"
[b1:c1000] = "=RAND()/100+0.5"
[a1:c1000] = [A1:C1000].Value
Debug.Print CDbl(Timer - t) ' 0.0546875 seconds
End Sub
Update
It's not that bad at all! This will work too even without Randomize
Sub ZoomRNGs() ' VBA.Rnd returns Single
t = Timer
For i = 1 To 1000
Cells(i, 1) = i
Cells(i, 2) = Rnd / 100 + 0.5
Cells(i, 3) = Rnd / 100 + 0.5
Next i
Debug.Print Timer - t ' 0.25 seconds
End Sub
Sub ZoomRNGd() ' the Excel Function RAND() returns Double
t = Timer
For i = 1 To 1000
Cells(i, 1) = i
Cells(i, 2) = [RAND()] / 100 + 0.5
Cells(i, 3) = [RAND()] / 100 + 0.5
Next i
Debug.Print Timer - t ' 0.625 seconds
End Sub
and Single has about half of the precision of Double :
s = Rnd: d = [RAND()]
Debug.Print s; d; Len(Str(s)); Len(Str(d)) ' " 0.2895625 0.580839555868045 9 17 "
Update 2
I found C alternative that is as fast as VBA Rnd.
C:\Windows\System32\msvcrt.dll is the Microsoft C Runtime Library:
Declare Function rand Lib "msvcrt" () As Long ' this in a VBA module
and then you can use it like this x = rand / 32767 in your code:
Sub ZoomRNG()
t = Timer
Dim i%, x#, y#, Found As Boolean
For i = 1 To 1000
Found = False
Do
x = rand / 32767 ' RAND_MAX = 32,767
y = rand / 32767
If ((x > 0.5) And (x < 0.51)) Then
If ((y > 0.5) And (y < 0.51)) Then
' Write if both x & y in a narrow range
Cells(i, 1) = i
Cells(i, 2) = x
Cells(i, 3) = y
Found = True
End If
End If
Loop While (Not Found)
Next i
Debug.Print Timer - t ' 2.875 seconds
End Sub

Answer 3

After reading this question I got curious and found the paper
"Assessing Excel VBA Suitability for Monte Carlo Simulation" by Alexei Botchkarev that is available here. Both RAND and RND functions are not recommended, but as pointed out in the paper the Mersenne Twister has been implemented in VBA by Jerry Wang.
A quick search led me to this nicely commented Version that has been updated the last 2015/2/28: http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/VERSIONS/BASIC/MTwister.xlsb
Source: http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/VERSIONS/BASIC/basic.html

Answer 4

All LCGs will generate hyperplanes. The quality of the LCG increases with decreasing distance between these hyperplanes. So, having more hyperplanes than RANDU is a good thing.
The MT plot looks much better because it is NOT an LCG. Indeed, any non-LCG pRNG could have a random looking plot and still be a bad.
To avoid the problem of 2D correlations, you could use the same LCG for x and y but have different seeds for x and y. Of course, this will not work with RND because you cannot have two separate streams. You will need an LCG pRNG that takes the seed as an argument by reference.

Answer 5

As a balance between speed and goodness, I was thinking of combining them like
for...
z = [rand()] ' good but slow.
for .. ' just a few
t = z + rnd()
t = t - int(t)
...
Remember that good entropy + bad entropy = better entropy.
That said, only 0.05ms per [rand()].