Rational functions are powerful because they can approximate functions closely near singularities, but the same power makes them difficult to work with. If you represent a rational function in the obvious way as a polynomial quotient $r = p/q$, then in interesting cases $p$ and $q$ will vary by many orders of magnitude over the region of interest. This can make computation in floating point arithmetic effectively impossible.
A solution to this problem is to represent $r$ in barycentric form instead, $r = N/D$, where $N$ and $D$ are partial fractions based on certain adaptively selected support points. This idea led to the Chebfun aaa
algorithm a few months ago [2], and now it has further led to an improvement in our capabilities for rational best approximation on an interval. The old remez
code has been replaced by a new and much more powerful minimax
command [1]. We believe this is the most powerful implementation of the Remez algorithm ever produced.
As a famous example in this area, let us consider type $(n,n)$ rational approximation of $x$ on $[1,1]$ for various $n$. Up to 2016, Chebfun's remez
code was only able to go up to $(8,8)$. For example, on p. 192 of [3], a plot is presented of errors up to type $(50,50)$, but they are taken from a table rather than computed on the fly.
Now, by contrast, here we go to type $(80,80)$. Chebfun has to work a bit, but the computation is successful.
x = chebfun('x'); f = abs(x); tic, [p,q,r] = minimax(f,80,80,'silent'); toc xx = linspace(1,1,3000).^3; LW = 'linewidth'; FS = 'fontsize'; plot(xx,f(xx)r(xx),LW,3) grid on, ylim(1e11*[1 1]) title('error curve for type (80,80) approximation',FS,36)
Elapsed time is 21.560009 seconds.
Computing such an approximations in ordinary 16digit arithmetic, so far as we know, is unprecedented. Varga, Carpenter, and Ruttan computed these approximations in the 1990s using 200digit extended precision [4].
The difficulty lies with the exponentially clustered equioscillation points (and poles along the imaginary axis, clustering near $x=0$). This clustering makes a $p/q$ representation out of the question. To show the exponential effect, we can plot the right half of the error curve on a semilogx scale:
xx = logspace(14,0,5000); semilogx(xx,f(xx)r(xx),LW,3) grid on, axis([1e14 1 1e11 1e11]) title('semilogx scale',FS,36)
Floatingpoint computing with rational functions like this has been effectively impossible in the past; we seem to be entering a new era.
References

B. Beckermann, S.I. Filip, Y. Nakatsukasa, and L. N. Trefethen, Rational minimax approximation via adaptive barycentric representations, to appear.

Y. Nakatsukasa, O. Sete, and L. N. Trefethen, The AAA algorithm for rational approximation, arXiv 2016:1612.00337.

L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, 2013.

R. S. Varga, A. Ruttan, and A. D. Carpenter, Numerical results on best uniform rational approximation of $x$ on $[1,+1]$. Mathematics of the USSRSbornik 74 (1993), 271.