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 r26298981 = 2.0;
        double r26298982 = x;
        double r26298983 = r26298981 * r26298982;
        double r26298984 = cos(r26298983);
        double r26298985 = cos;
        double r26298986 = pow(r26298985, r26298981);
        double r26298987 = sin;
        double r26298988 = pow(r26298987, r26298981);
        double r26298989 = r26298982 * r26298988;
        double r26298990 = r26298989 * r26298982;
        double r26298991 = r26298986 * r26298990;
        double r26298992 = r26298984 / r26298991;
        return r26298992;
}
↓
double f(double x, double cos, double sin) {
        double r26298993 = sin;
        double r26298994 = -9.428285242761686e-254;
        bool r26298995 = r26298993 <= r26298994;
        double r26298996 = 1.0;
        double r26298997 = r26298996 / r26298993;
        double r26298998 = cos;
        double r26298999 = r26298997 / r26298998;
        double r26299000 = x;
        double r26299001 = r26298996 / r26299000;
        double r26299002 = r26298999 * r26299001;
        double r26299003 = r26298998 * r26298993;
        double r26299004 = r26299000 * r26299003;
        double r26299005 = 2.0;
        double r26299006 = r26299005 * r26299000;
        double r26299007 = cos(r26299006);
        double r26299008 = r26299004 / r26299007;
        double r26299009 = r26299002 / r26299008;
        double r26299010 = r26299000 * r26298993;
        double r26299011 = r26298998 * r26299010;
        double r26299012 = r26299011 * r26299011;
        double r26299013 = r26299007 / r26299012;
        double r26299014 = r26298995 ? r26299009 : r26299013;
        return r26299014;
}

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 clear-num2.4

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

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

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

    9. Applied times-frac2.4

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

    10. Applied associate-/r*2.2

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

    12. Applied add-cube-cbrt2.2

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

    13. Applied times-frac2.2

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

    14. Applied add-sqr-sqrt2.2

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

    15. Applied times-frac2.3

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

    16. Simplified2.2

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

    17. 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 
(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))))