Average Error: 26.9 → 2.4
Time: 4.8m
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;sin \le -9.428285242761686 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{\frac{1}{sin}}{cos} \cdot \frac{1}{x}}{\frac{x \cdot \left(cos \cdot sin\right)}{\cos \left(2 \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot \left(x \cdot sin\right)\right) \cdot \left(cos \cdot \left(x \cdot sin\right)\right)}\\ \end{array}\]
\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 -9.428285242761686 \cdot 10^{-254}:\\
\;\;\;\;\frac{\frac{\frac{1}{sin}}{cos} \cdot \frac{1}{x}}{\frac{x \cdot \left(cos \cdot sin\right)}{\cos \left(2 \cdot x\right)}}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r26307249 = 2.0;
        double r26307250 = x;
        double r26307251 = r26307249 * r26307250;
        double r26307252 = cos(r26307251);
        double r26307253 = cos;
        double r26307254 = pow(r26307253, r26307249);
        double r26307255 = sin;
        double r26307256 = pow(r26307255, r26307249);
        double r26307257 = r26307250 * r26307256;
        double r26307258 = r26307257 * r26307250;
        double r26307259 = r26307254 * r26307258;
        double r26307260 = r26307252 / r26307259;
        return r26307260;
}
↓
double f(double x, double cos, double sin) {
        double r26307261 = sin;
        double r26307262 = -9.428285242761686e-254;
        bool r26307263 = r26307261 <= r26307262;
        double r26307264 = 1.0;
        double r26307265 = r26307264 / r26307261;
        double r26307266 = cos;
        double r26307267 = r26307265 / r26307266;
        double r26307268 = x;
        double r26307269 = r26307264 / r26307268;
        double r26307270 = r26307267 * r26307269;
        double r26307271 = r26307266 * r26307261;
        double r26307272 = r26307268 * r26307271;
        double r26307273 = 2.0;
        double r26307274 = r26307273 * r26307268;
        double r26307275 = cos(r26307274);
        double r26307276 = r26307272 / r26307275;
        double r26307277 = r26307270 / r26307276;
        double r26307278 = r26307268 * r26307261;
        double r26307279 = r26307266 * r26307278;
        double r26307280 = r26307279 * r26307279;
        double r26307281 = r26307275 / r26307280;
        double r26307282 = r26307263 ? r26307277 : r26307281;
        return r26307282;
}

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 < -9.428285242761686e-254

    1. Initial program 25.6

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

    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. Taylor expanded around inf 29.3

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

    4. Simplified2.4

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

    6. Applied *-un-lft-identity2.4

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

    7. Applied associate-/l*2.4

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

    9. Applied *-un-lft-identity2.4

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

    10. Applied times-frac2.4

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

    11. Applied associate-/r*2.2

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

    13. Applied add-cube-cbrt2.2

      \[\leadsto \frac{\frac{1}{\frac{\left(cos \cdot sin\right) \cdot x}{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}}}{\frac{\left(cos \cdot sin\right) \cdot x}{\cos \left(2 \cdot x\right)}}\]

    14. Applied times-frac2.2

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{cos \cdot sin}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{x}{\sqrt[3]{1}}}}}{\frac{\left(cos \cdot sin\right) \cdot x}{\cos \left(2 \cdot x\right)}}\]

    15. Applied add-sqr-sqrt2.2

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{cos \cdot sin}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{x}{\sqrt[3]{1}}}}{\frac{\left(cos \cdot sin\right) \cdot x}{\cos \left(2 \cdot x\right)}}\]

    16. Applied times-frac2.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\frac{cos \cdot sin}{\sqrt[3]{1} \cdot \sqrt[3]{1}}} \cdot \frac{\sqrt{1}}{\frac{x}{\sqrt[3]{1}}}}}{\frac{\left(cos \cdot sin\right) \cdot x}{\cos \left(2 \cdot x\right)}}\]

    17. Simplified2.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{sin}}{cos}} \cdot \frac{\sqrt{1}}{\frac{x}{\sqrt[3]{1}}}}{\frac{\left(cos \cdot sin\right) \cdot x}{\cos \left(2 \cdot x\right)}}\]

    18. Simplified2.2

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

    if -9.428285242761686e-254 < sin

    1. Initial program 28.1

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

    2. Simplified3.5

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

    3. Taylor expanded around -inf 31.4

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

    4. Simplified2.5

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;sin \le -9.428285242761686 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{\frac{1}{sin}}{cos} \cdot \frac{1}{x}}{\frac{x \cdot \left(cos \cdot sin\right)}{\cos \left(2 \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(cos \cdot \left(x \cdot sin\right)\right) \cdot \left(cos \cdot \left(x \cdot sin\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +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))))