Average Error: 1.6 → 1.6
Time: 33.4s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \cdot \left(\sqrt{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \cdot \sqrt{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}}\right)}\right)}\]
\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{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \cdot \left(\sqrt{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \cdot \sqrt{\sqrt[3]{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}}\right)}\right)}
double f(double l, double Om, double kx, double ky) {
        double r1794775 = 1.0;
        double r1794776 = 2.0;
        double r1794777 = r1794775 / r1794776;
        double r1794778 = l;
        double r1794779 = r1794776 * r1794778;
        double r1794780 = Om;
        double r1794781 = r1794779 / r1794780;
        double r1794782 = pow(r1794781, r1794776);
        double r1794783 = kx;
        double r1794784 = sin(r1794783);
        double r1794785 = pow(r1794784, r1794776);
        double r1794786 = ky;
        double r1794787 = sin(r1794786);
        double r1794788 = pow(r1794787, r1794776);
        double r1794789 = r1794785 + r1794788;
        double r1794790 = r1794782 * r1794789;
        double r1794791 = r1794775 + r1794790;
        double r1794792 = sqrt(r1794791);
        double r1794793 = r1794775 / r1794792;
        double r1794794 = r1794775 + r1794793;
        double r1794795 = r1794777 * r1794794;
        double r1794796 = sqrt(r1794795);
        return r1794796;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r1794797 = 1.0;
        double r1794798 = 2.0;
        double r1794799 = r1794797 / r1794798;
        double r1794800 = ky;
        double r1794801 = sin(r1794800);
        double r1794802 = pow(r1794801, r1794798);
        double r1794803 = kx;
        double r1794804 = sin(r1794803);
        double r1794805 = pow(r1794804, r1794798);
        double r1794806 = r1794802 + r1794805;
        double r1794807 = l;
        double r1794808 = r1794807 * r1794798;
        double r1794809 = Om;
        double r1794810 = r1794808 / r1794809;
        double r1794811 = pow(r1794810, r1794798);
        double r1794812 = r1794806 * r1794811;
        double r1794813 = r1794812 + r1794797;
        double r1794814 = cbrt(r1794813);
        double r1794815 = sqrt(r1794814);
        double r1794816 = r1794815 * r1794815;
        double r1794817 = r1794815 * r1794816;
        double r1794818 = r1794797 / r1794817;
        double r1794819 = r1794797 + r1794818;
        double r1794820 = r1794799 * r1794819;
        double r1794821 = sqrt(r1794820);
        return r1794821;
}

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 add-cube-cbrt1.6

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

  4. Applied sqrt-prod1.6

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

  6. Applied sqrt-prod1.6

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

  7. Final simplification1.6

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

Reproduce

herbie shell --seed 2019170 
(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))))))))))