Π day

[Albert Chan]

Here is getting C(x) = asin(√x)/(√x), using "half-angle" formula for asin

Code:

function C(x)   -- = asin(sqrt(x)) / sqrt(x)
    if x < 4e-4 then return 1+x/6*(1+x*9/20/(1-x*25/42)) end
    local y = 0.5/(sqrt(1-x)+1)
    y = 2*sqrt(y) * C(x*y)
    print('C( '..x..' ) = ',y)
    return y
end

Redo both previous examples in 1 go, with less calculations (2 sqrt per recursion)

Code:

lua> C(1)
C( 0.0006022718974138037 ) =    1.0001004058641836
C( 0.002407636663901557 ) =     1.0004017081549652
C( 0.009607359798384776 ) =     1.0016081890839748
C( 0.038060233744356624 ) =     1.0064545427995637
C( 0.14644660940672624 ) =      1.0261721529770307
C( 0.5 ) =      1.1107207345395913
C( 1 ) =        1.5707963267948963