Average Error: 32.8 → 24.6
Time: 44.3s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le 1.4828608859501 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot 2\right) \cdot n\right) \cdot U + \left(\left(2 \cdot \ell - \left(\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{Om} \cdot \left(U* - U\right)\right) \cdot \frac{\sqrt[3]{n}}{\frac{1}{\ell}}\right) \cdot \left(-2 \cdot U\right)\right) \cdot \frac{n}{\frac{Om}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)\right) \cdot n} \cdot \sqrt{U}\\ \end{array}\]
\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}\;U \le 1.4828608859501 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\left(\left(t \cdot 2\right) \cdot n\right) \cdot U + \left(\left(2 \cdot \ell - \left(\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{Om} \cdot \left(U* - U\right)\right) \cdot \frac{\sqrt[3]{n}}{\frac{1}{\ell}}\right) \cdot \left(-2 \cdot U\right)\right) \cdot \frac{n}{\frac{Om}{\ell}}}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2420569 = 2.0;
        double r2420570 = n;
        double r2420571 = r2420569 * r2420570;
        double r2420572 = U;
        double r2420573 = r2420571 * r2420572;
        double r2420574 = t;
        double r2420575 = l;
        double r2420576 = r2420575 * r2420575;
        double r2420577 = Om;
        double r2420578 = r2420576 / r2420577;
        double r2420579 = r2420569 * r2420578;
        double r2420580 = r2420574 - r2420579;
        double r2420581 = r2420575 / r2420577;
        double r2420582 = pow(r2420581, r2420569);
        double r2420583 = r2420570 * r2420582;
        double r2420584 = U_;
        double r2420585 = r2420572 - r2420584;
        double r2420586 = r2420583 * r2420585;
        double r2420587 = r2420580 - r2420586;
        double r2420588 = r2420573 * r2420587;
        double r2420589 = sqrt(r2420588);
        return r2420589;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2420590 = U;
        double r2420591 = 1.4828608859501e-310;
        bool r2420592 = r2420590 <= r2420591;
        double r2420593 = t;
        double r2420594 = 2.0;
        double r2420595 = r2420593 * r2420594;
        double r2420596 = n;
        double r2420597 = r2420595 * r2420596;
        double r2420598 = r2420597 * r2420590;
        double r2420599 = l;
        double r2420600 = r2420594 * r2420599;
        double r2420601 = cbrt(r2420596);
        double r2420602 = r2420601 * r2420601;
        double r2420603 = Om;
        double r2420604 = r2420602 / r2420603;
        double r2420605 = U_;
        double r2420606 = r2420605 - r2420590;
        double r2420607 = r2420604 * r2420606;
        double r2420608 = 1.0;
        double r2420609 = r2420608 / r2420599;
        double r2420610 = r2420601 / r2420609;
        double r2420611 = r2420607 * r2420610;
        double r2420612 = r2420600 - r2420611;
        double r2420613 = -2.0;
        double r2420614 = r2420613 * r2420590;
        double r2420615 = r2420612 * r2420614;
        double r2420616 = r2420603 / r2420599;
        double r2420617 = r2420596 / r2420616;
        double r2420618 = r2420615 * r2420617;
        double r2420619 = r2420598 + r2420618;
        double r2420620 = sqrt(r2420619);
        double r2420621 = r2420599 / r2420603;
        double r2420622 = r2420603 / r2420596;
        double r2420623 = r2420599 / r2420622;
        double r2420624 = r2420606 * r2420623;
        double r2420625 = r2420600 - r2420624;
        double r2420626 = r2420621 * r2420625;
        double r2420627 = r2420593 - r2420626;
        double r2420628 = r2420594 * r2420627;
        double r2420629 = r2420628 * r2420596;
        double r2420630 = sqrt(r2420629);
        double r2420631 = sqrt(r2420590);
        double r2420632 = r2420630 * r2420631;
        double r2420633 = r2420592 ? r2420620 : r2420632;
        return r2420633;
}

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 < 1.4828608859501e-310

    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

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

    4. Applied associate-*l*29.0

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

    6. Applied sub-neg29.0

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

    7. Applied distribute-rgt-in29.0

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

    8. Applied distribute-rgt-in29.0

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

    9. Applied distribute-lft-in29.0

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

    10. Simplified26.2

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

    12. Applied associate-*l*25.4

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

    14. Applied div-inv25.4

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

    15. Applied add-cube-cbrt25.5

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

    16. Applied times-frac25.8

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

    17. Applied associate-*r*27.4

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

    if 1.4828608859501e-310 < U

    1. Initial program 32.7

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

    2. Simplified29.9

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

    4. Applied associate-*l*29.0

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

    6. Applied sqrt-prod21.7

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

  4. Final simplification24.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 1.4828608859501 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot 2\right) \cdot n\right) \cdot U + \left(\left(2 \cdot \ell - \left(\frac{\sqrt[3]{n} \cdot \sqrt[3]{n}}{Om} \cdot \left(U* - U\right)\right) \cdot \frac{\sqrt[3]{n}}{\frac{1}{\ell}}\right) \cdot \left(-2 \cdot U\right)\right) \cdot \frac{n}{\frac{Om}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(U* - U\right) \cdot \frac{\ell}{\frac{Om}{n}}\right)\right)\right) \cdot n} \cdot \sqrt{U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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*))))))