Average Error: 27.6 → 2.9
Time: 20.7s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\left(\sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}} \cdot \sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}}\right) \cdot \sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\left(\sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}} \cdot \sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}}\right) \cdot \sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}}
double f(double x, double cos, double sin) {
        double r2164682 = 2.0;
        double r2164683 = x;
        double r2164684 = r2164682 * r2164683;
        double r2164685 = cos(r2164684);
        double r2164686 = cos;
        double r2164687 = pow(r2164686, r2164682);
        double r2164688 = sin;
        double r2164689 = pow(r2164688, r2164682);
        double r2164690 = r2164683 * r2164689;
        double r2164691 = r2164690 * r2164683;
        double r2164692 = r2164687 * r2164691;
        double r2164693 = r2164685 / r2164692;
        return r2164693;
}
↓
double f(double x, double cos, double sin) {
        double r2164694 = x;
        double r2164695 = 2.0;
        double r2164696 = r2164694 * r2164695;
        double r2164697 = cos(r2164696);
        double r2164698 = cos;
        double r2164699 = r2164694 * r2164698;
        double r2164700 = sin;
        double r2164701 = r2164699 * r2164700;
        double r2164702 = r2164697 / r2164701;
        double r2164703 = 1.0;
        double r2164704 = r2164703 / r2164700;
        double r2164705 = r2164704 / r2164699;
        double r2164706 = r2164702 * r2164705;
        double r2164707 = cbrt(r2164706);
        double r2164708 = r2164707 * r2164707;
        double r2164709 = r2164708 * r2164707;
        return r2164709;
}

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 27.6

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

  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 *-un-lft-identity2.7

    \[\leadsto \frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}\]

  5. Applied times-frac2.5

    \[\leadsto \color{blue}{\frac{1}{sin \cdot \left(x \cdot cos\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}\]
  6. Using strategy rm

  7. Applied associate-/r*2.5

    \[\leadsto \color{blue}{\frac{\frac{1}{sin}}{x \cdot cos}} \cdot \frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt2.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{1}{sin}}{x \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}} \cdot \sqrt[3]{\frac{\frac{1}{sin}}{x \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}\right) \cdot \sqrt[3]{\frac{\frac{1}{sin}}{x \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}}\]

  10. Final simplification2.9

    \[\leadsto \left(\sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}} \cdot \sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}}\right) \cdot \sqrt[3]{\frac{\cos \left(x \cdot 2\right)}{\left(x \cdot cos\right) \cdot sin} \cdot \frac{\frac{1}{sin}}{x \cdot cos}}\]

Reproduce

herbie shell --seed 2019164 +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))))