Average Error: 33.8 → 27.1
Time: 54.2s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le 8.515544881339301 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 - \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}} \cdot \left(\left(U* - U\right) \cdot \left(\frac{\sqrt[3]{n} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(t - \frac{\ell}{\frac{Om}{\ell \cdot 2 - \left(\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}} \cdot \left(\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}} \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}}\right)\right) \cdot \left(U* - U\right)}}\right)} \cdot \sqrt{U}\\ \end{array}\]
\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}\;U \le 8.515544881339301 \cdot 10^{-258}:\\
\;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 - \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}} \cdot \left(\left(U* - U\right) \cdot \left(\frac{\sqrt[3]{n} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)\right)\right)\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2059689 = 2.0;
        double r2059690 = n;
        double r2059691 = r2059689 * r2059690;
        double r2059692 = U;
        double r2059693 = r2059691 * r2059692;
        double r2059694 = t;
        double r2059695 = l;
        double r2059696 = r2059695 * r2059695;
        double r2059697 = Om;
        double r2059698 = r2059696 / r2059697;
        double r2059699 = r2059689 * r2059698;
        double r2059700 = r2059694 - r2059699;
        double r2059701 = r2059695 / r2059697;
        double r2059702 = pow(r2059701, r2059689);
        double r2059703 = r2059690 * r2059702;
        double r2059704 = U_;
        double r2059705 = r2059692 - r2059704;
        double r2059706 = r2059703 * r2059705;
        double r2059707 = r2059700 - r2059706;
        double r2059708 = r2059693 * r2059707;
        double r2059709 = sqrt(r2059708);
        return r2059709;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2059710 = U;
        double r2059711 = 8.515544881339301e-258;
        bool r2059712 = r2059710 <= r2059711;
        double r2059713 = n;
        double r2059714 = 2.0;
        double r2059715 = t;
        double r2059716 = l;
        double r2059717 = Om;
        double r2059718 = r2059716 / r2059717;
        double r2059719 = r2059716 * r2059714;
        double r2059720 = cbrt(r2059716);
        double r2059721 = cbrt(r2059717);
        double r2059722 = cbrt(r2059713);
        double r2059723 = r2059721 / r2059722;
        double r2059724 = r2059720 / r2059723;
        double r2059725 = U_;
        double r2059726 = r2059725 - r2059710;
        double r2059727 = r2059722 * r2059720;
        double r2059728 = r2059727 / r2059721;
        double r2059729 = r2059728 * r2059728;
        double r2059730 = r2059726 * r2059729;
        double r2059731 = r2059724 * r2059730;
        double r2059732 = r2059719 - r2059731;
        double r2059733 = r2059718 * r2059732;
        double r2059734 = r2059715 - r2059733;
        double r2059735 = r2059714 * r2059734;
        double r2059736 = r2059713 * r2059735;
        double r2059737 = r2059710 * r2059736;
        double r2059738 = sqrt(r2059737);
        double r2059739 = r2059713 * r2059714;
        double r2059740 = r2059724 * r2059724;
        double r2059741 = r2059724 * r2059740;
        double r2059742 = r2059741 * r2059726;
        double r2059743 = r2059719 - r2059742;
        double r2059744 = r2059717 / r2059743;
        double r2059745 = r2059716 / r2059744;
        double r2059746 = r2059715 - r2059745;
        double r2059747 = r2059739 * r2059746;
        double r2059748 = sqrt(r2059747);
        double r2059749 = sqrt(r2059710);
        double r2059750 = r2059748 * r2059749;
        double r2059751 = r2059712 ? r2059738 : r2059750;
        return r2059751;
}

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 U < 8.515544881339301e-258

    1. Initial program 34.9

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

    2. Simplified32.4

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

    4. Applied associate-*l*30.8

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

    6. Applied add-cube-cbrt30.9

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

    7. Applied add-cube-cbrt30.9

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

    8. Applied times-frac30.9

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

    9. Applied add-cube-cbrt30.9

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

    10. Applied times-frac30.6

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

    11. Applied associate-*r*30.6

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

    12. Simplified30.6

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

    if 8.515544881339301e-258 < U

    1. Initial program 32.3

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

    2. Simplified28.8

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

    4. Applied associate-*l*28.7

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

    6. Applied add-cube-cbrt28.8

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

    7. Applied add-cube-cbrt28.8

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

    8. Applied times-frac28.8

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

    9. Applied add-cube-cbrt28.8

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

    10. Applied times-frac28.5

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

    11. Applied associate-*r*28.5

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

    12. Simplified28.5

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

    14. Applied sqrt-prod21.7

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

    15. Simplified22.5

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

  4. Final simplification27.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 8.515544881339301 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(\ell \cdot 2 - \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}} \cdot \left(\left(U* - U\right) \cdot \left(\frac{\sqrt[3]{n} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot 2\right) \cdot \left(t - \frac{\ell}{\frac{Om}{\ell \cdot 2 - \left(\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}} \cdot \left(\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}} \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{n}}}\right)\right) \cdot \left(U* - U\right)}}\right)} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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*))))))