Average Error: 28.2 → 2.6
Time: 8.1s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left|\left|{\left({cos}^{1} \cdot {sin}^{1}\right)}^{1} \cdot x\right|\right|}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\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|}
double f(double x, double cos, double sin) {
        double r44861 = 2.0;
        double r44862 = x;
        double r44863 = r44861 * r44862;
        double r44864 = cos(r44863);
        double r44865 = cos;
        double r44866 = pow(r44865, r44861);
        double r44867 = sin;
        double r44868 = pow(r44867, r44861);
        double r44869 = r44862 * r44868;
        double r44870 = r44869 * r44862;
        double r44871 = r44866 * r44870;
        double r44872 = r44864 / r44871;
        return r44872;
}
↓
double f(double x, double cos, double sin) {
        double r44873 = 1.0;
        double r44874 = cos;
        double r44875 = 1.0;
        double r44876 = pow(r44874, r44875);
        double r44877 = sin;
        double r44878 = pow(r44877, r44875);
        double r44879 = r44876 * r44878;
        double r44880 = pow(r44879, r44875);
        double r44881 = x;
        double r44882 = r44880 * r44881;
        double r44883 = fabs(r44882);
        double r44884 = fabs(r44883);
        double r44885 = r44873 / r44884;
        double r44886 = 2.0;
        double r44887 = r44886 * r44881;
        double r44888 = cos(r44887);
        double r44889 = r44888 / r44884;
        double r44890 = r44885 * r44889;
        return r44890;
}

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 28.2

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

  3. Applied sqr-pow28.2

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

  4. Applied associate-*r*21.7

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

  6. Applied add-sqr-sqrt21.7

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

  7. Simplified21.7

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

  8. Simplified2.9

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

  9. Taylor expanded around inf 2.8

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

  11. Applied add-sqr-sqrt2.8

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

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

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

  13. Applied times-frac2.9

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

  14. Simplified2.9

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

  15. Simplified2.6

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

  16. Final simplification2.6

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

Reproduce

herbie shell --seed 2020033 
(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))))