Average Error: 1.1 → 0.8
Time: 5.9s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right) \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right)}}\right)}\]
\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{1 + \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right) \cdot \left({\left(\frac{2 \cdot \ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \sqrt{{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}}\right)}}\right)}
double f(double l, double Om, double kx, double ky) {
        double r36761 = 1.0;
        double r36762 = 2.0;
        double r36763 = r36761 / r36762;
        double r36764 = l;
        double r36765 = r36762 * r36764;
        double r36766 = Om;
        double r36767 = r36765 / r36766;
        double r36768 = pow(r36767, r36762);
        double r36769 = kx;
        double r36770 = sin(r36769);
        double r36771 = pow(r36770, r36762);
        double r36772 = ky;
        double r36773 = sin(r36772);
        double r36774 = pow(r36773, r36762);
        double r36775 = r36771 + r36774;
        double r36776 = r36768 * r36775;
        double r36777 = r36761 + r36776;
        double r36778 = sqrt(r36777);
        double r36779 = r36761 / r36778;
        double r36780 = r36761 + r36779;
        double r36781 = r36763 * r36780;
        double r36782 = sqrt(r36781);
        return r36782;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r36783 = 1.0;
        double r36784 = 2.0;
        double r36785 = r36783 / r36784;
        double r36786 = l;
        double r36787 = r36784 * r36786;
        double r36788 = Om;
        double r36789 = r36787 / r36788;
        double r36790 = 2.0;
        double r36791 = r36784 / r36790;
        double r36792 = pow(r36789, r36791);
        double r36793 = kx;
        double r36794 = sin(r36793);
        double r36795 = pow(r36794, r36784);
        double r36796 = ky;
        double r36797 = sin(r36796);
        double r36798 = pow(r36797, r36784);
        double r36799 = r36795 + r36798;
        double r36800 = sqrt(r36799);
        double r36801 = r36792 * r36800;
        double r36802 = r36801 * r36801;
        double r36803 = r36783 + r36802;
        double r36804 = sqrt(r36803);
        double r36805 = r36783 / r36804;
        double r36806 = r36783 + r36805;
        double r36807 = r36785 * r36806;
        double r36808 = sqrt(r36807);
        return r36808;
}

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 1.1

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

  3. Applied add-sqr-sqrt1.1

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

  4. Applied sqr-pow1.1

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

  5. Applied unswap-sqr0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))