Average Error: 34.6 → 29.9
Time: 1.1m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 8.48078266481481385711732205638014053282 \cdot 10^{128}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)}\\ \end{array}\]
\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 8.48078266481481385711732205638014053282 \cdot 10^{128}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r209804 = 2.0;
        double r209805 = n;
        double r209806 = r209804 * r209805;
        double r209807 = U;
        double r209808 = r209806 * r209807;
        double r209809 = t;
        double r209810 = l;
        double r209811 = r209810 * r209810;
        double r209812 = Om;
        double r209813 = r209811 / r209812;
        double r209814 = r209804 * r209813;
        double r209815 = r209809 - r209814;
        double r209816 = r209810 / r209812;
        double r209817 = pow(r209816, r209804);
        double r209818 = r209805 * r209817;
        double r209819 = U_;
        double r209820 = r209807 - r209819;
        double r209821 = r209818 * r209820;
        double r209822 = r209815 - r209821;
        double r209823 = r209808 * r209822;
        double r209824 = sqrt(r209823);
        return r209824;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r209825 = t;
        double r209826 = 8.480782664814814e+128;
        bool r209827 = r209825 <= r209826;
        double r209828 = 2.0;
        double r209829 = n;
        double r209830 = r209828 * r209829;
        double r209831 = U;
        double r209832 = l;
        double r209833 = Om;
        double r209834 = r209833 / r209832;
        double r209835 = r209832 / r209834;
        double r209836 = r209828 * r209835;
        double r209837 = r209825 - r209836;
        double r209838 = U_;
        double r209839 = r209831 - r209838;
        double r209840 = r209832 / r209833;
        double r209841 = 2.0;
        double r209842 = r209828 / r209841;
        double r209843 = r209841 * r209842;
        double r209844 = pow(r209840, r209843);
        double r209845 = r209829 * r209844;
        double r209846 = r209839 * r209845;
        double r209847 = r209837 - r209846;
        double r209848 = r209831 * r209847;
        double r209849 = r209830 * r209848;
        double r209850 = sqrt(r209849);
        double r209851 = r209830 * r209831;
        double r209852 = sqrt(r209851);
        double r209853 = sqrt(r209847);
        double r209854 = r209852 * r209853;
        double r209855 = r209827 ? r209850 : r209854;
        return r209855;
}

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*

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if t < 8.480782664814814e+128

    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 associate-/l*31.2

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

    5. Applied sqr-pow31.2

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

    6. Applied associate-*r*30.4

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

    8. Applied associate-*l*29.9

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

    9. Simplified31.0

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

    if 8.480782664814814e+128 < t

    1. Initial program 37.9

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

    3. Applied associate-/l*35.2

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

    5. Applied sqr-pow35.2

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

    6. Applied associate-*r*34.6

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

    8. Applied sqrt-prod22.9

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

    9. Simplified23.4

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

  4. Final simplification29.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 8.48078266481481385711732205638014053282 \cdot 10^{128}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))