Average Error: 34.6 → 27.4
Time: 2.2m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -1.084591956382188150200519977575983041249 \cdot 10^{140}:\\ \;\;\;\;\sqrt{\left(\left(t + \frac{\ell}{Om} \cdot \left(-\ell \cdot 2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;U \le 4.990611334173369244924830565155599718578 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right)}\\ \end{array}\]
\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 -1.084591956382188150200519977575983041249 \cdot 10^{140}:\\
\;\;\;\;\sqrt{\left(\left(t + \frac{\ell}{Om} \cdot \left(-\ell \cdot 2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\

\mathbf{elif}\;U \le 4.990611334173369244924830565155599718578 \cdot 10^{-293}:\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right) \cdot U\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3458428 = 2.0;
        double r3458429 = n;
        double r3458430 = r3458428 * r3458429;
        double r3458431 = U;
        double r3458432 = r3458430 * r3458431;
        double r3458433 = t;
        double r3458434 = l;
        double r3458435 = r3458434 * r3458434;
        double r3458436 = Om;
        double r3458437 = r3458435 / r3458436;
        double r3458438 = r3458428 * r3458437;
        double r3458439 = r3458433 - r3458438;
        double r3458440 = r3458434 / r3458436;
        double r3458441 = pow(r3458440, r3458428);
        double r3458442 = r3458429 * r3458441;
        double r3458443 = U_;
        double r3458444 = r3458431 - r3458443;
        double r3458445 = r3458442 * r3458444;
        double r3458446 = r3458439 - r3458445;
        double r3458447 = r3458432 * r3458446;
        double r3458448 = sqrt(r3458447);
        return r3458448;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r3458449 = U;
        double r3458450 = -1.0845919563821882e+140;
        bool r3458451 = r3458449 <= r3458450;
        double r3458452 = t;
        double r3458453 = l;
        double r3458454 = Om;
        double r3458455 = r3458453 / r3458454;
        double r3458456 = 2.0;
        double r3458457 = r3458453 * r3458456;
        double r3458458 = -r3458457;
        double r3458459 = r3458455 * r3458458;
        double r3458460 = r3458452 + r3458459;
        double r3458461 = n;
        double r3458462 = r3458456 * r3458461;
        double r3458463 = r3458460 * r3458462;
        double r3458464 = r3458463 * r3458449;
        double r3458465 = sqrt(r3458464);
        double r3458466 = 4.990611334173369e-293;
        bool r3458467 = r3458449 <= r3458466;
        double r3458468 = 2.0;
        double r3458469 = r3458456 / r3458468;
        double r3458470 = pow(r3458455, r3458469);
        double r3458471 = r3458461 * r3458470;
        double r3458472 = r3458471 * r3458470;
        double r3458473 = U_;
        double r3458474 = r3458473 - r3458449;
        double r3458475 = r3458472 * r3458474;
        double r3458476 = r3458455 * r3458457;
        double r3458477 = r3458475 - r3458476;
        double r3458478 = r3458477 + r3458452;
        double r3458479 = r3458478 * r3458449;
        double r3458480 = r3458462 * r3458479;
        double r3458481 = sqrt(r3458480);
        double r3458482 = sqrt(r3458449);
        double r3458483 = r3458462 * r3458478;
        double r3458484 = sqrt(r3458483);
        double r3458485 = r3458482 * r3458484;
        double r3458486 = r3458467 ? r3458481 : r3458485;
        double r3458487 = r3458451 ? r3458465 : r3458486;
        return r3458487;
}

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 3 regimes

  2. if U < -1.0845919563821882e+140

    1. Initial program 31.8

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

    2. Simplified32.8

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

    3. Taylor expanded around 0 33.7

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot \left(t + \left(\color{blue}{0} - \left(2 \cdot \ell\right) \cdot \frac{\ell}{Om}\right)\right)\right) \cdot U}\]

    if -1.0845919563821882e+140 < U < 4.990611334173369e-293

    1. Initial program 35.1

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

    2. Simplified32.1

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

    4. Applied sqr-pow32.1

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

    5. Applied associate-*r*30.9

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

    7. Applied associate-*l*29.4

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

    if 4.990611334173369e-293 < U

    1. Initial program 34.7

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

    2. Simplified32.3

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

    4. Applied sqr-pow32.3

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

    5. Applied associate-*r*31.2

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

    7. Applied sqrt-prod24.6

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

  4. Final simplification27.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -1.084591956382188150200519977575983041249 \cdot 10^{140}:\\ \;\;\;\;\sqrt{\left(\left(t + \frac{\ell}{Om} \cdot \left(-\ell \cdot 2\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;U \le 4.990611334173369244924830565155599718578 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{\left(2 \cdot n\right) \cdot \left(\left(\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U* - U\right) - \frac{\ell}{Om} \cdot \left(\ell \cdot 2\right)\right) + t\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))