transform_bbox is much faster when one does not need a TransformedBbox,
i.e. something that keeps track of later changes in the transform:

In [1]: from pylab import *
   ...: from matplotlib.transforms import *
   ...: tr = plt.gca().transAxes
   ...: bb = Bbox([[0, 0], [1, 1]])
   ...: %timeit TransformedBbox(bb, tr).x0
   ...: %timeit tr.transform_bbox(bb).x0
   ...: %timeit TransformedBbox(bb, tr).width
   ...: %timeit tr.transform_bbox(bb).width
11.8 µs ± 145 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
2.72 µs ± 31.5 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
11.9 µs ± 183 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
2.83 µs ± 16.6 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

(I don't know if any of the places changed here are actually performance-critical, but this just seems good practice...)

PR Summary

PR Checklist

• Has pytest style unit tests (and pytest passes).
• Is Flake 8 compliant (run flake8 on changed files to check).
• New features are documented, with examples if plot related.
• Documentation is sphinx and numpydoc compliant (the docs should build without error).
• Conforms to Matplotlib style conventions (install flake8-docstrings and run flake8 --docstring-convention=all).
• New features have an entry in doc/users/next_whats_new/ (follow instructions in README.rst there).
• API changes documented in doc/api/next_api_changes/ (follow instructions in README.rst there).