Average Error: 32.9 → 29.9
Time: 2.8m
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 r20243326 = 2.0;
        double r20243327 = n;
        double r20243328 = r20243326 * r20243327;
        double r20243329 = U;
        double r20243330 = r20243328 * r20243329;
        double r20243331 = t;
        double r20243332 = l;
        double r20243333 = r20243332 * r20243332;
        double r20243334 = Om;
        double r20243335 = r20243333 / r20243334;
        double r20243336 = r20243326 * r20243335;
        double r20243337 = r20243331 - r20243336;
        double r20243338 = r20243332 / r20243334;
        double r20243339 = pow(r20243338, r20243326);
        double r20243340 = r20243327 * r20243339;
        double r20243341 = U_;
        double r20243342 = r20243329 - r20243341;
        double r20243343 = r20243340 * r20243342;
        double r20243344 = r20243337 - r20243343;
        double r20243345 = r20243330 * r20243344;
        double r20243346 = sqrt(r20243345);
        return r20243346;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r20243347 = U_;
        double r20243348 = 2.53242417465238e-11;
        bool r20243349 = r20243347 <= r20243348;
        double r20243350 = U;
        double r20243351 = t;
        double r20243352 = l;
        double r20243353 = Om;
        double r20243354 = r20243352 / r20243353;
        double r20243355 = r20243354 * r20243352;
        double r20243356 = 2.0;
        double r20243357 = r20243355 * r20243356;
        double r20243358 = r20243351 - r20243357;
        double r20243359 = n;
        double r20243360 = r20243358 * r20243359;
        double r20243361 = r20243350 * r20243360;
        double r20243362 = r20243359 * r20243354;
        double r20243363 = r20243362 * r20243362;
        double r20243364 = r20243350 - r20243347;
        double r20243365 = r20243363 * r20243364;
        double r20243366 = -r20243350;
        double r20243367 = r20243365 * r20243366;
        double r20243368 = r20243361 + r20243367;
        double r20243369 = r20243368 * r20243356;
        double r20243370 = sqrt(r20243369);
        double r20243371 = sqrt(r20243370);
        double r20243372 = r20243371 * r20243371;
        double r20243373 = sqrt(r20243356);
        double r20243374 = r20243354 * r20243354;
        double r20243375 = r20243374 * r20243359;
        double r20243376 = r20243375 * r20243364;
        double r20243377 = r20243358 - r20243376;
        double r20243378 = r20243359 * r20243350;
        double r20243379 = r20243377 * r20243378;
        double r20243380 = sqrt(r20243379);
        double r20243381 = r20243373 * r20243380;
        double r20243382 = r20243349 ? r20243372 : r20243381;
        return r20243382;
}

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*))))))