Average Error: 1.6 → 1.6
Time: 33.2s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(\frac{1}{\left(\sqrt[3]{\sqrt{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2}{Om} \cdot \ell\right)}^{2} + 1}} \cdot \sqrt[3]{\sqrt{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2}{Om} \cdot \ell\right)}^{2} + 1}}\right) \cdot \sqrt[3]{\sqrt{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2}{Om} \cdot \ell\right)}^{2} + 1}}} + 1\right) \cdot \frac{1}{2}}\]
\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{\left(\frac{1}{\left(\sqrt[3]{\sqrt{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2}{Om} \cdot \ell\right)}^{2} + 1}} \cdot \sqrt[3]{\sqrt{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2}{Om} \cdot \ell\right)}^{2} + 1}}\right) \cdot \sqrt[3]{\sqrt{\left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{2}{Om} \cdot \ell\right)}^{2} + 1}}} + 1\right) \cdot \frac{1}{2}}
double f(double l, double Om, double kx, double ky) {
        double r101786 = 1.0;
        double r101787 = 2.0;
        double r101788 = r101786 / r101787;
        double r101789 = l;
        double r101790 = r101787 * r101789;
        double r101791 = Om;
        double r101792 = r101790 / r101791;
        double r101793 = pow(r101792, r101787);
        double r101794 = kx;
        double r101795 = sin(r101794);
        double r101796 = pow(r101795, r101787);
        double r101797 = ky;
        double r101798 = sin(r101797);
        double r101799 = pow(r101798, r101787);
        double r101800 = r101796 + r101799;
        double r101801 = r101793 * r101800;
        double r101802 = r101786 + r101801;
        double r101803 = sqrt(r101802);
        double r101804 = r101786 / r101803;
        double r101805 = r101786 + r101804;
        double r101806 = r101788 * r101805;
        double r101807 = sqrt(r101806);
        return r101807;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r101808 = 1.0;
        double r101809 = kx;
        double r101810 = sin(r101809);
        double r101811 = 2.0;
        double r101812 = pow(r101810, r101811);
        double r101813 = ky;
        double r101814 = sin(r101813);
        double r101815 = pow(r101814, r101811);
        double r101816 = r101812 + r101815;
        double r101817 = Om;
        double r101818 = r101811 / r101817;
        double r101819 = l;
        double r101820 = r101818 * r101819;
        double r101821 = pow(r101820, r101811);
        double r101822 = r101816 * r101821;
        double r101823 = r101822 + r101808;
        double r101824 = sqrt(r101823);
        double r101825 = cbrt(r101824);
        double r101826 = r101825 * r101825;
        double r101827 = r101826 * r101825;
        double r101828 = r101808 / r101827;
        double r101829 = r101828 + r101808;
        double r101830 = r101808 / r101811;
        double r101831 = r101829 * r101830;
        double r101832 = sqrt(r101831);
        return r101832;
}

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. Simplified1.6

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

  4. Applied add-cube-cbrt1.6

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

  5. Simplified1.6

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

  6. Simplified1.6

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

  7. Final simplification1.6

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

Reproduce

herbie shell --seed 2019174 
(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))))))))))