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 f(double g, double h) {
        double r133011 = 2.0;
        double r133012 = atan2(1.0, 0.0);
        double r133013 = r133011 * r133012;
        double r133014 = 3.0;
        double r133015 = r133013 / r133014;
        double r133016 = g;
        double r133017 = -r133016;
        double r133018 = h;
        double r133019 = r133017 / r133018;
        double r133020 = acos(r133019);
        double r133021 = r133020 / r133014;
        double r133022 = r133015 + r133021;
        double r133023 = cos(r133022);
        double r133024 = r133011 * r133023;
        return r133024;
}
↓
double f(double g, double h) {
        double r133025 = 2.0;
        double r133026 = 2.0;
        double r133027 = 3.0;
        double r133028 = g;
        double r133029 = -r133028;
        double r133030 = h;
        double r133031 = r133029 / r133030;
        double r133032 = acos(r133031);
        double r133033 = atan2(1.0, 0.0);
        double r133034 = r133025 * r133033;
        double r133035 = r133032 + r133034;
        double r133036 = r133027 * r133035;
        double r133037 = r133027 * r133027;
        double r133038 = r133036 / r133037;
        double r133039 = cos(r133038);
        double r133040 = exp(r133039);
        double r133041 = cbrt(r133040);
        double r133042 = log(r133041);
        double r133043 = r133026 * r133042;
        double r133044 = r133043 + r133042;
        double r133045 = r133025 * r133044;
        return r133045;
}

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)))))