You cannot define a shape preserving quadratic spline that is differentiable across the breaks.
So while you think you want it, you will be unhappy with the result, even if you did manage to formulate such an animal.
Why you cannot use pchip I don't know, because pchip will be extrmely fast. My guess is you are not using pchip efficiently, but I have no clue as to what it is you are doing with the code, so that is difficult to guess. But I will claim that any quadratic spline you do formulate will not provide any significant speedup over pchip, IF used properly. There will be many other places in your code where you would gain far more, even if you could find some insignificant speed gains over pchip.
