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

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

5. Applied associate-/l*1.0

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

7. Applied log1p-expm1-u1.0

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

9. Applied log1p-udef1.0

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

11. Applied flip-+0.1

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

12. Final simplification0.1

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

Runtime

Time bar (total: 19.3s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))