2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 3.9s

Precision: 64

Math

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

↓

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

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) (2.0 * ((double) log(((double) cbrt(((double) exp(((double) cos(((double) (((double) (3.0 * ((double) (((double) acos(((double) (((double) -(g)) / h)))) + ((double) (2.0 * ((double) M_PI))))))) / ((double) (3.0 * 3.0)))))))))))))) + ((double) log(((double) cbrt(((double) exp(((double) cos(((double) (((double) (3.0 * ((double) (((double) acos(((double) (((double) -(g)) / h)))) + ((double) (2.0 * ((double) M_PI))))))) / ((double) (3.0 * 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 frac-add 1.0

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

4. Simplified 1.0

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

6. Applied add-log-exp 1.0

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

8. Applied add-cube-cbrt 0.1

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

9. Applied log-prod 0.1

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

10. Simplified 0.1

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

11. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020123 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))