Average Error: 1.0 → 0.1
Time: 14.8s
Precision: 64
Internal Precision: 128
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left({\left(e^{\sqrt{\log_* (1 + (\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right))_*)}}\right)}^{\left(\sqrt{\log_* (1 + (\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right))_*)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\log_* (1 + (\frac{2}{3} \cdot \pi + \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right))_*)}\right) \cdot \sin 1\right)\]

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 expm1-log1p-u1.0

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

  6. Applied expm1-udef1.0

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

  7. Applied cos-diff1.0

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

  9. Applied add-sqr-sqrt1.0

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

  10. Applied exp-prod0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))