Average Error: 28.1 → 2.5
Time: 7.9s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;cos \le 4.19899225278698649 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|}\\ \mathbf{elif}\;cos \le 7.57685492731950486 \cdot 10^{145}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|{\left(\sqrt{cos}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sqrt{cos}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]
\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}\;cos \le 4.19899225278698649 \cdot 10^{-308}:\\
\;\;\;\;\frac{1}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|}\\

\mathbf{elif}\;cos \le 7.57685492731950486 \cdot 10^{145}:\\
\;\;\;\;\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 \left|{\left(\sqrt{cos}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sqrt{cos}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\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 r54256 = 2.0;
        double r54257 = x;
        double r54258 = r54256 * r54257;
        double r54259 = cos(r54258);
        double r54260 = cos;
        double r54261 = pow(r54260, r54256);
        double r54262 = sin;
        double r54263 = pow(r54262, r54256);
        double r54264 = r54257 * r54263;
        double r54265 = r54264 * r54257;
        double r54266 = r54261 * r54265;
        double r54267 = r54259 / r54266;
        return r54267;
}
↓
double f(double x, double cos, double sin) {
        double r54268 = cos;
        double r54269 = 4.1989922527869865e-308;
        bool r54270 = r54268 <= r54269;
        double r54271 = 1.0;
        double r54272 = 1.0;
        double r54273 = pow(r54268, r54272);
        double r54274 = sin;
        double r54275 = pow(r54274, r54272);
        double r54276 = r54273 * r54275;
        double r54277 = pow(r54276, r54272);
        double r54278 = x;
        double r54279 = r54277 * r54278;
        double r54280 = fabs(r54279);
        double r54281 = fabs(r54280);
        double r54282 = r54271 / r54281;
        double r54283 = 2.0;
        double r54284 = r54283 * r54278;
        double r54285 = cos(r54284);
        double r54286 = r54285 / r54281;
        double r54287 = r54282 * r54286;
        double r54288 = 7.576854927319505e+145;
        bool r54289 = r54268 <= r54288;
        double r54290 = 2.0;
        double r54291 = r54283 / r54290;
        double r54292 = pow(r54268, r54291);
        double r54293 = pow(r54274, r54291);
        double r54294 = r54278 * r54293;
        double r54295 = r54292 * r54294;
        double r54296 = fabs(r54295);
        double r54297 = sqrt(r54268);
        double r54298 = pow(r54297, r54291);
        double r54299 = r54298 * r54294;
        double r54300 = r54298 * r54299;
        double r54301 = fabs(r54300);
        double r54302 = r54296 * r54301;
        double r54303 = r54285 / r54302;
        double r54304 = pow(r54280, r54290);
        double r54305 = r54285 / r54304;
        double r54306 = r54289 ? r54303 : r54305;
        double r54307 = r54270 ? r54287 : r54306;
        return r54307;
}

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 < 4.1989922527869865e-308

    1. Initial program 28.3

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

    3. Applied sqr-pow28.3

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

    4. Applied associate-*r*22.6

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

    6. Applied add-sqr-sqrt22.7

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

    7. Simplified22.6

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

    8. Simplified3.0

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

    9. Taylor expanded around 0 2.8

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

    10. Simplified2.8

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

    12. Applied add-sqr-sqrt2.8

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

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

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

    14. Applied times-frac2.8

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

    15. Simplified2.8

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

    16. Simplified2.6

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

    if 4.1989922527869865e-308 < cos < 7.576854927319505e+145

    1. Initial program 29.7

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

    3. Applied sqr-pow29.7

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

    4. Applied associate-*r*21.8

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

    6. Applied add-sqr-sqrt21.8

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

    7. Simplified21.8

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

    8. Simplified2.4

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

    10. Applied add-sqr-sqrt2.5

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

    11. Applied unpow-prod-down2.5

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

    12. Applied associate-*l*2.4

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

    if 7.576854927319505e+145 < cos

    1. Initial program 24.7

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

    3. Applied sqr-pow24.7

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

    4. Applied associate-*r*21.1

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

    6. Applied add-sqr-sqrt21.1

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

    7. Simplified21.1

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

    8. Simplified3.4

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

    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. Simplified2.4

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;cos \le 4.19899225278698649 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|}\\ \mathbf{elif}\;cos \le 7.57685492731950486 \cdot 10^{145}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right| \cdot \left|{\left(\sqrt{cos}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sqrt{cos}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))))