Average Error: 1.0 → 0.0
Time: 2.6s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(-\cos \left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(\frac{g}{h}\right)}{\sqrt[3]{3}}\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(-\cos \left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(\frac{g}{h}\right)}{\sqrt[3]{3}}\right)\right)
(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))
↓
(FPCore (g h)
 :precision binary64
 (*
  2.0
  (-
   (cos (* (/ 1.0 (* (cbrt 3.0) (cbrt 3.0))) (/ (acos (/ g h)) (cbrt 3.0)))))))
double code(double g, double h) {
	return 2.0 * cos(((2.0 * ((double) M_PI)) / 3.0) + (acos(-g / h) / 3.0));
}
↓
double code(double g, double h) {
	return 2.0 * -cos((1.0 / (cbrt(3.0) * cbrt(3.0))) * (acos(g / h) / cbrt(3.0)));
}

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

    \[\leadsto \color{blue}{2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)}\]
  3. Using strategy rm

  4. Applied distribute-frac-neg_binary641.0

    \[\leadsto 2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \frac{\cos^{-1} \color{blue}{\left(-\frac{g}{h}\right)}}{3}\right)\]

  5. Applied acos-neg_binary641.0

    \[\leadsto 2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \frac{\color{blue}{\pi - \cos^{-1} \left(\frac{g}{h}\right)}}{3}\right)\]

  6. Applied div-sub_binary641.0

    \[\leadsto 2 \cdot \cos \left(\pi \cdot 0.6666666666666666 + \color{blue}{\left(\frac{\pi}{3} - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)\]

  7. Applied associate-+r-_binary641.0

    \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)}\]

  8. Applied cos-diff_binary640.0

    \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right) + \sin \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}\]

  9. Simplified0.0

    \[\leadsto 2 \cdot \left(\color{blue}{\left(-\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)} + \sin \left(\pi \cdot 0.6666666666666666 + \frac{\pi}{3}\right) \cdot \sin \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

  10. Simplified1.0

    \[\leadsto 2 \cdot \left(\left(-\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right) + \color{blue}{0}\right)\]
  11. Using strategy rm

  12. Applied add-cube-cbrt_binary641.0

    \[\leadsto 2 \cdot \left(\left(-\cos \left(\frac{\cos^{-1} \left(\frac{g}{h}\right)}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}\right)\right) + 0\right)\]

  13. Applied *-un-lft-identity_binary641.0

    \[\leadsto 2 \cdot \left(\left(-\cos \left(\frac{\color{blue}{1 \cdot \cos^{-1} \left(\frac{g}{h}\right)}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}\right)\right) + 0\right)\]

  14. Applied times-frac_binary640.0

    \[\leadsto 2 \cdot \left(\left(-\cos \color{blue}{\left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(\frac{g}{h}\right)}{\sqrt[3]{3}}\right)}\right) + 0\right)\]

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2020233 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))