Ramanujan's constant: Difference between revisions

=== 𝑒**(π*√x) ===

<pre>

Note 'citation for pi computation'

@MISC {3129700,

TITLE = {Series that converge to $\pi$ quickly},

AUTHOR = {El Ectric (https://math.stackexchange.com/users/301661/el-ectric)},

HOWPUBLISHED = {Mathematics Stack Exchange},

NOTE = {URL:https://math.stackexchange.com/q/3129700 (version: 2019-02-28)},

EPRINT = {https://math.stackexchange.com/q/3129700},

URL = {https://math.stackexchange.com/q/3129700}

}

)

NB. returns u to at least y significant digits

Digits=: adverb define NB. u Digits y u y is less accurate than u y+1

format=. ' _.' -.~ ((j.~ 50&+) y)&":

i =. 5

current=. format u i

whilst. last -.@-: current do.

last =. current

i =. i + 2

current=. format result =. u i

end.

result

)

rationalize_decimal=: 3 :0 NB. rationalize_decimal LITERAL

y=. y -. ' '

i=. y (, i. ]) '.'

I=. 'x' ,~ i {. y

F=. 'x' ,~ y }.~ >: i

I +&". (, ' % ' , 'x' ,~ '1' #!.'0'~ 1 j. <:@:#) F

)

NB. pi is sufficiently fast for partial series recomputation.

numerator=: (*&! +:) * _3 25&p.

denominator=: 2&^ * !@:(3&*)

pi=: (2 * [: +/ numerator % denominator)@:i.@:x: NB. use: pi TERMS

cf_sqrt=: 10&$: :(4 :0) NB. continued fraction approximation to square root. x sqrt y then x is the depth, y the square

a =. x: <. %: y NB. estimate

r =. y - *: a NB. remainder

a + %`+/ (+: x) $ r , +: a

)

exp=: (1x"_)`((($:~<:)+^%!@x:@])~)@.(0<[) NB. recursive Taylor series x exp y recursively sums x terms of Taylor series for Exp[y], memoization candidate.

</pre>

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NB. takes the constant beyond the repeat 9s.

S=: cf_sqrt&163 Digits 34

P=: pi Digits 34

Y=: rationalize_decimal 37j34":1r8*P*S

f=: exp&Y M. NB. memoize

59j40 ": 8 ^~ f Digits 34

262537412640768743.9999999999992500584460275237074973516438

NB. ^

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