How to 'normalize' the product of two variables from Gaussian distribution?

I have two variables, x1 and x2, which are sampled from two Gaussian distributions respectively.

I created an iteraction term x3 which is x1 multiply x2. Not surprisingly, x3 has very fat tail, ie., the value stretches far into right end and left end of x-axis.

Although linear regression does not equire x3 to be Gaussian, I still would 'cut' those very large values of x3

Obviously I can simply put a cap. But I am wondering if there is a way to systematically 'concentrate' all the x3 values towards 0

I have tried to take square root of x3 , for those negative x3, I use -1 * sqrt(x3*-1) . However, the result has two bells, one for positive and one for negative. I am wondering if there is a better way to 'contentrate' x3 that produces a distribution similar to Gaussian (I don't need it to be exactly Gaussian) , can someone share some idea?

Thanks

• If you are just going to force the data to be whatever you want it to be, why bother doing any analysis at all? – StatsStudent Apr 21 at 15:28
• What is the point of doing this? Once modified, it's not the interaction any longer. – Glen_b Apr 22 at 17:20