Average Error: 27.3 → 2.5
Time: 18.5s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.606769180032518 \cdot 10^{-270}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(cos \cdot sin\right)}}{x \cdot \left(cos \cdot sin\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(cos \cdot x\right) \cdot sin} \cdot \left(\frac{\cos \left(2 \cdot x\right)}{sin} \cdot \frac{\frac{1}{x}}{cos}\right)\\ \end{array}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.606769180032518 \cdot 10^{-270}:\\
\;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(cos \cdot sin\right)}}{x \cdot \left(cos \cdot sin\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\left(cos \cdot x\right) \cdot sin} \cdot \left(\frac{\cos \left(2 \cdot x\right)}{sin} \cdot \frac{\frac{1}{x}}{cos}\right)\\

\end{array}
double f(double x, double cos, double sin) {
        double r963866 = 2.0;
        double r963867 = x;
        double r963868 = r963866 * r963867;
        double r963869 = cos(r963868);
        double r963870 = cos;
        double r963871 = pow(r963870, r963866);
        double r963872 = sin;
        double r963873 = pow(r963872, r963866);
        double r963874 = r963867 * r963873;
        double r963875 = r963874 * r963867;
        double r963876 = r963871 * r963875;
        double r963877 = r963869 / r963876;
        return r963877;
}
↓
double f(double x, double cos, double sin) {
        double r963878 = x;
        double r963879 = -6.606769180032518e-270;
        bool r963880 = r963878 <= r963879;
        double r963881 = 2.0;
        double r963882 = r963881 * r963878;
        double r963883 = cos(r963882);
        double r963884 = cos;
        double r963885 = sin;
        double r963886 = r963884 * r963885;
        double r963887 = r963878 * r963886;
        double r963888 = r963883 / r963887;
        double r963889 = r963888 / r963887;
        double r963890 = 1.0;
        double r963891 = r963884 * r963878;
        double r963892 = r963891 * r963885;
        double r963893 = r963890 / r963892;
        double r963894 = r963883 / r963885;
        double r963895 = r963890 / r963878;
        double r963896 = r963895 / r963884;
        double r963897 = r963894 * r963896;
        double r963898 = r963893 * r963897;
        double r963899 = r963880 ? r963889 : r963898;
        return r963899;
}

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. Split input into 2 regimes

  2. if x < -6.606769180032518e-270

    1. Initial program 26.8

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

    2. Simplified2.4

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

    4. Applied associate-/r*2.1

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

    if -6.606769180032518e-270 < x

    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. Simplified3.6

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

    3. Taylor expanded around -inf 31.8

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

    4. Simplified3.0

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

    6. Applied *-un-lft-identity3.0

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

    7. Applied times-frac2.9

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

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

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

    10. Applied times-frac3.0

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

    12. Applied associate-/r*3.0

      \[\leadsto \frac{1}{\left(x \cdot cos\right) \cdot sin} \cdot \left(\color{blue}{\frac{\frac{1}{x}}{cos}} \cdot \frac{\cos \left(x \cdot 2\right)}{sin}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.606769180032518 \cdot 10^{-270}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(cos \cdot sin\right)}}{x \cdot \left(cos \cdot sin\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(cos \cdot x\right) \cdot sin} \cdot \left(\frac{\cos \left(2 \cdot x\right)}{sin} \cdot \frac{\frac{1}{x}}{cos}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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))))