The correct trick is to invert x and negate y if x is greater than 2,
since
((%x)^(-y)) = x^y
. Unfortunately this eats up quite a few characters due to the reliance
on both x and y. Here are two ways to implement it:
e=:1+[:+/[:*/\<:@[*(i.50)(-~%>:@[)]
exp=:e`(%@[e-@])@.(2<[)
or (all on one line)
exp=:(1+[:+/[:*/\<:@[*(i.50)(-~%>:@[)])/@((%@{.,-@{:)^:(2<{.))@,
Either is a complete solution to the given problem aside from the "no
array operations" restriction, although the parameter of 50 is too small
for some arguments.
Marshall
On Tue, Aug 05, 2014 at 12:12:29PM -0400, Marshall Lochbaum wrote:
> The binomial series can be implemented efficiently by grouping the terms
> like this:
>
> ((1+x)^y) = 1 + (x*y) * 1 + (x*2%~y-1) * 1 + (x*3%~y-2) ...
>
> that is,
>
> 1 + +/ */\ x * (y-i._)%(>:i._)
>
> Putting it together (and remembering to decrement x), we have
>
> exp =: 1 + [: +/ [: */\ <:@[ * (i.50) (-~%>:@[) ]
>
> 1.3 exp 9.6
> 12.4124
> 1.3^9.6
> 12.4124
> 1.3 (exp - ^) 9.6
> 5.32907e_15
>
> Unfortunately, this only converges for x where 1 >: |x-1 , that is, a
> disc in the complex plane around 1 of radius 1. To extend it to a
> complete solution, we need to rescale x to fit in that circle. For
> positive numbers, we can take the square root of x and double y until x
> is between zero and two. But that's already quite unwieldy. It's
> probably better to use the exp-multiply-log solution.
>
> Marshall
>
> On Tue, Aug 05, 2014 at 02:14:39PM +0100, Jon Hough wrote:
> > The J is a little out of my league, but for non-integers, youcould use
> > Binomial Theorem, as I said.(http://en.wikipedia.org/wiki/Binomial_series)
> > e.g.
> > e^pi = (1+(e-1))^pi = 1+ pi*e + pi*(pi - 1)*e*e/2! +...
> > There's no exponentiation and you can calculate to arbitrary precision.
> >
> > > Date: Tue, 5 Aug 2014 05:33:42 -0700
> > > From: [email protected]
> > > To: [email protected]
> > > Subject: Re: [Jprogramming] Power for the powerless
> > >
> > > "A really simple approach would be "
> > >
> > > for integer powers,
> > >
> > > pow =: [: */ #~
> > >
> > >
> > >
> > >
> > > ----- Original Message -----
> > > From: Raul Miller <[email protected]>
> > > To: Programming forum <[email protected]>
> > > Cc:
> > > Sent: Tuesday, August 5, 2014 2:36:53 AM
> > > Subject: Re: [Jprogramming] Power for the powerless
> > >
> > > A really simple approach would be to use T.
> > >
> > > pow=: ^ T. 99
> > > That gives you a polynomial expression
> > >
> > >
> > > Here's a shorter version:
> > >
> > > ^ T. 4
> > >
> > > 1 1 0.5 0.16666666666666666&p.
> > >
> > > Here's the more accurate version:
> > >
> > > (^ -: pow) 10 11 12
> > >
> > > 1
> > >
> > >
> > > It's not necessarily efficient, but it's really simple.
> > >
> > >
> > > Thanks,
> > >
> > >
> > > --
> > >
> > > Raul
> > >
> > >
> > >
> > > On Tue, Aug 5, 2014 at 1:01 AM, Dan Bron <[email protected]> wrote:
> > >
> > > > That's a long page, but in brief: can you calculate the power series
> > > > without using ^ explicitly or implicitly (e.g. via t. or #: etc)? Are
> > > > all
> > > > the ^s I see in those power series easily replaced by instances of
> > > > */@:#"0 ?
> > > >
> > > > In other words, does that page teach me how to do the trick when
> > > > literally
> > > > the only mathematical functions in my toolbox are (dyads) + - * % and
> > > > (monad) | ?
> > > >
> > > > -Dan
> > > >
> > > > ----- Original Message ---------------
> > > >
> > > > Subject: Re: [Jprogramming] Power for the powerless
> > > > From: Roger Hui <[email protected]>
> > > > Date: Mon, 4 Aug 2014 21:51:08 -0700
> > > > To: Programming forum <[email protected]>
> > > >
> > > > ?Can you not just use power series (for both exp and ln)? See
> > > > http://www.jsoftware.com/jwiki/Essays/Extended%20Precision%20Functions
> > > > .?
> > > >
> > > >
> > > > On Mon, Aug 4, 2014 at 9:39 PM, Dan Bron <[email protected]> wrote:
> > > >
> > > > > There's a StackExchange puzzle which challeges us to implement power
> > > > (i.e.
> > > > > dyad ^) using only the simple arithmetic dyads + - * % and monad |
> > > > > [1].
> > > > In
> > > > > other words, we may not use ^ or ^. or variants. There are still
> > > > > several
> > > > > open questions on the puzzle, not least of which involves the domain
> > > > > of
> > > > > the inputs (can the base be negative?) and range of the outputs (how
> > > > > much
> > > > > precision is required?), but neverthless we can make some assumptions
> > > > > and
> > > > > start to sketch an approach.
> > > >
> > > >
> > > > ----------------------------------------------------------------------
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> > >
> > >
> > >
> > > >
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> >
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