Average Error: 33.7 → 23.6
Time: 39.0s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 9.396757433575657 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-\left(2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right) + t \cdot \left(U \cdot n\right)\right)}\\ \end{array}\]
\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}\;\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)} \le 9.396757433575657 \cdot 10^{-152}:\\
\;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\sqrt[3]{n \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)} \cdot \left(\sqrt[3]{n \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)} \cdot \sqrt[3]{n \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)}\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3044442 = 2.0;
        double r3044443 = n;
        double r3044444 = r3044442 * r3044443;
        double r3044445 = U;
        double r3044446 = r3044444 * r3044445;
        double r3044447 = t;
        double r3044448 = l;
        double r3044449 = r3044448 * r3044448;
        double r3044450 = Om;
        double r3044451 = r3044449 / r3044450;
        double r3044452 = r3044442 * r3044451;
        double r3044453 = r3044447 - r3044452;
        double r3044454 = r3044448 / r3044450;
        double r3044455 = pow(r3044454, r3044442);
        double r3044456 = r3044443 * r3044455;
        double r3044457 = U_;
        double r3044458 = r3044445 - r3044457;
        double r3044459 = r3044456 * r3044458;
        double r3044460 = r3044453 - r3044459;
        double r3044461 = r3044446 * r3044460;
        double r3044462 = sqrt(r3044461);
        return r3044462;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3044463 = 2.0;
        double r3044464 = n;
        double r3044465 = r3044463 * r3044464;
        double r3044466 = U;
        double r3044467 = r3044465 * r3044466;
        double r3044468 = t;
        double r3044469 = l;
        double r3044470 = r3044469 * r3044469;
        double r3044471 = Om;
        double r3044472 = r3044470 / r3044471;
        double r3044473 = r3044472 * r3044463;
        double r3044474 = r3044468 - r3044473;
        double r3044475 = r3044469 / r3044471;
        double r3044476 = pow(r3044475, r3044463);
        double r3044477 = r3044464 * r3044476;
        double r3044478 = U_;
        double r3044479 = r3044466 - r3044478;
        double r3044480 = r3044477 * r3044479;
        double r3044481 = r3044474 - r3044480;
        double r3044482 = r3044467 * r3044481;
        double r3044483 = sqrt(r3044482);
        double r3044484 = 9.396757433575657e-152;
        bool r3044485 = r3044483 <= r3044484;
        double r3044486 = r3044463 * r3044469;
        double r3044487 = r3044478 - r3044466;
        double r3044488 = r3044464 * r3044487;
        double r3044489 = r3044475 * r3044488;
        double r3044490 = r3044486 - r3044489;
        double r3044491 = r3044490 * r3044475;
        double r3044492 = r3044468 - r3044491;
        double r3044493 = r3044464 * r3044492;
        double r3044494 = cbrt(r3044493);
        double r3044495 = r3044494 * r3044494;
        double r3044496 = r3044494 * r3044495;
        double r3044497 = r3044466 * r3044496;
        double r3044498 = r3044463 * r3044497;
        double r3044499 = sqrt(r3044498);
        double r3044500 = r3044464 * r3044475;
        double r3044501 = r3044466 * r3044500;
        double r3044502 = r3044500 * r3044487;
        double r3044503 = r3044486 - r3044502;
        double r3044504 = -r3044503;
        double r3044505 = r3044501 * r3044504;
        double r3044506 = r3044466 * r3044464;
        double r3044507 = r3044468 * r3044506;
        double r3044508 = r3044505 + r3044507;
        double r3044509 = r3044463 * r3044508;
        double r3044510 = sqrt(r3044509);
        double r3044511 = r3044485 ? r3044499 : r3044510;
        return r3044511;
}

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 (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))) < 9.396757433575657e-152

    1. Initial program 55.0

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

    2. Simplified39.3

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

    4. Applied add-cube-cbrt39.5

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

    if 9.396757433575657e-152 < (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))))

    1. Initial program 29.5

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

    2. Simplified30.3

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

    4. Applied associate-*r*27.3

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

    5. Taylor expanded around 0 28.4

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

    6. Simplified25.9

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

    8. Applied sub-neg25.9

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

    9. Applied distribute-rgt-in25.9

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

    10. Simplified22.8

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

    12. Applied associate-*r*20.5

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

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \le 9.396757433575657 \cdot 10^{-152}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \left(\sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt[3]{n \cdot \left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right)}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(U \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-\left(2 \cdot \ell - \left(n \cdot \frac{\ell}{Om}\right) \cdot \left(U* - U\right)\right)\right) + t \cdot \left(U \cdot n\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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*))))))