Average Error: 28.0 → 6.4
Time: 32.1s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(2 \cdot x\right)}{\left(\left({\left(\sqrt[3]{cos} \cdot \sqrt[3]{cos}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\sqrt[3]{cos}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot {cos}^{\left(\frac{2}{2}\right)}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(2 \cdot x\right)}{\left(\left({\left(\sqrt[3]{cos} \cdot \sqrt[3]{cos}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(\left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\sqrt[3]{cos}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot {cos}^{\left(\frac{2}{2}\right)}}
double f(double x, double cos, double sin) {
        double r2980747 = 2.0;
        double r2980748 = x;
        double r2980749 = r2980747 * r2980748;
        double r2980750 = cos(r2980749);
        double r2980751 = cos;
        double r2980752 = pow(r2980751, r2980747);
        double r2980753 = sin;
        double r2980754 = pow(r2980753, r2980747);
        double r2980755 = r2980748 * r2980754;
        double r2980756 = r2980755 * r2980748;
        double r2980757 = r2980752 * r2980756;
        double r2980758 = r2980750 / r2980757;
        return r2980758;
}
↓
double f(double x, double cos, double sin) {
        double r2980759 = 2.0;
        double r2980760 = x;
        double r2980761 = r2980759 * r2980760;
        double r2980762 = cos(r2980761);
        double r2980763 = cos;
        double r2980764 = cbrt(r2980763);
        double r2980765 = r2980764 * r2980764;
        double r2980766 = 2.0;
        double r2980767 = r2980759 / r2980766;
        double r2980768 = pow(r2980765, r2980767);
        double r2980769 = sin;
        double r2980770 = pow(r2980769, r2980767);
        double r2980771 = r2980760 * r2980770;
        double r2980772 = r2980768 * r2980771;
        double r2980773 = pow(r2980764, r2980767);
        double r2980774 = r2980771 * r2980773;
        double r2980775 = r2980772 * r2980774;
        double r2980776 = pow(r2980763, r2980767);
        double r2980777 = r2980775 * r2980776;
        double r2980778 = r2980762 / r2980777;
        return r2980778;
}

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 28.0

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

  3. Applied sqr-pow28.0

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

  4. Applied associate-*l*23.7

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

  6. Applied sqr-pow23.7

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

  7. Applied associate-*r*16.4

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

  9. Applied add-cube-cbrt16.6

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

  10. Applied unpow-prod-down16.6

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

  11. Applied associate-*l*16.6

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

  12. Simplified11.1

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

  14. Applied associate-*r*6.4

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

  15. Final simplification6.4

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2.0 x)) (* (pow cos 2.0) (* (* x (pow sin 2.0)) x))))