Question for Trig Gurus

[Albert Chan]

(07-31-2022 11:08 AM)Thomas Klemm Wrote:  Inverse Cosine

It turns out that we can not use \(d(2, 2)\) here as well to get 5 correct figures.
So we need to compute \(d(3, 3)\) ...

There is no difference in convergence rate, whether we pick atan, asin, or acos
All based on extrapolation of sequence a_k = (c1/2^k) / tan(c2/2^k) ⇒ a_∞ = c1/c2

Basing all from atan(x) is better, because acot(x) = atan(1/x), without using square roots.
Using d(3,3), and atan argument reduced to within to ±1/√3, we get 9+ correct digits.
(with simpler d(2,2), we have 6+ digits accuracy)

Code:

function atand(x) -- = deg(atan(x)), 9+ digits accuracy
    if signbit(x) then return -atand(-x) end
    if x > 1 then return 90 - atand(1/x) end
    if x > 1/sqrt(3) then return 45 - atand((1-x*x)/(2*x))/2 end
    local g = sqrt(1+x*x)   -- g0
    local a = (1+g)/2       -- a1
    local s = -1 + 84*a
    g = sqrt(a*g)           -- g1
    a = (a+g)/2             -- a2
    s = s + 704*a + 2048*sqrt(a*g)
    return x/s * 510300/pi
end

lua> function asind(x) return atand(x/sqrt(1-x*x)) end -- range = -90 .. 90
lua> function acosd(x) return 90 - asind(x) end -- range = 0 .. 180

lua> deg(atan(0.6)) , atand(0.6)
30.96375653207352  30.963756534561696
lua> deg(acos(0.6)) , acosd(0.6)
53.13010235415598  53.13010235413812
lua> deg(acos(0.7)) , acosd(0.7)
45.5729959991943    45.5729959991943