Average Error: 34.9 → 29.4
Time: 35.9s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le 1.096720098972556 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]
\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}\;n \le 1.096720098972556 \cdot 10^{-309}:\\
\;\;\;\;\sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r220653 = 2.0;
        double r220654 = n;
        double r220655 = r220653 * r220654;
        double r220656 = U;
        double r220657 = r220655 * r220656;
        double r220658 = t;
        double r220659 = l;
        double r220660 = r220659 * r220659;
        double r220661 = Om;
        double r220662 = r220660 / r220661;
        double r220663 = r220653 * r220662;
        double r220664 = r220658 - r220663;
        double r220665 = r220659 / r220661;
        double r220666 = pow(r220665, r220653);
        double r220667 = r220654 * r220666;
        double r220668 = U_;
        double r220669 = r220656 - r220668;
        double r220670 = r220667 * r220669;
        double r220671 = r220664 - r220670;
        double r220672 = r220657 * r220671;
        double r220673 = sqrt(r220672);
        return r220673;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r220674 = n;
        double r220675 = 1.096720098972556e-309;
        bool r220676 = r220674 <= r220675;
        double r220677 = 2.0;
        double r220678 = r220677 * r220674;
        double r220679 = U;
        double r220680 = t;
        double r220681 = l;
        double r220682 = Om;
        double r220683 = r220681 / r220682;
        double r220684 = r220681 * r220683;
        double r220685 = r220677 * r220684;
        double r220686 = r220680 - r220685;
        double r220687 = pow(r220683, r220677);
        double r220688 = U_;
        double r220689 = r220679 - r220688;
        double r220690 = r220687 * r220689;
        double r220691 = r220674 * r220690;
        double r220692 = r220686 - r220691;
        double r220693 = r220679 * r220692;
        double r220694 = r220678 * r220693;
        double r220695 = sqrt(r220694);
        double r220696 = sqrt(r220695);
        double r220697 = r220696 * r220696;
        double r220698 = sqrt(r220678);
        double r220699 = r220674 * r220687;
        double r220700 = r220699 * r220689;
        double r220701 = r220686 - r220700;
        double r220702 = r220679 * r220701;
        double r220703 = sqrt(r220702);
        double r220704 = r220698 * r220703;
        double r220705 = r220676 ? r220697 : r220704;
        return r220705;
}

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 n < 1.096720098972556e-309

    1. Initial program 34.8

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

    3. Applied *-un-lft-identity34.8

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

    4. Applied times-frac32.0

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

    5. Simplified32.0

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

    7. Applied associate-*l*32.4

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

    9. Applied associate-*l*32.7

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

    11. Applied add-sqr-sqrt32.9

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

    if 1.096720098972556e-309 < n

    1. Initial program 35.0

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

    3. Applied *-un-lft-identity35.0

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

    4. Applied times-frac32.4

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

    5. Simplified32.4

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

    7. Applied associate-*l*32.8

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

    9. Applied sqrt-prod25.8

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

  4. Final simplification29.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le 1.096720098972556 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}} \cdot \sqrt{\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - n \cdot \left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U - U*\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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*))))))