Imagine you have a soccer field, a ball with position x and y, a kick strength, and direction in an angle. Your job is to write a function that determines if the ball will end up in a goal. So that is 4 values. However the function itself will contain many intricacies, like trig functions, simulated drag, and so on.

In the contest of LLMs, you cant have these types of coded function. Your function has to be a mathematical equation that is smooth - i.e no discrete steps, no singularities. The reason for this is when any neural net is trained, you use backpropagation of the error to adjust weights, and how much you adjust them is directly proportional to the weights effect on the final output, and in order to compute this, you have to have smooth functions from start to finish.

So what you do instead is you add data to your 4 values, that capture different relationship between them. If your 4 values are x,y,k,and h, your first data point can be a1x + b1y + c1k + d1h. The second point can be a2x + b3y + c4k + d5h. And so on. You can have as many of those values as you want. And then you can add, combine, and scale those values in any way you chose.

This basically gives you a map of 4 values into a binary decision whether the ball will end up in a goal or not, after sufficient training. However, the total number of extra values that you chose has to be large enough to capture all possibilities - if you don't have enough, you will start to make mistakes for some initial conditions.