cos(2*x)/(cos^2(x)*sin^2(x))

Average Error: 28.3 → 2.2

Time: 8.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double cos, double sin) {
        double r52875 = 2.0;
        double r52876 = x;
        double r52877 = r52875 * r52876;
        double r52878 = cos(r52877);
        double r52879 = cos;
        double r52880 = pow(r52879, r52875);
        double r52881 = sin;
        double r52882 = pow(r52881, r52875);
        double r52883 = r52876 * r52882;
        double r52884 = r52883 * r52876;
        double r52885 = r52880 * r52884;
        double r52886 = r52878 / r52885;
        return r52886;
}

↓

double f(double x, double cos, double sin) {
        double r52887 = x;
        double r52888 = -3.8997952595148316e-188;
        bool r52889 = r52887 <= r52888;
        double r52890 = 2.0;
        double r52891 = r52890 * r52887;
        double r52892 = cos(r52891);
        double r52893 = cos;
        double r52894 = 1.0;
        double r52895 = pow(r52893, r52894);
        double r52896 = sin;
        double r52897 = pow(r52896, r52894);
        double r52898 = r52895 * r52897;
        double r52899 = pow(r52898, r52894);
        double r52900 = r52899 * r52887;
        double r52901 = fabs(r52900);
        double r52902 = r52892 / r52901;
        double r52903 = sqrt(r52901);
        double r52904 = 2.0;
        double r52905 = pow(r52903, r52904);
        double r52906 = r52902 / r52905;
        double r52907 = 3.0992080613360915e-150;
        bool r52908 = r52887 <= r52907;
        double r52909 = 1.0;
        double r52910 = r52890 / r52904;
        double r52911 = pow(r52893, r52910);
        double r52912 = pow(r52896, r52910);
        double r52913 = r52887 * r52912;
        double r52914 = r52911 * r52913;
        double r52915 = fabs(r52914);
        double r52916 = r52909 / r52915;
        double r52917 = r52892 / r52915;
        double r52918 = r52916 * r52917;
        double r52919 = pow(r52901, r52904);
        double r52920 = r52909 / r52919;
        double r52921 = r52892 * r52920;
        double r52922 = r52908 ? r52918 : r52921;
        double r52923 = r52889 ? r52906 : r52922;
        return r52923;
}

Error

Bits error versus x

Bits error versus cos

Bits error versus sin

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-pow 26.2

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

   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-sqrt 21.1

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

   7. Simplified 21.0

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

   8. Simplified 2.6

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

   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-sqrt 2.3

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

   12. Applied unpow-prod-down 2.3

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

   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. Simplified 1.8

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

   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-pow 44.2

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

   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-sqrt 30.5

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

   7. Simplified 30.4

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

   8. Simplified 4.8

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

   10. Applied *-un-lft-identity 4.8

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

   11. Applied times-frac 4.6

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

   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-pow 25.7

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

   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-sqrt 20.4

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

   7. Simplified 20.4

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

   8. Simplified 2.4

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

   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-inv 1.8

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

4. Final simplification 2.2

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

Reproduce

herbie shell --seed 2019352 
(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))))