Average Error: 1.0 → 0.0
Time: 4.3s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{2 \cdot \pi}{3}} \cdot \sqrt[3]{\frac{2 \cdot \pi}{3}}\right) \cdot \sqrt[3]{\frac{2 \cdot \pi}{3}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)}\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\sqrt[3]{\cos \left(\left(\sqrt[3]{\frac{2 \cdot \pi}{3}} \cdot \sqrt[3]{\frac{2 \cdot \pi}{3}}\right) \cdot \sqrt[3]{\frac{2 \cdot \pi}{3}} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \cdot \sqrt[3]{\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)}\right)
double f(double g, double h) {
        double r141173 = 2.0;
        double r141174 = atan2(1.0, 0.0);
        double r141175 = r141173 * r141174;
        double r141176 = 3.0;
        double r141177 = r141175 / r141176;
        double r141178 = g;
        double r141179 = -r141178;
        double r141180 = h;
        double r141181 = r141179 / r141180;
        double r141182 = acos(r141181);
        double r141183 = r141182 / r141176;
        double r141184 = r141177 + r141183;
        double r141185 = cos(r141184);
        double r141186 = r141173 * r141185;
        return r141186;
}
↓
double f(double g, double h) {
        double r141187 = 2.0;
        double r141188 = atan2(1.0, 0.0);
        double r141189 = r141187 * r141188;
        double r141190 = 3.0;
        double r141191 = r141189 / r141190;
        double r141192 = cbrt(r141191);
        double r141193 = r141192 * r141192;
        double r141194 = r141193 * r141192;
        double r141195 = g;
        double r141196 = -r141195;
        double r141197 = h;
        double r141198 = r141196 / r141197;
        double r141199 = acos(r141198);
        double r141200 = r141199 / r141190;
        double r141201 = r141194 + r141200;
        double r141202 = cos(r141201);
        double r141203 = r141191 + r141200;
        double r141204 = cos(r141203);
        double r141205 = r141202 * r141204;
        double r141206 = cbrt(r141205);
        double r141207 = cbrt(r141204);
        double r141208 = r141206 * r141207;
        double r141209 = r141187 * r141208;
        return r141209;
}

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 add-cbrt-cube1.6

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

  4. Simplified1.0

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

  6. Applied add-cube-cbrt1.0

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

  7. Applied unpow-prod-down1.0

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

  8. Applied cbrt-prod0.1

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

  9. Simplified0.1

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

  10. Simplified0.1

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

  12. Applied add-cube-cbrt0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))