Bignums would certainly be useful here. Regards, Elias
On 7 August 2014 14:00, Frederick H. Pitts <fred.pi...@comcast.net> wrote: > Elias, > > Juergen informed me that he had changed how monadic ! is > calculated in > in a reply with the above subject line. I assumed he made the change in > an effort to increase the range of exact results produced by dyadic ! > before the latter reverts to producing floating point numbers that are > limited to 15-decimal digit accuracy(4 digits less than the 64-bit > integer is capable of producing). > > So "No" my issue is not with monadic ! with integers arguments per > se. > My issue is that for increasing integer arguments dyadic ! starts > producing floating point results long before the limit imposed by a > signed 64-bit integer is reached. This situation results from a literal > implementation of the binomial formula n!/(k!(n-k)!). At n=21, GnuAPL > starts representing the value of n! as a floating point number and when > n=66, n! is so large that the 15 decimal digit precision is insufficient > for the calculation of an exact integer result of dyadic !. The BINOM > defined function gets around the limitation by canceling prime factors > between numerator and denominator before evaluating the factorials. > > BTW, for integer arguments monadic ! can produce a answer for n = > 0 to > n = 170. At n = 171 GnuAPL reports a DOMAIN ERROR because the result is > greater than the upper limit of double precision numbers. Also note > that monadic ! only produces exact integer results for n = 0 to n = 20. > Above n = 20, the results no longer fit in a 64-bit signed integer. I > have no desire to memoize a bunch of floating point numbers that are not > suited for the task at hand, calculating exact binomial coefficients > over a large a range as the 64-bit signed integer will allow. > > Regards, > > Fred > > > On Thu, 2014-08-07 at 11:36 +0800, Elias Mårtenson wrote: > > Frederick, > > > > > > Is you issue with monadic ! for integer arguments only? If so, since > > the are only 70 or so valid inputs to this function, the results can > > be precalculated and the implementation can simple dereference the > > result from an array. > > > > > > For fractional parameters to monadic !, the result of the Gamma > > function has to be computed which of course is much more expensive. > > > > > > Regards, > > Elias > > > > > > On 7 August 2014 11:15, Frederick H. Pitts <fred.pi...@comcast.net> > > wrote: > > Hello Juergen, > > > > I reran timing tests comparing the defined BINOM > > function with the > > dyadic ! function after upgrading to SVN rev. 422. I saw no > > difference > > in the BINOM and dyadic ! results and maybe a factor of 2 > > slowdown in > > the dyadic ! which I attributed to the change in how monadic ! > > is > > calculated. I was hoping to see the dyadic ! produce exact > > integer > > results up to the 9200000000000000000 upper limit on 64-bit > > GnuAPL > > integers after the change. :-(. > > > > Is the change to monadic ! actually in SVN rev. 422? > > Your email does > > not say. > > > > If the change is there, then the 15-digit precision of > > double precision > > numbers (even if the 15 digits are exact) simply isn't enough > > precision > > to exactly calculate the binomial ! if the right argument to > > the latter > > is 37 or greater. BINOM gives exact results for right > > arguments up to > > 66 with a result that approaches the upper limit of 64-bit > > integers. > > > > I'm reporting this only to let you know that the > > change to monadic ! > > did not impact the dyadic !, if in fact the change is in SVN, > > other than > > to slow it down. > > > > Regards, > > > > > > > > > > On Wed, 2014-08-06 at 17:29 +0200, Juergen Sauermann wrote: > > > > > Hi Fred, > > > > > > I did a small rework of the monadic ! function. It should > > now be > > > exact (well, as exact as ×/⍳N is) for integer arguments up > > to 170 > > > (i.e. the max integer before DOMAIN ERROR is thrown), > > > > > > Before it was calling tgamma() of libm. > > > > > > There are still many cases where GNU APL calls libm > > functions > > > and replacing them all by more exact versions would be > > rather tedious. > > > There are often speed-precision trade-offs involved. > > Normally the > > > precision of libm functions is good enough. Let's fix issues > > on a > > > case-by-case basis when we find them. > > > > > > /// Jürgen > > > > > > > > > On 08/06/2014 03:06 AM, Frederick H. Pitts wrote: > > > > > > > Gentle people, > > > > > > > > Please find attached binom.apl.tgz. It contains the > > source for BINOM, > > > > a defined function does what the primitive dyadic ! > > function does but > > > > produces exact results over a larger range. > > > > > > > > BINOM produces 1) the same results as ! over the > > range where ! > > > > produces exact integers, 2) exact integer results in range > > above that > > > > produced by the ! but below the 9200000000000000000 upper > > limit of GNU > > > > 64-bit integers and 3) floating point results that match > > those of ! > > > > above the 64-bit integer upper limit. > > > > > > > > In the interest of full disclosure: > > > > 1) The code is slow; 100 times slower than the primitive > > dyadic !. But > > > > then BINOM is interpreted while ! is compiled in to the > > interpreter. > > > > 2) The code was presented in 1996 on comp.lang.apl by Jim > > Weigang and > > > > others. > > > > > > > > Enjoy, > > > > > > > > Fred > > > > Retired Chemical Engineer > > > > > > > > > > > > > > > > > >
