Average Error: 32.9 → 29.9
Time: 1.8m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U* \le 2.53242417465238 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) \cdot n\right) + \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(-U\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) \cdot n\right) + \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(-U\right)\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) - \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)}\\ \end{array}\]
\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}\;U* \le 2.53242417465238 \cdot 10^{-11}:\\
\;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) \cdot n\right) + \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(-U\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) \cdot n\right) + \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(-U\right)\right) \cdot 2}}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r19722445 = 2.0;
        double r19722446 = n;
        double r19722447 = r19722445 * r19722446;
        double r19722448 = U;
        double r19722449 = r19722447 * r19722448;
        double r19722450 = t;
        double r19722451 = l;
        double r19722452 = r19722451 * r19722451;
        double r19722453 = Om;
        double r19722454 = r19722452 / r19722453;
        double r19722455 = r19722445 * r19722454;
        double r19722456 = r19722450 - r19722455;
        double r19722457 = r19722451 / r19722453;
        double r19722458 = pow(r19722457, r19722445);
        double r19722459 = r19722446 * r19722458;
        double r19722460 = U_;
        double r19722461 = r19722448 - r19722460;
        double r19722462 = r19722459 * r19722461;
        double r19722463 = r19722456 - r19722462;
        double r19722464 = r19722449 * r19722463;
        double r19722465 = sqrt(r19722464);
        return r19722465;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r19722466 = U_;
        double r19722467 = 2.53242417465238e-11;
        bool r19722468 = r19722466 <= r19722467;
        double r19722469 = U;
        double r19722470 = t;
        double r19722471 = l;
        double r19722472 = Om;
        double r19722473 = r19722471 / r19722472;
        double r19722474 = r19722473 * r19722471;
        double r19722475 = 2.0;
        double r19722476 = r19722474 * r19722475;
        double r19722477 = r19722470 - r19722476;
        double r19722478 = n;
        double r19722479 = r19722477 * r19722478;
        double r19722480 = r19722469 * r19722479;
        double r19722481 = r19722478 * r19722473;
        double r19722482 = r19722481 * r19722481;
        double r19722483 = r19722469 - r19722466;
        double r19722484 = r19722482 * r19722483;
        double r19722485 = -r19722469;
        double r19722486 = r19722484 * r19722485;
        double r19722487 = r19722480 + r19722486;
        double r19722488 = r19722487 * r19722475;
        double r19722489 = sqrt(r19722488);
        double r19722490 = sqrt(r19722489);
        double r19722491 = r19722490 * r19722490;
        double r19722492 = sqrt(r19722475);
        double r19722493 = r19722473 * r19722473;
        double r19722494 = r19722493 * r19722478;
        double r19722495 = r19722494 * r19722483;
        double r19722496 = r19722477 - r19722495;
        double r19722497 = r19722478 * r19722469;
        double r19722498 = r19722496 * r19722497;
        double r19722499 = sqrt(r19722498);
        double r19722500 = r19722492 * r19722499;
        double r19722501 = r19722468 ? r19722491 : r19722500;
        return r19722501;
}

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 U* < 2.53242417465238e-11

    1. Initial program 32.4

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

    2. Simplified32.4

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

    4. Applied *-un-lft-identity32.4

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

    5. Applied times-frac29.5

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

    6. Simplified29.5

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

    8. Applied associate-*l*30.0

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

    10. Applied sub-neg30.0

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

    11. Applied distribute-rgt-in30.0

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

    12. Applied distribute-rgt-in30.0

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

    13. Simplified28.9

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

    15. Applied add-sqr-sqrt29.1

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

    if 2.53242417465238e-11 < U*

    1. Initial program 34.2

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

    2. Simplified34.2

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

    4. Applied *-un-lft-identity34.2

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

    5. Applied times-frac32.1

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

    6. Simplified32.1

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

    8. Applied sqrt-prod32.2

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

  4. Final simplification29.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;U* \le 2.53242417465238 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) \cdot n\right) + \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(-U\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot \left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) \cdot n\right) + \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(-U\right)\right) \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \sqrt{\left(\left(t - \left(\frac{\ell}{Om} \cdot \ell\right) \cdot 2\right) - \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot n\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(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*))))))