Average Error: 1.0 → 0.0
Time: 3.6s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\left(\left(\sqrt[3]{\cos \left(\frac{2 \cdot \pi}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{2 \cdot \pi}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{2 \cdot \pi}{3}\right)}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) - \left(\sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)} \cdot \sqrt{\sin \left(\frac{2 \cdot \pi}{3}\right)}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)\]

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 cos-sum1.0

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

  5. Applied add-sqr-sqrt0.0

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

  7. Applied add-cube-cbrt0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020153 
(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)))))