Average Error: 28.3 → 2.2
Time: 8.4s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.899795259514831596491496327990136660459 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}}{{\left(\sqrt{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}\right)}^{2}}\\ \mathbf{elif}\;x \le 3.099208061336091548327514698736633602822 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{1}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]
\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 -3.899795259514831596491496327990136660459 \cdot 10^{-188}:\\
\;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}}{{\left(\sqrt{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}\right)}^{2}}\\

\mathbf{elif}\;x \le 3.099208061336091548327514698736633602822 \cdot 10^{-150}:\\
\;\;\;\;\frac{1}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \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|}\\

\mathbf{else}:\\
\;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{1}{{\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 r56026 = 2.0;
        double r56027 = x;
        double r56028 = r56026 * r56027;
        double r56029 = cos(r56028);
        double r56030 = cos;
        double r56031 = pow(r56030, r56026);
        double r56032 = sin;
        double r56033 = pow(r56032, r56026);
        double r56034 = r56027 * r56033;
        double r56035 = r56034 * r56027;
        double r56036 = r56031 * r56035;
        double r56037 = r56029 / r56036;
        return r56037;
}
↓
double f(double x, double cos, double sin) {
        double r56038 = x;
        double r56039 = -3.8997952595148316e-188;
        bool r56040 = r56038 <= r56039;
        double r56041 = 2.0;
        double r56042 = r56041 * r56038;
        double r56043 = cos(r56042);
        double r56044 = cos;
        double r56045 = 1.0;
        double r56046 = pow(r56044, r56045);
        double r56047 = sin;
        double r56048 = pow(r56047, r56045);
        double r56049 = r56046 * r56048;
        double r56050 = pow(r56049, r56045);
        double r56051 = r56050 * r56038;
        double r56052 = fabs(r56051);
        double r56053 = r56043 / r56052;
        double r56054 = sqrt(r56052);
        double r56055 = 2.0;
        double r56056 = pow(r56054, r56055);
        double r56057 = r56053 / r56056;
        double r56058 = 3.0992080613360915e-150;
        bool r56059 = r56038 <= r56058;
        double r56060 = 1.0;
        double r56061 = r56041 / r56055;
        double r56062 = pow(r56044, r56061);
        double r56063 = pow(r56047, r56061);
        double r56064 = r56038 * r56063;
        double r56065 = r56062 * r56064;
        double r56066 = fabs(r56065);
        double r56067 = r56060 / r56066;
        double r56068 = r56043 / r56066;
        double r56069 = r56067 * r56068;
        double r56070 = pow(r56052, r56055);
        double r56071 = r56060 / r56070;
        double r56072 = r56043 * r56071;
        double r56073 = r56059 ? r56069 : r56072;
        double r56074 = r56040 ? r56057 : r56073;
        return r56074;
}

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 x < -3.8997952595148316e-188

    1. Initial program 26.2

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

    3. Applied sqr-pow26.2

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

    4. Applied associate-*r*21.0

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

    6. Applied add-sqr-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. Simplified21.0

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

    8. Simplified2.6

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

    9. Taylor expanded around inf 2.1

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

    11. Applied add-sqr-sqrt2.3

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

    12. Applied unpow-prod-down2.3

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

    13. Applied associate-/r*1.9

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

    14. Simplified1.8

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

    if -3.8997952595148316e-188 < x < 3.0992080613360915e-150

    1. Initial program 44.2

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

    3. Applied sqr-pow44.2

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

    4. Applied associate-*r*30.4

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

    6. Applied add-sqr-sqrt30.5

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

    7. Simplified30.4

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

    8. Simplified4.8

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

    10. Applied *-un-lft-identity4.8

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

    11. Applied times-frac4.6

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

    if 3.0992080613360915e-150 < x

    1. Initial program 25.7

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

    3. Applied sqr-pow25.7

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

    4. Applied associate-*r*20.3

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

    6. Applied add-sqr-sqrt20.4

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

    7. Simplified20.4

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

    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. Taylor expanded around inf 1.8

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

    11. Applied div-inv1.8

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.899795259514831596491496327990136660459 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}}{{\left(\sqrt{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}\right)}^{2}}\\ \mathbf{elif}\;x \le 3.099208061336091548327514698736633602822 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{cos}^{\left(\frac{2}{2}\right)} \cdot \left(x \cdot {sin}^{\left(\frac{2}{2}\right)}\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot x\right) \cdot \frac{1}{{\left(\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +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))))