Average Error: 32.9 → 28.2
Time: 38.2s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 3.2328584205801053 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot U - \frac{U*}{\frac{Om}{\ell}}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 4.9836541702388693 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;t \le 3.780023435277137 \cdot 10^{-09}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot U - \frac{U*}{\frac{Om}{\ell}}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 3.750417619434724 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array}\]
\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 3.2328584205801053 \cdot 10^{-251}:\\
\;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot U - \frac{U*}{\frac{Om}{\ell}}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\

\mathbf{elif}\;t \le 4.9836541702388693 \cdot 10^{-157}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}\\

\mathbf{elif}\;t \le 3.780023435277137 \cdot 10^{-09}:\\
\;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot U - \frac{U*}{\frac{Om}{\ell}}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\

\mathbf{elif}\;t \le 3.750417619434724 \cdot 10^{+199}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r861846 = 2.0;
        double r861847 = n;
        double r861848 = r861846 * r861847;
        double r861849 = U;
        double r861850 = r861848 * r861849;
        double r861851 = t;
        double r861852 = l;
        double r861853 = r861852 * r861852;
        double r861854 = Om;
        double r861855 = r861853 / r861854;
        double r861856 = r861846 * r861855;
        double r861857 = r861851 - r861856;
        double r861858 = r861852 / r861854;
        double r861859 = pow(r861858, r861846);
        double r861860 = r861847 * r861859;
        double r861861 = U_;
        double r861862 = r861849 - r861861;
        double r861863 = r861860 * r861862;
        double r861864 = r861857 - r861863;
        double r861865 = r861850 * r861864;
        double r861866 = sqrt(r861865);
        return r861866;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r861867 = t;
        double r861868 = 3.2328584205801053e-251;
        bool r861869 = r861867 <= r861868;
        double r861870 = 2.0;
        double r861871 = l;
        double r861872 = Om;
        double r861873 = r861872 / r861871;
        double r861874 = r861871 / r861873;
        double r861875 = r861870 * r861874;
        double r861876 = r861867 - r861875;
        double r861877 = r861871 / r861872;
        double r861878 = n;
        double r861879 = r861877 * r861878;
        double r861880 = U;
        double r861881 = r861877 * r861880;
        double r861882 = U_;
        double r861883 = r861882 / r861873;
        double r861884 = r861881 - r861883;
        double r861885 = r861879 * r861884;
        double r861886 = r861876 - r861885;
        double r861887 = r861870 * r861878;
        double r861888 = r861887 * r861880;
        double r861889 = r861886 * r861888;
        double r861890 = sqrt(r861889);
        double r861891 = 4.9836541702388693e-157;
        bool r861892 = r861867 <= r861891;
        double r861893 = sqrt(r861888);
        double r861894 = r861877 * r861871;
        double r861895 = r861880 - r861882;
        double r861896 = r861879 * r861877;
        double r861897 = r861895 * r861896;
        double r861898 = fma(r861870, r861894, r861897);
        double r861899 = r861867 - r861898;
        double r861900 = sqrt(r861899);
        double r861901 = r861893 * r861900;
        double r861902 = 3.780023435277137e-09;
        bool r861903 = r861867 <= r861902;
        double r861904 = 3.750417619434724e+199;
        bool r861905 = r861867 <= r861904;
        double r861906 = r861899 * r861880;
        double r861907 = r861887 * r861906;
        double r861908 = sqrt(r861907);
        double r861909 = r861905 ? r861908 : r861901;
        double r861910 = r861903 ? r861890 : r861909;
        double r861911 = r861892 ? r861901 : r861910;
        double r861912 = r861869 ? r861890 : r861911;
        return r861912;
}

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 t < 3.2328584205801053e-251 or 4.9836541702388693e-157 < t < 3.780023435277137e-09

    1. Initial program 32.4

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

    3. Applied associate-/l*29.9

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

    5. Applied unpow229.9

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

    6. Applied associate-*r*29.0

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

    8. Applied associate-*l*28.9

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

    9. Taylor expanded around 0 31.0

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

    10. Simplified28.9

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

    if 3.2328584205801053e-251 < t < 4.9836541702388693e-157 or 3.750417619434724e+199 < t

    1. Initial program 38.0

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

    3. Applied associate-/l*35.9

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

    5. Applied unpow235.9

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

    6. Applied associate-*r*34.8

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

    8. Applied associate-*l*34.7

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

    10. Applied sqrt-prod27.5

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

    11. Simplified27.2

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

    if 3.780023435277137e-09 < t < 3.750417619434724e+199

    1. Initial program 30.2

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

    3. Applied associate-/l*26.9

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

    5. Applied unpow226.9

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

    6. Applied associate-*r*26.6

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

    8. Applied associate-*l*26.7

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

    10. Applied associate-*l*26.6

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

    11. Simplified26.3

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

  4. Final simplification28.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.2328584205801053 \cdot 10^{-251}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot U - \frac{U*}{\frac{Om}{\ell}}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 4.9836541702388693 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;t \le 3.780023435277137 \cdot 10^{-09}:\\ \;\;\;\;\sqrt{\left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot U - \frac{U*}{\frac{Om}{\ell}}\right)\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;t \le 3.750417619434724 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{t - \mathsf{fma}\left(2, \frac{\ell}{Om} \cdot \ell, \left(U - U*\right) \cdot \left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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*))))))