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. Initial simplification1.0

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

4. Applied add-log-exp1.0

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

6. Applied add-cube-cbrt0.1

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

7. Applied log-prod0.1

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

9. Applied log1p-expm1-u0.1

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

11. Applied fma-udef0.1

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

12. Applied cos-sum0.1

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

13. Final simplification0.1

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

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))

Details

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