What is the Fastest Multiprecision Package for Trigonometric Functions?

[CRGreathouse]

I'm looking for a tool to solve a problem I've been working on. In particular, I want something that implements subquadratic trig functions -- actually just the tangent would work for me. :) I've been using Pari, but its tangent routine seems to be $\mathcal{O}(n^2)$ or $\mathcal{O}(n^2\log n)$ based on timing, and for high precision this simply takes too long. 10,000 digits in Pari takes half a second, but 100,000 takes more than a minute, and ten million takes over a week, which isn't feasible.

Any suggestions? Can anyone test their preferred platform (Math'ca, Maple, etc.) to see how long these take? I was testing tan(1) if you want comparability. I tested Maxima, but it seemed to be inappropriate for the task: in addition to apparently also using a quadratic method, it took ~30 times longer than Pari for similar precisions.

 

[Dragonfall]

I could to test Maple (11) for you, if you told me how.

So what you want is the amount of time (in seconds) that it takes maple to calculate tan(1) to the 10 000th, 100 000th, 1 000 000th, and 10 000 000th digit?

 

[CRGreathouse]

I could to test Maple (11) for you, if you told me how.

So what you want is the amount of time (in seconds) that it takes maple to calculate tan(1) to the 10 000th, 100 000th, 1 000 000th, and 10 000 000th digit?

That would be great. I'd just test it to 10 000, 100 000, 200 000, and 300 000 digits, though -- 10 000 000 would probably take a long time.

The commands should be something like this:

digits := 10000; settime := time(); tan(1): time() − settime;
digits := 100000; settime := time(); tan(1): time() − settime;
digits := 200000; settime := time(); tan(1): time() − settime;
digits := 300000; settime := time(); tan(1): time() − settime;

 

[Dragonfall]

1 000: 0.078s
10 000: 0.280s
100 000: 24.031s
200 000: 97.719s
300 000: 216.750s
1 000 000: > 900s.

It does not seem Maple can be of much help.

 

[slider142]

From Mathematica 6.0 on single core Athlon64 3400+, 2GB RAM, Vista Ultimate OS:

Timing[N[Tan[1], 100000]][[1]]
2.14
Timing[N[Tan[1], 200000]][[1]]
5.422
Timing[N[Tan[1], 300000]][[1]]
9.203
Timing[N[Tan[1], 1000000]][[1]]
44.531

Times are in seconds. Note that Mathematica cautions that it tables previous results for like calculations, so it probably used the previous tangents as stepping stones for the larger numbers.

 

[CRGreathouse]

Yes, Mathematica seems like the solution for me. It took only (!) 1462 seconds to calculate tan(1) to 10,000,000 places:

Timing[N[Tan[1], 10000000];]

 

[CRGreathouse]

1 000: 0.078s
10 000: 0.280s
100 000: 24.031s
200 000: 97.719s
300 000: 216.750s
1 000 000: > 900s.

It does not seem Maple can be of much help.

Thank you for running the tests! I appreciate it. It looks like Maple also uses a quadratic algorithm.