Average Error: 33.9 → 28.4
Time: 4.4m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -2.8826762190597844 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(n \cdot \frac{U \cdot \left(-2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}} + t \cdot \left(U \cdot n\right)\right) \cdot 2}\\ \mathbf{elif}\;\ell \le 1.1266507648201761 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot \frac{U \cdot \left(-2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}} + t \cdot \left(U \cdot n\right)\right) \cdot 2}\\ \end{array}\]
\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}\;\ell \le -2.8826762190597844 \cdot 10^{-302}:\\
\;\;\;\;\sqrt{\left(n \cdot \frac{U \cdot \left(-2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}} + t \cdot \left(U \cdot n\right)\right) \cdot 2}\\

\mathbf{elif}\;\ell \le 1.1266507648201761 \cdot 10^{-116}:\\
\;\;\;\;\sqrt{\left(\left(n \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot 2}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r28985891 = 2.0;
        double r28985892 = n;
        double r28985893 = r28985891 * r28985892;
        double r28985894 = U;
        double r28985895 = r28985893 * r28985894;
        double r28985896 = t;
        double r28985897 = l;
        double r28985898 = r28985897 * r28985897;
        double r28985899 = Om;
        double r28985900 = r28985898 / r28985899;
        double r28985901 = r28985891 * r28985900;
        double r28985902 = r28985896 - r28985901;
        double r28985903 = r28985897 / r28985899;
        double r28985904 = pow(r28985903, r28985891);
        double r28985905 = r28985892 * r28985904;
        double r28985906 = U_;
        double r28985907 = r28985894 - r28985906;
        double r28985908 = r28985905 * r28985907;
        double r28985909 = r28985902 - r28985908;
        double r28985910 = r28985895 * r28985909;
        double r28985911 = sqrt(r28985910);
        return r28985911;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r28985912 = l;
        double r28985913 = -2.8826762190597844e-302;
        bool r28985914 = r28985912 <= r28985913;
        double r28985915 = n;
        double r28985916 = U;
        double r28985917 = -2.0;
        double r28985918 = r28985917 * r28985912;
        double r28985919 = Om;
        double r28985920 = r28985919 / r28985912;
        double r28985921 = r28985915 / r28985920;
        double r28985922 = U_;
        double r28985923 = r28985916 - r28985922;
        double r28985924 = r28985921 * r28985923;
        double r28985925 = r28985918 - r28985924;
        double r28985926 = r28985916 * r28985925;
        double r28985927 = r28985926 / r28985920;
        double r28985928 = r28985915 * r28985927;
        double r28985929 = t;
        double r28985930 = r28985916 * r28985915;
        double r28985931 = r28985929 * r28985930;
        double r28985932 = r28985928 + r28985931;
        double r28985933 = 2.0;
        double r28985934 = r28985932 * r28985933;
        double r28985935 = sqrt(r28985934);
        double r28985936 = 1.1266507648201761e-116;
        bool r28985937 = r28985912 <= r28985936;
        double r28985938 = r28985912 / r28985919;
        double r28985939 = r28985912 * r28985938;
        double r28985940 = r28985933 * r28985939;
        double r28985941 = r28985929 - r28985940;
        double r28985942 = r28985921 / r28985920;
        double r28985943 = r28985923 * r28985942;
        double r28985944 = r28985941 - r28985943;
        double r28985945 = r28985915 * r28985944;
        double r28985946 = r28985945 * r28985916;
        double r28985947 = r28985946 * r28985933;
        double r28985948 = sqrt(r28985947);
        double r28985949 = r28985937 ? r28985948 : r28985935;
        double r28985950 = r28985914 ? r28985935 : r28985949;
        return r28985950;
}

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 l < -2.8826762190597844e-302 or 1.1266507648201761e-116 < l

    1. Initial program 36.2

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

    2. Simplified36.2

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

    4. Applied *-un-lft-identity36.2

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

    5. Applied times-frac33.1

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

    6. Simplified33.1

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

    7. Taylor expanded around -inf 38.9

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

    8. Simplified32.4

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

    10. Applied sub-neg32.4

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

    11. Applied associate--l+32.4

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

    12. Applied distribute-lft-in32.4

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

    13. Simplified32.0

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

    15. Applied associate-*l/31.7

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

    16. Applied sub-div31.7

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

    17. Applied associate-*r/29.3

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

    if -2.8826762190597844e-302 < l < 1.1266507648201761e-116

    1. Initial program 24.5

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

    2. Simplified24.4

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

    4. Applied *-un-lft-identity24.4

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

    5. Applied times-frac24.4

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

    6. Simplified24.4

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

    7. Taylor expanded around -inf 31.4

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

    8. Simplified23.8

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

    10. Applied associate-*l*24.9

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

  4. Final simplification28.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.8826762190597844 \cdot 10^{-302}:\\ \;\;\;\;\sqrt{\left(n \cdot \frac{U \cdot \left(-2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}} + t \cdot \left(U \cdot n\right)\right) \cdot 2}\\ \mathbf{elif}\;\ell \le 1.1266507648201761 \cdot 10^{-116}:\\ \;\;\;\;\sqrt{\left(\left(n \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(U - U*\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}\right)\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n \cdot \frac{U \cdot \left(-2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right)}{\frac{Om}{\ell}} + t \cdot \left(U \cdot n\right)\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(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*))))))