Average Error: 1.8 → 1.8
Time: 24.5s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \left(\sqrt[3]{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}} \cdot \sqrt[3]{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right) \cdot \sqrt[3]{\frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right)}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\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 + \left(\sqrt[3]{\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)}}} \cdot \sqrt[3]{\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) \cdot \sqrt[3]{\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)}
double f(double l, double Om, double kx, double ky) {
        double r38767 = 1.0;
        double r38768 = 2.0;
        double r38769 = r38767 / r38768;
        double r38770 = l;
        double r38771 = r38768 * r38770;
        double r38772 = Om;
        double r38773 = r38771 / r38772;
        double r38774 = pow(r38773, r38768);
        double r38775 = kx;
        double r38776 = sin(r38775);
        double r38777 = pow(r38776, r38768);
        double r38778 = ky;
        double r38779 = sin(r38778);
        double r38780 = pow(r38779, r38768);
        double r38781 = r38777 + r38780;
        double r38782 = r38774 * r38781;
        double r38783 = r38767 + r38782;
        double r38784 = sqrt(r38783);
        double r38785 = r38767 / r38784;
        double r38786 = r38767 + r38785;
        double r38787 = r38769 * r38786;
        double r38788 = sqrt(r38787);
        return r38788;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r38789 = 1.0;
        double r38790 = 2.0;
        double r38791 = r38789 / r38790;
        double r38792 = l;
        double r38793 = r38790 * r38792;
        double r38794 = Om;
        double r38795 = r38793 / r38794;
        double r38796 = pow(r38795, r38790);
        double r38797 = kx;
        double r38798 = sin(r38797);
        double r38799 = pow(r38798, r38790);
        double r38800 = ky;
        double r38801 = sin(r38800);
        double r38802 = pow(r38801, r38790);
        double r38803 = r38799 + r38802;
        double r38804 = r38796 * r38803;
        double r38805 = r38789 + r38804;
        double r38806 = sqrt(r38805);
        double r38807 = r38789 / r38806;
        double r38808 = cbrt(r38807);
        double r38809 = r38808 * r38808;
        double r38810 = r38809 * r38808;
        double r38811 = r38789 + r38810;
        double r38812 = r38791 * r38811;
        double r38813 = sqrt(r38812);
        return r38813;
}

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 1.8

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

  3. Applied add-cube-cbrt1.8

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

  4. Final simplification1.8

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

Reproduce

herbie shell --seed 2019303 
(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))))))))))