Average Error: 1.7 → 1.3
Time: 26.1s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 6.255042022075511028897152922160877393 \cdot 10^{-314}:\\ \;\;\;\;\sqrt{\left(1 + \frac{1}{\sqrt{1}}\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}} \cdot \frac{1}{\left(\sqrt[3]{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 6.255042022075511028897152922160877393 \cdot 10^{-314}:\\
\;\;\;\;\sqrt{\left(1 + \frac{1}{\sqrt{1}}\right) \cdot \frac{1}{2}}\\

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

\end{array}
double f(double l, double Om, double kx, double ky) {
        double r45815 = 1.0;
        double r45816 = 2.0;
        double r45817 = r45815 / r45816;
        double r45818 = l;
        double r45819 = r45816 * r45818;
        double r45820 = Om;
        double r45821 = r45819 / r45820;
        double r45822 = pow(r45821, r45816);
        double r45823 = kx;
        double r45824 = sin(r45823);
        double r45825 = pow(r45824, r45816);
        double r45826 = ky;
        double r45827 = sin(r45826);
        double r45828 = pow(r45827, r45816);
        double r45829 = r45825 + r45828;
        double r45830 = r45822 * r45829;
        double r45831 = r45815 + r45830;
        double r45832 = sqrt(r45831);
        double r45833 = r45815 / r45832;
        double r45834 = r45815 + r45833;
        double r45835 = r45817 * r45834;
        double r45836 = sqrt(r45835);
        return r45836;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r45837 = kx;
        double r45838 = sin(r45837);
        double r45839 = 2.0;
        double r45840 = pow(r45838, r45839);
        double r45841 = ky;
        double r45842 = sin(r45841);
        double r45843 = pow(r45842, r45839);
        double r45844 = r45840 + r45843;
        double r45845 = 6.2550420220755e-314;
        bool r45846 = r45844 <= r45845;
        double r45847 = 1.0;
        double r45848 = sqrt(r45847);
        double r45849 = r45847 / r45848;
        double r45850 = r45847 + r45849;
        double r45851 = r45847 / r45839;
        double r45852 = r45850 * r45851;
        double r45853 = sqrt(r45852);
        double r45854 = 1.0;
        double r45855 = l;
        double r45856 = r45839 * r45855;
        double r45857 = Om;
        double r45858 = r45856 / r45857;
        double r45859 = pow(r45858, r45839);
        double r45860 = r45859 * r45844;
        double r45861 = r45847 + r45860;
        double r45862 = sqrt(r45861);
        double r45863 = cbrt(r45862);
        double r45864 = r45863 * r45863;
        double r45865 = r45854 / r45864;
        double r45866 = cbrt(r45863);
        double r45867 = r45866 * r45866;
        double r45868 = r45867 * r45866;
        double r45869 = r45847 / r45868;
        double r45870 = r45865 * r45869;
        double r45871 = r45847 + r45870;
        double r45872 = r45851 * r45871;
        double r45873 = sqrt(r45872);
        double r45874 = r45846 ? r45853 : r45873;
        return r45874;
}

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. Split input into 2 regimes

  2. if (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)) < 6.2550420220755e-314

    1. Initial program 17.0

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

    2. Taylor expanded around 0 11.0

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{0}}}\right)}\]

    if 6.2550420220755e-314 < (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))

    1. Initial program 0.7

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

    3. Applied add-cube-cbrt0.7

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

    4. Applied *-un-lft-identity0.7

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

    5. Applied times-frac0.7

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

    7. Applied add-cube-cbrt0.7

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2} \le 6.255042022075511028897152922160877393 \cdot 10^{-314}:\\ \;\;\;\;\sqrt{\left(1 + \frac{1}{\sqrt{1}}\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}} \cdot \frac{1}{\left(\sqrt[3]{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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))))))))))