Toniolo and Linder, Equation (13)

Average Error: 33.4 → 24.4

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \frac{n \cdot -2}{\frac{\frac{Om}{\ell}}{\ell}} + \left(U \cdot 2\right) \cdot \left(n \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 8.046145761530675 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\frac{1}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{Om} \cdot \left(\frac{\frac{n}{\frac{Om}{\sqrt[3]{\ell}}}}{\frac{1}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right) + \left(\left(\frac{\sqrt[3]{n}}{\frac{\frac{\sqrt[3]{Om}}{\ell}}{\sqrt[3]{\sqrt[3]{\ell}}}} \cdot \left(U \cdot 2\right)\right) \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}}\right) \cdot \frac{-2}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3757708 = 2.0;
        double r3757709 = n;
        double r3757710 = r3757708 * r3757709;
        double r3757711 = U;
        double r3757712 = r3757710 * r3757711;
        double r3757713 = t;
        double r3757714 = l;
        double r3757715 = r3757714 * r3757714;
        double r3757716 = Om;
        double r3757717 = r3757715 / r3757716;
        double r3757718 = r3757708 * r3757717;
        double r3757719 = r3757713 - r3757718;
        double r3757720 = r3757714 / r3757716;
        double r3757721 = pow(r3757720, r3757708);
        double r3757722 = r3757709 * r3757721;
        double r3757723 = U_;
        double r3757724 = r3757711 - r3757723;
        double r3757725 = r3757722 * r3757724;
        double r3757726 = r3757719 - r3757725;
        double r3757727 = r3757712 * r3757726;
        double r3757728 = sqrt(r3757727);
        return r3757728;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3757729 = 2.0;
        double r3757730 = n;
        double r3757731 = r3757729 * r3757730;
        double r3757732 = U;
        double r3757733 = r3757731 * r3757732;
        double r3757734 = t;
        double r3757735 = l;
        double r3757736 = r3757735 * r3757735;
        double r3757737 = Om;
        double r3757738 = r3757736 / r3757737;
        double r3757739 = r3757738 * r3757729;
        double r3757740 = r3757734 - r3757739;
        double r3757741 = r3757735 / r3757737;
        double r3757742 = pow(r3757741, r3757729);
        double r3757743 = r3757730 * r3757742;
        double r3757744 = U_;
        double r3757745 = r3757732 - r3757744;
        double r3757746 = r3757743 * r3757745;
        double r3757747 = r3757740 - r3757746;
        double r3757748 = r3757733 * r3757747;
        double r3757749 = 0.0;
        bool r3757750 = r3757748 <= r3757749;
        double r3757751 = r3757732 * r3757729;
        double r3757752 = -2.0;
        double r3757753 = r3757730 * r3757752;
        double r3757754 = r3757737 / r3757735;
        double r3757755 = r3757754 / r3757735;
        double r3757756 = r3757753 / r3757755;
        double r3757757 = r3757751 * r3757756;
        double r3757758 = r3757730 / r3757754;
        double r3757759 = r3757758 * r3757745;
        double r3757760 = r3757759 / r3757754;
        double r3757761 = r3757734 - r3757760;
        double r3757762 = r3757730 * r3757761;
        double r3757763 = r3757751 * r3757762;
        double r3757764 = r3757757 + r3757763;
        double r3757765 = sqrt(r3757764);
        double r3757766 = 8.046145761530675e+302;
        bool r3757767 = r3757748 <= r3757766;
        double r3757768 = sqrt(r3757748);
        double r3757769 = 1.0;
        double r3757770 = cbrt(r3757735);
        double r3757771 = r3757770 * r3757770;
        double r3757772 = r3757769 / r3757771;
        double r3757773 = r3757769 / r3757772;
        double r3757774 = r3757773 / r3757737;
        double r3757775 = r3757737 / r3757770;
        double r3757776 = r3757730 / r3757775;
        double r3757777 = r3757769 / r3757735;
        double r3757778 = r3757776 / r3757777;
        double r3757779 = r3757778 * r3757745;
        double r3757780 = r3757774 * r3757779;
        double r3757781 = r3757734 - r3757780;
        double r3757782 = r3757781 * r3757730;
        double r3757783 = r3757782 * r3757751;
        double r3757784 = cbrt(r3757730);
        double r3757785 = cbrt(r3757737);
        double r3757786 = r3757785 / r3757735;
        double r3757787 = cbrt(r3757770);
        double r3757788 = r3757786 / r3757787;
        double r3757789 = r3757784 / r3757788;
        double r3757790 = r3757789 * r3757751;
        double r3757791 = r3757784 * r3757784;
        double r3757792 = r3757785 * r3757785;
        double r3757793 = r3757787 * r3757787;
        double r3757794 = r3757792 / r3757793;
        double r3757795 = r3757791 / r3757794;
        double r3757796 = r3757790 * r3757795;
        double r3757797 = r3757752 / r3757772;
        double r3757798 = r3757796 * r3757797;
        double r3757799 = r3757783 + r3757798;
        double r3757800 = sqrt(r3757799);
        double r3757801 = r3757767 ? r3757768 : r3757800;
        double r3757802 = r3757750 ? r3757765 : r3757801;
        return r3757802;
}

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*

Derivation

1. Split input into 3 regimes

2. if (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 0.0

   1. Initial program 56.7

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

   2. Simplified 38.2

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

   4. Applied distribute-lft-in 38.2

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

   5. Applied distribute-lft-in 38.2

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

   6. Applied distribute-rgt-in 38.2

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

   7. Simplified 37.1

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

   8. Simplified 37.1

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

   10. Applied associate-*l/ 35.7

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

   if 0.0 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 8.046145761530675e+302

   1. Initial program 2.0

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

   if 8.046145761530675e+302 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

   1. Initial program 60.2

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

   2. Simplified 57.8

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

   4. Applied distribute-lft-in 57.8

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

   5. Applied distribute-lft-in 57.8

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

   6. Applied distribute-rgt-in 57.8

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

   7. Simplified 50.5

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

   8. Simplified 50.5

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

   10. Applied div-inv 50.5

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

   11. Applied add-cube-cbrt 50.5

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

   12. Applied *-un-lft-identity 50.5

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

   13. Applied times-frac 50.5

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

   14. Applied *-un-lft-identity 50.5

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

   15. Applied times-frac 50.5

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

   16. Applied times-frac 50.5

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

   17. Applied associate-*l* 50.0

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

   19. Applied add-cube-cbrt 50.1

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

   20. Applied *-un-lft-identity 50.1

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

   21. Applied times-frac 50.1

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

   22. Applied times-frac 46.5

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

   23. Applied associate-*l* 45.9

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

   25. Applied add-cube-cbrt 45.9

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

   26. Applied *-un-lft-identity 45.9

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

   27. Applied add-cube-cbrt 45.9

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

   28. Applied times-frac 45.9

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

   29. Applied times-frac 45.9

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

   30. Applied add-cube-cbrt 46.0

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

   31. Applied times-frac 45.5

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

   32. Applied associate-*l* 46.0

       \[\leadsto \sqrt{\frac{-2}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \color{blue}{\left(\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1}}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}} \cdot \left(\frac{\sqrt[3]{n}}{\frac{\frac{\sqrt[3]{Om}}{\ell}}{\sqrt[3]{\sqrt[3]{\ell}}}} \cdot \left(U \cdot 2\right)\right)\right)} + \left(U \cdot 2\right) \cdot \left(n \cdot \left(t - \frac{\frac{1}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{Om} \cdot \left(\frac{\frac{n}{\frac{Om}{\sqrt[3]{\ell}}}}{\frac{1}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 24.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \frac{n \cdot -2}{\frac{\frac{Om}{\ell}}{\ell}} + \left(U \cdot 2\right) \cdot \left(n \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 8.046145761530675 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\frac{1}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{Om} \cdot \left(\frac{\frac{n}{\frac{Om}{\sqrt[3]{\ell}}}}{\frac{1}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right) + \left(\left(\frac{\sqrt[3]{n}}{\frac{\frac{\sqrt[3]{Om}}{\ell}}{\sqrt[3]{\sqrt[3]{\ell}}}} \cdot \left(U \cdot 2\right)\right) \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}}\right) \cdot \frac{-2}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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*))))))