Adaptive step size Runge Kutta: getting a specific value A system of differential equations can be solved using a numerical integrator with adaptive stepsize $ h$, e.g. Runge Kutta Fehlberg

Now suppose I want to integrate a set of differential equations $\{\dot{x}, \dot{y}, \dot{z} \}$ numerically (e.g. the 3D trajectory of some particle). Each differential equation uses the same adaptive stepsize $h$.

I want to evaluate $x,y$ at some specific target value of $z = z_t$ 

Now, due to the adaptive stepsize, it is unlikely that $z$ will ever be exactly $z_t$. That is to say, I might get for integration step $n$ and step $n+1$, $z_n=z_t + \epsilon$, $z_{n+1}=z_t - \delta$ for small values $\epsilon, \delta$

Obviously, by setting a more strict tolerance on the integrator, I can get close to $z_t$, but this would be at the expense of speed.

Given that I know a priori that I want to evaluate $x,y$ at $z=z_t$, is there a method which permits both an adaptive stepsize, and an exact $z=z_t$ result?

Thanks
Once you choose a method, it has an order which approximates how the error depends on stepsize.  The popular fourth order Runge-Kutta has an error that scales as $h^4$.  You can get an approximation to the accuracy by doing your step once as a full step and once as two half steps.  You would expect the error to be reduced by a factor $16$ so the difference between them is about $15$ times the error of the two step approach.  You can use this to update your stepsize.  There is a discussion in chapter 17.2 of Numerical Recipes and probably in many other numerical analysis texts.
Once you choose a method, it has an order which approximates how the error depends on stepsize.  The popular fourth order Runge-Kutta has an error that scales as $h^4$.  You can get an approximation to the accuracy by doing your step once as a full step and once as two half steps.  You would expect the error to be reduced by a factor $16$ so the difference between them is about $15$ times the error of the two step approach.  You can use this to update your stepsize.  There is a discussion in chapter 17.2 of Numerical Recipes and probably in many other numerical analysis texts.