Average Error: 27.9 → 2.6
Time: 23.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(cos \cdot x\right) \cdot sin} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\frac{1}{\sqrt[3]{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}}}{\left(cos \cdot x\right) \cdot sin}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\left(cos \cdot x\right) \cdot sin} \cdot \frac{\sqrt[3]{\cos \left(2 \cdot x\right)}}{\frac{1}{\sqrt[3]{\left(\sqrt[3]{\cos \left(2 \cdot x\right)} \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}\right) \cdot \sqrt[3]{\cos \left(2 \cdot x\right)}}}}}{\left(cos \cdot x\right) \cdot sin}
double f(double x, double cos, double sin) {
        double r2281887 = 2.0;
        double r2281888 = x;
        double r2281889 = r2281887 * r2281888;
        double r2281890 = cos(r2281889);
        double r2281891 = cos;
        double r2281892 = pow(r2281891, r2281887);
        double r2281893 = sin;
        double r2281894 = pow(r2281893, r2281887);
        double r2281895 = r2281888 * r2281894;
        double r2281896 = r2281895 * r2281888;
        double r2281897 = r2281892 * r2281896;
        double r2281898 = r2281890 / r2281897;
        return r2281898;
}
↓
double f(double x, double cos, double sin) {
        double r2281899 = 2.0;
        double r2281900 = x;
        double r2281901 = r2281899 * r2281900;
        double r2281902 = cos(r2281901);
        double r2281903 = cbrt(r2281902);
        double r2281904 = cos;
        double r2281905 = r2281904 * r2281900;
        double r2281906 = sin;
        double r2281907 = r2281905 * r2281906;
        double r2281908 = r2281903 / r2281907;
        double r2281909 = 1.0;
        double r2281910 = r2281903 * r2281903;
        double r2281911 = r2281910 * r2281903;
        double r2281912 = cbrt(r2281911);
        double r2281913 = r2281909 / r2281912;
        double r2281914 = r2281903 / r2281913;
        double r2281915 = r2281908 * r2281914;
        double r2281916 = r2281915 / r2281907;
        return r2281916;
}

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 27.9

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

  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*2.4

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

  6. Applied add-cube-cbrt2.6

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

  7. Applied associate-/l*2.6

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

  9. Applied div-inv2.6

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

  10. Applied times-frac2.6

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

  12. Applied add-cbrt-cube2.6

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

  13. Final simplification2.6

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

Reproduce

herbie shell --seed 2019163 
(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))))