Average Error: 1.0 → 0.0
Time: 1.3m
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\sqrt[3]{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right))_*\right) \cdot \cos \left((\frac{2}{3} \cdot \pi + \left(\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right))_*\right)} \cdot \sqrt[3]{\cos \left((\frac{2}{3} \cdot \pi + \left(\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right))_*\right)}\right) \cdot 2\]

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 add-sqr-sqrt1.0

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

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

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

  6. Applied times-frac1.0

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

  8. Applied add-cbrt-cube1.0

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

  10. Applied cbrt-prod0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))