\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

↓

\[2 \cdot \left(\left(\left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \cos \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\]

Derivation

Initial program 1.0
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
Applied simplify1.0
\[\leadsto \color{blue}{2 \cdot \cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right))_*\right)}\]
Using strategy rm
Applied fma-udef1.0
\[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{2}{3} \cdot \pi + \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)}\]
Applied cos-sum1.0
\[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)}\]
Using strategy rm
Applied add-cube-cbrt0.0
\[\leadsto 2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right)} \cdot \cos \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\]

Runtime

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))

Error

Derivation

Runtime