Toniolo and Linder, Equation (13)

Average Error: 34.6 → 30.6

Time: 1.2m

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 r204981 = 2.0;
        double r204982 = n;
        double r204983 = r204981 * r204982;
        double r204984 = U;
        double r204985 = r204983 * r204984;
        double r204986 = t;
        double r204987 = l;
        double r204988 = r204987 * r204987;
        double r204989 = Om;
        double r204990 = r204988 / r204989;
        double r204991 = r204981 * r204990;
        double r204992 = r204986 - r204991;
        double r204993 = r204987 / r204989;
        double r204994 = pow(r204993, r204981);
        double r204995 = r204982 * r204994;
        double r204996 = U_;
        double r204997 = r204984 - r204996;
        double r204998 = r204995 * r204997;
        double r204999 = r204992 - r204998;
        double r205000 = r204985 * r204999;
        double r205001 = sqrt(r205000);
        return r205001;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r205002 = t;
        double r205003 = -2.91138552679685e+178;
        bool r205004 = r205002 <= r205003;
        double r205005 = 2.0;
        double r205006 = n;
        double r205007 = r205005 * r205006;
        double r205008 = U;
        double r205009 = r205007 * r205008;
        double r205010 = l;
        double r205011 = Om;
        double r205012 = r205010 / r205011;
        double r205013 = r205010 * r205012;
        double r205014 = r205005 * r205013;
        double r205015 = r205002 - r205014;
        double r205016 = 2.0;
        double r205017 = r205005 / r205016;
        double r205018 = pow(r205012, r205017);
        double r205019 = r205006 * r205018;
        double r205020 = r205019 * r205018;
        double r205021 = U_;
        double r205022 = r205008 - r205021;
        double r205023 = r205020 * r205022;
        double r205024 = r205015 - r205023;
        double r205025 = r205009 * r205024;
        double r205026 = sqrt(r205025);
        double r205027 = -2.992074399044745e+97;
        bool r205028 = r205002 <= r205027;
        double r205029 = r205010 * r205010;
        double r205030 = r205029 / r205011;
        double r205031 = pow(r205012, r205005);
        double r205032 = r205031 * r205022;
        double r205033 = r205032 * r205006;
        double r205034 = fma(r205005, r205030, r205033);
        double r205035 = r205002 - r205034;
        double r205036 = r205008 * r205035;
        double r205037 = r205036 * r205007;
        double r205038 = sqrt(r205037);
        double r205039 = sqrt(r205038);
        double r205040 = r205039 * r205039;
        double r205041 = 1.6642681464261513e-294;
        bool r205042 = r205002 <= r205041;
        double r205043 = 2.653751305673715e-145;
        bool r205044 = r205002 <= r205043;
        double r205045 = 1.2482956029550216e-08;
        bool r205046 = r205002 <= r205045;
        double r205047 = !r205046;
        bool r205048 = r205044 || r205047;
        double r205049 = sqrt(r205009);
        double r205050 = r205005 * r205010;
        double r205051 = r205016 * r205017;
        double r205052 = pow(r205012, r205051);
        double r205053 = r205006 * r205052;
        double r205054 = r205022 * r205053;
        double r205055 = fma(r205050, r205012, r205054);
        double r205056 = r205002 - r205055;
        double r205057 = sqrt(r205056);
        double r205058 = r205049 * r205057;
        double r205059 = pow(r205010, r205016);
        double r205060 = r205059 / r205011;
        double r205061 = r205005 * r205060;
        double r205062 = r205002 - r205061;
        double r205063 = r205008 * r205062;
        double r205064 = r205007 * r205063;
        double r205065 = cbrt(r205006);
        double r205066 = 3.0;
        double r205067 = pow(r205065, r205066);
        double r205068 = r205067 * r205052;
        double r205069 = -r205068;
        double r205070 = r205069 * r205009;
        double r205071 = r205022 * r205070;
        double r205072 = r205064 + r205071;
        double r205073 = sqrt(r205072);
        double r205074 = r205048 ? r205058 : r205073;
        double r205075 = r205042 ? r205026 : r205074;
        double r205076 = r205028 ? r205040 : r205075;
        double r205077 = r205004 ? r205026 : r205076;
        return r205077;
}

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*))))))