Toniolo and Linder, Equation (13)

Average Error: 34.4 → 31.2

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -9.014626965436397950093167243230168496464 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U* - U\right) \cdot n\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 1.008725107202353820896882228886919841361 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right)\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;n \le 31641216734603.953125:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(2 \cdot n\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2}\right) \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{2}\right) + \left(-\left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(2 \cdot n\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right) \cdot \sqrt[3]{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot {\left(\sqrt[3]{U* - U}\right)}^{3} - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell - t\right)\right)} \cdot \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot {\left(\sqrt[3]{U* - U}\right)}^{3} - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell - t\right)\right)}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r207053 = 2.0;
        double r207054 = n;
        double r207055 = r207053 * r207054;
        double r207056 = U;
        double r207057 = r207055 * r207056;
        double r207058 = t;
        double r207059 = l;
        double r207060 = r207059 * r207059;
        double r207061 = Om;
        double r207062 = r207060 / r207061;
        double r207063 = r207053 * r207062;
        double r207064 = r207058 - r207063;
        double r207065 = r207059 / r207061;
        double r207066 = pow(r207065, r207053);
        double r207067 = r207054 * r207066;
        double r207068 = U_;
        double r207069 = r207056 - r207068;
        double r207070 = r207067 * r207069;
        double r207071 = r207064 - r207070;
        double r207072 = r207057 * r207071;
        double r207073 = sqrt(r207072);
        return r207073;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r207074 = n;
        double r207075 = -9.014626965436398e-81;
        bool r207076 = r207074 <= r207075;
        double r207077 = 2.0;
        double r207078 = r207077 * r207074;
        double r207079 = l;
        double r207080 = Om;
        double r207081 = r207079 / r207080;
        double r207082 = pow(r207081, r207077);
        double r207083 = U_;
        double r207084 = U;
        double r207085 = r207083 - r207084;
        double r207086 = r207085 * r207074;
        double r207087 = r207082 * r207086;
        double r207088 = r207081 * r207079;
        double r207089 = r207088 * r207077;
        double r207090 = t;
        double r207091 = r207089 - r207090;
        double r207092 = r207087 - r207091;
        double r207093 = r207092 * r207084;
        double r207094 = r207078 * r207093;
        double r207095 = sqrt(r207094);
        double r207096 = 1.0087251072023538e-199;
        bool r207097 = r207074 <= r207096;
        double r207098 = -r207077;
        double r207099 = r207098 * r207079;
        double r207100 = r207081 * r207099;
        double r207101 = r207090 + r207100;
        double r207102 = r207101 * r207078;
        double r207103 = r207084 * r207102;
        double r207104 = sqrt(r207103);
        double r207105 = 31641216734603.953;
        bool r207106 = r207074 <= r207105;
        double r207107 = cbrt(r207084);
        double r207108 = r207107 * r207107;
        double r207109 = r207108 * r207078;
        double r207110 = r207109 * r207085;
        double r207111 = cbrt(r207079);
        double r207112 = cbrt(r207080);
        double r207113 = r207111 / r207112;
        double r207114 = pow(r207113, r207077);
        double r207115 = r207074 * r207114;
        double r207116 = r207111 * r207111;
        double r207117 = r207112 * r207112;
        double r207118 = r207116 / r207117;
        double r207119 = pow(r207118, r207077);
        double r207120 = r207115 * r207119;
        double r207121 = r207110 * r207120;
        double r207122 = r207077 * r207079;
        double r207123 = r207081 * r207122;
        double r207124 = r207123 - r207090;
        double r207125 = r207109 * r207124;
        double r207126 = -r207125;
        double r207127 = r207121 + r207126;
        double r207128 = r207127 * r207107;
        double r207129 = sqrt(r207128);
        double r207130 = r207084 * r207078;
        double r207131 = r207082 * r207074;
        double r207132 = cbrt(r207085);
        double r207133 = 3.0;
        double r207134 = pow(r207132, r207133);
        double r207135 = r207131 * r207134;
        double r207136 = r207077 * r207081;
        double r207137 = r207136 * r207079;
        double r207138 = r207137 - r207090;
        double r207139 = r207135 - r207138;
        double r207140 = r207130 * r207139;
        double r207141 = sqrt(r207140);
        double r207142 = r207141 * r207141;
        double r207143 = sqrt(r207142);
        double r207144 = r207106 ? r207129 : r207143;
        double r207145 = r207097 ? r207104 : r207144;
        double r207146 = r207076 ? r207095 : r207145;
        return r207146;
}

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 4 regimes

2. if n < -9.014626965436398e-81

   1. Initial program 33.2

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

   2. Simplified 34.0

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

   4. Applied associate-*l* 31.1

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

   5. Simplified 35.2

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

   if -9.014626965436398e-81 < n < 1.0087251072023538e-199

   1. Initial program 37.1

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

   2. Simplified 30.3

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

   3. Taylor expanded around 0 30.9

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \left(\color{blue}{0} - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U}\]

   if 1.0087251072023538e-199 < n < 31641216734603.953

   1. Initial program 31.7

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

   2. Simplified 26.0

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

   4. Applied add-cube-cbrt 26.0

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

   5. Applied associate-*r* 26.0

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

   6. Simplified 26.0

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

   8. Applied add-cube-cbrt 26.4

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

   9. Applied associate-*r* 26.4

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

   10. Simplified 26.5

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

   12. Applied sub-neg 26.5

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

   13. Applied distribute-lft-in 26.5

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

   14. Simplified 27.6

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

   15. Simplified 27.6

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

   17. Applied add-cube-cbrt 27.7

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

   18. Applied add-cube-cbrt 27.7

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

   19. Applied times-frac 27.7

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

   20. Applied unpow-prod-down 27.7

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

   21. Applied associate-*l* 26.7

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

   22. Simplified 26.7

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

   if 31641216734603.953 < n

   1. Initial program 33.8

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

   2. Simplified 36.8

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

   4. Applied add-cube-cbrt 36.8

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

   5. Applied associate-*r* 36.8

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

   6. Simplified 36.8

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

   8. Applied add-sqr-sqrt 36.8

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

   9. Simplified 38.3

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

   10. Simplified 31.0

       \[\leadsto \sqrt{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left({\left(\sqrt[3]{U* - U}\right)}^{3} \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)} \cdot \color{blue}{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left({\left(\sqrt[3]{U* - U}\right)}^{3} \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 31.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -9.014626965436397950093167243230168496464 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U* - U\right) \cdot n\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 1.008725107202353820896882228886919841361 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t + \frac{\ell}{Om} \cdot \left(\left(-2\right) \cdot \ell\right)\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;n \le 31641216734603.953125:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(2 \cdot n\right)\right) \cdot \left(U* - U\right)\right) \cdot \left(\left(n \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2}\right) \cdot {\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{2}\right) + \left(-\left(\left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right) \cdot \left(2 \cdot n\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right) \cdot \sqrt[3]{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot {\left(\sqrt[3]{U* - U}\right)}^{3} - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell - t\right)\right)} \cdot \sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) \cdot {\left(\sqrt[3]{U* - U}\right)}^{3} - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell - t\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))