Toniolo and Linder, Equation (13)

Average Error: 32.8 → 29.7

Time: 47.8s

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 -1.6760288117773952 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \frac{\left(\frac{\ell}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right) \cdot \left(U - U*\right)}{Om} + \left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)}\\ \mathbf{elif}\;t \le -1.908160256120522 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot U}\\ \mathbf{elif}\;t \le -6.039684361220293 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2 \cdot t, U \cdot n, \left(\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \frac{U}{Om}\right) \cdot -4\right) + \left(\left(\frac{\ell}{Om} \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{elif}\;t \le 3.0731526450190984 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot U}\\ \mathbf{elif}\;t \le 83934.69895366188:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 2.2801453703210546 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(\left(\sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n} \cdot U\right) \cdot \left(\sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n} \cdot \sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n}\right)\right) \cdot \frac{\ell}{Om}\right)\right) + \left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3134874 = 2.0;
        double r3134875 = n;
        double r3134876 = r3134874 * r3134875;
        double r3134877 = U;
        double r3134878 = r3134876 * r3134877;
        double r3134879 = t;
        double r3134880 = l;
        double r3134881 = r3134880 * r3134880;
        double r3134882 = Om;
        double r3134883 = r3134881 / r3134882;
        double r3134884 = r3134874 * r3134883;
        double r3134885 = r3134879 - r3134884;
        double r3134886 = r3134880 / r3134882;
        double r3134887 = pow(r3134886, r3134874);
        double r3134888 = r3134875 * r3134887;
        double r3134889 = U_;
        double r3134890 = r3134877 - r3134889;
        double r3134891 = r3134888 * r3134890;
        double r3134892 = r3134885 - r3134891;
        double r3134893 = r3134878 * r3134892;
        double r3134894 = sqrt(r3134893);
        return r3134894;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3134895 = t;
        double r3134896 = -1.6760288117773952e+183;
        bool r3134897 = r3134895 <= r3134896;
        double r3134898 = n;
        double r3134899 = -2.0;
        double r3134900 = r3134898 * r3134899;
        double r3134901 = l;
        double r3134902 = Om;
        double r3134903 = r3134901 / r3134902;
        double r3134904 = r3134898 * r3134901;
        double r3134905 = U;
        double r3134906 = r3134904 * r3134905;
        double r3134907 = r3134903 * r3134906;
        double r3134908 = U_;
        double r3134909 = r3134905 - r3134908;
        double r3134910 = r3134907 * r3134909;
        double r3134911 = r3134910 / r3134902;
        double r3134912 = r3134900 * r3134911;
        double r3134913 = 2.0;
        double r3134914 = r3134913 * r3134898;
        double r3134915 = r3134901 * r3134903;
        double r3134916 = fma(r3134899, r3134915, r3134895);
        double r3134917 = r3134916 * r3134905;
        double r3134918 = r3134914 * r3134917;
        double r3134919 = r3134912 + r3134918;
        double r3134920 = sqrt(r3134919);
        double r3134921 = -1.908160256120522e+56;
        bool r3134922 = r3134895 <= r3134921;
        double r3134923 = r3134903 * r3134903;
        double r3134924 = r3134923 * r3134909;
        double r3134925 = r3134924 * r3134898;
        double r3134926 = fma(r3134915, r3134913, r3134925);
        double r3134927 = r3134895 - r3134926;
        double r3134928 = r3134914 * r3134927;
        double r3134929 = r3134928 * r3134905;
        double r3134930 = sqrt(r3134929);
        double r3134931 = -6.039684361220293e-180;
        bool r3134932 = r3134895 <= r3134931;
        double r3134933 = r3134913 * r3134895;
        double r3134934 = r3134905 * r3134898;
        double r3134935 = r3134901 * r3134904;
        double r3134936 = r3134905 / r3134902;
        double r3134937 = r3134935 * r3134936;
        double r3134938 = -4.0;
        double r3134939 = r3134937 * r3134938;
        double r3134940 = fma(r3134933, r3134934, r3134939);
        double r3134941 = r3134898 * r3134903;
        double r3134942 = r3134941 * r3134905;
        double r3134943 = r3134903 * r3134942;
        double r3134944 = r3134943 * r3134909;
        double r3134945 = r3134944 * r3134900;
        double r3134946 = r3134940 + r3134945;
        double r3134947 = sqrt(r3134946);
        double r3134948 = 3.0731526450190984e-256;
        bool r3134949 = r3134895 <= r3134948;
        double r3134950 = 83934.69895366188;
        bool r3134951 = r3134895 <= r3134950;
        double r3134952 = sqrt(r3134927);
        double r3134953 = r3134914 * r3134905;
        double r3134954 = sqrt(r3134953);
        double r3134955 = r3134952 * r3134954;
        double r3134956 = 2.2801453703210546e+171;
        bool r3134957 = r3134895 <= r3134956;
        double r3134958 = -r3134903;
        double r3134959 = r3134958 * r3134898;
        double r3134960 = cbrt(r3134959);
        double r3134961 = r3134960 * r3134905;
        double r3134962 = r3134960 * r3134960;
        double r3134963 = r3134961 * r3134962;
        double r3134964 = r3134963 * r3134903;
        double r3134965 = r3134909 * r3134964;
        double r3134966 = r3134914 * r3134965;
        double r3134967 = r3134966 + r3134918;
        double r3134968 = sqrt(r3134967);
        double r3134969 = r3134957 ? r3134968 : r3134955;
        double r3134970 = r3134951 ? r3134955 : r3134969;
        double r3134971 = r3134949 ? r3134930 : r3134970;
        double r3134972 = r3134932 ? r3134947 : r3134971;
        double r3134973 = r3134922 ? r3134930 : r3134972;
        double r3134974 = r3134897 ? r3134920 : r3134973;
        return r3134974;
}

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 5 regimes

2. if t < -1.6760288117773952e+183

   1. Initial program 35.6

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

   3. Applied sub-neg 35.6

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

   4. Applied distribute-lft-in 35.7

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

   5. Simplified 34.5

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

   6. Simplified 32.4

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

   8. Applied *-un-lft-identity 32.4

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

   9. Applied associate-*r* 32.4

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

   10. Simplified 31.9

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

   12. Applied associate-*l/ 32.2

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

   13. Applied distribute-neg-frac 32.2

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

   14. Applied associate-*l/ 33.0

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

   15. Applied associate-*r/ 33.0

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

   16. Applied associate-*l/ 33.0

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

   if -1.6760288117773952e+183 < t < -1.908160256120522e+56 or -6.039684361220293e-180 < t < 3.0731526450190984e-256

   1. Initial program 34.1

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

   3. Applied pow1 34.1

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

   4. Applied pow1 34.1

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

   5. Applied pow1 34.1

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

   6. Applied pow1 34.1

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

   7. Applied pow-prod-down 34.1

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

   8. Applied pow-prod-down 34.1

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

   9. Applied pow-prod-down 34.1

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

   10. Simplified 32.8

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

   if -1.908160256120522e+56 < t < -6.039684361220293e-180

   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 sub-neg 31.6

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

   4. Applied distribute-lft-in 31.6

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

   5. Simplified 29.7

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

   6. Simplified 28.1

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

   8. Applied *-un-lft-identity 28.1

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

   9. Applied associate-*r* 28.1

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

   10. Simplified 27.6

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

   11. Taylor expanded around inf 30.7

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}\right)} + \left(2 \cdot n\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \left(\left(-\frac{\ell}{Om} \cdot n\right) \cdot U\right)\right) \cdot \left(U - U*\right)\right)}\]

   12. Simplified 29.5

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(t \cdot 2, U \cdot n, -4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)\right)} + \left(2 \cdot n\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \left(\left(-\frac{\ell}{Om} \cdot n\right) \cdot U\right)\right) \cdot \left(U - U*\right)\right)}\]

   if 3.0731526450190984e-256 < t < 83934.69895366188 or 2.2801453703210546e+171 < t

   1. Initial program 33.4

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

   3. Applied sqrt-prod 30.2

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

   4. Simplified 28.6

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

   if 83934.69895366188 < t < 2.2801453703210546e+171

   1. Initial program 28.6

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

   3. Applied sub-neg 28.6

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

   4. Applied distribute-lft-in 28.6

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

   5. Simplified 26.1

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

   6. Simplified 24.9

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

   8. Applied *-un-lft-identity 24.9

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

   9. Applied associate-*r* 24.9

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

   10. Simplified 24.4

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

   12. Applied add-cube-cbrt 24.5

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

   13. Applied associate-*l* 24.5

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \mathsf{fma}\left(-2, \frac{\ell}{Om} \cdot \ell, t\right)\right) + \left(2 \cdot n\right) \cdot \left(\left(\frac{\ell}{Om} \cdot \color{blue}{\left(\left(\sqrt[3]{-\frac{\ell}{Om} \cdot n} \cdot \sqrt[3]{-\frac{\ell}{Om} \cdot n}\right) \cdot \left(\sqrt[3]{-\frac{\ell}{Om} \cdot n} \cdot U\right)\right)}\right) \cdot \left(U - U*\right)\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 29.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.6760288117773952 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \frac{\left(\frac{\ell}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right) \cdot \left(U - U*\right)}{Om} + \left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)}\\ \mathbf{elif}\;t \le -1.908160256120522 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot U}\\ \mathbf{elif}\;t \le -6.039684361220293 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2 \cdot t, U \cdot n, \left(\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \frac{U}{Om}\right) \cdot -4\right) + \left(\left(\frac{\ell}{Om} \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{elif}\;t \le 3.0731526450190984 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot U}\\ \mathbf{elif}\;t \le 83934.69895366188:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 2.2801453703210546 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(\left(\sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n} \cdot U\right) \cdot \left(\sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n} \cdot \sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n}\right)\right) \cdot \frac{\ell}{Om}\right)\right) + \left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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*))))))