MiniPi

Mini Pi approximations in JS (WIP)

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Gregory-Leibniz series:

π = (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + (4/13) - (4/15) …

(exponential and very inefficient: ~3h to get 6 decimals, ~18h to get 7 decimals, …)

• commented
• golfed (256b)

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Nilakantha series:

π = 3 + (4/(2*3*4)) - (4/(4*5*6)) + (4/(6*7*8)) - …

(exponential: computes 10+ decimals in a few seconds then it gets very long)

• commented
• golfed (220b)

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Machin formula:

π = 16*atan(1/5) - 4*atan(1/239)
with atan(x) = x - x^3/3 + x^5/5 - x^7/7…

which is equal to:

π = (16 * (1/5 ^ 1 / 1) - 4 * (1/239 ^ 1 / 1)) + (- 16 * (1/5 ^ 3 / 3) + 4 * (1/239 ^ 3 / 3)) + (16 * (1/5 ^ 5 / 5) - 4 * (1/239 ^ 5 / 5)) + (- 16 * (1/5 ^ 7 / 7) + 4 * (1/239 ^ 7 / 7)) …

(linear! computes ~1,4 digit per iteration, inspired by http://stephenbrooks.org/misc/jspi/decimal_animated.html)

• readable
• golfed (397b)

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Other “simple” formulae to test maybe:

• π = x * sin(180 / x), in degrees, for x as big as possible
• π = 2 * (Arcsin(sqrt(1 - x^2)) + abs(Arcsin(x))), for x in [-1;1]
• π = (k_6 sqrt(k_3))/(S), where k_1 = 545140134; k_2 = 13591409; k_3 = 640320; k_4 = 100100025; k_5 = 327843840; k_6 = 53360; S = sum_(n = 0)^oo (-1)^n ((6n)!(k_2 + nk_1))/(n!^3(3n)!(8k_4k_5)^n)
• π = SUM (k=0 to infinity) 16^-k [ 4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6) ]. (Bailey, Borwein, Plouffe formula)
• π/2 = (2*2)/(1*3)*(4*4)/(3*5)*(6*6)/(5*7)… (Walli’s formula)
• and many more!