Average Error: 32.9 → 27.9
Time: 40.8s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.141827696680125 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{U \cdot n}\\ \end{array}\]
\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 1.141827696680125 \cdot 10^{+46}:\\
\;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2041371 = 2.0;
        double r2041372 = n;
        double r2041373 = r2041371 * r2041372;
        double r2041374 = U;
        double r2041375 = r2041373 * r2041374;
        double r2041376 = t;
        double r2041377 = l;
        double r2041378 = r2041377 * r2041377;
        double r2041379 = Om;
        double r2041380 = r2041378 / r2041379;
        double r2041381 = r2041371 * r2041380;
        double r2041382 = r2041376 - r2041381;
        double r2041383 = r2041377 / r2041379;
        double r2041384 = pow(r2041383, r2041371);
        double r2041385 = r2041372 * r2041384;
        double r2041386 = U_;
        double r2041387 = r2041374 - r2041386;
        double r2041388 = r2041385 * r2041387;
        double r2041389 = r2041382 - r2041388;
        double r2041390 = r2041375 * r2041389;
        double r2041391 = sqrt(r2041390);
        return r2041391;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2041392 = t;
        double r2041393 = 1.141827696680125e+46;
        bool r2041394 = r2041392 <= r2041393;
        double r2041395 = U;
        double r2041396 = n;
        double r2041397 = 2.0;
        double r2041398 = l;
        double r2041399 = r2041397 * r2041398;
        double r2041400 = Om;
        double r2041401 = r2041400 / r2041398;
        double r2041402 = r2041396 / r2041401;
        double r2041403 = U_;
        double r2041404 = r2041403 - r2041395;
        double r2041405 = r2041402 * r2041404;
        double r2041406 = r2041399 - r2041405;
        double r2041407 = r2041398 / r2041400;
        double r2041408 = r2041406 * r2041407;
        double r2041409 = r2041392 - r2041408;
        double r2041410 = r2041397 * r2041409;
        double r2041411 = r2041396 * r2041410;
        double r2041412 = r2041395 * r2041411;
        double r2041413 = sqrt(r2041412);
        double r2041414 = sqrt(r2041410);
        double r2041415 = r2041395 * r2041396;
        double r2041416 = sqrt(r2041415);
        double r2041417 = r2041414 * r2041416;
        double r2041418 = r2041394 ? r2041413 : r2041417;
        return r2041418;
}

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 < 1.141827696680125e+46

    1. Initial program 32.5

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

    2. Simplified28.6

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

    4. Applied associate-*l*28.8

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

    if 1.141827696680125e+46 < t

    1. Initial program 34.4

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

    2. Simplified30.7

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

    4. Applied sqrt-prod25.0

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

  4. Final simplification27.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.141827696680125 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{U \cdot \left(n \cdot \left(2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(t - \left(2 \cdot \ell - \frac{n}{\frac{Om}{\ell}} \cdot \left(U* - U\right)\right) \cdot \frac{\ell}{Om}\right)} \cdot \sqrt{U \cdot n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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*))))))