Average Error: 27.3 → 2.7
Time: 50.3s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\left(\frac{1}{cos} \cdot \frac{\frac{1}{sin}}{x}\right) \cdot \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(x \cdot sin\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\left(\frac{1}{cos} \cdot \frac{\frac{1}{sin}}{x}\right) \cdot \frac{\cos \left(2 \cdot x\right)}{cos \cdot \left(x \cdot sin\right)}
double f(double x, double cos, double sin) {
        double r2733824 = 2.0;
        double r2733825 = x;
        double r2733826 = r2733824 * r2733825;
        double r2733827 = cos(r2733826);
        double r2733828 = cos;
        double r2733829 = pow(r2733828, r2733824);
        double r2733830 = sin;
        double r2733831 = pow(r2733830, r2733824);
        double r2733832 = r2733825 * r2733831;
        double r2733833 = r2733832 * r2733825;
        double r2733834 = r2733829 * r2733833;
        double r2733835 = r2733827 / r2733834;
        return r2733835;
}
↓
double f(double x, double cos, double sin) {
        double r2733836 = 1.0;
        double r2733837 = cos;
        double r2733838 = r2733836 / r2733837;
        double r2733839 = sin;
        double r2733840 = r2733836 / r2733839;
        double r2733841 = x;
        double r2733842 = r2733840 / r2733841;
        double r2733843 = r2733838 * r2733842;
        double r2733844 = 2.0;
        double r2733845 = r2733844 * r2733841;
        double r2733846 = cos(r2733845);
        double r2733847 = r2733841 * r2733839;
        double r2733848 = r2733837 * r2733847;
        double r2733849 = r2733846 / r2733848;
        double r2733850 = r2733843 * r2733849;
        return r2733850;
}

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 27.3

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

  2. Simplified2.8

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

  4. Applied *-un-lft-identity2.8

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

  5. Applied times-frac2.6

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

  7. Applied *-un-lft-identity2.6

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

  8. Applied times-frac2.7

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

  10. Applied *-un-lft-identity2.7

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

  11. Applied *-un-lft-identity2.7

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

  12. Applied times-frac2.7

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

  13. Applied associate-*r*2.7

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

  14. Simplified2.7

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

  15. Final simplification2.7

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

Reproduce

herbie shell --seed 2019152 
(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))))