Toniolo and Linder, Equation (13)

Average Error: 34.6 → 27.0

Time: 2.7m

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}\;U \le -7.411448247083684376522735829909480721372 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}\\ \mathbf{elif}\;U \le 2.469810196486197144576887320962391000635 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r756211 = 2.0;
        double r756212 = n;
        double r756213 = r756211 * r756212;
        double r756214 = U;
        double r756215 = r756213 * r756214;
        double r756216 = t;
        double r756217 = l;
        double r756218 = r756217 * r756217;
        double r756219 = Om;
        double r756220 = r756218 / r756219;
        double r756221 = r756211 * r756220;
        double r756222 = r756216 - r756221;
        double r756223 = r756217 / r756219;
        double r756224 = pow(r756223, r756211);
        double r756225 = r756212 * r756224;
        double r756226 = U_;
        double r756227 = r756214 - r756226;
        double r756228 = r756225 * r756227;
        double r756229 = r756222 - r756228;
        double r756230 = r756215 * r756229;
        double r756231 = sqrt(r756230);
        return r756231;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r756232 = U;
        double r756233 = -7.411448247083684e-46;
        bool r756234 = r756232 <= r756233;
        double r756235 = 2.0;
        double r756236 = n;
        double r756237 = r756235 * r756236;
        double r756238 = l;
        double r756239 = Om;
        double r756240 = r756238 / r756239;
        double r756241 = 2.0;
        double r756242 = r756235 / r756241;
        double r756243 = pow(r756240, r756242);
        double r756244 = r756236 * r756243;
        double r756245 = r756244 * r756243;
        double r756246 = U_;
        double r756247 = r756246 - r756232;
        double r756248 = r756245 * r756247;
        double r756249 = r756235 * r756238;
        double r756250 = r756240 * r756249;
        double r756251 = r756248 - r756250;
        double r756252 = t;
        double r756253 = r756251 + r756252;
        double r756254 = r756237 * r756253;
        double r756255 = r756232 * r756254;
        double r756256 = sqrt(r756255);
        double r756257 = 2.469810196486197e-303;
        bool r756258 = r756232 <= r756257;
        double r756259 = cbrt(r756239);
        double r756260 = r756238 / r756259;
        double r756261 = pow(r756260, r756242);
        double r756262 = r756261 * r756243;
        double r756263 = 1.0;
        double r756264 = r756259 * r756259;
        double r756265 = r756263 / r756264;
        double r756266 = pow(r756265, r756242);
        double r756267 = r756236 * r756266;
        double r756268 = r756262 * r756267;
        double r756269 = r756268 * r756247;
        double r756270 = r756250 - r756252;
        double r756271 = r756269 - r756270;
        double r756272 = r756232 * r756271;
        double r756273 = r756237 * r756272;
        double r756274 = cbrt(r756273);
        double r756275 = r756274 * r756274;
        double r756276 = r756275 * r756274;
        double r756277 = sqrt(r756276);
        double r756278 = r756271 * r756237;
        double r756279 = sqrt(r756278);
        double r756280 = sqrt(r756232);
        double r756281 = r756279 * r756280;
        double r756282 = r756258 ? r756277 : r756281;
        double r756283 = r756234 ? r756256 : r756282;
        return r756283;
}

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 U < -7.411448247083684e-46

   1. Initial program 29.8

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

   2. Simplified 28.4

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

   4. Applied sqr-pow 28.4

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

   5. Applied associate-*r* 27.3

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

   if -7.411448247083684e-46 < U < 2.469810196486197e-303

   1. Initial program 39.3

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

   2. Simplified 35.6

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

   4. Applied sqr-pow 35.6

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

   5. Applied associate-*r* 34.6

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

   7. Applied add-cube-cbrt 34.7

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

   8. Applied *-un-lft-identity 34.7

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

   9. Applied times-frac 34.7

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

   10. Applied unpow-prod-down 34.7

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

   11. Applied associate-*r* 34.9

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

   12. Simplified 34.9

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

   14. Applied add-cube-cbrt 35.1

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

   15. Simplified 37.2

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

   16. Simplified 32.9

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

   if 2.469810196486197e-303 < U

   1. Initial program 34.2

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

   2. Simplified 31.3

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

   4. Applied sqr-pow 31.3

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

   5. Applied associate-*r* 30.2

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

   7. Applied add-cube-cbrt 30.3

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

   8. Applied *-un-lft-identity 30.3

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

   9. Applied times-frac 30.3

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

   10. Applied unpow-prod-down 30.3

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

   11. Applied associate-*r* 30.3

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

   12. Simplified 30.3

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

   14. Applied sqrt-prod 23.2

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

   15. Simplified 23.7

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

4. Final simplification 27.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -7.411448247083684376522735829909480721372 \cdot 10^{-46}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(2 \cdot n\right) \cdot \left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)\right) + t\right)\right)}\\ \mathbf{elif}\;U \le 2.469810196486197144576887320962391000635 \cdot 10^{-303}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)} \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}\right) \cdot \sqrt[3]{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left({\left(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \left(U* - U\right) - \left(\frac{\ell}{Om} \cdot \left(2 \cdot \ell\right) - t\right)\right) \cdot \left(2 \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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*))))))