Hi Fredrik,

These papers might provide benchmark problems
for QQbar/AA arithmetic which originate in actual
math problems, though not in Sage.

Pierre-Vincent Koseleff, Fabrice Rouillier, Cuong Tran
On the Sign of a Trigonometric Expression
ISSAC’15, July 6–9, 2015, Bath, United Kingdom.
http://dx.doi.org/10.1145/2755996.2756664.

Koseleff, P.-V., Pecker, D., and Rouillier, F.
Computing Chebyshev knot diagrams.
http://arxiv.org/abs/1001.5192

Koseleff, P.-V., Pecker, D., and Rouillier, F.
The first rational Chebyshev knots.
Journal of Symbolic Computation 45 (2010), 1341–1358. 

See Pierre-Vincent Koseleff's publications page:
https://webusers.imj-prg.fr/~pierre-vincent.koseleff/publications.html

There are some interesting algebraic numbers in Theorem 3 of

Shigeki Akiyama
Minimum polyhedron with n vertices
https://arxiv.org/abs/1704.03149

which maybe can also provide benchmark problems.

Happy benchmarking!   --Samuel

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