Toniolo and Linder, Equation (13)

Average Error: 33.4 → 24.4

Time: 2.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \frac{n \cdot -2}{\frac{\frac{Om}{\ell}}{\ell}} + \left(U \cdot 2\right) \cdot \left(n \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 8.046145761530675 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\frac{1}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{Om} \cdot \left(\frac{\frac{n}{\frac{Om}{\sqrt[3]{\ell}}}}{\frac{1}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right) + \left(\left(\frac{\sqrt[3]{n}}{\frac{\frac{\sqrt[3]{Om}}{\ell}}{\sqrt[3]{\sqrt[3]{\ell}}}} \cdot \left(U \cdot 2\right)\right) \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}}\right) \cdot \frac{-2}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r4236907 = 2.0;
        double r4236908 = n;
        double r4236909 = r4236907 * r4236908;
        double r4236910 = U;
        double r4236911 = r4236909 * r4236910;
        double r4236912 = t;
        double r4236913 = l;
        double r4236914 = r4236913 * r4236913;
        double r4236915 = Om;
        double r4236916 = r4236914 / r4236915;
        double r4236917 = r4236907 * r4236916;
        double r4236918 = r4236912 - r4236917;
        double r4236919 = r4236913 / r4236915;
        double r4236920 = pow(r4236919, r4236907);
        double r4236921 = r4236908 * r4236920;
        double r4236922 = U_;
        double r4236923 = r4236910 - r4236922;
        double r4236924 = r4236921 * r4236923;
        double r4236925 = r4236918 - r4236924;
        double r4236926 = r4236911 * r4236925;
        double r4236927 = sqrt(r4236926);
        return r4236927;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r4236928 = 2.0;
        double r4236929 = n;
        double r4236930 = r4236928 * r4236929;
        double r4236931 = U;
        double r4236932 = r4236930 * r4236931;
        double r4236933 = t;
        double r4236934 = l;
        double r4236935 = r4236934 * r4236934;
        double r4236936 = Om;
        double r4236937 = r4236935 / r4236936;
        double r4236938 = r4236937 * r4236928;
        double r4236939 = r4236933 - r4236938;
        double r4236940 = r4236934 / r4236936;
        double r4236941 = pow(r4236940, r4236928);
        double r4236942 = r4236929 * r4236941;
        double r4236943 = U_;
        double r4236944 = r4236931 - r4236943;
        double r4236945 = r4236942 * r4236944;
        double r4236946 = r4236939 - r4236945;
        double r4236947 = r4236932 * r4236946;
        double r4236948 = 0.0;
        bool r4236949 = r4236947 <= r4236948;
        double r4236950 = r4236931 * r4236928;
        double r4236951 = -2.0;
        double r4236952 = r4236929 * r4236951;
        double r4236953 = r4236936 / r4236934;
        double r4236954 = r4236953 / r4236934;
        double r4236955 = r4236952 / r4236954;
        double r4236956 = r4236950 * r4236955;
        double r4236957 = r4236929 / r4236953;
        double r4236958 = r4236957 * r4236944;
        double r4236959 = r4236958 / r4236953;
        double r4236960 = r4236933 - r4236959;
        double r4236961 = r4236929 * r4236960;
        double r4236962 = r4236950 * r4236961;
        double r4236963 = r4236956 + r4236962;
        double r4236964 = sqrt(r4236963);
        double r4236965 = 8.046145761530675e+302;
        bool r4236966 = r4236947 <= r4236965;
        double r4236967 = sqrt(r4236947);
        double r4236968 = 1.0;
        double r4236969 = cbrt(r4236934);
        double r4236970 = r4236969 * r4236969;
        double r4236971 = r4236968 / r4236970;
        double r4236972 = r4236968 / r4236971;
        double r4236973 = r4236972 / r4236936;
        double r4236974 = r4236936 / r4236969;
        double r4236975 = r4236929 / r4236974;
        double r4236976 = r4236968 / r4236934;
        double r4236977 = r4236975 / r4236976;
        double r4236978 = r4236977 * r4236944;
        double r4236979 = r4236973 * r4236978;
        double r4236980 = r4236933 - r4236979;
        double r4236981 = r4236980 * r4236929;
        double r4236982 = r4236981 * r4236950;
        double r4236983 = cbrt(r4236929);
        double r4236984 = cbrt(r4236936);
        double r4236985 = r4236984 / r4236934;
        double r4236986 = cbrt(r4236969);
        double r4236987 = r4236985 / r4236986;
        double r4236988 = r4236983 / r4236987;
        double r4236989 = r4236988 * r4236950;
        double r4236990 = r4236983 * r4236983;
        double r4236991 = r4236984 * r4236984;
        double r4236992 = r4236986 * r4236986;
        double r4236993 = r4236991 / r4236992;
        double r4236994 = r4236990 / r4236993;
        double r4236995 = r4236989 * r4236994;
        double r4236996 = r4236951 / r4236971;
        double r4236997 = r4236995 * r4236996;
        double r4236998 = r4236982 + r4236997;
        double r4236999 = sqrt(r4236998);
        double r4237000 = r4236966 ? r4236967 : r4236999;
        double r4237001 = r4236949 ? r4236964 : r4237000;
        return r4237001;
}

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

   1. Initial program 56.7

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

   2. Simplified 38.2

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

   4. Applied distribute-rgt-in 38.2

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

   5. Applied distribute-rgt-in 38.2

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

   6. Applied distribute-rgt-in 38.2

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

   7. Simplified 37.1

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

   8. Simplified 37.1

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

   10. Applied associate-*l/ 35.7

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

   if 0.0 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 8.046145761530675e+302

   1. Initial program 2.0

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

   if 8.046145761530675e+302 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

   1. Initial program 60.2

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

   2. Simplified 57.8

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

   4. Applied distribute-rgt-in 57.8

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

   5. Applied distribute-rgt-in 57.8

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

   6. Applied distribute-rgt-in 57.8

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

   7. Simplified 50.5

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

   8. Simplified 50.5

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

   10. Applied div-inv 50.5

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

   11. Applied add-cube-cbrt 50.5

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

   12. Applied *-un-lft-identity 50.5

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

   13. Applied times-frac 50.5

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

   14. Applied *-un-lft-identity 50.5

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

   15. Applied times-frac 50.5

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

   16. Applied times-frac 50.5

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

   17. Applied associate-*l* 50.0

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

   19. Applied add-cube-cbrt 50.1

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

   20. Applied *-un-lft-identity 50.1

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

   21. Applied times-frac 50.1

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

   22. Applied times-frac 46.5

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

   23. Applied associate-*l* 45.9

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

   25. Applied add-cube-cbrt 45.9

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

   26. Applied *-un-lft-identity 45.9

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

   27. Applied add-cube-cbrt 45.9

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

   28. Applied times-frac 45.9

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

   29. Applied times-frac 45.9

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

   30. Applied add-cube-cbrt 46.0

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

   31. Applied times-frac 45.5

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

   32. Applied associate-*l* 46.0

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

4. Final simplification 24.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \frac{n \cdot -2}{\frac{\frac{Om}{\ell}}{\ell}} + \left(U \cdot 2\right) \cdot \left(n \cdot \left(t - \frac{\frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)}{\frac{Om}{\ell}}\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 8.046145761530675 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - \frac{\frac{1}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{Om} \cdot \left(\frac{\frac{n}{\frac{Om}{\sqrt[3]{\ell}}}}{\frac{1}{\ell}} \cdot \left(U - U*\right)\right)\right) \cdot n\right) \cdot \left(U \cdot 2\right) + \left(\left(\frac{\sqrt[3]{n}}{\frac{\frac{\sqrt[3]{Om}}{\ell}}{\sqrt[3]{\sqrt[3]{\ell}}}} \cdot \left(U \cdot 2\right)\right) \cdot \frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}}\right) \cdot \frac{-2}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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*))))))