Average Error: 1.7 → 0.6
Time: 30.0s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} + \sqrt[3]{\frac{\frac{\frac{1}{8}}{\left(\left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) + \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right) \cdot \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right)\right) + 1}}{\sqrt{\left(\left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) + \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right) \cdot \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right)\right) + 1}}}}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{1}{2} + \sqrt[3]{\frac{\frac{\frac{1}{8}}{\left(\left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) + \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right) \cdot \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right)\right) + 1}}{\sqrt{\left(\left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) + \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right) \cdot \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right)\right) + 1}}}}
double f(double l, double Om, double kx, double ky) {
        double r1801845 = 1.0;
        double r1801846 = 2.0;
        double r1801847 = r1801845 / r1801846;
        double r1801848 = l;
        double r1801849 = r1801846 * r1801848;
        double r1801850 = Om;
        double r1801851 = r1801849 / r1801850;
        double r1801852 = pow(r1801851, r1801846);
        double r1801853 = kx;
        double r1801854 = sin(r1801853);
        double r1801855 = pow(r1801854, r1801846);
        double r1801856 = ky;
        double r1801857 = sin(r1801856);
        double r1801858 = pow(r1801857, r1801846);
        double r1801859 = r1801855 + r1801858;
        double r1801860 = r1801852 * r1801859;
        double r1801861 = r1801845 + r1801860;
        double r1801862 = sqrt(r1801861);
        double r1801863 = r1801845 / r1801862;
        double r1801864 = r1801845 + r1801863;
        double r1801865 = r1801847 * r1801864;
        double r1801866 = sqrt(r1801865);
        return r1801866;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1801867 = 0.5;
        double r1801868 = 0.125;
        double r1801869 = l;
        double r1801870 = r1801869 + r1801869;
        double r1801871 = Om;
        double r1801872 = r1801870 / r1801871;
        double r1801873 = ky;
        double r1801874 = sin(r1801873);
        double r1801875 = r1801872 * r1801874;
        double r1801876 = r1801875 * r1801875;
        double r1801877 = kx;
        double r1801878 = sin(r1801877);
        double r1801879 = r1801878 * r1801872;
        double r1801880 = r1801879 * r1801879;
        double r1801881 = r1801876 + r1801880;
        double r1801882 = 1.0;
        double r1801883 = r1801881 + r1801882;
        double r1801884 = r1801868 / r1801883;
        double r1801885 = sqrt(r1801883);
        double r1801886 = r1801884 / r1801885;
        double r1801887 = cbrt(r1801886);
        double r1801888 = r1801867 + r1801887;
        double r1801889 = sqrt(r1801888);
        return r1801889;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.7

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]

  2. Simplified1.7

    \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}\right) + 1}} + \frac{1}{2}}}\]
  3. Using strategy rm

  4. Applied insert-posit165.9

    \[\leadsto \sqrt{\color{blue}{\left(\left(\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}\right) + 1}}\right)\right)} + \frac{1}{2}}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube5.9

    \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\left(\left(\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}\right) + 1}}\right)\right) \cdot \left(\left(\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}\right) + 1}}\right)\right)\right) \cdot \left(\left(\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}\right) + 1}}\right)\right)}} + \frac{1}{2}}\]

  7. Simplified0.6

    \[\leadsto \sqrt{\sqrt[3]{\color{blue}{\frac{\frac{\frac{1}{8}}{\left(\left(\frac{\ell + \ell}{Om} \cdot \sin kx\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin kx\right) + \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right)\right) + 1}}{\sqrt{\left(\left(\frac{\ell + \ell}{Om} \cdot \sin kx\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin kx\right) + \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right)\right) + 1}}}} + \frac{1}{2}}\]

  8. Final simplification0.6

    \[\leadsto \sqrt{\frac{1}{2} + \sqrt[3]{\frac{\frac{\frac{1}{8}}{\left(\left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) + \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right) \cdot \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right)\right) + 1}}{\sqrt{\left(\left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) \cdot \left(\frac{\ell + \ell}{Om} \cdot \sin ky\right) + \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right) \cdot \left(\sin kx \cdot \frac{\ell + \ell}{Om}\right)\right) + 1}}}}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))