Average Error: 33.0 → 26.5
Time: 2.1m
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.3439225546940806 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(\left(\left(t - \frac{1}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \left(\left(U - U*\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{\ell}}}\right)\right) + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right) \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.0215309585777349 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(t - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right) \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]
\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}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.3439225546940806 \cdot 10^{-297}:\\
\;\;\;\;\sqrt{\left(2 \cdot \left(\left(\left(t - \frac{1}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \left(\left(U - U*\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{\ell}}}\right)\right) + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right) \cdot n\right)\right) \cdot U}\\

\mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.0215309585777349 \cdot 10^{+298}:\\
\;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r4961287 = 2.0;
        double r4961288 = n;
        double r4961289 = r4961287 * r4961288;
        double r4961290 = U;
        double r4961291 = r4961289 * r4961290;
        double r4961292 = t;
        double r4961293 = l;
        double r4961294 = r4961293 * r4961293;
        double r4961295 = Om;
        double r4961296 = r4961294 / r4961295;
        double r4961297 = r4961287 * r4961296;
        double r4961298 = r4961292 - r4961297;
        double r4961299 = r4961293 / r4961295;
        double r4961300 = pow(r4961299, r4961287);
        double r4961301 = r4961288 * r4961300;
        double r4961302 = U_;
        double r4961303 = r4961290 - r4961302;
        double r4961304 = r4961301 * r4961303;
        double r4961305 = r4961298 - r4961304;
        double r4961306 = r4961291 * r4961305;
        double r4961307 = sqrt(r4961306);
        return r4961307;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r4961308 = 2.0;
        double r4961309 = n;
        double r4961310 = r4961308 * r4961309;
        double r4961311 = U;
        double r4961312 = r4961310 * r4961311;
        double r4961313 = t;
        double r4961314 = l;
        double r4961315 = r4961314 * r4961314;
        double r4961316 = Om;
        double r4961317 = r4961315 / r4961316;
        double r4961318 = r4961317 * r4961308;
        double r4961319 = r4961313 - r4961318;
        double r4961320 = r4961314 / r4961316;
        double r4961321 = pow(r4961320, r4961308);
        double r4961322 = r4961309 * r4961321;
        double r4961323 = U_;
        double r4961324 = r4961311 - r4961323;
        double r4961325 = r4961322 * r4961324;
        double r4961326 = r4961319 - r4961325;
        double r4961327 = r4961312 * r4961326;
        double r4961328 = 1.3439225546940806e-297;
        bool r4961329 = r4961327 <= r4961328;
        double r4961330 = 1.0;
        double r4961331 = cbrt(r4961316);
        double r4961332 = r4961331 * r4961331;
        double r4961333 = cbrt(r4961314);
        double r4961334 = r4961333 * r4961333;
        double r4961335 = r4961332 / r4961334;
        double r4961336 = r4961330 / r4961335;
        double r4961337 = r4961316 / r4961314;
        double r4961338 = r4961309 / r4961337;
        double r4961339 = r4961331 / r4961333;
        double r4961340 = r4961338 / r4961339;
        double r4961341 = r4961324 * r4961340;
        double r4961342 = r4961336 * r4961341;
        double r4961343 = r4961313 - r4961342;
        double r4961344 = -2.0;
        double r4961345 = r4961337 / r4961314;
        double r4961346 = r4961344 / r4961345;
        double r4961347 = r4961343 + r4961346;
        double r4961348 = r4961347 * r4961309;
        double r4961349 = r4961308 * r4961348;
        double r4961350 = r4961349 * r4961311;
        double r4961351 = sqrt(r4961350);
        double r4961352 = 1.0215309585777349e+298;
        bool r4961353 = r4961327 <= r4961352;
        double r4961354 = sqrt(r4961327);
        double r4961355 = r4961338 / r4961337;
        double r4961356 = r4961355 * r4961324;
        double r4961357 = r4961313 - r4961356;
        double r4961358 = r4961357 + r4961346;
        double r4961359 = r4961358 * r4961309;
        double r4961360 = r4961308 * r4961359;
        double r4961361 = sqrt(r4961360);
        double r4961362 = sqrt(r4961311);
        double r4961363 = r4961361 * r4961362;
        double r4961364 = r4961353 ? r4961354 : r4961363;
        double r4961365 = r4961329 ? r4961351 : r4961364;
        return r4961365;
}

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 (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 1.3439225546940806e-297

    1. Initial program 54.5

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

    2. Simplified38.0

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

    4. Applied associate-/r*36.7

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

    6. Applied add-cube-cbrt36.7

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

    7. Applied add-cube-cbrt36.7

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

    8. Applied times-frac36.7

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

    9. Applied *-un-lft-identity36.7

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

    10. Applied times-frac36.7

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

    11. Applied associate-*l*35.9

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

    if 1.3439225546940806e-297 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 1.0215309585777349e+298

    1. Initial program 1.4

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

    if 1.0215309585777349e+298 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

    1. Initial program 59.8

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

    2. Simplified57.4

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

    4. Applied associate-/r*51.1

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

    6. Applied sqrt-prod51.3

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

  4. Final simplification26.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.3439225546940806 \cdot 10^{-297}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(\left(\left(t - \frac{1}{\frac{\sqrt[3]{Om} \cdot \sqrt[3]{Om}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \left(\left(U - U*\right) \cdot \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{\sqrt[3]{Om}}{\sqrt[3]{\ell}}}\right)\right) + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right) \cdot n\right)\right) \cdot U}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.0215309585777349 \cdot 10^{+298}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\left(t - \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} \cdot \left(U - U*\right)\right) + \frac{-2}{\frac{\frac{Om}{\ell}}{\ell}}\right) \cdot n\right)} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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*))))))