2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 4.0s

Precision: binary64

Math

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

↓

\[2 \cdot \left(\sqrt[3]{\cos \left(\pi \cdot \frac{2}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{\cos \left(\pi \cdot \frac{2}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)}\right)}^{6}}\right)\]

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-cube 1.6

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

4. Simplified 1.0

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

6. Applied add-cbrt-cube 2.0

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

7. Applied add-cbrt-cube 2.0

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

8. Applied cbrt-undiv 1.0

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

9. Applied add-cbrt-cube 1.0

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

10. Applied cbrt-unprod 1.0

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

11. Simplified 1.0

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

13. Applied add-cube-cbrt 1.0

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

14. Applied unpow-prod-down 1.0

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

15. Applied cbrt-prod 0.1

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

16. Simplified 0.1

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

17. Simplified 0.1

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

18. Final simplification 0.1

    \[\leadsto 2 \cdot \left(\sqrt[3]{\cos \left(\pi \cdot \frac{2}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \cdot \sqrt[3]{{\left(\sqrt[3]{\cos \left(\pi \cdot \frac{2}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)}\right)}^{6}}\right)\]

Reproduce

herbie shell --seed 2020182 
(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)))))