  In this paper, we revisit the classic approximate All-Pairs Shortest Paths
(APSP) problem in undirected graphs. For unweighted graphs, we provide an
algorithm for $2$-approximate APSP in $\tilde O(n^{2.5-r}+n^{\omega(r)})$ time,
for any $r\in[0,1]$. This is $O(n^{2.032})$ time, using known bounds for
rectangular matrix multiplication $n^{\omega(r)}$ [Le Gall, Urrutia, SODA
2018]. Our result improves on the $\tilde{O}(n^{2.25})$ bound of [Roditty, STOC
2023], and on the $\tilde{O}(m\sqrt n+n^2)$ bound of [Baswana, Kavitha, SICOMP
2010] for graphs with $m\geq n^{1.532}$ edges.
  For weighted graphs, we obtain $(2+\epsilon)$-approximate APSP in $\tilde
O(n^{3-r}+n^{\omega(r)})$ time, for any $r\in [0,1]$. This is $O(n^{2.214})$
time using known bounds for $\omega(r)$. It improves on the state of the art
bound of $O(n^{2.25})$ by [Kavitha, Algorithmica 2012]. Our techniques further
lead to improved bounds in a wide range of density for weighted graphs. In
particular, for the sparse regime we construct a distance oracle in $\tilde
O(mn^{2/3})$ time that supports $2$-approximate queries in constant time. For
sparse graphs, the preprocessing time of the algorithm matches conditional
lower bounds [Patrascu, Roditty, Thorup, FOCS 2012; Abboud, Bringmann, Fischer,
STOC 2023]. To the best of our knowledge, this is the first 2-approximate
distance oracle that has subquadratic preprocessing time in sparse graphs.
  We also obtain new bounds in the near additive regime for unweighted graphs.
We give faster algorithms for $(1+\epsilon,k)$-approximate APSP, for
$k=2,4,6,8$.
  We obtain these results by incorporating fast rectangular matrix
multiplications into various combinatorial algorithms that carefully balance
out distance computation on layers of sparse graphs preserving certain distance
information.
