Average Error: 1.0 → 0.1
Time: 6.9s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\sqrt[3]{\cos \left(\frac{\mathsf{fma}\left(\pi, 3, \frac{3}{2} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)}{\frac{3}{2} \cdot 3}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\pi, 3, \frac{3}{2} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)}{\frac{3}{2} \cdot 3}\right)} \cdot \sqrt[3]{\cos \left(\frac{\mathsf{fma}\left(\pi, 3, \frac{3}{2} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)}{\frac{3}{2} \cdot 3}\right)}\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\sqrt[3]{\cos \left(\frac{\mathsf{fma}\left(\pi, 3, \frac{3}{2} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)}{\frac{3}{2} \cdot 3}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\pi, 3, \frac{3}{2} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)}{\frac{3}{2} \cdot 3}\right)} \cdot \sqrt[3]{\cos \left(\frac{\mathsf{fma}\left(\pi, 3, \frac{3}{2} \cdot \cos^{-1} \left(\frac{-g}{h}\right)\right)}{\frac{3}{2} \cdot 3}\right)}\right)
double code(double g, double h) {
	return ((double) (2.0 * ((double) cos(((double) (((double) (((double) (2.0 * ((double) M_PI))) / 3.0)) + ((double) (((double) acos(((double) (((double) -(g)) / h)))) / 3.0))))))));
}
↓
double code(double g, double h) {
	return ((double) (2.0 * ((double) (((double) cbrt(((double) (((double) cos(((double) (((double) fma(((double) M_PI), 3.0, ((double) (((double) (3.0 / 2.0)) * ((double) acos(((double) (((double) -(g)) / h)))))))) / ((double) (((double) (3.0 / 2.0)) * 3.0)))))) * ((double) cos(((double) (((double) fma(((double) M_PI), 3.0, ((double) (((double) (3.0 / 2.0)) * ((double) acos(((double) (((double) -(g)) / h)))))))) / ((double) (((double) (3.0 / 2.0)) * 3.0)))))))))) * ((double) cbrt(((double) cos(((double) (((double) fma(((double) M_PI), 3.0, ((double) (((double) (3.0 / 2.0)) * ((double) acos(((double) (((double) -(g)) / h)))))))) / ((double) (((double) (3.0 / 2.0)) * 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. Using strategy rm

  3. Applied *-commutative1.0

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

  4. Applied associate-/l*1.0

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

  5. Applied frac-add1.0

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

  6. Simplified1.0

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

  8. Applied add-cbrt-cube1.5

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

  9. Simplified1.0

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

  11. Applied add-cube-cbrt1.0

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

  12. Applied unpow-prod-down1.0

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

  13. Applied cbrt-prod0.1

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

  14. Simplified0.1

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))