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

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

4. Applied add-sqr-sqrt1.0

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

6. Applied add-cbrt-cube1.6

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

8. Applied cbrt-prod0.1

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

9. Simplified0.1

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

10. Final simplification0.1

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

Runtime

Time bar (total: 20.5s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))