Average Error: 33.6 → 27.6
Time: 54.4s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1.5658292974647863 \cdot 10^{-05}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \mathbf{elif}\;n \le 7.058910794888959 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \frac{\ell}{\frac{Om}{n}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \end{array}\]
\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}\;n \le -1.5658292974647863 \cdot 10^{-05}:\\
\;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\

\mathbf{elif}\;n \le 7.058910794888959 \cdot 10^{-223}:\\
\;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \frac{\ell}{\frac{Om}{n}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1895303 = 2.0;
        double r1895304 = n;
        double r1895305 = r1895303 * r1895304;
        double r1895306 = U;
        double r1895307 = r1895305 * r1895306;
        double r1895308 = t;
        double r1895309 = l;
        double r1895310 = r1895309 * r1895309;
        double r1895311 = Om;
        double r1895312 = r1895310 / r1895311;
        double r1895313 = r1895303 * r1895312;
        double r1895314 = r1895308 - r1895313;
        double r1895315 = r1895309 / r1895311;
        double r1895316 = pow(r1895315, r1895303);
        double r1895317 = r1895304 * r1895316;
        double r1895318 = U_;
        double r1895319 = r1895306 - r1895318;
        double r1895320 = r1895317 * r1895319;
        double r1895321 = r1895314 - r1895320;
        double r1895322 = r1895307 * r1895321;
        double r1895323 = sqrt(r1895322);
        return r1895323;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1895324 = n;
        double r1895325 = -1.5658292974647863e-05;
        bool r1895326 = r1895324 <= r1895325;
        double r1895327 = U;
        double r1895328 = r1895327 * r1895324;
        double r1895329 = t;
        double r1895330 = l;
        double r1895331 = 2.0;
        double r1895332 = r1895330 * r1895331;
        double r1895333 = U_;
        double r1895334 = r1895333 - r1895327;
        double r1895335 = cbrt(r1895330);
        double r1895336 = Om;
        double r1895337 = cbrt(r1895336);
        double r1895338 = r1895335 / r1895337;
        double r1895339 = r1895338 * r1895338;
        double r1895340 = r1895337 / r1895324;
        double r1895341 = r1895335 / r1895340;
        double r1895342 = r1895339 * r1895341;
        double r1895343 = r1895334 * r1895342;
        double r1895344 = r1895332 - r1895343;
        double r1895345 = r1895330 / r1895336;
        double r1895346 = r1895344 * r1895345;
        double r1895347 = r1895329 - r1895346;
        double r1895348 = r1895347 * r1895331;
        double r1895349 = r1895328 * r1895348;
        double r1895350 = sqrt(r1895349);
        double r1895351 = 7.058910794888959e-223;
        bool r1895352 = r1895324 <= r1895351;
        double r1895353 = r1895336 / r1895324;
        double r1895354 = r1895330 / r1895353;
        double r1895355 = r1895354 * r1895334;
        double r1895356 = r1895332 - r1895355;
        double r1895357 = r1895356 * r1895345;
        double r1895358 = r1895329 - r1895357;
        double r1895359 = r1895358 * r1895331;
        double r1895360 = r1895359 * r1895324;
        double r1895361 = r1895327 * r1895360;
        double r1895362 = sqrt(r1895361);
        double r1895363 = r1895352 ? r1895362 : r1895350;
        double r1895364 = r1895326 ? r1895350 : r1895363;
        return r1895364;
}

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 n < -1.5658292974647863e-05 or 7.058910794888959e-223 < n

    1. Initial program 32.0

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

    2. Simplified29.1

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

    4. Applied *-un-lft-identity29.1

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

    5. Applied add-cube-cbrt29.2

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{\color{blue}{\left(\sqrt[3]{Om} \cdot \sqrt[3]{Om}\right) \cdot \sqrt[3]{Om}}}{1 \cdot n}}\right)\right)\right)}\]

    6. Applied times-frac29.2

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\color{blue}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1} \cdot \frac{\sqrt[3]{Om}}{n}}}\right)\right)\right)}\]

    7. Applied add-cube-cbrt29.2

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1} \cdot \frac{\sqrt[3]{Om}}{n}}\right)\right)\right)}\]

    8. Applied times-frac28.3

      \[\leadsto \sqrt{\left(U \cdot n\right) \cdot \left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{1}} \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)}\right)\right)\right)}\]

    9. Simplified28.3

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

    if -1.5658292974647863e-05 < n < 7.058910794888959e-223

    1. Initial program 36.0

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

    2. Simplified31.7

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

    4. Applied associate-*l*26.6

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

  4. Final simplification27.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.5658292974647863 \cdot 10^{-05}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \mathbf{elif}\;n \le 7.058910794888959 \cdot 10^{-223}:\\ \;\;\;\;\sqrt{U \cdot \left(\left(\left(t - \left(\ell \cdot 2 - \frac{\ell}{\frac{Om}{n}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot n\right) \cdot \left(\left(t - \left(\ell \cdot 2 - \left(U* - U\right) \cdot \left(\left(\frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{Om}}\right) \cdot \frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{Om}}{n}}\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(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*))))))