2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.0

Time: 6.2s

Precision: 64

Math

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

↓

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

C

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) (((double) (2.0 * 3.0)) + ((double) (((double) (3.0 / ((double) M_PI))) * ((double) acos(((double) (((double) -(g)) / h)))))))) / ((double) (((double) (3.0 / ((double) M_PI))) * 3.0)))))) * ((double) cos(((double) (((double) (((double) (2.0 * 3.0)) + ((double) (((double) (3.0 / ((double) M_PI))) * ((double) acos(((double) (((double) -(g)) / h)))))))) / ((double) (((double) (3.0 / ((double) M_PI))) * 3.0)))))))))) * ((double) cbrt(((double) cos(((double) (((double) (((double) (2.0 * 3.0)) + ((double) (((double) (3.0 / ((double) M_PI))) * ((double) acos(((double) (((double) -(g)) / h)))))))) / ((double) (((double) (3.0 / ((double) M_PI))) * 3.0))))))))))));
}

Error

Bits error versus g

Bits error versus h

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 associate-/l* 1.0

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

4. Applied frac-add 1.0

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

6. Applied add-cbrt-cube 1.0

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

7. Simplified 1.0

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

9. Applied add-cube-cbrt 1.0

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

10. Applied unpow-prod-down 1.0

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

11. Applied cbrt-prod 0.1

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

12. Simplified 0.1

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

13. Simplified 0.0

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

14. Final simplification 0.0

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

Reproduce

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