Average Error: 1.0 → 0.0
Time: 2.9s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\sqrt[3]{{\left(\cos \left(\pi \cdot \frac{2 + 1}{3} - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}^{2}} \cdot \sqrt[3]{\cos \left(\pi \cdot \frac{2 + 1}{3} - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\sqrt[3]{{\left(\cos \left(\pi \cdot \frac{2 + 1}{3} - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)}^{2}} \cdot \sqrt[3]{\cos \left(\pi \cdot \frac{2 + 1}{3} - \frac{\cos^{-1} \left(\frac{g}{h}\right)}{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
  (*
   (cbrt (pow (cos (- (* PI (/ (+ 2.0 1.0) 3.0)) (/ (acos (/ g h)) 3.0))) 2.0))
   (cbrt (cos (- (* PI (/ (+ 2.0 1.0) 3.0)) (/ (acos (/ g h)) 3.0)))))))
double code(double g, double h) {
	return ((double) (2.0 * ((double) cos(((double) ((((double) (2.0 * ((double) M_PI))) / 3.0) + (((double) acos((((double) -(g)) / h))) / 3.0)))))));
}
↓
double code(double g, double h) {
	return ((double) (2.0 * ((double) (((double) cbrt(((double) pow(((double) cos(((double) (((double) (((double) M_PI) * (((double) (2.0 + 1.0)) / 3.0))) - (((double) acos((g / h))) / 3.0))))), 2.0)))) * ((double) cbrt(((double) cos(((double) (((double) (((double) M_PI) * (((double) (2.0 + 1.0)) / 3.0))) - (((double) acos((g / h))) / 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 Error: 1.0 bits

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

  3. Applied distribute-frac-negError: 1.0 bits

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

  4. Applied acos-negError: 1.0 bits

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

  5. Applied div-subError: 1.0 bits

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

  6. Applied associate-+r-Error: 1.0 bits

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

  7. SimplifiedError: 1.0 bits

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

  9. Applied add-cbrt-cubeError: 1.0 bits

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

  10. SimplifiedError: 1.0 bits

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

  12. Applied add-cube-cbrtError: 1.0 bits

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

  13. Applied unpow-prod-downError: 1.0 bits

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

  14. Applied cbrt-prodError: 0.1 bits

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

  15. SimplifiedError: 0.0 bits

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

  16. SimplifiedError: 0.0 bits

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

  17. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))