Average Error: 27.4 → 3.1
Time: 20.5s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}} \cdot \left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}} \cdot \sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}\right)}{\left(cos \cdot x\right) \cdot sin}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}} \cdot \left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}} \cdot \sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot x\right) \cdot sin}}\right)}{\left(cos \cdot x\right) \cdot sin}
double f(double x, double cos, double sin) {
        double r2325799 = 2.0;
        double r2325800 = x;
        double r2325801 = r2325799 * r2325800;
        double r2325802 = cos(r2325801);
        double r2325803 = cos;
        double r2325804 = pow(r2325803, r2325799);
        double r2325805 = sin;
        double r2325806 = pow(r2325805, r2325799);
        double r2325807 = r2325800 * r2325806;
        double r2325808 = r2325807 * r2325800;
        double r2325809 = r2325804 * r2325808;
        double r2325810 = r2325802 / r2325809;
        return r2325810;
}
↓
double f(double x, double cos, double sin) {
        double r2325811 = 2.0;
        double r2325812 = x;
        double r2325813 = r2325811 * r2325812;
        double r2325814 = cos(r2325813);
        double r2325815 = cos;
        double r2325816 = r2325815 * r2325812;
        double r2325817 = sin;
        double r2325818 = r2325816 * r2325817;
        double r2325819 = r2325814 / r2325818;
        double r2325820 = cbrt(r2325819);
        double r2325821 = r2325820 * r2325820;
        double r2325822 = r2325820 * r2325821;
        double r2325823 = r2325822 / r2325818;
        return r2325823;
}

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 27.4

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

  2. Simplified3.0

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

  4. Applied associate-/r*2.7

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

  6. Applied add-cube-cbrt3.1

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

  7. Final simplification3.1

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

Reproduce

herbie shell --seed 2019168 +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))))