Average Error: 27.7 → 2.7
Time: 18.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;cos \le -2.047820410589973 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right) \cdot \frac{1}{x}}{cos \cdot sin}}{\left(cos \cdot sin\right) \cdot x}\\ \mathbf{elif}\;cos \le 5.117094630849011 \cdot 10^{-162}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{cos \cdot sin} \cdot \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{cos}}{x}\\ \end{array}\]
\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 -2.047820410589973 \cdot 10^{+33}:\\
\;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right) \cdot \frac{1}{x}}{cos \cdot sin}}{\left(cos \cdot sin\right) \cdot x}\\

\mathbf{elif}\;cos \le 5.117094630849011 \cdot 10^{-162}:\\
\;\;\;\;\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)}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r1228968 = 2.0;
        double r1228969 = x;
        double r1228970 = r1228968 * r1228969;
        double r1228971 = cos(r1228970);
        double r1228972 = cos;
        double r1228973 = pow(r1228972, r1228968);
        double r1228974 = sin;
        double r1228975 = pow(r1228974, r1228968);
        double r1228976 = r1228969 * r1228975;
        double r1228977 = r1228976 * r1228969;
        double r1228978 = r1228973 * r1228977;
        double r1228979 = r1228971 / r1228978;
        return r1228979;
}
↓
double f(double x, double cos, double sin) {
        double r1228980 = cos;
        double r1228981 = -2.047820410589973e+33;
        bool r1228982 = r1228980 <= r1228981;
        double r1228983 = 2.0;
        double r1228984 = x;
        double r1228985 = r1228983 * r1228984;
        double r1228986 = cos(r1228985);
        double r1228987 = 1.0;
        double r1228988 = r1228987 / r1228984;
        double r1228989 = r1228986 * r1228988;
        double r1228990 = sin;
        double r1228991 = r1228980 * r1228990;
        double r1228992 = r1228989 / r1228991;
        double r1228993 = r1228991 * r1228984;
        double r1228994 = r1228992 / r1228993;
        double r1228995 = 5.117094630849011e-162;
        bool r1228996 = r1228980 <= r1228995;
        double r1228997 = r1228984 * r1228980;
        double r1228998 = r1228990 * r1228997;
        double r1228999 = r1228998 * r1228998;
        double r1229000 = r1228986 / r1228999;
        double r1229001 = r1228988 / r1228991;
        double r1229002 = r1228986 / r1228990;
        double r1229003 = r1229002 / r1228980;
        double r1229004 = r1229003 / r1228984;
        double r1229005 = r1229001 * r1229004;
        double r1229006 = r1228996 ? r1229000 : r1229005;
        double r1229007 = r1228982 ? r1228994 : r1229006;
        return r1229007;
}

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 3 regimes

  2. if cos < -2.047820410589973e+33

    1. Initial program 23.4

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

    2. Simplified2.8

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

    4. Applied associate-/r*2.6

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

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

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

    7. Applied times-frac2.5

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

    9. Applied associate-*r/2.5

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

    if -2.047820410589973e+33 < cos < 5.117094630849011e-162

    1. Initial program 41.1

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

    2. Simplified3.5

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

    3. Taylor expanded around 0 43.7

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

    4. Simplified3.7

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

    if 5.117094630849011e-162 < cos

    1. Initial program 21.9

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

    2. Simplified2.4

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

    4. Applied associate-/r*2.1

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

    6. Applied *-un-lft-identity2.1

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

    7. Applied times-frac2.1

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

    9. Applied associate-*l/2.1

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

    10. Simplified2.1

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

    12. Applied div-inv2.1

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

    13. Applied times-frac2.1

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le -2.047820410589973 \cdot 10^{+33}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right) \cdot \frac{1}{x}}{cos \cdot sin}}{\left(cos \cdot sin\right) \cdot x}\\ \mathbf{elif}\;cos \le 5.117094630849011 \cdot 10^{-162}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left(sin \cdot \left(x \cdot cos\right)\right) \cdot \left(sin \cdot \left(x \cdot cos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{cos \cdot sin} \cdot \frac{\frac{\frac{\cos \left(2 \cdot x\right)}{sin}}{cos}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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))))