According to Simulation by Ross, 5th edition p 72, it discusses how to generate a pair of exponentials with mean 1.

Step 1: Generate U1,U2 from U(0,1) .

Step 2 : Let t:=log(U1U2)

Step 3 : Generate U3 from U(0,1)

Step 4: X=tU3,Y=tX then they follow exp(1) .

The book's guidance is to use the fact : XX+Y=tU(0,t) when X,Yiidexp(1).

I could prove this fact but I don't know how it helps me sample two exponential variables.