Average Error: 33.7 → 27.6
Time: 43.9s
Precision: 64
\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.1063631272675845 \cdot 10^{-282}:\\ \;\;\;\;{\left(\left(\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.51509684742799 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot n} \cdot \sqrt{\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2}\\ \end{array}\]
\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}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.1063631272675845 \cdot 10^{-282}:\\
\;\;\;\;{\left(\left(\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)}^{\frac{1}{2}}\\

\mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.51509684742799 \cdot 10^{+287}:\\
\;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\

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

\end{array}
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2504602 = 2.0;
        double r2504603 = n;
        double r2504604 = r2504602 * r2504603;
        double r2504605 = U;
        double r2504606 = r2504604 * r2504605;
        double r2504607 = t;
        double r2504608 = l;
        double r2504609 = r2504608 * r2504608;
        double r2504610 = Om;
        double r2504611 = r2504609 / r2504610;
        double r2504612 = r2504602 * r2504611;
        double r2504613 = r2504607 - r2504612;
        double r2504614 = r2504608 / r2504610;
        double r2504615 = pow(r2504614, r2504602);
        double r2504616 = r2504603 * r2504615;
        double r2504617 = U_;
        double r2504618 = r2504605 - r2504617;
        double r2504619 = r2504616 * r2504618;
        double r2504620 = r2504613 - r2504619;
        double r2504621 = r2504606 * r2504620;
        double r2504622 = sqrt(r2504621);
        return r2504622;
}
↓
double f(double n, double U, double t, double l, double Om, double U_) {
        double r2504623 = 2.0;
        double r2504624 = n;
        double r2504625 = r2504623 * r2504624;
        double r2504626 = U;
        double r2504627 = r2504625 * r2504626;
        double r2504628 = t;
        double r2504629 = l;
        double r2504630 = r2504629 * r2504629;
        double r2504631 = Om;
        double r2504632 = r2504630 / r2504631;
        double r2504633 = r2504632 * r2504623;
        double r2504634 = r2504628 - r2504633;
        double r2504635 = r2504629 / r2504631;
        double r2504636 = pow(r2504635, r2504623);
        double r2504637 = r2504624 * r2504636;
        double r2504638 = U_;
        double r2504639 = r2504626 - r2504638;
        double r2504640 = r2504637 * r2504639;
        double r2504641 = r2504634 - r2504640;
        double r2504642 = r2504627 * r2504641;
        double r2504643 = 1.1063631272675845e-282;
        bool r2504644 = r2504642 <= r2504643;
        double r2504645 = r2504623 * r2504629;
        double r2504646 = r2504638 - r2504626;
        double r2504647 = r2504624 * r2504646;
        double r2504648 = r2504635 * r2504647;
        double r2504649 = r2504645 - r2504648;
        double r2504650 = r2504649 * r2504635;
        double r2504651 = r2504628 - r2504650;
        double r2504652 = r2504651 * r2504625;
        double r2504653 = r2504652 * r2504626;
        double r2504654 = 0.5;
        double r2504655 = pow(r2504653, r2504654);
        double r2504656 = 6.51509684742799e+287;
        bool r2504657 = r2504642 <= r2504656;
        double r2504658 = sqrt(r2504642);
        double r2504659 = r2504626 * r2504624;
        double r2504660 = sqrt(r2504659);
        double r2504661 = r2504651 * r2504623;
        double r2504662 = sqrt(r2504661);
        double r2504663 = r2504660 * r2504662;
        double r2504664 = r2504657 ? r2504658 : r2504663;
        double r2504665 = r2504644 ? r2504655 : r2504664;
        return r2504665;
}

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 (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 1.1063631272675845e-282

    1. Initial program 53.5

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

    2. Simplified48.1

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

    4. Applied pow1/248.1

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

    6. Applied associate-*l*35.9

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

    7. Simplified36.8

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

    if 1.1063631272675845e-282 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 6.51509684742799e+287

    1. Initial program 1.6

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

    if 6.51509684742799e+287 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

    1. Initial program 58.9

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

    2. Simplified51.9

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

    4. Applied pow1/251.9

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

    6. Applied unpow-prod-down51.0

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

    7. Simplified51.0

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

    8. Simplified51.4

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

  4. Final simplification27.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 1.1063631272675845 \cdot 10^{-282}:\\ \;\;\;\;{\left(\left(\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot \left(2 \cdot n\right)\right) \cdot U\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 6.51509684742799 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U \cdot n} \cdot \sqrt{\left(t - \left(2 \cdot \ell - \frac{\ell}{Om} \cdot \left(n \cdot \left(U* - U\right)\right)\right) \cdot \frac{\ell}{Om}\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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*))))))