Average Error: 33.9 → 28.4
Time: 7.2m
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 r29615713 = 2.0;
        double r29615714 = n;
        double r29615715 = r29615713 * r29615714;
        double r29615716 = U;
        double r29615717 = r29615715 * r29615716;
        double r29615718 = t;
        double r29615719 = l;
        double r29615720 = r29615719 * r29615719;
        double r29615721 = Om;
        double r29615722 = r29615720 / r29615721;
        double r29615723 = r29615713 * r29615722;
        double r29615724 = r29615718 - r29615723;
        double r29615725 = r29615719 / r29615721;
        double r29615726 = pow(r29615725, r29615713);
        double r29615727 = r29615714 * r29615726;
        double r29615728 = U_;
        double r29615729 = r29615716 - r29615728;
        double r29615730 = r29615727 * r29615729;
        double r29615731 = r29615724 - r29615730;
        double r29615732 = r29615717 * r29615731;
        double r29615733 = sqrt(r29615732);
        return r29615733;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r29615734 = l;
        double r29615735 = -2.8826762190597844e-302;
        bool r29615736 = r29615734 <= r29615735;
        double r29615737 = n;
        double r29615738 = U;
        double r29615739 = -2.0;
        double r29615740 = r29615739 * r29615734;
        double r29615741 = Om;
        double r29615742 = r29615741 / r29615734;
        double r29615743 = r29615737 / r29615742;
        double r29615744 = U_;
        double r29615745 = r29615738 - r29615744;
        double r29615746 = r29615743 * r29615745;
        double r29615747 = r29615740 - r29615746;
        double r29615748 = r29615738 * r29615747;
        double r29615749 = r29615748 / r29615742;
        double r29615750 = r29615737 * r29615749;
        double r29615751 = t;
        double r29615752 = r29615738 * r29615737;
        double r29615753 = r29615751 * r29615752;
        double r29615754 = r29615750 + r29615753;
        double r29615755 = 2.0;
        double r29615756 = r29615754 * r29615755;
        double r29615757 = sqrt(r29615756);
        double r29615758 = 1.1266507648201761e-116;
        bool r29615759 = r29615734 <= r29615758;
        double r29615760 = r29615734 / r29615741;
        double r29615761 = r29615734 * r29615760;
        double r29615762 = r29615755 * r29615761;
        double r29615763 = r29615751 - r29615762;
        double r29615764 = r29615743 / r29615742;
        double r29615765 = r29615745 * r29615764;
        double r29615766 = r29615763 - r29615765;
        double r29615767 = r29615737 * r29615766;
        double r29615768 = r29615767 * r29615738;
        double r29615769 = r29615768 * r29615755;
        double r29615770 = sqrt(r29615769);
        double r29615771 = r29615759 ? r29615770 : r29615757;
        double r29615772 = r29615736 ? r29615757 : r29615771;
        return r29615772;
}

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 0 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 0 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*))))))