First an error. If you look at the last code box in my posting you see that two errors have appeared even though I just pasted the code from a notebook.
Sin[ x*y*z*w ] somehow became Sin[ x*y*z*q ] and {z,-5,5} became {z,{-5,5}. Usually I would dismiss things like this as operator error, but it seems extremely unlikely that I would have made two errors like that when doing a simple select, copy, paste. Unfortunately it is almost certainly impossible for me to track down and pinpoint the source of this kind of error.
Next, to try to answer your question about the number of points for 95% confidence in 0.01. My first impression would be this would be impossible to answer. I assumed from the size of the error estimate that the calculation would need to run for many years to get to an error of 0.01. But I tried the following.
In[1]:= f[x_, y_, z_, w_] := Piecewise[{{Exp[x] Sin[x*y*z*w], x*y*z*w <= 1}}, 0];
NIntegrate[f[x, y, z, w], {w, -5, 5}, {z, -5, 5}, {y, -5, 5}, {x, -5, 5}, MaxPoints -> 10^1, Method -> "MonteCarlo"]
During evaluation of In[1]:= NIntegrate::maxp: The integral failed to converge after 100 integrand evaluations.
NIntegrate obtained 33747.63448633355` and 18817.447915103512` for the integral and error estimates. >>
Out[2]= 33747.6
In[3]:= NIntegrate[f[x, y, z, w], {w, -5, 5}, {z, -5, 5}, {y, -5, 5}, {x, -5, 5}, MaxPoints -> 10^2, Method -> "MonteCarlo"]
During evaluation of In[3]:= NIntegrate::maxp: The integral failed to converge after 200 integrand evaluations.
NIntegrate obtained 3840.5470856443385` and 13915.684872403956` for the integral and error estimates. >>
Out[3]= 3840.55
In[4]:= NIntegrate[f[x, y, z, w], {w, -5, 5}, {z, -5, 5}, {y, -5, 5}, {x, -5, 5}, MaxPoints -> 10^3, Method -> "MonteCarlo"]
During evaluation of In[4]:= NIntegrate::maxp: The integral failed to converge after 1100 integrand evaluations.
NIntegrate obtained -892.994 and 4556.737738794749` for the integral and error estimates. >>
Out[4]= -892.994
In[5]:= NIntegrate[f[x, y, z, w], {w, -5, 5}, {z, -5, 5}, {y, -5, 5}, {x, -5, 5}, MaxPoints -> 10^4, Method -> "MonteCarlo"]
During evaluation of In[5]:= NIntegrate::maxp: The integral failed to converge after 10100 integrand evaluations.
NIntegrate obtained -2101.01 and 1601.1200343423657` for the integral and error estimates. >>
Out[5]= -2101.01
In[6]:= NIntegrate[f[x, y, z, w], {w, -5, 5}, {z, -5, 5}, {y, -5, 5}, {x, -5, 5}, MaxPoints -> 10^5, Method -> "MonteCarlo"]
During evaluation of In[6]:= NIntegrate::maxp: The integral failed to converge after 100100 integrand evaluations.
NIntegrate obtained -2001.8 and 523.7733991685055` for the integral and error estimates. >>
Out[6]= -2001.8
In[7]:= NIntegrate[f[x, y, z, w], {w, -5, 5}, {z, -5, 5}, {y, -5, 5}, {x, -5, 5}, MaxPoints -> 10^6, Method -> "MonteCarlo"]
During evaluation of In[7]:= NIntegrate::maxp: The integral failed to converge after 1000100 integrand evaluations.
NIntegrate obtained -2035.14 and 166.86159178863022` for the integral and error estimates. >>
Out[7]= -2035.14
In[8]:= NIntegrate[f[x, y, z, w], {w, -5, 5}, {z, -5, 5}, {y, -5, 5}, {x, -5, 5}, MaxPoints -> 10^7, Method -> "MonteCarlo"]
During evaluation of In[8]:= NIntegrate::maxp: The integral failed to converge after 10000100 integrand evaluations.
NIntegrate obtained -1898.9 and 52.75810129576391` for the integral and error estimates. >>
Out[8]= -1898.9
In[9]:= NIntegrate[f[x, y, z, w], {w, -5, 5}, {z, -5, 5}, {y, -5, 5}, {x, -5, 5}, MaxPoints -> 10^8, Method -> "MonteCarlo"]
Out[9]= -1916.95
After some initial variation it appeared that the error was decreasing by a factor of three when increasing the number of points by a factor of ten. I was about to calculate how many powers of ten would be needed to decrease the error to 0.01 when the last result appeared.
I do not know that the possible error in the Monte Carlo NIntegrate is precisely documented, but if you believe this calculation is sufficient then I think you are done and the calculation does not take as much time as I feared it would.
Please check all this very carefully for any more errors that have appeared after a single simple copy and paste.
