Average Error: 1.0 → 0.1
Time: 17.2s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\sqrt[3]{\log_* (1 + (e^{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{3} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right))_*\right)} - 1)^*) \cdot \sqrt[3]{\log_* (1 + (e^{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{3} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right))_*\right)} - 1)^*) \cdot \log_* (1 + (e^{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{3} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right))_*\right)} - 1)^*)}} \cdot \sqrt[3]{\sqrt[3]{\log_* (1 + (e^{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{3} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right))_*\right)} - 1)^*) \cdot \log_* (1 + (e^{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{3} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right))_*\right)} - 1)^*)} \cdot \sqrt[3]{\log_* (1 + (e^{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{3} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right))_*\right)} - 1)^*) \cdot \log_* (1 + (e^{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{3} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right))_*\right)} - 1)^*)}}\right) \cdot 2\]

Error

Bits error versus g

Bits error versus h

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

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

  4. Applied *-un-lft-identity1.0

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

  5. Applied add-sqr-sqrt1.0

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

  6. Applied times-frac1.0

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

  7. Simplified1.0

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

  9. Applied log1p-expm1-u1.0

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

  11. Applied add-cbrt-cube1.6

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

  13. Applied add-cube-cbrt0.1

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

  14. Applied associate-*l*0.1

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

  15. Applied cbrt-prod0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))