Average Error: 33.8 → 27.1
Time: 53.4s
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 r3322691 = 2.0;
        double r3322692 = n;
        double r3322693 = r3322691 * r3322692;
        double r3322694 = U;
        double r3322695 = r3322693 * r3322694;
        double r3322696 = t;
        double r3322697 = l;
        double r3322698 = r3322697 * r3322697;
        double r3322699 = Om;
        double r3322700 = r3322698 / r3322699;
        double r3322701 = r3322691 * r3322700;
        double r3322702 = r3322696 - r3322701;
        double r3322703 = r3322697 / r3322699;
        double r3322704 = pow(r3322703, r3322691);
        double r3322705 = r3322692 * r3322704;
        double r3322706 = U_;
        double r3322707 = r3322694 - r3322706;
        double r3322708 = r3322705 * r3322707;
        double r3322709 = r3322702 - r3322708;
        double r3322710 = r3322695 * r3322709;
        double r3322711 = sqrt(r3322710);
        return r3322711;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3322712 = U;
        double r3322713 = 8.515544881339301e-258;
        bool r3322714 = r3322712 <= r3322713;
        double r3322715 = n;
        double r3322716 = 2.0;
        double r3322717 = t;
        double r3322718 = l;
        double r3322719 = Om;
        double r3322720 = r3322718 / r3322719;
        double r3322721 = r3322718 * r3322716;
        double r3322722 = cbrt(r3322718);
        double r3322723 = cbrt(r3322719);
        double r3322724 = cbrt(r3322715);
        double r3322725 = r3322723 / r3322724;
        double r3322726 = r3322722 / r3322725;
        double r3322727 = U_;
        double r3322728 = r3322727 - r3322712;
        double r3322729 = r3322724 * r3322722;
        double r3322730 = r3322729 / r3322723;
        double r3322731 = r3322730 * r3322730;
        double r3322732 = r3322728 * r3322731;
        double r3322733 = r3322726 * r3322732;
        double r3322734 = r3322721 - r3322733;
        double r3322735 = r3322720 * r3322734;
        double r3322736 = r3322717 - r3322735;
        double r3322737 = r3322716 * r3322736;
        double r3322738 = r3322715 * r3322737;
        double r3322739 = r3322712 * r3322738;
        double r3322740 = sqrt(r3322739);
        double r3322741 = r3322715 * r3322716;
        double r3322742 = r3322726 * r3322726;
        double r3322743 = r3322726 * r3322742;
        double r3322744 = r3322743 * r3322728;
        double r3322745 = r3322721 - r3322744;
        double r3322746 = r3322719 / r3322745;
        double r3322747 = r3322718 / r3322746;
        double r3322748 = r3322717 - r3322747;
        double r3322749 = r3322741 * r3322748;
        double r3322750 = sqrt(r3322749);
        double r3322751 = sqrt(r3322712);
        double r3322752 = r3322750 * r3322751;
        double r3322753 = r3322714 ? r3322740 : r3322752;
        return r3322753;
}

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*))))))