Average Error: 26.9 → 2.8
Time: 3.3m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\left(\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}}{\sqrt[3]{\left(cos \cdot x\right) \cdot sin}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}{\left(cos \cdot x\right) \cdot sin}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}{\left(cos \cdot x\right) \cdot sin}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\left(\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}}{\sqrt[3]{\left(cos \cdot x\right) \cdot sin}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}{\left(cos \cdot x\right) \cdot sin}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}{\left(cos \cdot x\right) \cdot sin}}
double f(double x, double cos, double sin) {
        double r21395498 = 2.0;
        double r21395499 = x;
        double r21395500 = r21395498 * r21395499;
        double r21395501 = cos(r21395500);
        double r21395502 = cos;
        double r21395503 = pow(r21395502, r21395498);
        double r21395504 = sin;
        double r21395505 = pow(r21395504, r21395498);
        double r21395506 = r21395499 * r21395505;
        double r21395507 = r21395506 * r21395499;
        double r21395508 = r21395503 * r21395507;
        double r21395509 = r21395501 / r21395508;
        return r21395509;
}
↓
double f(double x, double cos, double sin) {
        double r21395510 = 2.0;
        double r21395511 = x;
        double r21395512 = r21395510 * r21395511;
        double r21395513 = cos(r21395512);
        double r21395514 = cos;
        double r21395515 = r21395514 * r21395511;
        double r21395516 = sin;
        double r21395517 = r21395515 * r21395516;
        double r21395518 = r21395513 / r21395517;
        double r21395519 = cbrt(r21395518);
        double r21395520 = cbrt(r21395517);
        double r21395521 = r21395519 / r21395520;
        double r21395522 = r21395518 / r21395517;
        double r21395523 = cbrt(r21395522);
        double r21395524 = r21395521 * r21395523;
        double r21395525 = r21395524 * r21395523;
        return r21395525;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 26.9

    \[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]

  2. Simplified2.7

    \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*2.5

    \[\leadsto \color{blue}{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt2.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}}\]
  7. Using strategy rm

  8. Applied cbrt-div2.8

    \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}}{\sqrt[3]{sin \cdot \left(x \cdot cos\right)}}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}}\]

  9. Final simplification2.8

    \[\leadsto \left(\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}}{\sqrt[3]{\left(cos \cdot x\right) \cdot sin}} \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}{\left(cos \cdot x\right) \cdot sin}}\right) \cdot \sqrt[3]{\frac{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}{\left(cos \cdot x\right) \cdot sin}}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))