Average Error: 27.8 → 2.7
Time: 36.8s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\left(\frac{1}{x} \cdot \frac{1}{sin}\right) \cdot \frac{\cos \left(x \cdot 2\right)}{cos}}{cos \cdot \left(x \cdot sin\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\left(\frac{1}{x} \cdot \frac{1}{sin}\right) \cdot \frac{\cos \left(x \cdot 2\right)}{cos}}{cos \cdot \left(x \cdot sin\right)}
double f(double x, double cos, double sin) {
        double r2825672 = 2.0;
        double r2825673 = x;
        double r2825674 = r2825672 * r2825673;
        double r2825675 = cos(r2825674);
        double r2825676 = cos;
        double r2825677 = pow(r2825676, r2825672);
        double r2825678 = sin;
        double r2825679 = pow(r2825678, r2825672);
        double r2825680 = r2825673 * r2825679;
        double r2825681 = r2825680 * r2825673;
        double r2825682 = r2825677 * r2825681;
        double r2825683 = r2825675 / r2825682;
        return r2825683;
}
↓
double f(double x, double cos, double sin) {
        double r2825684 = 1.0;
        double r2825685 = x;
        double r2825686 = r2825684 / r2825685;
        double r2825687 = sin;
        double r2825688 = r2825684 / r2825687;
        double r2825689 = r2825686 * r2825688;
        double r2825690 = 2.0;
        double r2825691 = r2825685 * r2825690;
        double r2825692 = cos(r2825691);
        double r2825693 = cos;
        double r2825694 = r2825692 / r2825693;
        double r2825695 = r2825689 * r2825694;
        double r2825696 = r2825685 * r2825687;
        double r2825697 = r2825693 * r2825696;
        double r2825698 = r2825695 / r2825697;
        return r2825698;
}

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 27.8

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

  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 associate-/r*2.5

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

  6. Applied *-un-lft-identity2.5

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

  7. Applied times-frac2.7

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

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

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

  10. Applied times-frac2.7

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

  11. Final simplification2.7

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

Reproduce

herbie shell --seed 2019142 +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))))