Average Error: 1.0 → 0.0
Time: 4.1s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\left(\left(\sqrt[3]{\cos \left(\sqrt[3]{{\left(\frac{2 \cdot \pi}{3}\right)}^{3}}\right)} \cdot \sqrt[3]{\cos \left(\sqrt[3]{{\left(\frac{2 \cdot \pi}{3}\right)}^{3}}\right)}\right) \cdot \sqrt[3]{\cos \left(\sqrt[3]{{\left(\frac{2 \cdot \pi}{3}\right)}^{3}}\right)}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) - \sin \left(\sqrt[3]{{\left(\frac{2 \cdot \pi}{3}\right)}^{3}}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\]

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. Using strategy rm

  3. Applied add-cbrt-cube1.0

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

  4. Applied add-cbrt-cube1.0

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

  5. Applied add-cbrt-cube1.0

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

  6. Applied cbrt-unprod1.0

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

  7. Applied cbrt-undiv1.0

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

  8. Simplified1.0

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

  10. Applied cos-sum1.0

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

  12. Applied add-cube-cbrt0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020155 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (neg g) h)) 3.0)))))