According to Simulation by Ross, 5th edition p 72, it discusses how to generate a pair of exponentials with mean 1.
Step 1: Generate U_1, U_2 from U(0,1).
Step 2 : Let t := -log(U_1U_2)
Step 3 : Generate U_3 from U(0,1)
Step 4: X = tU_3, Y = t - X then they follow exp(1).
The book says that X|X+Y = t ~ U(0, t) when X and Y are independent and follow exp(1) but I don't know how to proceed.