Average Error: 1.0 → 0.1
Time: 3.8s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\cos \left(\frac{3 \cdot \left(\cos^{-1} \left(\frac{-g}{h}\right) + 2 \cdot \pi\right)}{3 \cdot 3}\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\frac{3 \cdot \left(\cos^{-1} \left(\frac{-g}{h}\right) + 2 \cdot \pi\right)}{3 \cdot 3}\right)}}\right)\right)\]
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)
double code(double g, double h) {
	return (2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0))));
}
↓
double code(double g, double h) {
	return (2.0 * ((2.0 * log(cbrt(exp(cos(((3.0 * (acos((-g / h)) + (2.0 * ((double) M_PI)))) / (3.0 * 3.0))))))) + log(cbrt(exp(cos(((3.0 * (acos((-g / h)) + (2.0 * ((double) M_PI)))) / (3.0 * 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. Using strategy rm

  3. Applied frac-add1.0

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

  4. Simplified1.0

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

  6. Applied add-log-exp1.0

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

  8. Applied add-cube-cbrt0.1

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

  9. Applied log-prod0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020049 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))