Average Error: 27.9 → 2.6
Time: 22.9s
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 r2341712 = 2.0;
        double r2341713 = x;
        double r2341714 = r2341712 * r2341713;
        double r2341715 = cos(r2341714);
        double r2341716 = cos;
        double r2341717 = pow(r2341716, r2341712);
        double r2341718 = sin;
        double r2341719 = pow(r2341718, r2341712);
        double r2341720 = r2341713 * r2341719;
        double r2341721 = r2341720 * r2341713;
        double r2341722 = r2341717 * r2341721;
        double r2341723 = r2341715 / r2341722;
        return r2341723;
}
↓
double f(double x, double cos, double sin) {
        double r2341724 = 2.0;
        double r2341725 = x;
        double r2341726 = r2341724 * r2341725;
        double r2341727 = cos(r2341726);
        double r2341728 = cbrt(r2341727);
        double r2341729 = cos;
        double r2341730 = r2341729 * r2341725;
        double r2341731 = sin;
        double r2341732 = r2341730 * r2341731;
        double r2341733 = r2341728 / r2341732;
        double r2341734 = 1.0;
        double r2341735 = r2341728 * r2341728;
        double r2341736 = r2341735 * r2341728;
        double r2341737 = cbrt(r2341736);
        double r2341738 = r2341734 / r2341737;
        double r2341739 = r2341728 / r2341738;
        double r2341740 = r2341733 * r2341739;
        double r2341741 = r2341740 / r2341732;
        return r2341741;
}

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 +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))))