Average Error: 1.0 → 0.0
Time: 21.0s
Precision: 64
Internal Precision: 320
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\left(\left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)} \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{2}{3} \cdot \pi\right)}}\right) \cdot \cos \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)\]

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. Applied simplify1.0

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

  4. Applied fma-udef1.0

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

  5. Applied cos-sum1.0

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

  7. Applied add-cube-cbrt0.0

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

  9. Applied add-cube-cbrt0.0

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

Runtime

Time bar (total: 21.0s)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))