Would scaling numerical coefficient in YALMIP lead to different model send to the solver ?

[JackJack]

Dear Professor Johan,

I need to optimize this term T with some constraints, and suppose that I have a global solver.

Implementation S_1: Just straight up, enter this term exactly in to yalmip




The red symbol $b_m,p_11,p_12$ are my decision variables and the blue symbols is my numerical data. From my expertise, I know that these blue ones in the bracket are badly scaled.

However, I know that this can be fixed by do the following step

Implementation S_2: precompute this badly scaled term, I know from my expertise that this division will give nice values to A_1 and A_2

Here, it is trivial to see that  Implementation S_1 and  Implementation S_2 are practically the same symbolically. Therefore, my questions are

1/Is the model sent to the solver different between  Implementation S_1 and  Implementation S_2? 

2/ Is  Implementation S_2 preferred over  Implementation S_1, and if it is preferred, then preferred over which criteria

Thank you and best regards!