# What are the all possible ways to increase the precision of this numerical integral?

I have to evaluate the following code

``````me = SetPrecision(1, 50);
mp = SetPrecision(400, 50);
M = mp + me;
(Omega) = SetPrecision(1000, 50);
(Gamma) = SetPrecision(0.5*M*(Omega), 50);
{(Alpha), (Beta)} =
SetPrecision({0.8086371511120481`, 495.09178203818004`}, 50);
chvar((Alpha)_, (Beta)_, re_, rp_) :=
E^(-(((me re + mp rp)^2 (Gamma))/(me + mp)^2) - (Alpha) RealAbs(
re - rp) - (Beta) RealAbs(re - rp)^2);
overlap((Alpha)_, (Beta)_) :=
NIntegrate(
re^2 rp^2 (E^(-(((me re + mp rp)^2 (Gamma))/(me +
mp)^2) - (Alpha) RealAbs(re - rp) - (Beta) RealAbs(
re - rp)^2))^2, {re, 0, 10}, {rp, 0, 10});
lapP((Alpha)_, (Beta)_, re_, rp_) :=
E^(-(((me re + mp rp)^2 (Gamma))/(me + mp)^2) - (Alpha) RealAbs(
re - rp) - (Beta) RealAbs(re - rp)^2) (-2 (Beta) - (
2 mp^2 (Gamma))/(me + mp)^2 + ((re - rp)^2 (Alpha))/
RealAbs(re - rp)^3 - (Alpha)/RealAbs(re - rp)) + (
2 E^(-(((me re + mp rp)^2 (Gamma))/(me +
mp)^2) - (Alpha) RealAbs(re - rp) - (Beta) RealAbs(
re - rp)^2) (2 (re - rp) (Beta) - (
2 mp (me re + mp rp) (Gamma))/(me +
mp)^2 + ((re - rp) (Alpha))/RealAbs(re - rp)))/rp +
E^(-(((me re + mp rp)^2 (Gamma))/(me + mp)^2) - (Alpha) RealAbs(
re - rp) - (Beta) RealAbs(re - rp)^2) (2 (re - rp) (Beta) - (
2 mp (me re + mp rp) (Gamma))/(me +
mp)^2 + ((re - rp) (Alpha))/RealAbs(re - rp))^2;
kinP((Alpha)_, (Beta)_) := -1/(2* mp)*
NIntegrate(
re^2*rp^2*chvar((Alpha), (Beta), re, rp)*
lapP((Alpha), (Beta), re, rp), {re, 0, 10}, {rp, 0, 10});
TP = kinP((Alpha), (Beta))/overlap((Alpha), (Beta))
``````

Evaluating this code gives `749.661702852` for TP while I know the true value is `750.0221`. Since obtained value is close to true one, seems that it’s just a matter of precision (also Mathematica gives warnings during evaluation), so I added

``````WorkingPrecision -> 60, MaxRecursion -> 50, PrecisionGoal -> 10
``````

for two included `NIntegrate` commands in the code, but it didn’t improve the result as I expected, Any idea to get true result?
Any help would be appreciated