Toniolo and Linder, Equation (13)

Average Error: 34.4 → 31.2

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -9.014626965436397950093167243230168496464 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U* - U\right) \cdot n\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 1.008725107202353820896882228886919841361 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t + \left(0 \cdot \left(U* - U\right) - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;n \le 31641216734603.953125:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left({\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{2} \cdot \left(n \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2}\right)\right) + \left(-\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right) \cdot \sqrt[3]{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left({\left(\sqrt[3]{U* - U}\right)}^{3} \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r207177 = 2.0;
        double r207178 = n;
        double r207179 = r207177 * r207178;
        double r207180 = U;
        double r207181 = r207179 * r207180;
        double r207182 = t;
        double r207183 = l;
        double r207184 = r207183 * r207183;
        double r207185 = Om;
        double r207186 = r207184 / r207185;
        double r207187 = r207177 * r207186;
        double r207188 = r207182 - r207187;
        double r207189 = r207183 / r207185;
        double r207190 = pow(r207189, r207177);
        double r207191 = r207178 * r207190;
        double r207192 = U_;
        double r207193 = r207180 - r207192;
        double r207194 = r207191 * r207193;
        double r207195 = r207188 - r207194;
        double r207196 = r207181 * r207195;
        double r207197 = sqrt(r207196);
        return r207197;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r207198 = n;
        double r207199 = -9.014626965436398e-81;
        bool r207200 = r207198 <= r207199;
        double r207201 = 2.0;
        double r207202 = r207201 * r207198;
        double r207203 = l;
        double r207204 = Om;
        double r207205 = r207203 / r207204;
        double r207206 = pow(r207205, r207201);
        double r207207 = U_;
        double r207208 = U;
        double r207209 = r207207 - r207208;
        double r207210 = r207209 * r207198;
        double r207211 = r207206 * r207210;
        double r207212 = r207205 * r207203;
        double r207213 = r207212 * r207201;
        double r207214 = t;
        double r207215 = r207213 - r207214;
        double r207216 = r207211 - r207215;
        double r207217 = r207216 * r207208;
        double r207218 = r207202 * r207217;
        double r207219 = sqrt(r207218);
        double r207220 = 1.0087251072023538e-199;
        bool r207221 = r207198 <= r207220;
        double r207222 = 0.0;
        double r207223 = r207222 * r207209;
        double r207224 = r207201 * r207203;
        double r207225 = r207224 * r207205;
        double r207226 = r207223 - r207225;
        double r207227 = r207214 + r207226;
        double r207228 = r207227 * r207202;
        double r207229 = r207208 * r207228;
        double r207230 = sqrt(r207229);
        double r207231 = 31641216734603.953;
        bool r207232 = r207198 <= r207231;
        double r207233 = cbrt(r207208);
        double r207234 = r207233 * r207233;
        double r207235 = r207202 * r207234;
        double r207236 = r207235 * r207209;
        double r207237 = cbrt(r207203);
        double r207238 = r207237 * r207237;
        double r207239 = cbrt(r207204);
        double r207240 = r207239 * r207239;
        double r207241 = r207238 / r207240;
        double r207242 = pow(r207241, r207201);
        double r207243 = r207237 / r207239;
        double r207244 = pow(r207243, r207201);
        double r207245 = r207198 * r207244;
        double r207246 = r207242 * r207245;
        double r207247 = r207236 * r207246;
        double r207248 = r207225 - r207214;
        double r207249 = -r207248;
        double r207250 = r207249 * r207235;
        double r207251 = r207247 + r207250;
        double r207252 = r207251 * r207233;
        double r207253 = sqrt(r207252);
        double r207254 = cbrt(r207209);
        double r207255 = 3.0;
        double r207256 = pow(r207254, r207255);
        double r207257 = r207206 * r207198;
        double r207258 = r207256 * r207257;
        double r207259 = r207201 * r207205;
        double r207260 = r207203 * r207259;
        double r207261 = r207260 - r207214;
        double r207262 = r207258 - r207261;
        double r207263 = r207208 * r207202;
        double r207264 = r207262 * r207263;
        double r207265 = sqrt(r207264);
        double r207266 = r207232 ? r207253 : r207265;
        double r207267 = r207221 ? r207230 : r207266;
        double r207268 = r207200 ? r207219 : r207267;
        return r207268;
}

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

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

   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 pow1 36.8

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

   9. Applied pow1 36.8

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

   10. Applied pow1 36.8

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

   11. Applied pow1 36.8

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

   12. Applied pow-prod-down 36.8

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

   13. Applied pow-prod-down 36.8

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

   14. Applied pow-prod-down 36.8

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

   15. Simplified 31.0

       \[\leadsto \sqrt{{\color{blue}{\left(\left(U \cdot \left(2 \cdot n\right)\right) \cdot \left({\left(\sqrt[3]{U* - U}\right)}^{3} \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right)\right)}}^{1}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 31.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -9.014626965436397950093167243230168496464 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(\left(U* - U\right) \cdot n\right) - \left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2 - t\right)\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 1.008725107202353820896882228886919841361 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(t + \left(0 \cdot \left(U* - U\right) - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right) \cdot \left(2 \cdot n\right)\right)}\\ \mathbf{elif}\;n \le 31641216734603.953125:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right) \cdot \left(U* - U\right)\right) \cdot \left({\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{2} \cdot \left(n \cdot {\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right)}^{2}\right)\right) + \left(-\left(\left(2 \cdot \ell\right) \cdot \frac{\ell}{Om} - t\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot \left(\sqrt[3]{U} \cdot \sqrt[3]{U}\right)\right)\right) \cdot \sqrt[3]{U}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left({\left(\sqrt[3]{U* - U}\right)}^{3} \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot n\right) - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) - t\right)\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)}\\ \end{array}\]

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*))))))