Average Error: 34.7 → 30.5
Time: 34.4s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 5.921596680002772760806493524139888367658 \cdot 10^{-305} \lor \neg \left(t \le 1.070028174441901986804731946878056420476 \cdot 10^{-172} \lor \neg \left(t \le 2.724465956200713647886827411142398031389 \cdot 10^{192}\right)\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]
\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 5.921596680002772760806493524139888367658 \cdot 10^{-305} \lor \neg \left(t \le 1.070028174441901986804731946878056420476 \cdot 10^{-172} \lor \neg \left(t \le 2.724465956200713647886827411142398031389 \cdot 10^{192}\right)\right):\\
\;\;\;\;\sqrt{\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)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r159591 = 2.0;
        double r159592 = n;
        double r159593 = r159591 * r159592;
        double r159594 = U;
        double r159595 = r159593 * r159594;
        double r159596 = t;
        double r159597 = l;
        double r159598 = r159597 * r159597;
        double r159599 = Om;
        double r159600 = r159598 / r159599;
        double r159601 = r159591 * r159600;
        double r159602 = r159596 - r159601;
        double r159603 = r159597 / r159599;
        double r159604 = pow(r159603, r159591);
        double r159605 = r159592 * r159604;
        double r159606 = U_;
        double r159607 = r159594 - r159606;
        double r159608 = r159605 * r159607;
        double r159609 = r159602 - r159608;
        double r159610 = r159595 * r159609;
        double r159611 = sqrt(r159610);
        return r159611;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r159612 = t;
        double r159613 = 5.921596680002773e-305;
        bool r159614 = r159612 <= r159613;
        double r159615 = 1.070028174441902e-172;
        bool r159616 = r159612 <= r159615;
        double r159617 = 2.7244659562007136e+192;
        bool r159618 = r159612 <= r159617;
        double r159619 = !r159618;
        bool r159620 = r159616 || r159619;
        double r159621 = !r159620;
        bool r159622 = r159614 || r159621;
        double r159623 = 2.0;
        double r159624 = n;
        double r159625 = r159623 * r159624;
        double r159626 = U;
        double r159627 = l;
        double r159628 = Om;
        double r159629 = r159627 / r159628;
        double r159630 = r159627 * r159629;
        double r159631 = r159623 * r159630;
        double r159632 = r159612 - r159631;
        double r159633 = pow(r159629, r159623);
        double r159634 = r159624 * r159633;
        double r159635 = U_;
        double r159636 = r159626 - r159635;
        double r159637 = r159634 * r159636;
        double r159638 = r159632 - r159637;
        double r159639 = r159626 * r159638;
        double r159640 = r159625 * r159639;
        double r159641 = sqrt(r159640);
        double r159642 = r159625 * r159626;
        double r159643 = sqrt(r159642);
        double r159644 = r159628 / r159627;
        double r159645 = r159627 / r159644;
        double r159646 = r159623 * r159645;
        double r159647 = r159612 - r159646;
        double r159648 = r159647 - r159637;
        double r159649 = sqrt(r159648);
        double r159650 = r159643 * r159649;
        double r159651 = r159622 ? r159641 : r159650;
        return r159651;
}

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 < 5.921596680002773e-305 or 1.070028174441902e-172 < t < 2.7244659562007136e+192

    1. Initial program 33.9

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

    3. Applied *-un-lft-identity33.9

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

    4. Applied times-frac31.2

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

    5. Simplified31.2

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

    7. Applied associate-*l*31.1

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

    if 5.921596680002773e-305 < t < 1.070028174441902e-172 or 2.7244659562007136e+192 < 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*34.5

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

    5. Applied sqrt-prod28.0

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

  4. Final simplification30.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 5.921596680002772760806493524139888367658 \cdot 10^{-305} \lor \neg \left(t \le 1.070028174441901986804731946878056420476 \cdot 10^{-172} \lor \neg \left(t \le 2.724465956200713647886827411142398031389 \cdot 10^{192}\right)\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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*))))))