Average Error: 33.6 → 27.6
Time: 54.0s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1.5658292974647863 \cdot 10^{-05}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \mathbf{elif}\;n \le 7.058910794888959 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \frac{\ell}{\frac{Om}{n}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \end{array}\]
\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -1.5658292974647863 \cdot 10^{-05}:\\
\;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\

\mathbf{elif}\;n \le 7.058910794888959 \cdot 10^{-223}:\\
\;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \frac{\ell}{\frac{Om}{n}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3133806 = 2.0;
        double r3133807 = n;
        double r3133808 = r3133806 * r3133807;
        double r3133809 = U;
        double r3133810 = r3133808 * r3133809;
        double r3133811 = t;
        double r3133812 = l;
        double r3133813 = r3133812 * r3133812;
        double r3133814 = Om;
        double r3133815 = r3133813 / r3133814;
        double r3133816 = r3133806 * r3133815;
        double r3133817 = r3133811 - r3133816;
        double r3133818 = r3133812 / r3133814;
        double r3133819 = pow(r3133818, r3133806);
        double r3133820 = r3133807 * r3133819;
        double r3133821 = U_;
        double r3133822 = r3133809 - r3133821;
        double r3133823 = r3133820 * r3133822;
        double r3133824 = r3133817 - r3133823;
        double r3133825 = r3133810 * r3133824;
        double r3133826 = sqrt(r3133825);
        return r3133826;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3133827 = n;
        double r3133828 = -1.5658292974647863e-05;
        bool r3133829 = r3133827 <= r3133828;
        double r3133830 = U;
        double r3133831 = r3133830 * r3133827;
        double r3133832 = t;
        double r3133833 = l;
        double r3133834 = 2.0;
        double r3133835 = r3133833 * r3133834;
        double r3133836 = U_;
        double r3133837 = r3133836 - r3133830;
        double r3133838 = cbrt(r3133833);
        double r3133839 = Om;
        double r3133840 = cbrt(r3133839);
        double r3133841 = r3133838 / r3133840;
        double r3133842 = r3133841 * r3133841;
        double r3133843 = r3133840 / r3133827;
        double r3133844 = r3133838 / r3133843;
        double r3133845 = r3133842 * r3133844;
        double r3133846 = r3133837 * r3133845;
        double r3133847 = r3133835 - r3133846;
        double r3133848 = r3133833 / r3133839;
        double r3133849 = r3133847 * r3133848;
        double r3133850 = r3133832 - r3133849;
        double r3133851 = r3133850 * r3133834;
        double r3133852 = r3133831 * r3133851;
        double r3133853 = sqrt(r3133852);
        double r3133854 = 7.058910794888959e-223;
        bool r3133855 = r3133827 <= r3133854;
        double r3133856 = r3133839 / r3133827;
        double r3133857 = r3133833 / r3133856;
        double r3133858 = r3133857 * r3133837;
        double r3133859 = r3133835 - r3133858;
        double r3133860 = r3133859 * r3133848;
        double r3133861 = r3133832 - r3133860;
        double r3133862 = r3133861 * r3133834;
        double r3133863 = r3133862 * r3133827;
        double r3133864 = r3133830 * r3133863;
        double r3133865 = sqrt(r3133864);
        double r3133866 = r3133855 ? r3133865 : r3133853;
        double r3133867 = r3133829 ? r3133853 : r3133866;
        return r3133867;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if n < -1.5658292974647863e-05 or 7.058910794888959e-223 < n

    1. Initial program 32.0

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

    2. Simplified29.1

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)\right)}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity29.1

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{Om}{\color{blue}{1 \cdot n}}}\right)\right)\right)}\]

    5. Applied add-cube-cbrt29.2

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{\color{blue}{\left(\sqrt[3]{Om} \cdot \sqrt[3]{Om}\right) \cdot \sqrt[3]{Om}}}{1 \cdot n}}\right)\right)\right)}\]

    6. Applied times-frac29.2

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\color{blue}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1} \cdot \frac{\sqrt[3]{Om}}{n}}}\right)\right)\right)}\]

    7. Applied add-cube-cbrt29.2

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1} \cdot \frac{\sqrt[3]{Om}}{n}}\right)\right)\right)}\]

    8. Applied times-frac28.3

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1}} \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)}\right)\right)\right)}\]

    9. Simplified28.3

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)} \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right)\right)\right)}\]

    if -1.5658292974647863e-05 < n < 7.058910794888959e-223

    1. Initial program 36.0

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

    2. Simplified31.7

      \[\leadsto \color{blue}{\sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)\right)}}\]
    3. Using strategy rm

    4. Applied associate-*l*26.6

      \[\leadsto \sqrt{\color{blue}{U \cdot \left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification27.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.5658292974647863 \cdot 10^{-05}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \mathbf{elif}\;n \le 7.058910794888959 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \frac{\ell}{\frac{Om}{n}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))