Average Error: 34.7 → 30.5
Time: 33.5s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 5.921596680002772760806493524139888367658 \cdot 10^{-305} \lor \neg \left(t \le 1.070028174441901986804731946878056420476 \cdot 10^{-172} \lor \neg \left(t \le 2.724465956200713647886827411142398031389 \cdot 10^{192}\right)\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]
\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}\;t \le 5.921596680002772760806493524139888367658 \cdot 10^{-305} \lor \neg \left(t \le 1.070028174441901986804731946878056420476 \cdot 10^{-172} \lor \neg \left(t \le 2.724465956200713647886827411142398031389 \cdot 10^{192}\right)\right):\\
\;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r199312 = 2.0;
        double r199313 = n;
        double r199314 = r199312 * r199313;
        double r199315 = U;
        double r199316 = r199314 * r199315;
        double r199317 = t;
        double r199318 = l;
        double r199319 = r199318 * r199318;
        double r199320 = Om;
        double r199321 = r199319 / r199320;
        double r199322 = r199312 * r199321;
        double r199323 = r199317 - r199322;
        double r199324 = r199318 / r199320;
        double r199325 = pow(r199324, r199312);
        double r199326 = r199313 * r199325;
        double r199327 = U_;
        double r199328 = r199315 - r199327;
        double r199329 = r199326 * r199328;
        double r199330 = r199323 - r199329;
        double r199331 = r199316 * r199330;
        double r199332 = sqrt(r199331);
        return r199332;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r199333 = t;
        double r199334 = 5.921596680002773e-305;
        bool r199335 = r199333 <= r199334;
        double r199336 = 1.070028174441902e-172;
        bool r199337 = r199333 <= r199336;
        double r199338 = 2.7244659562007136e+192;
        bool r199339 = r199333 <= r199338;
        double r199340 = !r199339;
        bool r199341 = r199337 || r199340;
        double r199342 = !r199341;
        bool r199343 = r199335 || r199342;
        double r199344 = 2.0;
        double r199345 = n;
        double r199346 = r199344 * r199345;
        double r199347 = U;
        double r199348 = l;
        double r199349 = Om;
        double r199350 = r199348 / r199349;
        double r199351 = r199348 * r199350;
        double r199352 = r199344 * r199351;
        double r199353 = r199333 - r199352;
        double r199354 = pow(r199350, r199344);
        double r199355 = r199345 * r199354;
        double r199356 = U_;
        double r199357 = r199347 - r199356;
        double r199358 = r199355 * r199357;
        double r199359 = r199353 - r199358;
        double r199360 = r199347 * r199359;
        double r199361 = r199346 * r199360;
        double r199362 = sqrt(r199361);
        double r199363 = r199346 * r199347;
        double r199364 = sqrt(r199363);
        double r199365 = r199349 / r199348;
        double r199366 = r199348 / r199365;
        double r199367 = r199344 * r199366;
        double r199368 = r199333 - r199367;
        double r199369 = r199368 - r199358;
        double r199370 = sqrt(r199369);
        double r199371 = r199364 * r199370;
        double r199372 = r199343 ? r199362 : r199371;
        return r199372;
}

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 t < 5.921596680002773e-305 or 1.070028174441902e-172 < t < 2.7244659562007136e+192

    1. Initial program 33.9

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

    3. Applied *-un-lft-identity33.9

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

    4. Applied times-frac31.2

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

    5. Simplified31.2

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

    7. Applied associate-*l*31.1

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

    if 5.921596680002773e-305 < t < 1.070028174441902e-172 or 2.7244659562007136e+192 < t

    1. Initial program 37.9

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

    3. Applied associate-/l*34.5

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

    5. Applied sqrt-prod28.0

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

  4. Final simplification30.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 5.921596680002772760806493524139888367658 \cdot 10^{-305} \lor \neg \left(t \le 1.070028174441901986804731946878056420476 \cdot 10^{-172} \lor \neg \left(t \le 2.724465956200713647886827411142398031389 \cdot 10^{192}\right)\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2 \cdot \left(\ell \cdot \frac{\ell}{Om}\right)\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))