Average Error: 1.0 → 0.0
Time: 2.2s
Precision: binary64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\left(-\frac{\sqrt{3}}{2}\right) \cdot \cos \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right) - 0.5 \cdot \sin \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\left(-\frac{\sqrt{3}}{2}\right) \cdot \cos \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right) - 0.5 \cdot \sin \left(\frac{\sin^{-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
  (-
   (* (- (/ (sqrt 3.0) 2.0)) (cos (/ (asin (/ g h)) 3.0)))
   (* 0.5 (sin (/ (asin (/ 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) (((double) -((((double) sqrt(3.0)) / 2.0))) * ((double) cos((((double) asin((g / h))) / 3.0))))) - ((double) (0.5 * ((double) sin((((double) asin((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 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. Simplified1.0

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

  10. Applied acos-asin_binary641.0

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

  11. Applied div-sub_binary641.0

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

  12. Applied associate--r-_binary641.0

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

  13. Applied cos-sum_binary641.0

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

  14. Simplified0.0

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

  15. Simplified0.0

    \[\leadsto 2 \cdot \left(\left(-\frac{\sqrt{3}}{2}\right) \cdot \cos \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right) - \color{blue}{0.5 \cdot \sin \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)}\right)\]

  16. Final simplification0.0

    \[\leadsto 2 \cdot \left(\left(-\frac{\sqrt{3}}{2}\right) \cdot \cos \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right) - 0.5 \cdot \sin \left(\frac{\sin^{-1} \left(\frac{g}{h}\right)}{3}\right)\right)\]

Reproduce

herbie shell --seed 2020210 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))