Toniolo and Linder, Equation (13)

Average Error: 33.1 → 26.8

Time: 2.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot \left(\frac{-2}{\frac{Om}{\ell \cdot \ell}} + \left(t - \frac{\ell \cdot \frac{n}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot U}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 5.7155522405275665 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(\left(t + \frac{-2}{\frac{Om}{\ell} \cdot \frac{1}{\ell}}\right) - \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{Om} \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{1}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}}\right) \cdot \frac{\frac{\sqrt[3]{n}}{\frac{Om}{\sqrt[3]{\sqrt[3]{\ell}}}} \cdot \left(U - U*\right)}{\frac{1}{\ell}}\right)\right) \cdot 2} \cdot \sqrt{U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r4837278 = 2.0;
        double r4837279 = n;
        double r4837280 = r4837278 * r4837279;
        double r4837281 = U;
        double r4837282 = r4837280 * r4837281;
        double r4837283 = t;
        double r4837284 = l;
        double r4837285 = r4837284 * r4837284;
        double r4837286 = Om;
        double r4837287 = r4837285 / r4837286;
        double r4837288 = r4837278 * r4837287;
        double r4837289 = r4837283 - r4837288;
        double r4837290 = r4837284 / r4837286;
        double r4837291 = pow(r4837290, r4837278);
        double r4837292 = r4837279 * r4837291;
        double r4837293 = U_;
        double r4837294 = r4837281 - r4837293;
        double r4837295 = r4837292 * r4837294;
        double r4837296 = r4837289 - r4837295;
        double r4837297 = r4837282 * r4837296;
        double r4837298 = sqrt(r4837297);
        return r4837298;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r4837299 = 2.0;
        double r4837300 = n;
        double r4837301 = r4837299 * r4837300;
        double r4837302 = U;
        double r4837303 = r4837301 * r4837302;
        double r4837304 = t;
        double r4837305 = l;
        double r4837306 = r4837305 * r4837305;
        double r4837307 = Om;
        double r4837308 = r4837306 / r4837307;
        double r4837309 = r4837308 * r4837299;
        double r4837310 = r4837304 - r4837309;
        double r4837311 = r4837305 / r4837307;
        double r4837312 = pow(r4837311, r4837299);
        double r4837313 = r4837300 * r4837312;
        double r4837314 = U_;
        double r4837315 = r4837302 - r4837314;
        double r4837316 = r4837313 * r4837315;
        double r4837317 = r4837310 - r4837316;
        double r4837318 = r4837303 * r4837317;
        double r4837319 = sqrt(r4837318);
        double r4837320 = 0.0;
        bool r4837321 = r4837319 <= r4837320;
        double r4837322 = -2.0;
        double r4837323 = r4837307 / r4837306;
        double r4837324 = r4837322 / r4837323;
        double r4837325 = r4837300 / r4837307;
        double r4837326 = r4837305 * r4837325;
        double r4837327 = r4837307 / r4837305;
        double r4837328 = r4837326 / r4837327;
        double r4837329 = r4837328 * r4837315;
        double r4837330 = r4837304 - r4837329;
        double r4837331 = r4837324 + r4837330;
        double r4837332 = r4837300 * r4837331;
        double r4837333 = r4837299 * r4837332;
        double r4837334 = r4837333 * r4837302;
        double r4837335 = sqrt(r4837334);
        double r4837336 = 5.7155522405275665e+152;
        bool r4837337 = r4837319 <= r4837336;
        double r4837338 = 1.0;
        double r4837339 = r4837338 / r4837305;
        double r4837340 = r4837327 * r4837339;
        double r4837341 = r4837322 / r4837340;
        double r4837342 = r4837304 + r4837341;
        double r4837343 = cbrt(r4837305);
        double r4837344 = r4837343 * r4837343;
        double r4837345 = r4837344 / r4837307;
        double r4837346 = cbrt(r4837300);
        double r4837347 = r4837346 * r4837346;
        double r4837348 = cbrt(r4837343);
        double r4837349 = r4837348 * r4837348;
        double r4837350 = r4837338 / r4837349;
        double r4837351 = r4837347 / r4837350;
        double r4837352 = r4837345 * r4837351;
        double r4837353 = r4837307 / r4837348;
        double r4837354 = r4837346 / r4837353;
        double r4837355 = r4837354 * r4837315;
        double r4837356 = r4837355 / r4837339;
        double r4837357 = r4837352 * r4837356;
        double r4837358 = r4837342 - r4837357;
        double r4837359 = r4837300 * r4837358;
        double r4837360 = r4837359 * r4837299;
        double r4837361 = sqrt(r4837360);
        double r4837362 = sqrt(r4837302);
        double r4837363 = r4837361 * r4837362;
        double r4837364 = r4837337 ? r4837319 : r4837363;
        double r4837365 = r4837321 ? r4837335 : r4837364;
        return r4837365;
}

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

   1. Initial program 57.1

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

   2. Simplified 39.4

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

   4. Applied associate-/r/ 39.4

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

   if 0.0 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 5.7155522405275665e+152

   1. Initial program 1.7

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

   if 5.7155522405275665e+152 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

   1. Initial program 60.5

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

   2. Simplified 56.7

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

   4. Applied div-inv 56.7

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

   5. Applied add-cube-cbrt 56.7

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

   6. Applied *-un-lft-identity 56.7

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

   7. Applied times-frac 56.7

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

   8. Applied *-un-lft-identity 56.7

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

   9. Applied times-frac 56.7

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

   10. Applied times-frac 56.7

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

   11. Applied associate-*l* 56.0

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

   13. Applied *-un-lft-identity 56.0

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

   14. Applied times-frac 48.9

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

   16. Applied *-un-lft-identity 48.9

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

   17. Applied add-cube-cbrt 48.9

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

   18. Applied *-un-lft-identity 48.9

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

   19. Applied times-frac 48.9

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

   20. Applied add-cube-cbrt 48.9

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

   21. Applied times-frac 48.9

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

   22. Applied times-frac 48.9

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

   23. Applied associate-*l* 49.0

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

   25. Applied sqrt-prod 50.9

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

   26. Simplified 51.2

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

4. Final simplification 26.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 0.0:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(n \cdot \left(\frac{-2}{\frac{Om}{\ell \cdot \ell}} + \left(t - \frac{\ell \cdot \frac{n}{Om}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)\right)\right)\right) \cdot U}\\ \mathbf{elif}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 5.7155522405275665 \cdot 10^{+152}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot \left(\left(t + \frac{-2}{\frac{Om}{\ell} \cdot \frac{1}{\ell}}\right) - \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{Om} \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{1}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}}\right) \cdot \frac{\frac{\sqrt[3]{n}}{\frac{Om}{\sqrt[3]{\sqrt[3]{\ell}}}} \cdot \left(U - U*\right)}{\frac{1}{\ell}}\right)\right) \cdot 2} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(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*))))))