Average Error: 32.9 → 28.0
Time: 34.7s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 4.583682591988209 \cdot 10^{-251}:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 6.552094729632033 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\ \mathbf{elif}\;t \le 2.539192710887033 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \frac{\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\ \end{array}\]
\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 4.583682591988209 \cdot 10^{-251}:\\
\;\;\;\;{\left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}^{\frac{1}{2}}\\

\mathbf{elif}\;t \le 6.552094729632033 \cdot 10^{-157}:\\
\;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\

\mathbf{elif}\;t \le 2.539192710887033 \cdot 10^{+96}:\\
\;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \frac{\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot n\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1404282 = 2.0;
        double r1404283 = n;
        double r1404284 = r1404282 * r1404283;
        double r1404285 = U;
        double r1404286 = r1404284 * r1404285;
        double r1404287 = t;
        double r1404288 = l;
        double r1404289 = r1404288 * r1404288;
        double r1404290 = Om;
        double r1404291 = r1404289 / r1404290;
        double r1404292 = r1404282 * r1404291;
        double r1404293 = r1404287 - r1404292;
        double r1404294 = r1404288 / r1404290;
        double r1404295 = pow(r1404294, r1404282);
        double r1404296 = r1404283 * r1404295;
        double r1404297 = U_;
        double r1404298 = r1404285 - r1404297;
        double r1404299 = r1404296 * r1404298;
        double r1404300 = r1404293 - r1404299;
        double r1404301 = r1404286 * r1404300;
        double r1404302 = sqrt(r1404301);
        return r1404302;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r1404303 = t;
        double r1404304 = 4.583682591988209e-251;
        bool r1404305 = r1404303 <= r1404304;
        double r1404306 = l;
        double r1404307 = 2.0;
        double r1404308 = Om;
        double r1404309 = r1404306 / r1404308;
        double r1404310 = r1404307 * r1404309;
        double r1404311 = r1404306 * r1404310;
        double r1404312 = U;
        double r1404313 = U_;
        double r1404314 = r1404312 - r1404313;
        double r1404315 = r1404309 * r1404314;
        double r1404316 = n;
        double r1404317 = r1404309 * r1404316;
        double r1404318 = r1404315 * r1404317;
        double r1404319 = r1404311 + r1404318;
        double r1404320 = r1404303 - r1404319;
        double r1404321 = r1404312 * r1404307;
        double r1404322 = r1404321 * r1404316;
        double r1404323 = r1404320 * r1404322;
        double r1404324 = 0.5;
        double r1404325 = pow(r1404323, r1404324);
        double r1404326 = 6.552094729632033e-157;
        bool r1404327 = r1404303 <= r1404326;
        double r1404328 = sqrt(r1404322);
        double r1404329 = r1404303 - r1404318;
        double r1404330 = r1404307 * r1404306;
        double r1404331 = r1404309 * r1404330;
        double r1404332 = r1404329 - r1404331;
        double r1404333 = sqrt(r1404332);
        double r1404334 = r1404328 * r1404333;
        double r1404335 = 2.539192710887033e+96;
        bool r1404336 = r1404303 <= r1404335;
        double r1404337 = r1404317 * r1404306;
        double r1404338 = r1404337 * r1404314;
        double r1404339 = r1404338 / r1404308;
        double r1404340 = r1404311 + r1404339;
        double r1404341 = r1404303 - r1404340;
        double r1404342 = r1404341 * r1404316;
        double r1404343 = r1404321 * r1404342;
        double r1404344 = sqrt(r1404343);
        double r1404345 = r1404336 ? r1404344 : r1404334;
        double r1404346 = r1404327 ? r1404334 : r1404345;
        double r1404347 = r1404305 ? r1404325 : r1404346;
        return r1404347;
}

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 t < 4.583682591988209e-251

    1. Initial program 32.8

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

    2. 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 pow1/229.5

      \[\leadsto \color{blue}{{\left(\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right) + \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)\right)\right)\right)}^{\frac{1}{2}}}\]
    5. Using strategy rm

    6. Applied associate-*r*29.3

      \[\leadsto {\color{blue}{\left(\left(\left(U \cdot 2\right) \cdot n\right) \cdot \left(t - \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right) + \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)\right)\right)}}^{\frac{1}{2}}\]
    7. Using strategy rm

    8. Applied associate-*l*29.1

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

    if 4.583682591988209e-251 < t < 6.552094729632033e-157 or 2.539192710887033e+96 < t

    1. Initial program 35.6

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

    2. Simplified30.8

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

    4. Applied pow1/230.8

      \[\leadsto \color{blue}{{\left(\left(U \cdot 2\right) \cdot \left(n \cdot \left(t - \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right) + \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)\right)\right)\right)}^{\frac{1}{2}}}\]
    5. Using strategy rm

    6. Applied associate-*r*32.4

      \[\leadsto {\color{blue}{\left(\left(\left(U \cdot 2\right) \cdot n\right) \cdot \left(t - \left(\left(\left(n \cdot \frac{\ell}{Om}\right) \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right) + \left(\frac{\ell}{Om} \cdot 2\right) \cdot \ell\right)\right)\right)}}^{\frac{1}{2}}\]
    7. Using strategy rm

    8. Applied unpow-prod-down26.3

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

    9. Simplified26.3

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

    10. Simplified26.6

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

    if 6.552094729632033e-157 < t < 2.539192710887033e+96

    1. Initial program 30.2

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

    2. Simplified26.4

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

    4. Applied associate-*r/26.7

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

    5. Applied associate-*l/26.8

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

  4. Final simplification28.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 4.583682591988209 \cdot 10^{-251}:\\ \;\;\;\;{\left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right)\right) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 6.552094729632033 \cdot 10^{-157}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\ \mathbf{elif}\;t \le 2.539192710887033 \cdot 10^{+96}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot \left(\left(t - \left(\ell \cdot \left(2 \cdot \frac{\ell}{Om}\right) + \frac{\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot \ell\right) \cdot \left(U - U*\right)}{Om}\right)\right) \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \sqrt{\left(t - \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right) \cdot \left(\frac{\ell}{Om} \cdot n\right)\right) - \frac{\ell}{Om} \cdot \left(2 \cdot \ell\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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*))))))