Average Error: 27.9 → 2.2
Time: 8.3s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 5.4433641186682693 \cdot 10^{-168} \lor \neg \left(x \le 2.73621970403785706 \cdot 10^{219}\right):\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{\frac{1}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]
\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}\;x \le 5.4433641186682693 \cdot 10^{-168} \lor \neg \left(x \le 2.73621970403785706 \cdot 10^{219}\right):\\
\;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{\frac{1}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r59814 = 2.0;
        double r59815 = x;
        double r59816 = r59814 * r59815;
        double r59817 = cos(r59816);
        double r59818 = cos;
        double r59819 = pow(r59818, r59814);
        double r59820 = sin;
        double r59821 = pow(r59820, r59814);
        double r59822 = r59815 * r59821;
        double r59823 = r59822 * r59815;
        double r59824 = r59819 * r59823;
        double r59825 = r59817 / r59824;
        return r59825;
}
↓
double f(double x, double cos, double sin) {
        double r59826 = x;
        double r59827 = 5.443364118668269e-168;
        bool r59828 = r59826 <= r59827;
        double r59829 = 2.736219704037857e+219;
        bool r59830 = r59826 <= r59829;
        double r59831 = !r59830;
        bool r59832 = r59828 || r59831;
        double r59833 = 2.0;
        double r59834 = r59833 * r59826;
        double r59835 = cos(r59834);
        double r59836 = 1.0;
        double r59837 = cos;
        double r59838 = 2.0;
        double r59839 = r59833 / r59838;
        double r59840 = pow(r59837, r59839);
        double r59841 = sin;
        double r59842 = pow(r59841, r59839);
        double r59843 = r59826 * r59842;
        double r59844 = r59840 * r59843;
        double r59845 = fabs(r59844);
        double r59846 = r59836 / r59845;
        double r59847 = r59846 / r59845;
        double r59848 = r59835 * r59847;
        double r59849 = 1.0;
        double r59850 = pow(r59837, r59849);
        double r59851 = pow(r59841, r59849);
        double r59852 = r59850 * r59851;
        double r59853 = pow(r59852, r59849);
        double r59854 = r59853 * r59826;
        double r59855 = fabs(r59854);
        double r59856 = pow(r59855, r59838);
        double r59857 = r59835 / r59856;
        double r59858 = r59832 ? r59848 : r59857;
        return r59858;
}

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 x < 5.443364118668269e-168 or 2.736219704037857e+219 < x

    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. Using strategy rm

    3. Applied sqr-pow29.6

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

    4. Applied associate-*r*22.0

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

    6. Applied add-sqr-sqrt22.0

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

    7. Simplified22.0

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

    8. Simplified3.1

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

    10. Applied associate-/r*2.8

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

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

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

    13. Applied div-inv2.8

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

    14. Applied times-frac2.8

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

    15. Simplified2.8

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

    if 5.443364118668269e-168 < x < 2.736219704037857e+219

    1. Initial program 24.4

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

    3. Applied sqr-pow24.4

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

    4. Applied associate-*r*20.9

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

    6. Applied add-sqr-sqrt20.9

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

    7. Simplified20.9

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

    8. Simplified2.3

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

    9. Taylor expanded around inf 0.9

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5.4433641186682693 \cdot 10^{-168} \lor \neg \left(x \le 2.73621970403785706 \cdot 10^{219}\right):\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{\frac{1}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  :precision binary64
  (/ (cos (* 2 x)) (* (pow cos 2) (* (* x (pow sin 2)) x))))