Average Error: 1.0 → 0.0
Time: 32.7s
Precision: 64
Internal Precision: 384
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot (e^{\sqrt[3]{{\left(\sqrt[3]{{\left(\log_* (1 + \cos \left((\left(\frac{2}{3}\right) \cdot \pi + \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right))_*\right))\right)}^{3}}\right)}^{3}}} - 1)^*\]

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 expm1-log1p-u1.0

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

  4. Applied simplify1.0

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

  6. Applied add-cbrt-cube1.0

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

  7. Applied simplify1.0

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

  9. Applied add-cbrt-cube0.0

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

  10. Applied simplify0.0

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

Runtime

Time bar (total: 32.7s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))