Average Error: 1.6 → 1.6
Time: 1.1m
Precision: 64
\[\sqrt{\frac{1.0}{2.0} \cdot \left(1.0 + \frac{1.0}{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}\right)}\]
↓
\[\sqrt{\frac{1.0}{2.0} \cdot \left(\frac{\frac{1.0}{\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}}\right)}}{\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}}\right)} + 1.0\right)}\]
\sqrt{\frac{1.0}{2.0} \cdot \left(1.0 + \frac{1.0}{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}\right)}
↓
\sqrt{\frac{1.0}{2.0} \cdot \left(\frac{\frac{1.0}{\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}}\right)}}{\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}}\right)} + 1.0\right)}
double f(double l, double Om, double kx, double ky) {
        double r2109833 = 1.0;
        double r2109834 = 2.0;
        double r2109835 = r2109833 / r2109834;
        double r2109836 = l;
        double r2109837 = r2109834 * r2109836;
        double r2109838 = Om;
        double r2109839 = r2109837 / r2109838;
        double r2109840 = pow(r2109839, r2109834);
        double r2109841 = kx;
        double r2109842 = sin(r2109841);
        double r2109843 = pow(r2109842, r2109834);
        double r2109844 = ky;
        double r2109845 = sin(r2109844);
        double r2109846 = pow(r2109845, r2109834);
        double r2109847 = r2109843 + r2109846;
        double r2109848 = r2109840 * r2109847;
        double r2109849 = r2109833 + r2109848;
        double r2109850 = sqrt(r2109849);
        double r2109851 = r2109833 / r2109850;
        double r2109852 = r2109833 + r2109851;
        double r2109853 = r2109835 * r2109852;
        double r2109854 = sqrt(r2109853);
        return r2109854;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r2109855 = 1.0;
        double r2109856 = 2.0;
        double r2109857 = r2109855 / r2109856;
        double r2109858 = ky;
        double r2109859 = sin(r2109858);
        double r2109860 = pow(r2109859, r2109856);
        double r2109861 = kx;
        double r2109862 = sin(r2109861);
        double r2109863 = pow(r2109862, r2109856);
        double r2109864 = r2109860 + r2109863;
        double r2109865 = l;
        double r2109866 = r2109865 * r2109856;
        double r2109867 = Om;
        double r2109868 = r2109866 / r2109867;
        double r2109869 = pow(r2109868, r2109856);
        double r2109870 = r2109864 * r2109869;
        double r2109871 = r2109870 + r2109855;
        double r2109872 = sqrt(r2109871);
        double r2109873 = sqrt(r2109872);
        double r2109874 = cbrt(r2109873);
        double r2109875 = r2109874 * r2109874;
        double r2109876 = r2109874 * r2109875;
        double r2109877 = r2109855 / r2109876;
        double r2109878 = r2109877 / r2109876;
        double r2109879 = r2109878 + r2109855;
        double r2109880 = r2109857 * r2109879;
        double r2109881 = sqrt(r2109880);
        return r2109881;
}

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 1.6

    \[\sqrt{\frac{1.0}{2.0} \cdot \left(1.0 + \frac{1.0}{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}\right)}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt1.6

    \[\leadsto \sqrt{\frac{1.0}{2.0} \cdot \left(1.0 + \frac{1.0}{\sqrt{\color{blue}{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)} \cdot \sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}}\right)}\]

  4. Applied sqrt-prod1.6

    \[\leadsto \sqrt{\frac{1.0}{2.0} \cdot \left(1.0 + \frac{1.0}{\color{blue}{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}} \cdot \sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}}\right)}\]

  5. Applied associate-/r*1.6

    \[\leadsto \sqrt{\frac{1.0}{2.0} \cdot \left(1.0 + \color{blue}{\frac{\frac{1.0}{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}}{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}}\right)}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt1.6

    \[\leadsto \sqrt{\frac{1.0}{2.0} \cdot \left(1.0 + \frac{\frac{1.0}{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}}{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}}}\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt1.6

    \[\leadsto \sqrt{\frac{1.0}{2.0} \cdot \left(1.0 + \frac{\frac{1.0}{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}}}}{\left(\sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{1.0 + {\left(\frac{2.0 \cdot \ell}{Om}\right)}^{2.0} \cdot \left({\left(\sin kx\right)}^{2.0} + {\left(\sin ky\right)}^{2.0}\right)}}}}\right)}\]

  10. Final simplification1.6

    \[\leadsto \sqrt{\frac{1.0}{2.0} \cdot \left(\frac{\frac{1.0}{\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}}\right)}}{\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}} \cdot \sqrt[3]{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2.0} + {\left(\sin kx\right)}^{2.0}\right) \cdot {\left(\frac{\ell \cdot 2.0}{Om}\right)}^{2.0} + 1.0}}}\right)} + 1.0\right)}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))