implement cos and sin of power series

[chapoton]

As we already have exp and log.

[chapoton]

This has been extracted from #14878.

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New commits:

​20163eb	trac #14878 sin for power series, step1
​ed55e28	trac #14878 step 2, cos for multi-variable power series
​0a0921c	Merge branch 'u/chapoton/14878' into 7.1.b0
​2e093f6	trac #14878 now sin and cos for all power series
​a7cb277	Merge branch 'u/chapoton/14878' into 7.1.b2

[rws]

You're practically substituting into the series for x. This is fast for small order but may not be optimal for say 1000 coefficients.

You may be interested in general in a timing comparison. For Symengine there is a benchmark to compute sin(x+x^2)*cos(x+x^2) to order 1000 which needs 120ms using the Flint implementation on my machine. Your branch at the moment takes 2.1s which is not too bad.

[rws]

Apart from the not optimal performance at higher order this looks good. Please cc me with any further tickets like this.

sage: R.set_default_prec(100)
sage: time _ = sin(x*exp(x))
CPU times: user 44 ms, sys: 0 ns, total: 44 ms
Wall time: 44.7 ms
sage: R.set_default_prec(200)
sage: time _ = sin(x*exp(x))
CPU times: user 216 ms, sys: 0 ns, total: 216 ms
Wall time: 214 ms
sage: R.set_default_prec(300)
sage: time _ = sin(x*exp(x))
CPU times: user 660 ms, sys: 8 ms, total: 668 ms
Wall time: 667 ms
sage: R.set_default_prec(400)
sage: time _ = sin(x*exp(x))
CPU times: user 1.61 s, sys: 4 ms, total: 1.62 s
Wall time: 1.61 s
sage: R.set_default_prec(500)
sage: time _ = sin(x*exp(x))
CPU times: user 3.19 s, sys: 4 ms, total: 3.2 s
Wall time: 3.19 s

[rws]

The cases in comment:3 are computed about 10x faster by pynac-0.7.0.