I wrote:
> [log*(n)] can be expressed directly in J using agenda (@.):
>
> logstar =: 0:`(1+$:@^.)@.(1<:])"0
>
> But this feels fairly "procedural". All the work is in the recursion.
> Is there a more elegant way to express it?
Roger responded:
> In practice log* (at least for base 2) can have only a very few values,
> so something involving I. should work quite well.
>
> f=: 1&<: + (^^:(1 2 3)1)&I.
This is neat! Thanks for the idea.
How would you extend it to handle large numbers? For example, Wikipedia
has a table of values up to 2^65536 (that is, 2^2^2^2^2 or ^/5#2), but:
f 2^65536x
|domain error: f
| f 2^2^16
whereas the less-than-satisfying recursive approach can tackle it:
logstar 2^65536x
4
Anyway, I think what bugs me about logstar, more than anything else, is the
recursion. So I suppose we could stick with a fundamentally iterative
approach, but hide some of the infrastructure by using one of J's more
elegant tools, the adverb ^:a: :
# <.@^.^:(>&1)^:a: 2^65536x
5
Though because extended integers and floating point values don't mix, I had
to cheat a little and take the floor of the log. I can't imagine a
scenario where this would effect the outcome (unless logstar is later
extended to take one non-integral values, such that log*(5) would be
[infinitesimally] larger than log*(4) ).
-Dan
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