Average Error: 27.6 → 3.1
Time: 36.2s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\frac{\left(x \cdot sin\right) \cdot cos}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}}}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\frac{\left(x \cdot sin\right) \cdot cos}{\frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}}}
double f(double x, double cos, double sin) {
        double r2732910 = 2.0;
        double r2732911 = x;
        double r2732912 = r2732910 * r2732911;
        double r2732913 = cos(r2732912);
        double r2732914 = cos;
        double r2732915 = pow(r2732914, r2732910);
        double r2732916 = sin;
        double r2732917 = pow(r2732916, r2732910);
        double r2732918 = r2732911 * r2732917;
        double r2732919 = r2732918 * r2732911;
        double r2732920 = r2732915 * r2732919;
        double r2732921 = r2732913 / r2732920;
        return r2732921;
}
↓
double f(double x, double cos, double sin) {
        double r2732922 = 1.0;
        double r2732923 = x;
        double r2732924 = sin;
        double r2732925 = r2732923 * r2732924;
        double r2732926 = cos;
        double r2732927 = r2732925 * r2732926;
        double r2732928 = 2.0;
        double r2732929 = r2732928 * r2732923;
        double r2732930 = cos(r2732929);
        double r2732931 = r2732930 / r2732927;
        double r2732932 = r2732927 / r2732931;
        double r2732933 = r2732922 / r2732932;
        return r2732933;
}

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 27.6

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

  2. Simplified3.1

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

  4. Applied associate-/r*2.8

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

  6. Applied *-un-lft-identity2.8

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

  7. Applied associate-/l*3.1

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

  8. Final simplification3.1

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

Reproduce

herbie shell --seed 2019141 +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))))