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

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

4. Applied acos-asin1.0

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

5. Applied div-sub1.0

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

6. Applied associate-+l-1.0

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

7. Applied cos-diff1.0

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

9. Applied add-cube-cbrt0.1

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

10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019050 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))