Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.3

Time: 2.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le -7122680.873158953:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le -1.8663328166415404 \cdot 10^{-262}:\\ \;\;\;\;{\left(\left(U \cdot \left(t - \mathsf{fma}\left(\left(\ell \cdot 2\right), \left(\frac{\ell}{Om}\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le 2.2093515262227777 \cdot 10^{-301}:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot {\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\left(\ell \cdot 2\right), \left(\frac{\ell}{Om}\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r17575142 = 2.0;
        double r17575143 = n;
        double r17575144 = r17575142 * r17575143;
        double r17575145 = U;
        double r17575146 = r17575144 * r17575145;
        double r17575147 = t;
        double r17575148 = l;
        double r17575149 = r17575148 * r17575148;
        double r17575150 = Om;
        double r17575151 = r17575149 / r17575150;
        double r17575152 = r17575142 * r17575151;
        double r17575153 = r17575147 - r17575152;
        double r17575154 = r17575148 / r17575150;
        double r17575155 = pow(r17575154, r17575142);
        double r17575156 = r17575143 * r17575155;
        double r17575157 = U_;
        double r17575158 = r17575145 - r17575157;
        double r17575159 = r17575156 * r17575158;
        double r17575160 = r17575153 - r17575159;
        double r17575161 = r17575146 * r17575160;
        double r17575162 = sqrt(r17575161);
        return r17575162;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r17575163 = U;
        double r17575164 = -7122680.873158953;
        bool r17575165 = r17575163 <= r17575164;
        double r17575166 = t;
        double r17575167 = l;
        double r17575168 = 2.0;
        double r17575169 = r17575167 * r17575168;
        double r17575170 = Om;
        double r17575171 = r17575167 / r17575170;
        double r17575172 = U_;
        double r17575173 = r17575163 - r17575172;
        double r17575174 = r17575171 * r17575173;
        double r17575175 = n;
        double r17575176 = r17575174 * r17575175;
        double r17575177 = r17575169 + r17575176;
        double r17575178 = r17575177 * r17575171;
        double r17575179 = r17575166 - r17575178;
        double r17575180 = r17575168 * r17575175;
        double r17575181 = r17575163 * r17575180;
        double r17575182 = r17575179 * r17575181;
        double r17575183 = 0.5;
        double r17575184 = pow(r17575182, r17575183);
        double r17575185 = -1.8663328166415404e-262;
        bool r17575186 = r17575163 <= r17575185;
        double r17575187 = r17575175 * r17575171;
        double r17575188 = r17575171 * r17575187;
        double r17575189 = r17575173 * r17575188;
        double r17575190 = fma(r17575169, r17575171, r17575189);
        double r17575191 = r17575166 - r17575190;
        double r17575192 = r17575163 * r17575191;
        double r17575193 = r17575192 * r17575180;
        double r17575194 = pow(r17575193, r17575183);
        double r17575195 = 2.2093515262227777e-301;
        bool r17575196 = r17575163 <= r17575195;
        double r17575197 = sqrt(r17575163);
        double r17575198 = r17575180 * r17575191;
        double r17575199 = pow(r17575198, r17575183);
        double r17575200 = r17575197 * r17575199;
        double r17575201 = r17575196 ? r17575184 : r17575200;
        double r17575202 = r17575186 ? r17575194 : r17575201;
        double r17575203 = r17575165 ? r17575184 : r17575202;
        return r17575203;
}

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 < -7122680.873158953 or -1.8663328166415404e-262 < U < 2.2093515262227777e-301

   1. Initial program 32.3

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

   3. Applied *-un-lft-identity 32.3

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

   4. Applied times-frac 29.3

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

   5. Simplified 29.3

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

   7. Applied pow1 29.3

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

   8. Applied pow1 29.3

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

   9. Applied pow1 29.3

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

   10. Applied pow-prod-down 29.3

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

   11. Applied pow-prod-down 29.3

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

   12. Applied sqrt-pow1 29.3

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

   13. Simplified 30.1

       \[\leadsto {\color{blue}{\left(U \cdot \left(\left(t - \mathsf{fma}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{Om}\right), \left(\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right)}}^{\left(\frac{1}{2}\right)}\]
   14. Using strategy rm

   15. Applied pow1 30.1

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

   16. Applied pow1 30.1

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

   17. Applied pow-prod-down 30.1

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

   18. Applied pow1 30.1

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

   19. Applied pow-prod-down 30.1

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

   20. Applied pow1 30.1

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

   21. Applied pow-prod-down 30.1

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

   22. Applied pow-pow 30.1

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

   23. Simplified 29.2

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

   if -7122680.873158953 < U < -1.8663328166415404e-262

   1. Initial program 34.7

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

   3. Applied *-un-lft-identity 34.7

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

   4. Applied times-frac 31.8

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

   5. Simplified 31.8

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

   7. Applied pow1 31.8

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

   8. Applied pow1 31.8

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

   9. Applied pow1 31.8

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

   10. Applied pow-prod-down 31.8

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

   11. Applied pow-prod-down 31.8

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

   12. Applied sqrt-pow1 31.8

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

   13. Simplified 30.4

       \[\leadsto {\color{blue}{\left(U \cdot \left(\left(t - \mathsf{fma}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{Om}\right), \left(\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right)}}^{\left(\frac{1}{2}\right)}\]
   14. Using strategy rm

   15. Applied associate-*r* 27.1

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

   if 2.2093515262227777e-301 < U

   1. Initial program 32.8

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

   3. Applied *-un-lft-identity 32.8

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

   4. Applied times-frac 30.1

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

   5. Simplified 30.1

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

   7. Applied pow1 30.1

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

   8. Applied pow1 30.1

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

   9. Applied pow1 30.1

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

   10. Applied pow-prod-down 30.1

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

   11. Applied pow-prod-down 30.1

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

   12. Applied sqrt-pow1 30.1

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

   13. Simplified 29.1

       \[\leadsto {\color{blue}{\left(U \cdot \left(\left(t - \mathsf{fma}\left(\left(2 \cdot \ell\right), \left(\frac{\ell}{Om}\right), \left(\left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right)\right)}}^{\left(\frac{1}{2}\right)}\]
   14. Using strategy rm

   15. Applied unpow-prod-down 22.4

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

   16. Simplified 22.4

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

4. Final simplification 25.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -7122680.873158953:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le -1.8663328166415404 \cdot 10^{-262}:\\ \;\;\;\;{\left(\left(U \cdot \left(t - \mathsf{fma}\left(\left(\ell \cdot 2\right), \left(\frac{\ell}{Om}\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right) \cdot \left(2 \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;U \le 2.2093515262227777 \cdot 10^{-301}:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot 2 + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U \cdot \left(2 \cdot n\right)\right)\right)}^{\frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot {\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\left(\ell \cdot 2\right), \left(\frac{\ell}{Om}\right), \left(\left(U - U*\right) \cdot \left(\frac{\ell}{Om} \cdot \left(n \cdot \frac{\ell}{Om}\right)\right)\right)\right)\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(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*))))))