2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 3.5s

Precision: binary64

Math

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

↓

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

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

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

6. Applied add-log-exp 1.0

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

7. Simplified 1.0

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

9. Applied add-cube-cbrt 1.0

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

10. Applied unpow-prod-down 1.0

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

11. Applied exp-prod 1.0

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

12. Applied log-pow 1.0

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

13. Applied cbrt-prod 0.1

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

14. Simplified 0.1

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

15. Simplified 0.1

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

16. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020179 
(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)))))