Average Error: 26.9 → 2.8
Time: 30.2s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}{\cos \left(2 \cdot x\right)}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}{\cos \left(2 \cdot x\right)}}
double f(double x, double cos, double sin) {
        double r2347730 = 2.0;
        double r2347731 = x;
        double r2347732 = r2347730 * r2347731;
        double r2347733 = cos(r2347732);
        double r2347734 = cos;
        double r2347735 = pow(r2347734, r2347730);
        double r2347736 = sin;
        double r2347737 = pow(r2347736, r2347730);
        double r2347738 = r2347731 * r2347737;
        double r2347739 = r2347738 * r2347731;
        double r2347740 = r2347735 * r2347739;
        double r2347741 = r2347733 / r2347740;
        return r2347741;
}
↓
double f(double x, double cos, double sin) {
        double r2347742 = 1.0;
        double r2347743 = sin;
        double r2347744 = x;
        double r2347745 = cos;
        double r2347746 = r2347744 * r2347745;
        double r2347747 = r2347743 * r2347746;
        double r2347748 = r2347747 * r2347747;
        double r2347749 = 2.0;
        double r2347750 = r2347749 * r2347744;
        double r2347751 = cos(r2347750);
        double r2347752 = r2347748 / r2347751;
        double r2347753 = r2347742 / r2347752;
        return r2347753;
}

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 26.9

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

  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 *-un-lft-identity2.9

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

  5. Applied associate-/l*2.9

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

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

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

  8. Applied associate-/l*2.9

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

  9. Simplified2.8

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

  11. Applied clear-num2.8

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

  12. Simplified2.8

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

  13. Final simplification2.8

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

Reproduce

herbie shell --seed 2019139 +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))))