Average Error: 27.8 → 4.2
Time: 40.0s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;cos \le -6.168883927619536 \cdot 10^{-205}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(x \cdot \left(sin \cdot cos\right)\right) \cdot \left(sin \cdot cos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)}{\cos \left(2 \cdot x\right)}}\\ \end{array}\]
\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}\;cos \le -6.168883927619536 \cdot 10^{-205}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(x \cdot \left(sin \cdot cos\right)\right) \cdot \left(sin \cdot cos\right)\right)}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r2567255 = 2.0;
        double r2567256 = x;
        double r2567257 = r2567255 * r2567256;
        double r2567258 = cos(r2567257);
        double r2567259 = cos;
        double r2567260 = pow(r2567259, r2567255);
        double r2567261 = sin;
        double r2567262 = pow(r2567261, r2567255);
        double r2567263 = r2567256 * r2567262;
        double r2567264 = r2567263 * r2567256;
        double r2567265 = r2567260 * r2567264;
        double r2567266 = r2567258 / r2567265;
        return r2567266;
}
↓
double f(double x, double cos, double sin) {
        double r2567267 = cos;
        double r2567268 = -6.168883927619536e-205;
        bool r2567269 = r2567267 <= r2567268;
        double r2567270 = 2.0;
        double r2567271 = x;
        double r2567272 = r2567270 * r2567271;
        double r2567273 = cos(r2567272);
        double r2567274 = sin;
        double r2567275 = r2567274 * r2567267;
        double r2567276 = r2567271 * r2567275;
        double r2567277 = r2567276 * r2567275;
        double r2567278 = r2567271 * r2567277;
        double r2567279 = r2567273 / r2567278;
        double r2567280 = 1.0;
        double r2567281 = r2567271 * r2567267;
        double r2567282 = r2567281 * r2567274;
        double r2567283 = r2567282 * r2567282;
        double r2567284 = r2567283 / r2567273;
        double r2567285 = r2567280 / r2567284;
        double r2567286 = r2567269 ? r2567279 : r2567285;
        return r2567286;
}

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 cos < -6.168883927619536e-205

    1. Initial program 24.6

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

    2. Simplified2.3

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

    4. Applied *-commutative2.3

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

    5. Applied associate-*r*4.2

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

    6. Applied associate-*r*6.9

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

    7. Simplified5.3

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

    if -6.168883927619536e-205 < cos

    1. Initial program 30.5

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

    2. Simplified3.2

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

    4. Applied clear-num3.2

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le -6.168883927619536 \cdot 10^{-205}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(x \cdot \left(sin \cdot cos\right)\right) \cdot \left(sin \cdot cos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(\left(x \cdot cos\right) \cdot sin\right) \cdot \left(\left(x \cdot cos\right) \cdot sin\right)}{\cos \left(2 \cdot x\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))))