Error

Derivation

1. Initial program 1.0

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

2. Simplified1.0

   \[\leadsto \color{blue}{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right))_*\right) \cdot 2}\]
3. Using strategy rm

4. Applied fma-udef1.0

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

5. Applied cos-sum1.0

   \[\leadsto \color{blue}{\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)} \cdot 2\]
6. Using strategy rm

7. Applied add-sqr-sqrt0.0

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

8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))