Average Error: 28.1 → 2.4
Time: 7.9s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|}
double f(double x, double cos, double sin) {
        double r70900 = 2.0;
        double r70901 = x;
        double r70902 = r70900 * r70901;
        double r70903 = cos(r70902);
        double r70904 = cos;
        double r70905 = pow(r70904, r70900);
        double r70906 = sin;
        double r70907 = pow(r70906, r70900);
        double r70908 = r70901 * r70907;
        double r70909 = r70908 * r70901;
        double r70910 = r70905 * r70909;
        double r70911 = r70903 / r70910;
        return r70911;
}
↓
double f(double x, double cos, double sin) {
        double r70912 = 1.0;
        double r70913 = cos;
        double r70914 = 1.0;
        double r70915 = pow(r70913, r70914);
        double r70916 = sin;
        double r70917 = pow(r70916, r70914);
        double r70918 = r70915 * r70917;
        double r70919 = pow(r70918, r70914);
        double r70920 = x;
        double r70921 = r70919 * r70920;
        double r70922 = fabs(r70921);
        double r70923 = r70912 / r70922;
        double r70924 = 2.0;
        double r70925 = r70924 * r70920;
        double r70926 = cos(r70925);
        double r70927 = r70926 / r70922;
        double r70928 = r70923 * r70927;
        return r70928;
}

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. Initial program 28.1

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

  3. Applied sqr-pow28.1

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

  4. Applied associate-*r*21.9

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

  6. Applied add-sqr-sqrt21.9

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

  7. Simplified21.9

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

  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 inf 2.6

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

  11. Applied add-sqr-sqrt2.8

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

  12. Applied unpow-prod-down2.8

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

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

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

  14. Applied times-frac2.6

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

  15. Simplified2.5

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

  16. Simplified2.4

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

  17. Final simplification2.4

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

Reproduce

herbie shell --seed 2020003 
(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))))