Average Error: 1.0 → 1.0
Time: 17.7s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \cos \left((\frac{2}{3} \cdot \pi + \left(\frac{\sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}}{3} \cdot \sqrt{\cos^{-1} \left(\frac{-g}{h}\right)}\right))_*\right)\]

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. 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

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

  5. Applied associate-/l*1.0

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

  7. Applied associate-/r/1.0

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

  8. Final simplification1.0

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))