Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.3

Time: 1.7m

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 r15011034 = 2.0;
        double r15011035 = n;
        double r15011036 = r15011034 * r15011035;
        double r15011037 = U;
        double r15011038 = r15011036 * r15011037;
        double r15011039 = t;
        double r15011040 = l;
        double r15011041 = r15011040 * r15011040;
        double r15011042 = Om;
        double r15011043 = r15011041 / r15011042;
        double r15011044 = r15011034 * r15011043;
        double r15011045 = r15011039 - r15011044;
        double r15011046 = r15011040 / r15011042;
        double r15011047 = pow(r15011046, r15011034);
        double r15011048 = r15011035 * r15011047;
        double r15011049 = U_;
        double r15011050 = r15011037 - r15011049;
        double r15011051 = r15011048 * r15011050;
        double r15011052 = r15011045 - r15011051;
        double r15011053 = r15011038 * r15011052;
        double r15011054 = sqrt(r15011053);
        return r15011054;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r15011055 = U;
        double r15011056 = -7122680.873158953;
        bool r15011057 = r15011055 <= r15011056;
        double r15011058 = t;
        double r15011059 = l;
        double r15011060 = 2.0;
        double r15011061 = r15011059 * r15011060;
        double r15011062 = Om;
        double r15011063 = r15011059 / r15011062;
        double r15011064 = U_;
        double r15011065 = r15011055 - r15011064;
        double r15011066 = r15011063 * r15011065;
        double r15011067 = n;
        double r15011068 = r15011066 * r15011067;
        double r15011069 = r15011061 + r15011068;
        double r15011070 = r15011069 * r15011063;
        double r15011071 = r15011058 - r15011070;
        double r15011072 = r15011060 * r15011067;
        double r15011073 = r15011055 * r15011072;
        double r15011074 = r15011071 * r15011073;
        double r15011075 = 0.5;
        double r15011076 = pow(r15011074, r15011075);
        double r15011077 = -1.8663328166415404e-262;
        bool r15011078 = r15011055 <= r15011077;
        double r15011079 = r15011067 * r15011063;
        double r15011080 = r15011063 * r15011079;
        double r15011081 = r15011065 * r15011080;
        double r15011082 = fma(r15011061, r15011063, r15011081);
        double r15011083 = r15011058 - r15011082;
        double r15011084 = r15011055 * r15011083;
        double r15011085 = r15011084 * r15011072;
        double r15011086 = pow(r15011085, r15011075);
        double r15011087 = 2.2093515262227777e-301;
        bool r15011088 = r15011055 <= r15011087;
        double r15011089 = sqrt(r15011055);
        double r15011090 = r15011072 * r15011083;
        double r15011091 = pow(r15011090, r15011075);
        double r15011092 = r15011089 * r15011091;
        double r15011093 = r15011088 ? r15011076 : r15011092;
        double r15011094 = r15011078 ? r15011086 : r15011093;
        double r15011095 = r15011057 ? r15011076 : r15011094;
        return r15011095;
}

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

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

   16. Applied pow-pow 30.1

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

   17. 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*))))))