Average Error: 33.1 → 25.4
Time: 42.5s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -1.1626446501105 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot U}\\ \end{array}\]
\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 -1.1626446501105 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r909338 = 2.0;
        double r909339 = n;
        double r909340 = r909338 * r909339;
        double r909341 = U;
        double r909342 = r909340 * r909341;
        double r909343 = t;
        double r909344 = l;
        double r909345 = r909344 * r909344;
        double r909346 = Om;
        double r909347 = r909345 / r909346;
        double r909348 = r909338 * r909347;
        double r909349 = r909343 - r909348;
        double r909350 = r909344 / r909346;
        double r909351 = pow(r909350, r909338);
        double r909352 = r909339 * r909351;
        double r909353 = U_;
        double r909354 = r909341 - r909353;
        double r909355 = r909352 * r909354;
        double r909356 = r909349 - r909355;
        double r909357 = r909342 * r909356;
        double r909358 = sqrt(r909357);
        return r909358;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r909359 = U;
        double r909360 = -1.1626446501105e-310;
        bool r909361 = r909359 <= r909360;
        double r909362 = 2.0;
        double r909363 = r909362 * r909359;
        double r909364 = n;
        double r909365 = t;
        double r909366 = l;
        double r909367 = Om;
        double r909368 = r909366 / r909367;
        double r909369 = r909362 * r909368;
        double r909370 = r909369 * r909366;
        double r909371 = r909368 * r909364;
        double r909372 = r909368 * r909371;
        double r909373 = U_;
        double r909374 = r909359 - r909373;
        double r909375 = r909372 * r909374;
        double r909376 = r909370 + r909375;
        double r909377 = r909365 - r909376;
        double r909378 = r909364 * r909377;
        double r909379 = r909363 * r909378;
        double r909380 = sqrt(r909379);
        double r909381 = sqrt(r909378);
        double r909382 = sqrt(r909363);
        double r909383 = r909381 * r909382;
        double r909384 = r909361 ? r909380 : r909383;
        return r909384;
}

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 < -1.1626446501105e-310

    1. Initial program 33.2

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

    2. Simplified29.5

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

    4. Applied associate-*r*29.6

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

    6. Applied associate-*l*29.5

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

    if -1.1626446501105e-310 < U

    1. Initial program 33.0

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

    2. Simplified28.9

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

    4. Applied sqrt-prod21.2

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

  4. Final simplification25.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -1.1626446501105 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) \cdot \left(U - U*\right)\right)\right)} \cdot \sqrt{2 \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(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*))))))