Let's run adaptive RK on 
.
. f = @(t,u) exp(t-u*sin(u));
[t,u] = rk23(f,[0,5],0,1e-5);
plot(t,u,'.')
xlabel('t'), ylabel('u(t)') % ignore this line
title('Adaptive IVP solution') % ignore this line
The solution makes a very abrupt change near 
. The resulting time steps vary over three orders of magnitude.
. The resulting time steps vary over three orders of magnitude.semilogy(t(1:end-1),diff(t))
axis tight, title('Adaptive step sizes') % ignore this line
xlabel('t'), ylabel('step size') % ignore this line
If we had to run with a uniform step size to get this accuracy, it would be
h_min = min(diff(t))
On the other hand, the average step size that was actually taken was
h_avg = mean(diff(t))
We took fewer steps by a factor of almost 1000! Even accounting for the extra stage per step and the occasional rejected step, the savings are clear.