Average Error: 28.0 → 6.3
Time: 55.3s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left(\sqrt[3]{{cos}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{cos}^{\left(\frac{2}{2}\right)}}\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right) \cdot \left(\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot \sqrt[3]{{cos}^{\left(\frac{2}{2}\right)}}\right)\right)}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{\cos \left(x \cdot 2\right)}{{cos}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\left(\sqrt[3]{{cos}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{cos}^{\left(\frac{2}{2}\right)}}\right) \cdot \left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right)\right) \cdot \left(\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot \sqrt[3]{{cos}^{\left(\frac{2}{2}\right)}}\right)\right)}
double f(double x, double cos, double sin) {
        double r2942217 = 2.0;
        double r2942218 = x;
        double r2942219 = r2942217 * r2942218;
        double r2942220 = cos(r2942219);
        double r2942221 = cos;
        double r2942222 = pow(r2942221, r2942217);
        double r2942223 = sin;
        double r2942224 = pow(r2942223, r2942217);
        double r2942225 = r2942218 * r2942224;
        double r2942226 = r2942225 * r2942218;
        double r2942227 = r2942222 * r2942226;
        double r2942228 = r2942220 / r2942227;
        return r2942228;
}
↓
double f(double x, double cos, double sin) {
        double r2942229 = x;
        double r2942230 = 2.0;
        double r2942231 = r2942229 * r2942230;
        double r2942232 = cos(r2942231);
        double r2942233 = cos;
        double r2942234 = 2.0;
        double r2942235 = r2942230 / r2942234;
        double r2942236 = pow(r2942233, r2942235);
        double r2942237 = cbrt(r2942236);
        double r2942238 = r2942237 * r2942237;
        double r2942239 = sin;
        double r2942240 = pow(r2942239, r2942235);
        double r2942241 = r2942240 * r2942229;
        double r2942242 = r2942238 * r2942241;
        double r2942243 = r2942241 * r2942237;
        double r2942244 = r2942242 * r2942243;
        double r2942245 = r2942236 * r2942244;
        double r2942246 = r2942232 / r2942245;
        return r2942246;
}

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*23.4

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

  6. Applied sqr-pow23.4

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

  7. Applied associate-*r*16.6

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

  9. Applied add-cube-cbrt16.8

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

  10. Applied associate-*l*16.8

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

  11. Simplified11.1

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

  13. Applied associate-*r*6.3

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

  14. Final simplification6.3

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

Reproduce

herbie shell --seed 2019200 
(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))))