Average Error: 27.0 → 3.0
Time: 33.6s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}} \cdot \left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}} \cdot \sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}}\right)}{\left(x \cdot sin\right) \cdot cos}\]
\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(x \cdot sin\right) \cdot cos}} \cdot \left(\sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}} \cdot \sqrt[3]{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}}\right)}{\left(x \cdot sin\right) \cdot cos}
double f(double x, double cos, double sin) {
        double r2524809 = 2.0;
        double r2524810 = x;
        double r2524811 = r2524809 * r2524810;
        double r2524812 = cos(r2524811);
        double r2524813 = cos;
        double r2524814 = pow(r2524813, r2524809);
        double r2524815 = sin;
        double r2524816 = pow(r2524815, r2524809);
        double r2524817 = r2524810 * r2524816;
        double r2524818 = r2524817 * r2524810;
        double r2524819 = r2524814 * r2524818;
        double r2524820 = r2524812 / r2524819;
        return r2524820;
}
↓
double f(double x, double cos, double sin) {
        double r2524821 = 2.0;
        double r2524822 = x;
        double r2524823 = r2524821 * r2524822;
        double r2524824 = cos(r2524823);
        double r2524825 = sin;
        double r2524826 = r2524822 * r2524825;
        double r2524827 = cos;
        double r2524828 = r2524826 * r2524827;
        double r2524829 = r2524824 / r2524828;
        double r2524830 = cbrt(r2524829);
        double r2524831 = r2524830 * r2524830;
        double r2524832 = r2524830 * r2524831;
        double r2524833 = r2524832 / r2524828;
        return r2524833;
}

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 27.0

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

  2. Simplified2.9

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

  4. Applied associate-/r*2.7

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

  6. Applied add-cube-cbrt3.0

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

  7. Final simplification3.0

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

Reproduce

herbie shell --seed 2019146 
(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))))