Average Error: 27.8 → 2.4
Time: 18.3s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;sin \le 1.4141501599718892 \cdot 10^{-225}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot cos}}{sin}}{sin \cdot \left(x \cdot cos\right)}\\ \end{array}\]
\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 1.4141501599718892 \cdot 10^{-225}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r1389802 = 2.0;
        double r1389803 = x;
        double r1389804 = r1389802 * r1389803;
        double r1389805 = cos(r1389804);
        double r1389806 = cos;
        double r1389807 = pow(r1389806, r1389802);
        double r1389808 = sin;
        double r1389809 = pow(r1389808, r1389802);
        double r1389810 = r1389803 * r1389809;
        double r1389811 = r1389810 * r1389803;
        double r1389812 = r1389807 * r1389811;
        double r1389813 = r1389805 / r1389812;
        return r1389813;
}
↓
double f(double x, double cos, double sin) {
        double r1389814 = sin;
        double r1389815 = 1.4141501599718892e-225;
        bool r1389816 = r1389814 <= r1389815;
        double r1389817 = 2.0;
        double r1389818 = x;
        double r1389819 = r1389817 * r1389818;
        double r1389820 = cos(r1389819);
        double r1389821 = r1389818 * r1389814;
        double r1389822 = cos;
        double r1389823 = r1389821 * r1389822;
        double r1389824 = r1389823 * r1389823;
        double r1389825 = r1389820 / r1389824;
        double r1389826 = r1389818 * r1389822;
        double r1389827 = r1389820 / r1389826;
        double r1389828 = r1389827 / r1389814;
        double r1389829 = r1389814 * r1389826;
        double r1389830 = r1389828 / r1389829;
        double r1389831 = r1389816 ? r1389825 : r1389830;
        return r1389831;
}

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 < 1.4141501599718892e-225

    1. Initial program 29.6

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

    2. Simplified4.0

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

    3. Taylor expanded around inf 33.0

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

    4. Simplified3.8

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

    6. Applied associate-/r*3.9

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

    7. Taylor expanded around inf 32.9

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

    8. Simplified2.9

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

    if 1.4141501599718892e-225 < 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. Simplified2.1

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

    3. Taylor expanded around inf 29.3

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

    4. Simplified1.8

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

    6. Applied associate-/r*1.9

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;sin \le 1.4141501599718892 \cdot 10^{-225}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot sin\right) \cdot cos\right) \cdot \left(\left(x \cdot sin\right) \cdot cos\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot cos}}{sin}}{sin \cdot \left(x \cdot cos\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(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))))