The approximation is only around 4.28*10^-101 off from pi.
https://www.wolframalpha.com/input?i=integral+from+-a+to+a+of+%28integral+from+-a+to+a+of+e%5E-%28x%5E2%2By%5E2%29dy%29dx
https://www.wolframalpha.com/input?i=%281-erf%28e%5Ee%29%5E2%29*pi
And because it always bears repeating;
According to JPL’s Chief Engineer for Mission Operations and Science, Marc Rayman-
Let's go to the largest size there is: the known universe. The radius of the universe is about 46 billion light years. Now let me ask (and answer!) a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom, the simplest atom? It turns out that 37 decimal places (38 digits, including the number 3 to the left of the decimal point) would be quite sufficient.
Technically you need another 20 digits if you want to get down to a Planck length. (57 digits in total)
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