Fortran (Was: The "does Python have variables?" debate)

[Mark H Harris]

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 π 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.