Average Error: 28.4 → 2.1
Time: 7.1s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.42323363288671218 \cdot 10^{183}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({sin}^{1}\right)}^{1} \cdot \left({\left({cos}^{1}\right)}^{1} \cdot x\right)\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{\left({sin}^{1} \cdot {cos}^{1}\right)}^{1} \cdot x\right|}}{{\left(\left|{\left({sin}^{1} \cdot {cos}^{1}\right)}^{1} \cdot x\right|\right)}^{1}}\\ \end{array}\]
\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 -1.42323363288671218 \cdot 10^{183}:\\
\;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({sin}^{1}\right)}^{1} \cdot \left({\left({cos}^{1}\right)}^{1} \cdot x\right)\right|\right)}^{2}}\\

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

\end{array}
double f(double x, double cos, double sin) {
        double r42168 = 2.0;
        double r42169 = x;
        double r42170 = r42168 * r42169;
        double r42171 = cos(r42170);
        double r42172 = cos;
        double r42173 = pow(r42172, r42168);
        double r42174 = sin;
        double r42175 = pow(r42174, r42168);
        double r42176 = r42169 * r42175;
        double r42177 = r42176 * r42169;
        double r42178 = r42173 * r42177;
        double r42179 = r42171 / r42178;
        return r42179;
}
↓
double f(double x, double cos, double sin) {
        double r42180 = x;
        double r42181 = -1.4232336328867122e+183;
        bool r42182 = r42180 <= r42181;
        double r42183 = 2.0;
        double r42184 = r42183 * r42180;
        double r42185 = cos(r42184);
        double r42186 = sin;
        double r42187 = 1.0;
        double r42188 = pow(r42186, r42187);
        double r42189 = pow(r42188, r42187);
        double r42190 = cos;
        double r42191 = pow(r42190, r42187);
        double r42192 = pow(r42191, r42187);
        double r42193 = r42192 * r42180;
        double r42194 = r42189 * r42193;
        double r42195 = fabs(r42194);
        double r42196 = 2.0;
        double r42197 = pow(r42195, r42196);
        double r42198 = r42185 / r42197;
        double r42199 = r42188 * r42191;
        double r42200 = pow(r42199, r42187);
        double r42201 = r42200 * r42180;
        double r42202 = fabs(r42201);
        double r42203 = r42185 / r42202;
        double r42204 = 1.0;
        double r42205 = pow(r42202, r42204);
        double r42206 = r42203 / r42205;
        double r42207 = r42182 ? r42198 : r42206;
        return r42207;
}

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 < -1.4232336328867122e+183

    1. Initial program 26.5

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

    3. Applied sqr-pow26.5

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

    4. Applied associate-*r*16.5

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

    6. Applied add-sqr-sqrt16.5

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

    7. Simplified16.5

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

    8. Simplified1.9

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

    9. Taylor expanded around 0 4.0

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

    11. Applied unpow-prod-down4.0

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

    12. Applied associate-*l*2.0

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

    if -1.4232336328867122e+183 < x

    1. Initial program 28.7

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

    3. Applied sqr-pow28.7

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

    4. Applied associate-*r*23.1

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

    6. Applied add-sqr-sqrt23.2

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

    7. Simplified23.1

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

    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. Taylor expanded around 0 2.4

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

    11. Applied sqr-pow2.4

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

    12. Applied associate-/r*2.1

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

    13. Simplified2.1

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.42323363288671218 \cdot 10^{183}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({sin}^{1}\right)}^{1} \cdot \left({\left({cos}^{1}\right)}^{1} \cdot x\right)\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{\left({sin}^{1} \cdot {cos}^{1}\right)}^{1} \cdot x\right|}}{{\left(\left|{\left({sin}^{1} \cdot {cos}^{1}\right)}^{1} \cdot x\right|\right)}^{1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(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))))