Re: fastest way to add up a billion numbers
- To: mathgroup at smc.vnet.net
- Subject: [mg74230] Re: fastest way to add up a billion numbers
- From: hemphill at hemphills.net
- Date: Thu, 15 Mar 2007 04:56:56 -0500 (EST)
- References: <es92b2$3oj$1@smc.vnet.net> <200703030555.AAA02643@smc.vnet.net> <200703040453.XAA25305@smc.vnet.net> <200703050946.EAA19233@smc.vnet.net> <45EC435A.30700@wolfram.com> <et54hq$atu$1@smc.vnet.net>
- Reply-to: hemphill at alumni.caltech.edu
Andrzej Kozlowski <akoz at mimuw.edu.pl> writes: > Certianly for small vaues explicit addition is performed, as can be > seen from: > > Trace[Sum[i, {i, 1, n}] /. n -> 5] > > Out[9]= > {{HoldForm[Sum[i, {i, 1, n}]], > HoldForm[(1/2)*n*(n + 1)]}, > HoldForm[(1/2)*n*(n + 1) /. n -> 5], > HoldForm[(5*(1 + 5))/2], {HoldForm[1 + 5], > HoldForm[6]}, HoldForm[(5*6)/2], HoldForm[15]} > > In[10]:= > Trace[Sum[i, {i, 1, 5}]] > > Out[10]= > {HoldForm[Sum[i, {i, 1, 5}]], {HoldForm[i], HoldForm[1]}, > {HoldForm[i], HoldForm[2]}, {HoldForm[i], HoldForm[3]}, > {HoldForm[i], HoldForm[4]}, {HoldForm[i], HoldForm[5]}, > HoldForm[1 + 2 + 3 + 4 + 5], HoldForm[15]} > > The second Trace certianly suggests explicit summation. > > On the other hand, indeed we have: > > In[22]:= > Trace[Sum[i, {i, 1, 10^9}]] > > Out[22]= > {HoldForm[Sum[i, {i, 1, 10^9}]], {HoldForm[10^9], > HoldForm[1000000000]}, HoldForm[500000000500000000]} > > > If this is right, it sugges that the algorithm switches at some point > form explicit addition to using a formula. One could obviously find > out the point at which the algorithm switches form one approach to > the other, I suspect the number would b eunder 1000. I have version 5.1 for Linux. The algorithm switches approaches at 10^6+1. Scott -- Scott Hemphill hemphill at alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear
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- From: Murray Eisenberg <murray@math.umass.edu>
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- From: Andrzej Kozlowski <akoz@mimuw.edu.pl>
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