Toniolo and Linder, Equation (13)

Average Error: 34.6 → 30.6

Time: 1.1m

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}\;t \le -2.91138552679685006846860711181604876416 \cdot 10^{178}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \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)\right)}\\ \mathbf{elif}\;t \le -2.99207439904474513830909419640009574747 \cdot 10^{97}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)}} \cdot \sqrt{\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)}}\\ \mathbf{elif}\;t \le 1.664268146426151312546296416098443579856 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \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)\right)}\\ \mathbf{elif}\;t \le 2.6537513056737148695584217694138294538 \cdot 10^{-145} \lor \neg \left(t \le 1.248295602955021596545459106660652670584 \cdot 10^{-8}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + \left(U - U*\right) \cdot \left(\left(-{\left(\sqrt[3]{n}\right)}^{3} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r189242 = 2.0;
        double r189243 = n;
        double r189244 = r189242 * r189243;
        double r189245 = U;
        double r189246 = r189244 * r189245;
        double r189247 = t;
        double r189248 = l;
        double r189249 = r189248 * r189248;
        double r189250 = Om;
        double r189251 = r189249 / r189250;
        double r189252 = r189242 * r189251;
        double r189253 = r189247 - r189252;
        double r189254 = r189248 / r189250;
        double r189255 = pow(r189254, r189242);
        double r189256 = r189243 * r189255;
        double r189257 = U_;
        double r189258 = r189245 - r189257;
        double r189259 = r189256 * r189258;
        double r189260 = r189253 - r189259;
        double r189261 = r189246 * r189260;
        double r189262 = sqrt(r189261);
        return r189262;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r189263 = t;
        double r189264 = -2.91138552679685e+178;
        bool r189265 = r189263 <= r189264;
        double r189266 = 2.0;
        double r189267 = n;
        double r189268 = r189266 * r189267;
        double r189269 = U;
        double r189270 = r189268 * r189269;
        double r189271 = l;
        double r189272 = Om;
        double r189273 = r189271 / r189272;
        double r189274 = r189271 * r189273;
        double r189275 = r189266 * r189274;
        double r189276 = r189263 - r189275;
        double r189277 = 2.0;
        double r189278 = r189266 / r189277;
        double r189279 = pow(r189273, r189278);
        double r189280 = r189267 * r189279;
        double r189281 = r189280 * r189279;
        double r189282 = U_;
        double r189283 = r189269 - r189282;
        double r189284 = r189281 * r189283;
        double r189285 = r189276 - r189284;
        double r189286 = r189270 * r189285;
        double r189287 = sqrt(r189286);
        double r189288 = -2.992074399044745e+97;
        bool r189289 = r189263 <= r189288;
        double r189290 = r189271 * r189271;
        double r189291 = r189290 / r189272;
        double r189292 = pow(r189273, r189266);
        double r189293 = r189292 * r189283;
        double r189294 = r189293 * r189267;
        double r189295 = fma(r189266, r189291, r189294);
        double r189296 = r189263 - r189295;
        double r189297 = r189269 * r189296;
        double r189298 = r189297 * r189268;
        double r189299 = sqrt(r189298);
        double r189300 = sqrt(r189299);
        double r189301 = r189300 * r189300;
        double r189302 = 1.6642681464261513e-294;
        bool r189303 = r189263 <= r189302;
        double r189304 = 2.653751305673715e-145;
        bool r189305 = r189263 <= r189304;
        double r189306 = 1.2482956029550216e-08;
        bool r189307 = r189263 <= r189306;
        double r189308 = !r189307;
        bool r189309 = r189305 || r189308;
        double r189310 = sqrt(r189270);
        double r189311 = r189266 * r189271;
        double r189312 = r189277 * r189278;
        double r189313 = pow(r189273, r189312);
        double r189314 = r189267 * r189313;
        double r189315 = r189283 * r189314;
        double r189316 = fma(r189311, r189273, r189315);
        double r189317 = r189263 - r189316;
        double r189318 = sqrt(r189317);
        double r189319 = r189310 * r189318;
        double r189320 = pow(r189271, r189277);
        double r189321 = r189320 / r189272;
        double r189322 = r189266 * r189321;
        double r189323 = r189263 - r189322;
        double r189324 = r189269 * r189323;
        double r189325 = r189268 * r189324;
        double r189326 = cbrt(r189267);
        double r189327 = 3.0;
        double r189328 = pow(r189326, r189327);
        double r189329 = r189328 * r189313;
        double r189330 = -r189329;
        double r189331 = r189330 * r189270;
        double r189332 = r189283 * r189331;
        double r189333 = r189325 + r189332;
        double r189334 = sqrt(r189333);
        double r189335 = r189309 ? r189319 : r189334;
        double r189336 = r189303 ? r189287 : r189335;
        double r189337 = r189289 ? r189301 : r189336;
        double r189338 = r189265 ? r189287 : r189337;
        return r189338;
}

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 t < -2.91138552679685e+178 or -2.992074399044745e+97 < t < 1.6642681464261513e-294

   1. Initial program 34.3

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

   3. Applied *-un-lft-identity 34.3

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

   4. Applied times-frac 31.7

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

   5. Simplified 31.7

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

   7. Applied sqr-pow 31.7

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

   8. Applied associate-*r* 30.7

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

   if -2.91138552679685e+178 < t < -2.992074399044745e+97

   1. Initial program 35.1

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

   3. Applied add-sqr-sqrt 35.3

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

   4. Simplified 39.1

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

   5. Simplified 34.1

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

   if 1.6642681464261513e-294 < t < 2.653751305673715e-145 or 1.2482956029550216e-08 < t

   1. Initial program 35.8

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

   3. Applied *-un-lft-identity 35.8

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

   4. Applied times-frac 33.5

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

   5. Simplified 33.5

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

   7. Applied sqr-pow 33.5

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

   8. Applied associate-*r* 32.6

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

   10. Applied sqrt-prod 28.8

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

   11. Simplified 29.4

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

   if 2.653751305673715e-145 < t < 1.2482956029550216e-08

   1. Initial program 31.6

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

   3. Applied *-un-lft-identity 31.6

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

   4. Applied times-frac 28.9

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

   5. Simplified 28.9

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

   7. Applied sqr-pow 28.9

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

   8. Applied associate-*r* 28.3

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

   10. Applied add-cube-cbrt 28.4

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

   11. Applied associate-*l* 28.3

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

   13. Applied sub-neg 28.3

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

   14. Applied distribute-lft-in 28.3

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

   15. Simplified 31.0

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

   16. Simplified 32.1

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

4. Final simplification 30.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.91138552679685006846860711181604876416 \cdot 10^{178}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \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)\right)}\\ \mathbf{elif}\;t \le -2.99207439904474513830909419640009574747 \cdot 10^{97}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)}} \cdot \sqrt{\sqrt{\left(U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{Om}, \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot \left(2 \cdot n\right)}}\\ \mathbf{elif}\;t \le 1.664268146426151312546296416098443579856 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \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)\right)}\\ \mathbf{elif}\;t \le 2.6537513056737148695584217694138294538 \cdot 10^{-145} \lor \neg \left(t \le 1.248295602955021596545459106660652670584 \cdot 10^{-8}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2 \cdot \ell, \frac{\ell}{Om}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) + \left(U - U*\right) \cdot \left(\left(-{\left(\sqrt[3]{n}\right)}^{3} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))