Average Error: 33.3 → 26.3
Time: 42.1s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le 1.0337592973629363 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot U}\\ \end{array}\]
\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}\;U \le 1.0337592973629363 \cdot 10^{-288}:\\
\;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r903480 = 2.0;
        double r903481 = n;
        double r903482 = r903480 * r903481;
        double r903483 = U;
        double r903484 = r903482 * r903483;
        double r903485 = t;
        double r903486 = l;
        double r903487 = r903486 * r903486;
        double r903488 = Om;
        double r903489 = r903487 / r903488;
        double r903490 = r903480 * r903489;
        double r903491 = r903485 - r903490;
        double r903492 = r903486 / r903488;
        double r903493 = pow(r903492, r903480);
        double r903494 = r903481 * r903493;
        double r903495 = U_;
        double r903496 = r903483 - r903495;
        double r903497 = r903494 * r903496;
        double r903498 = r903491 - r903497;
        double r903499 = r903484 * r903498;
        double r903500 = sqrt(r903499);
        return r903500;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r903501 = U;
        double r903502 = 1.0337592973629363e-288;
        bool r903503 = r903501 <= r903502;
        double r903504 = 2.0;
        double r903505 = r903504 * r903501;
        double r903506 = n;
        double r903507 = t;
        double r903508 = l;
        double r903509 = Om;
        double r903510 = r903508 / r903509;
        double r903511 = r903504 * r903510;
        double r903512 = r903511 * r903508;
        double r903513 = r903510 * r903506;
        double r903514 = U_;
        double r903515 = r903501 - r903514;
        double r903516 = r903510 * r903515;
        double r903517 = r903513 * r903516;
        double r903518 = r903512 + r903517;
        double r903519 = r903507 - r903518;
        double r903520 = r903506 * r903519;
        double r903521 = r903505 * r903520;
        double r903522 = sqrt(r903521);
        double r903523 = sqrt(r903520);
        double r903524 = sqrt(r903505);
        double r903525 = r903523 * r903524;
        double r903526 = r903503 ? r903522 : r903525;
        return r903526;
}

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 U < 1.0337592973629363e-288

    1. Initial program 33.5

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

    2. Simplified30.0

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

    4. Applied associate-*l*30.3

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

    if 1.0337592973629363e-288 < U

    1. Initial program 33.1

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

    2. Simplified29.2

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

    4. Applied associate-*l*29.3

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

    6. Applied sqrt-prod22.0

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 1.0337592973629363 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{\left(2 \cdot U\right) \cdot \left(n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(t - \left(\left(2 \cdot \frac{\ell}{Om}\right) \cdot \ell + \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(\frac{\ell}{Om} \cdot \left(U - U*\right)\right)\right)\right)} \cdot \sqrt{2 \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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*))))))