complex log

[Albert Chan]

Code:

dd = require'dd' -- David Bailey's double-double
function ratio_relerr(n, lo, hi)
    local e1, e2, x = 0, 0
    for i = 1,n do
        repeat x = lo + (hi-lo)*random() until x~=1
        local g = dd.new(x)
        g = g*g-1; e1 = e1 + abs(((g-(x+1)*(x-1))/g):tonumber())
        x = x - 1; e2 = e2 + abs(((g-(2*x + x*x))/g):tonumber())
    end
    return e1/e2
end

.5 < h < 3
min(|x|) = sqrt(.5/2) = 0.5
max(|x|) = sqrt(3/1) ≈ 1.732

lua> ratio_relerr(1e6, .5, 1)
1.1272738159812026
lua> ratio_relerr(1e6, 1, 1.732)
1.4886689516224851
lua> ratio_relerr(1e6, .5, 1.732)
1.3195779095764026

OP Clog, (h=|x|-1, 2*h + h*h) statistically more accurate than (x+1)*(x-1)