Average Error: 1.0 → 0.1
Time: 10.3s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\sqrt[3]{\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \cdot \sqrt[3]{{\left(\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}^{2}}\right)\]
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(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)} \cdot \sqrt[3]{{\left(\cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}^{2}}\right)
double f(double g, double h) {
        double r150405 = 2.0;
        double r150406 = atan2(1.0, 0.0);
        double r150407 = r150405 * r150406;
        double r150408 = 3.0;
        double r150409 = r150407 / r150408;
        double r150410 = g;
        double r150411 = -r150410;
        double r150412 = h;
        double r150413 = r150411 / r150412;
        double r150414 = acos(r150413);
        double r150415 = r150414 / r150408;
        double r150416 = r150409 + r150415;
        double r150417 = cos(r150416);
        double r150418 = r150405 * r150417;
        return r150418;
}
↓
double f(double g, double h) {
        double r150419 = 2.0;
        double r150420 = atan2(1.0, 0.0);
        double r150421 = r150419 * r150420;
        double r150422 = 3.0;
        double r150423 = r150421 / r150422;
        double r150424 = g;
        double r150425 = -r150424;
        double r150426 = h;
        double r150427 = r150425 / r150426;
        double r150428 = acos(r150427);
        double r150429 = r150428 / r150422;
        double r150430 = r150423 + r150429;
        double r150431 = cos(r150430);
        double r150432 = cbrt(r150431);
        double r150433 = 2.0;
        double r150434 = pow(r150431, r150433);
        double r150435 = cbrt(r150434);
        double r150436 = r150432 * r150435;
        double r150437 = r150419 * r150436;
        return r150437;
}

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 cube-mult1.6

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

  7. Applied cbrt-prod0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))