The latter is a bit slower than Genz and PDE approach.
If we require to compute something like M(x<=0.4, y<=0.12, rho=0.9) with 7 digits precision how long wil Tanh, Genz and PDE take? The PDE is *a lot* slower than Tanh or Genz in this case. You need to explain how you measure PDE speed and what it's precision is and why my simple example conflicts with that.
I think Genz pretty much does what's stated, translate it to a 1d integral.
Your analysis is incorrect. Your intuition tells you it is slower. I have done all the tests and will post on the other thread.
And this thread on "best method", Efficiency is only one of the metrics.
BTW have you ever used these methods? What's your run time for Tanh for example? Is it 10, 20, 30 times slower than Genz?
Your results show me it's slower, there is no intuition.You need very large grid to get a couple of digits precision.
Shall we do a defined bet then? I bet that:
1) to compute M(x<=0.4, y<=0.12, rho=0.9) with 7 digits precision we have Tahn and Genz being *much* faster than PDE. Probably at least a factor 100. You claim as quoted that PDE is faster and so I bet it's it slower by at leat a factor 10. You are allowed to use Richardson extrapolation but need to of course include that in your timing. The Genz answer to this bet is 5.195173365E-1
To answer your question: depending on the required precision you need to pick a number of quadrature points for the 2d numerical tanh integral . In my experience both Gauss Lengedre and Tanh quatrature converge very fast (as a function of the number of nodes) fro smooth function like the bivariate Guassian. A 2d numerical Tanh integral is more generic than the specialized Genz method and I expect Genz to be at least 5 times fasters for similar precision.
With the low precision 7 digits requirements in the bet the PDE method will also be at least factor 1000 less accurate. So its' both slower *and* less accurate by orders of magnitudes!