Average Error: 34.4 → 31.0
Time: 2.3m
Precision: 64
\[\sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2.9726791114426635 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le -4.618776174471877 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\frac{U}{\frac{\frac{Om}{n} \cdot \frac{Om}{n}}{U* \cdot \left(\ell \cdot \ell\right)}} + n \cdot \left(t \cdot U\right)\right) - \frac{U \cdot 4.0}{\frac{\frac{Om}{\ell \cdot \ell}}{n}}}\\ \mathbf{elif}\;n \le 4.9836847106930725 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 9.66698558162677 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - \left(\frac{\ell}{Om} \cdot 2.0\right) \cdot \ell\right) - {\left(\frac{\ell}{Om}\right)}^{2.0} \cdot \left(n \cdot \left(U - U*\right)\right)\right) \cdot U\right) \cdot \left(2.0 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\ \end{array}\]
\sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}
↓
\begin{array}{l}
\mathbf{if}\;n \le -2.9726791114426635 \cdot 10^{-160}:\\
\;\;\;\;\sqrt{\left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\

\mathbf{elif}\;n \le -4.618776174471877 \cdot 10^{-273}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(\frac{U}{\frac{\frac{Om}{n} \cdot \frac{Om}{n}}{U* \cdot \left(\ell \cdot \ell\right)}} + n \cdot \left(t \cdot U\right)\right) - \frac{U \cdot 4.0}{\frac{\frac{Om}{\ell \cdot \ell}}{n}}}\\

\mathbf{elif}\;n \le 4.9836847106930725 \cdot 10^{-288}:\\
\;\;\;\;\sqrt{\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\

\mathbf{elif}\;n \le 9.66698558162677 \cdot 10^{-75}:\\
\;\;\;\;\sqrt{\left(\left(\left(t - \left(\frac{\ell}{Om} \cdot 2.0\right) \cdot \ell\right) - {\left(\frac{\ell}{Om}\right)}^{2.0} \cdot \left(n \cdot \left(U - U*\right)\right)\right) \cdot U\right) \cdot \left(2.0 \cdot n\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2786261 = 2.0;
        double r2786262 = n;
        double r2786263 = r2786261 * r2786262;
        double r2786264 = U;
        double r2786265 = r2786263 * r2786264;
        double r2786266 = t;
        double r2786267 = l;
        double r2786268 = r2786267 * r2786267;
        double r2786269 = Om;
        double r2786270 = r2786268 / r2786269;
        double r2786271 = r2786261 * r2786270;
        double r2786272 = r2786266 - r2786271;
        double r2786273 = r2786267 / r2786269;
        double r2786274 = pow(r2786273, r2786261);
        double r2786275 = r2786262 * r2786274;
        double r2786276 = U_;
        double r2786277 = r2786264 - r2786276;
        double r2786278 = r2786275 * r2786277;
        double r2786279 = r2786272 - r2786278;
        double r2786280 = r2786265 * r2786279;
        double r2786281 = sqrt(r2786280);
        return r2786281;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2786282 = n;
        double r2786283 = -2.9726791114426635e-160;
        bool r2786284 = r2786282 <= r2786283;
        double r2786285 = t;
        double r2786286 = 2.0;
        double r2786287 = l;
        double r2786288 = Om;
        double r2786289 = r2786288 / r2786287;
        double r2786290 = r2786287 / r2786289;
        double r2786291 = r2786286 * r2786290;
        double r2786292 = r2786285 - r2786291;
        double r2786293 = r2786287 / r2786288;
        double r2786294 = 2.0;
        double r2786295 = r2786286 / r2786294;
        double r2786296 = pow(r2786293, r2786295);
        double r2786297 = r2786282 * r2786296;
        double r2786298 = U;
        double r2786299 = U_;
        double r2786300 = r2786298 - r2786299;
        double r2786301 = r2786300 * r2786296;
        double r2786302 = r2786297 * r2786301;
        double r2786303 = r2786292 - r2786302;
        double r2786304 = r2786286 * r2786282;
        double r2786305 = r2786304 * r2786298;
        double r2786306 = r2786303 * r2786305;
        double r2786307 = sqrt(r2786306);
        double r2786308 = -4.618776174471877e-273;
        bool r2786309 = r2786282 <= r2786308;
        double r2786310 = r2786288 / r2786282;
        double r2786311 = r2786310 * r2786310;
        double r2786312 = r2786287 * r2786287;
        double r2786313 = r2786299 * r2786312;
        double r2786314 = r2786311 / r2786313;
        double r2786315 = r2786298 / r2786314;
        double r2786316 = r2786285 * r2786298;
        double r2786317 = r2786282 * r2786316;
        double r2786318 = r2786315 + r2786317;
        double r2786319 = r2786286 * r2786318;
        double r2786320 = 4.0;
        double r2786321 = r2786298 * r2786320;
        double r2786322 = r2786288 / r2786312;
        double r2786323 = r2786322 / r2786282;
        double r2786324 = r2786321 / r2786323;
        double r2786325 = r2786319 - r2786324;
        double r2786326 = sqrt(r2786325);
        double r2786327 = 4.9836847106930725e-288;
        bool r2786328 = r2786282 <= r2786327;
        double r2786329 = r2786292 * r2786305;
        double r2786330 = sqrt(r2786329);
        double r2786331 = 9.66698558162677e-75;
        bool r2786332 = r2786282 <= r2786331;
        double r2786333 = r2786293 * r2786286;
        double r2786334 = r2786333 * r2786287;
        double r2786335 = r2786285 - r2786334;
        double r2786336 = pow(r2786293, r2786286);
        double r2786337 = r2786282 * r2786300;
        double r2786338 = r2786336 * r2786337;
        double r2786339 = r2786335 - r2786338;
        double r2786340 = r2786339 * r2786298;
        double r2786341 = r2786340 * r2786304;
        double r2786342 = sqrt(r2786341);
        double r2786343 = r2786332 ? r2786342 : r2786307;
        double r2786344 = r2786328 ? r2786330 : r2786343;
        double r2786345 = r2786309 ? r2786326 : r2786344;
        double r2786346 = r2786284 ? r2786307 : r2786345;
        return r2786346;
}

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 4 regimes

  2. if n < -2.9726791114426635e-160 or 9.66698558162677e-75 < n

    1. Initial program 32.5

      \[\sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied associate-/l*29.9

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm

    5. Applied sqr-pow29.9

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)}\]

    6. Applied associate-*r*29.0

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)} \cdot \left(U - U*\right)\right)}\]
    7. Using strategy rm

    8. Applied associate-*l*28.1

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)} \cdot \left(U - U*\right)\right)}\right)}\]

    if -2.9726791114426635e-160 < n < -4.618776174471877e-273

    1. Initial program 38.2

      \[\sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied associate-/l*35.3

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm

    5. Applied sqr-pow35.3

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)}\]

    6. Applied associate-*r*34.3

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)} \cdot \left(U - U*\right)\right)}\]

    7. Taylor expanded around inf 42.1

      \[\leadsto \sqrt{\color{blue}{\left(2.0 \cdot \left(t \cdot \left(U \cdot n\right)\right) + 2.0 \cdot \frac{U \cdot \left({n}^{2} \cdot \left(U* \cdot {\ell}^{2}\right)\right)}{{Om}^{2}}\right) - 4.0 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\]

    8. Simplified37.9

      \[\leadsto \sqrt{\color{blue}{\left(\left(U \cdot t\right) \cdot n + \frac{U}{\frac{\frac{Om}{n} \cdot \frac{Om}{n}}{U* \cdot \left(\ell \cdot \ell\right)}}\right) \cdot 2.0 - \frac{4.0 \cdot U}{\frac{\frac{Om}{\ell \cdot \ell}}{n}}}}\]

    if -4.618776174471877e-273 < n < 4.9836847106930725e-288

    1. Initial program 39.6

      \[\sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied associate-/l*37.9

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]

    4. Taylor expanded around 0 37.5

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{0}\right)}\]

    if 4.9836847106930725e-288 < n < 9.66698558162677e-75

    1. Initial program 36.1

      \[\sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]
    2. Using strategy rm

    3. Applied associate-/l*33.7

      \[\leadsto \sqrt{\left(\left(2.0 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2.0 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)}\]
    4. Using strategy rm

    5. Applied associate-*l*33.5

      \[\leadsto \sqrt{\color{blue}{\left(2.0 \cdot n\right) \cdot \left(U \cdot \left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right) \cdot \left(U - U*\right)\right)\right)}}\]

    6. Simplified33.5

      \[\leadsto \sqrt{\left(2.0 \cdot n\right) \cdot \color{blue}{\left(U \cdot \left(\left(t - \left(2.0 \cdot \frac{\ell}{Om}\right) \cdot \ell\right) - \left(n \cdot \left(U - U*\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2.0}\right)\right)}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification31.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2.9726791114426635 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le -4.618776174471877 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\frac{U}{\frac{\frac{Om}{n} \cdot \frac{Om}{n}}{U* \cdot \left(\ell \cdot \ell\right)}} + n \cdot \left(t \cdot U\right)\right) - \frac{U \cdot 4.0}{\frac{\frac{Om}{\ell \cdot \ell}}{n}}}\\ \mathbf{elif}\;n \le 4.9836847106930725 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\ \mathbf{elif}\;n \le 9.66698558162677 \cdot 10^{-75}:\\ \;\;\;\;\sqrt{\left(\left(\left(t - \left(\frac{\ell}{Om} \cdot 2.0\right) \cdot \ell\right) - {\left(\frac{\ell}{Om}\right)}^{2.0} \cdot \left(n \cdot \left(U - U*\right)\right)\right) \cdot U\right) \cdot \left(2.0 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t - 2.0 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right) \cdot \left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2.0}{2}\right)}\right)\right) \cdot \left(\left(2.0 \cdot n\right) \cdot U\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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*))))))