Re: Re: Floating-Point Computing

*To*: mathgroup at smc.vnet.net*Subject*: [mg93952] Re: [mg93927] Re: Floating-Point Computing*From*: "Hobbs, Sylvia (DPH)" <Sylvia.Hobbs at state.ma.us>*Date*: Sat, 29 Nov 2008 04:32:37 -0500 (EST)*References*: <ggofnk$rsh$1@smc.vnet.net> <200811281211.HAA01366@smc.vnet.net>

List is my formal name. You can just call me Sylvia. > -----Original Message----- > From: mathgroup-adm at smc.vnet.net [mailto:mathgroup-adm at smc.vnet.net] = On Behalf > Of Szabolcs Horv=E1t > Sent: Friday, November 28, 2008 7:11 AM > To: mathgroup at smc.vnet.net > Subject: [mg93927] Re: Floating-Point Computing > > Antonio wrote: > > Dear List, > > > > I came across this Sun article on floating point: > > http://developers.sun.com/solaris/articles/fp_errors.html > > > > which calculates the following code: > > > > main() > > { > > register i; > > double f=0.0; > > double sqrt(); > > > > for(i=0;i<=1000000000;i++) > > { > > f+=sqrt( (double) i); > > } > > printf("Finished %20.14f\n",f); > > } > > > > whose correct answer using Euler-Maclaurin formula is: > > f=21081851083600.37596259382529338 > > > > Tried to implement in mathematica, just for fun, so that later I = could > > play with precision, but the kernel has no more memory and shuts = down. > > > > N[Total[Sqrt[Range[1000000000]]]] > > > > If all those numbers are stored as 32-bit integers in a packed array, > they would take up about 4*10^9/1024^3 ~= 3.7 GB space. That is = already > too much for a 32-bit machine, and more than most present-day personal > computers can handle. Mapping Sqrt to the list unpacks it, eating up > even more memory. Creating a symbolic sum with Total and only > numericising at the end doesn't help either. > > The only reasonable approach here is to write code that is equivalent = to > the C version, i.e. doesn't pre-compute the list of numbers to be > summed. But that will run too slowly in Mathematica. So Mathematica = is > not really the right tool for summing 10^9 numbers. > > Two possible versions: > > fun1[n_Integer] := Module[{f = 0.}, > Do[f += N@Sqrt[i], {i, n}]; > f > ] > > fun2 = Compile[{{n, _Integer}}, Module[{f = 0.}, > Do[f += N@Sqrt[i], {i, n}]; > f > ] > ] > > fun2[10^9] can finish in a reasonable time, but it only works with > machine precision numbers (because of Compile). > > If you sum fewer numbers, use Total@Sqrt@N@Range[n] instead of > N@Total@Sqrt@Range[n].

**References**:**Re: Floating-Point Computing***From:*Szabolcs Horvát <szhorvat@gmail.com>