HP50g simplifing a root

[Albert Chan]

I was being stupid. Why solve cubic equation for a
This is the way to get cube roots simplified:

\(\sqrt[3]{A ± \sqrt{R}} = a ± \sqrt{r} \quad ⇒ \quad a = \large\frac{\sqrt[3]{A+\sqrt{R}}
\;+\; \sqrt[3]{A-\sqrt{R}}}{2} \)

Round a to closest halves, get r, and double check if it round-trip back to (A, R)

Code:

function simp_cbrt4(A,B,k)  -- simplify cbrt(A + B * sqrt(k))
    local R = B*B*k
    local a = sqrt(abs(R))
    if R<0 then a = (A*A-R)^(1/6) * cos(atan2(a,A)/3)
    else        a = (cbrt(A+a) + cbrt(A-a))/2
    end
    a = a + 0x3p50 - 0x3p50 -- round to halves
    local r = (A/a-a*a)/3
    local b = B/(3*a*a+r)
    if r == b*b*k then return a,b,k end
end

lua> simp_cbrt4(1859814842094, -59687820010, 415)
11589      -145         415
lua> simp_cbrt4(300940299,103940300,101)
99           100          101
lua> simp_cbrt(180, 23, 157)   -- (a,b) can be halves
1.5          0.5          157
lua> simp_cbrt4(-36, 20, -7)    -- work with complex roots too.
3             1             -7
lua> simp_cbrt4(81,30,-3)       -- this is simplest, see comment from previous post.
4.5          0.5          -3