Average Error: 1.6 → 1.3
Time: 19.1s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\left({\left(\frac{\ell \cdot 2}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right)\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{\left(\frac{2}{2}\right)} + 1}}\right)}\]
\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} \cdot \left(1 + \frac{1}{\sqrt{\left({\left(\frac{\ell \cdot 2}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right)\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{\left(\frac{2}{2}\right)} + 1}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r1146852 = 1.0;
        double r1146853 = 2.0;
        double r1146854 = r1146852 / r1146853;
        double r1146855 = l;
        double r1146856 = r1146853 * r1146855;
        double r1146857 = Om;
        double r1146858 = r1146856 / r1146857;
        double r1146859 = pow(r1146858, r1146853);
        double r1146860 = kx;
        double r1146861 = sin(r1146860);
        double r1146862 = pow(r1146861, r1146853);
        double r1146863 = ky;
        double r1146864 = sin(r1146863);
        double r1146865 = pow(r1146864, r1146853);
        double r1146866 = r1146862 + r1146865;
        double r1146867 = r1146859 * r1146866;
        double r1146868 = r1146852 + r1146867;
        double r1146869 = sqrt(r1146868);
        double r1146870 = r1146852 / r1146869;
        double r1146871 = r1146852 + r1146870;
        double r1146872 = r1146854 * r1146871;
        double r1146873 = sqrt(r1146872);
        return r1146873;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1146874 = 1.0;
        double r1146875 = 2.0;
        double r1146876 = r1146874 / r1146875;
        double r1146877 = l;
        double r1146878 = r1146877 * r1146875;
        double r1146879 = Om;
        double r1146880 = r1146878 / r1146879;
        double r1146881 = 2.0;
        double r1146882 = r1146875 / r1146881;
        double r1146883 = pow(r1146880, r1146882);
        double r1146884 = ky;
        double r1146885 = sin(r1146884);
        double r1146886 = pow(r1146885, r1146875);
        double r1146887 = kx;
        double r1146888 = sin(r1146887);
        double r1146889 = pow(r1146888, r1146875);
        double r1146890 = r1146886 + r1146889;
        double r1146891 = r1146883 * r1146890;
        double r1146892 = r1146891 * r1146883;
        double r1146893 = r1146892 + r1146874;
        double r1146894 = sqrt(r1146893);
        double r1146895 = r1146874 / r1146894;
        double r1146896 = r1146874 + r1146895;
        double r1146897 = r1146876 * r1146896;
        double r1146898 = sqrt(r1146897);
        return r1146898;
}

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

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

  3. Applied sqr-pow1.6

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

  4. Applied associate-*l*1.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2019171 
(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))))))))))