Average Error: 1.0 → 0.1
Time: 3.5s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\sqrt[3]{{\left(\cos \left(\pi \cdot \frac{2}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}^{2}} \cdot \sqrt[3]{\cos \left(\pi \cdot \frac{2}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)}\right)\]

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-cube1.5

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

  4. Simplified1.0

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

  6. Applied add-cube-cbrt1.0

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

  7. Applied unpow-prod-down1.0

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

  8. Applied cbrt-prod0.1

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

  9. Simplified0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020184 
(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)))))