Average Error: 28.2 → 2.5
Time: 31.1s
Precision: 64
\[\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}\]
↓
\[\frac{1}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot {cos}^{\left(\frac{2}{2}\right)}} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}\]
\frac{\cos \left(2 \cdot x\right)}{{cos}^{2} \cdot \left(\left(x \cdot {sin}^{2}\right) \cdot x\right)}
↓
\frac{1}{\left({sin}^{\left(\frac{2}{2}\right)} \cdot x\right) \cdot {cos}^{\left(\frac{2}{2}\right)}} \cdot \frac{\frac{\cos \left(2 \cdot x\right)}{{cos}^{\left(\frac{2}{2}\right)}}}{{sin}^{\left(\frac{2}{2}\right)} \cdot x}
double f(double x, double cos, double sin) {
        double r3674846 = 2.0;
        double r3674847 = x;
        double r3674848 = r3674846 * r3674847;
        double r3674849 = cos(r3674848);
        double r3674850 = cos;
        double r3674851 = pow(r3674850, r3674846);
        double r3674852 = sin;
        double r3674853 = pow(r3674852, r3674846);
        double r3674854 = r3674847 * r3674853;
        double r3674855 = r3674854 * r3674847;
        double r3674856 = r3674851 * r3674855;
        double r3674857 = r3674849 / r3674856;
        return r3674857;
}
↓
double f(double x, double cos, double sin) {
        double r3674858 = 1.0;
        double r3674859 = sin;
        double r3674860 = 2.0;
        double r3674861 = 2.0;
        double r3674862 = r3674860 / r3674861;
        double r3674863 = pow(r3674859, r3674862);
        double r3674864 = x;
        double r3674865 = r3674863 * r3674864;
        double r3674866 = cos;
        double r3674867 = pow(r3674866, r3674862);
        double r3674868 = r3674865 * r3674867;
        double r3674869 = r3674858 / r3674868;
        double r3674870 = r3674860 * r3674864;
        double r3674871 = cos(r3674870);
        double r3674872 = r3674871 / r3674867;
        double r3674873 = r3674872 / r3674865;
        double r3674874 = r3674869 * r3674873;
        return r3674874;
}

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 sqr-pow21.7

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

  7. Applied associate-*l*16.1

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

  8. Simplified6.1

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

  10. Applied associate-/r*5.9

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

  12. Applied *-un-lft-identity5.9

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

  13. Applied times-frac2.5

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

  14. Final simplification2.5

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x cos sin)
  :name "cos(2*x)/(cos^2(x)*sin^2(x))"
  (/ (cos (* 2.0 x)) (* (pow cos 2.0) (* (* x (pow sin 2.0)) x))))