Average Error: 33.7 → 24.0
Time: 43.4s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.108446529047814 \cdot 10^{-226}:\\ \;\;\;\;{\left(\left(t \cdot n + \left(2 \cdot \ell - \left(U - U*\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.0427927580921139 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t \cdot n + \left(n \cdot \left(2 \cdot \ell - \left(n \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \end{array}\]
\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}\;\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 6.108446529047814 \cdot 10^{-226}:\\
\;\;\;\;{\left(\left(t \cdot n + \left(2 \cdot \ell - \left(U - U*\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\

\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 1.0427927580921139 \cdot 10^{+304}:\\
\;\;\;\;\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}:\\
\;\;\;\;{\left(\left(t \cdot n + \left(n \cdot \left(2 \cdot \ell - \left(n \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2612769 = 2.0;
        double r2612770 = n;
        double r2612771 = r2612769 * r2612770;
        double r2612772 = U;
        double r2612773 = r2612771 * r2612772;
        double r2612774 = t;
        double r2612775 = l;
        double r2612776 = r2612775 * r2612775;
        double r2612777 = Om;
        double r2612778 = r2612776 / r2612777;
        double r2612779 = r2612769 * r2612778;
        double r2612780 = r2612774 - r2612779;
        double r2612781 = r2612775 / r2612777;
        double r2612782 = pow(r2612781, r2612769);
        double r2612783 = r2612770 * r2612782;
        double r2612784 = U_;
        double r2612785 = r2612772 - r2612784;
        double r2612786 = r2612783 * r2612785;
        double r2612787 = r2612780 - r2612786;
        double r2612788 = r2612773 * r2612787;
        double r2612789 = sqrt(r2612788);
        return r2612789;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2612790 = 2.0;
        double r2612791 = n;
        double r2612792 = r2612790 * r2612791;
        double r2612793 = U;
        double r2612794 = r2612792 * r2612793;
        double r2612795 = t;
        double r2612796 = l;
        double r2612797 = r2612796 * r2612796;
        double r2612798 = Om;
        double r2612799 = r2612797 / r2612798;
        double r2612800 = r2612799 * r2612790;
        double r2612801 = r2612795 - r2612800;
        double r2612802 = r2612796 / r2612798;
        double r2612803 = pow(r2612802, r2612790);
        double r2612804 = r2612791 * r2612803;
        double r2612805 = U_;
        double r2612806 = r2612793 - r2612805;
        double r2612807 = r2612804 * r2612806;
        double r2612808 = r2612801 - r2612807;
        double r2612809 = r2612794 * r2612808;
        double r2612810 = 6.108446529047814e-226;
        bool r2612811 = r2612809 <= r2612810;
        double r2612812 = r2612795 * r2612791;
        double r2612813 = r2612790 * r2612796;
        double r2612814 = -r2612791;
        double r2612815 = r2612814 * r2612796;
        double r2612816 = r2612815 / r2612798;
        double r2612817 = r2612806 * r2612816;
        double r2612818 = r2612813 - r2612817;
        double r2612819 = r2612818 * r2612816;
        double r2612820 = r2612812 + r2612819;
        double r2612821 = r2612793 * r2612790;
        double r2612822 = r2612820 * r2612821;
        double r2612823 = 0.5;
        double r2612824 = pow(r2612822, r2612823);
        double r2612825 = 1.0427927580921139e+304;
        bool r2612826 = r2612809 <= r2612825;
        double r2612827 = sqrt(r2612809);
        double r2612828 = -r2612802;
        double r2612829 = r2612791 * r2612828;
        double r2612830 = r2612829 * r2612806;
        double r2612831 = r2612813 - r2612830;
        double r2612832 = r2612791 * r2612831;
        double r2612833 = r2612832 * r2612828;
        double r2612834 = r2612812 + r2612833;
        double r2612835 = r2612834 * r2612821;
        double r2612836 = pow(r2612835, r2612823);
        double r2612837 = r2612826 ? r2612827 : r2612836;
        double r2612838 = r2612811 ? r2612824 : r2612837;
        return r2612838;
}

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 3 regimes

  2. if (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 6.108446529047814e-226

    1. Initial program 46.6

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

    2. Simplified33.5

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

    4. Applied sub-neg33.5

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

    5. Applied distribute-rgt-in33.5

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

    6. Simplified31.9

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

    8. Applied associate-*r*33.0

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

    10. Applied pow1/233.0

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

    12. Applied *-un-lft-identity33.0

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

    13. Applied pow-unpow33.0

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

    14. Simplified32.6

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

    if 6.108446529047814e-226 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 1.0427927580921139e+304

    1. Initial program 1.4

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

    if 1.0427927580921139e+304 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

    1. Initial program 60.4

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

    2. Simplified52.2

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

    4. Applied sub-neg52.2

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

    5. Applied distribute-rgt-in52.2

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

    6. Simplified51.0

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

    8. Applied associate-*r*43.1

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

    10. Applied pow1/243.1

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

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.108446529047814 \cdot 10^{-226}:\\ \;\;\;\;{\left(\left(t \cdot n + \left(2 \cdot \ell - \left(U - U*\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \frac{\left(-n\right) \cdot \ell}{Om}\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.0427927580921139 \cdot 10^{+304}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t \cdot n + \left(n \cdot \left(2 \cdot \ell - \left(n \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(-\frac{\ell}{Om}\right)\right) \cdot \left(U \cdot 2\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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*))))))