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

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

  4. SimplifiedError: 1.0 bits

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

  5. SimplifiedError: 1.0 bits

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

  7. Applied add-sqr-sqrtError: 0.0 bits

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

  8. Applied associate-*l*Error: 0.0 bits

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

  9. SimplifiedError: 0.0 bits

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

  10. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))