Average Error: 27.9 → 2.3
Time: 21.8s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;sin \le 3.4016721265247604 \cdot 10^{-241}:\\ \;\;\;\;\frac{1}{\left(x \cdot sin\right) \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}\\ \end{array}\]
\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}\;sin \le 3.4016721265247604 \cdot 10^{-241}:\\
\;\;\;\;\frac{1}{\left(x \cdot sin\right) \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r2089839 = 2.0;
        double r2089840 = x;
        double r2089841 = r2089839 * r2089840;
        double r2089842 = cos(r2089841);
        double r2089843 = cos;
        double r2089844 = pow(r2089843, r2089839);
        double r2089845 = sin;
        double r2089846 = pow(r2089845, r2089839);
        double r2089847 = r2089840 * r2089846;
        double r2089848 = r2089847 * r2089840;
        double r2089849 = r2089844 * r2089848;
        double r2089850 = r2089842 / r2089849;
        return r2089850;
}
↓
double f(double x, double cos, double sin) {
        double r2089851 = sin;
        double r2089852 = 3.4016721265247604e-241;
        bool r2089853 = r2089851 <= r2089852;
        double r2089854 = 1.0;
        double r2089855 = x;
        double r2089856 = r2089855 * r2089851;
        double r2089857 = cos;
        double r2089858 = r2089856 * r2089857;
        double r2089859 = r2089854 / r2089858;
        double r2089860 = 2.0;
        double r2089861 = r2089860 * r2089855;
        double r2089862 = cos(r2089861);
        double r2089863 = r2089862 / r2089858;
        double r2089864 = r2089859 * r2089863;
        double r2089865 = r2089855 * r2089857;
        double r2089866 = r2089851 * r2089865;
        double r2089867 = r2089862 / r2089866;
        double r2089868 = r2089867 / r2089866;
        double r2089869 = r2089853 ? r2089864 : r2089868;
        return r2089869;
}

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 sin < 3.4016721265247604e-241

    1. Initial program 29.8

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

    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 div-inv2.9

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

    if 3.4016721265247604e-241 < sin

    1. Initial program 25.9

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

    2. Simplified2.9

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

    4. Applied associate-/r*2.7

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

    5. Taylor expanded around inf 29.5

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

    6. Simplified1.8

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;sin \le 3.4016721265247604 \cdot 10^{-241}:\\ \;\;\;\;\frac{1}{\left(x \cdot sin\right) \cdot cos} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot sin\right) \cdot cos}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{sin \cdot \left(x \cdot cos\right)}}{sin \cdot \left(x \cdot cos\right)}\\ \end{array}\]

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))))