In article <lkoi5v$vfj$1...@speranza.aioe.org>, Mark H Harris <harrismh...@gmail.com> wrote: >On 5/11/14 1:59 PM, Chris Angelico wrote: >>>> julia> prec=524288 >>>> 524288 >>> >>>> julia> with_bigfloat_precision(prec) do >>>> println(atan(BigFloat(1)/5)*16 - atan(BigFloat(1)/239)*4) >>>> end >> >> Would it be quicker (and no less accurate) to represent pi as >> atan(BigFloat(1))*4 instead? That's how I originally met a >> pi-calculation (as opposed to "PI = 3.14" extended to however much >> accuracy someone cared to do). > > No. Simple experiment will show you. The atan(x<=1) will converge >faster. For 524288 bits atan(1) formula converged in 3 seconds, and >Machin's formula atan(x<1) converged in 2 seconds. Where it becomes very >apparent is 10K and 100K or above. Also, the difference is much more >noticeable in Python than in Julia, but it is there no-the-less. > > But here is the cool part: what if your Ï function could be broken >down into three very fast converging atan(x<1) functions like this one: > > > pi = 24*atan(1/8) + 8*atan(1/57) + 4*atan(1/239) (Shanks used this) > > >... and then, you have julia send each piece to a separate >processor|core (it does this at its center) and they converge together, >then julia pieces them together at the end. Then things get incredibly >faster.
I know now how to interpret your posts. Using "incredible" for a mere factor of at most 3. Balanced views are more convincing. Groetjes Albert > -- Albert van der Horst, UTRECHT,THE NETHERLANDS Economic growth -- being exponential -- ultimately falters. albert@spe&ar&c.xs4all.nl &=n http://home.hccnet.nl/a.w.m.van.der.horst
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