Average Error: 32.8 → 28.8
Time: 47.9s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -9.804527147066395 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \ell - \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;t \le 5.990765886638682 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot 2} \cdot \sqrt{U \cdot n}\\ \mathbf{elif}\;t \le 1.1941332647520435 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \ell - \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot 2} \cdot \sqrt{U \cdot n}\\ \end{array}\]
\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 -9.804527147066395 \cdot 10^{-276}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \ell - \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\

\mathbf{elif}\;t \le 5.990765886638682 \cdot 10^{-192}:\\
\;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot 2} \cdot \sqrt{U \cdot n}\\

\mathbf{elif}\;t \le 1.1941332647520435 \cdot 10^{+46}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \ell - \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2559406 = 2.0;
        double r2559407 = n;
        double r2559408 = r2559406 * r2559407;
        double r2559409 = U;
        double r2559410 = r2559408 * r2559409;
        double r2559411 = t;
        double r2559412 = l;
        double r2559413 = r2559412 * r2559412;
        double r2559414 = Om;
        double r2559415 = r2559413 / r2559414;
        double r2559416 = r2559406 * r2559415;
        double r2559417 = r2559411 - r2559416;
        double r2559418 = r2559412 / r2559414;
        double r2559419 = pow(r2559418, r2559406);
        double r2559420 = r2559407 * r2559419;
        double r2559421 = U_;
        double r2559422 = r2559409 - r2559421;
        double r2559423 = r2559420 * r2559422;
        double r2559424 = r2559417 - r2559423;
        double r2559425 = r2559410 * r2559424;
        double r2559426 = sqrt(r2559425);
        return r2559426;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2559427 = t;
        double r2559428 = -9.804527147066395e-276;
        bool r2559429 = r2559427 <= r2559428;
        double r2559430 = 2.0;
        double r2559431 = n;
        double r2559432 = r2559430 * r2559431;
        double r2559433 = U;
        double r2559434 = l;
        double r2559435 = r2559430 * r2559434;
        double r2559436 = U_;
        double r2559437 = r2559436 - r2559433;
        double r2559438 = Om;
        double r2559439 = r2559434 / r2559438;
        double r2559440 = r2559431 * r2559439;
        double r2559441 = r2559437 * r2559440;
        double r2559442 = r2559435 - r2559441;
        double r2559443 = r2559442 * r2559439;
        double r2559444 = r2559427 - r2559443;
        double r2559445 = r2559433 * r2559444;
        double r2559446 = r2559432 * r2559445;
        double r2559447 = sqrt(r2559446);
        double r2559448 = 5.990765886638682e-192;
        bool r2559449 = r2559427 <= r2559448;
        double r2559450 = r2559431 * r2559437;
        double r2559451 = r2559439 * r2559450;
        double r2559452 = r2559435 - r2559451;
        double r2559453 = r2559439 * r2559452;
        double r2559454 = r2559427 - r2559453;
        double r2559455 = r2559454 * r2559430;
        double r2559456 = sqrt(r2559455);
        double r2559457 = r2559433 * r2559431;
        double r2559458 = sqrt(r2559457);
        double r2559459 = r2559456 * r2559458;
        double r2559460 = 1.1941332647520435e+46;
        bool r2559461 = r2559427 <= r2559460;
        double r2559462 = r2559461 ? r2559447 : r2559459;
        double r2559463 = r2559449 ? r2559459 : r2559462;
        double r2559464 = r2559429 ? r2559447 : r2559463;
        return r2559464;
}

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 < -9.804527147066395e-276 or 5.990765886638682e-192 < t < 1.1941332647520435e+46

    1. Initial program 31.6

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

    2. Simplified28.4

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

    4. Applied associate-*l*28.1

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

    5. Simplified30.1

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

    7. Applied associate-*r*27.9

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

    9. Applied associate-*r*27.8

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

    if -9.804527147066395e-276 < t < 5.990765886638682e-192 or 1.1941332647520435e+46 < t

    1. Initial program 35.3

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

    2. Simplified32.0

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

    4. Applied sqrt-prod29.5

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

    5. Simplified30.6

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

  4. Final simplification28.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.804527147066395 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \ell - \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{elif}\;t \le 5.990765886638682 \cdot 10^{-192}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot 2} \cdot \sqrt{U \cdot n}\\ \mathbf{elif}\;t \le 1.1941332647520435 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(t - \left(2 \cdot \ell - \left(U* - U\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \frac{\ell}{Om}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right)\right) \cdot 2} \cdot \sqrt{U \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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*))))))