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This program solve x^3 = a*x + b, for x

Code:
cubic_ab(a,b) :=  BEGIN LOCAL d; a, b := a/3, b/2;    /* x^3 = (3a)*x + (2b) */ d := sqrt(b^2-a^3) * (-1)^(sign(re(b))==-1); b := surd(b+d, 3); return [a/b, b];     /* alpha, beta */ END;
Code:
cubic(a,b) := BEGIN LOCAL w := exp(2*pi/3*i); [[1,1],[w,conj(w)],[conj(w),w]] * cubic_ab(a,b); END;

Michael Penn's example

Let x = sin(θ) + cos(θ), it reduced to cubic: x^3 = 3*x - 11/8

Cas> r := cubic(3, -11/8)
Cas> simplify(r)                 → [1/2,    (3*√5-1)/4,       (-3*√5-1)/4]
Cas> float(Ans)                  → [0.5, 1.42705098312, −1.92705098312]
Cas> float(r)                      → [1.42705098312, −1.92705098312, 0.5]

We have Cas simplify bug here! symbolic and approximate numbers don't match.
Luckily, the algorithm doesn't care which cube roots is used (no need for principle root).

But, this bug should be fixed.

Cas> b := cubic_ab(3, -11/8) [2]     → (1/256*(-√15*48*i-176))^(1/3)
Cas> polar(float(b))                         → 1., −0.77627903074
Cas> polar(simplify(b))                    → 1 , atan(√15)

Simplify bug had phase angle off by 2*pi/3:

Cas> atan(√15) - 2*pi/3.0               → −0.77627903074

Update: cubic_ab(a,b) code explained here
Root simplify returned wrong (not principle) root bug existed for XCas 1.5.0

But, it is worse on XCas 1.9.0-31 (mingw32 win32)
Result does not match phase *and* magnitude.

XCas> b := (1/256*(-√15*48*i-176))^(1/3)
XCas> polar(float(b))

1.0, -0.77627903074

XCas> simplify(polar(b))                     // good

$$\displaystyle 1\;,\;\frac{-\pi +\arctan \left(\frac{3}{11} \sqrt{15}\right)}{3}$$

XCas> simplify(polar(simplify(b)))       // bad

$$256\;,\;-\pi +\arctan \left(\frac{3}{11} \sqrt{15}\right)$$
Yes, there is an additional check in recent versions of Xcas, and here it failed because an argument was not reduced mod 2*pi.
https://github.com/geogebra/giac/commit/...ea8fd94c9d
Reference URL's
• HP Forums: https://www.hpmuseum.org/forum/index.php
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