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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. Initial simplification1.0

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

4. Applied distribute-frac-neg1.0

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

5. Applied acos-neg1.0

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

6. Applied div-sub1.0

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

7. Applied associate-+l-1.0

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

8. Applied cos-diff0.1

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

9. Simplified0.1

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

11. Applied acos-asin0.1

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

12. Applied div-sub0.1

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

13. Applied associate--l-0.0

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

14. Simplified0.0

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

15. Simplified0.0

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

16. Final simplification0.0

    \[\leadsto \sqrt{3} \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3} - \frac{\pi}{\frac{3}{2}}\right) + \cos \left(\left(\frac{\pi}{6} - \frac{\pi}{\frac{3}{2}}\right) - \frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)\]

Reproduce

herbie shell --seed 2018362 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))

Details

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