Toniolo and Linder, Equation (13)

Average Error: 33.2 → 25.5

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.2929690545772815 \cdot 10^{+195}:\\ \;\;\;\;{\left(\left(U \cdot \left(n \cdot t + \frac{\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell - \frac{U - U*}{\frac{Om}{\ell \cdot n}}\right)}{Om}\right)\right) \cdot 2\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 4.7931236222139514 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2 + \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \cdot \left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)}\\ \mathbf{elif}\;t \le 9.503639693249915 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(-\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \le 8.989646100831337 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\\ \mathbf{elif}\;t \le 2.3441672195036816 \cdot 10^{+280}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(-\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7992562 = 2.0;
        double r7992563 = n;
        double r7992564 = r7992562 * r7992563;
        double r7992565 = U;
        double r7992566 = r7992564 * r7992565;
        double r7992567 = t;
        double r7992568 = l;
        double r7992569 = r7992568 * r7992568;
        double r7992570 = Om;
        double r7992571 = r7992569 / r7992570;
        double r7992572 = r7992562 * r7992571;
        double r7992573 = r7992567 - r7992572;
        double r7992574 = r7992568 / r7992570;
        double r7992575 = pow(r7992574, r7992562);
        double r7992576 = r7992563 * r7992575;
        double r7992577 = U_;
        double r7992578 = r7992565 - r7992577;
        double r7992579 = r7992576 * r7992578;
        double r7992580 = r7992573 - r7992579;
        double r7992581 = r7992566 * r7992580;
        double r7992582 = sqrt(r7992581);
        return r7992582;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r7992583 = t;
        double r7992584 = -2.2929690545772815e+195;
        bool r7992585 = r7992583 <= r7992584;
        double r7992586 = U;
        double r7992587 = n;
        double r7992588 = r7992587 * r7992583;
        double r7992589 = l;
        double r7992590 = r7992589 * r7992587;
        double r7992591 = -2.0;
        double r7992592 = r7992591 * r7992589;
        double r7992593 = U_;
        double r7992594 = r7992586 - r7992593;
        double r7992595 = Om;
        double r7992596 = r7992595 / r7992590;
        double r7992597 = r7992594 / r7992596;
        double r7992598 = r7992592 - r7992597;
        double r7992599 = r7992590 * r7992598;
        double r7992600 = r7992599 / r7992595;
        double r7992601 = r7992588 + r7992600;
        double r7992602 = r7992586 * r7992601;
        double r7992603 = 2.0;
        double r7992604 = r7992602 * r7992603;
        double r7992605 = 0.5;
        double r7992606 = pow(r7992604, r7992605);
        double r7992607 = 4.7931236222139514e-300;
        bool r7992608 = r7992583 <= r7992607;
        double r7992609 = r7992589 / r7992595;
        double r7992610 = r7992609 * r7992587;
        double r7992611 = r7992610 * r7992586;
        double r7992612 = r7992610 * r7992594;
        double r7992613 = r7992592 - r7992612;
        double r7992614 = r7992611 * r7992613;
        double r7992615 = r7992614 * r7992603;
        double r7992616 = r7992603 * r7992587;
        double r7992617 = r7992616 * r7992586;
        double r7992618 = r7992617 * r7992583;
        double r7992619 = cbrt(r7992618);
        double r7992620 = r7992619 * r7992619;
        double r7992621 = r7992619 * r7992620;
        double r7992622 = r7992615 + r7992621;
        double r7992623 = sqrt(r7992622);
        double r7992624 = 9.503639693249915e-246;
        bool r7992625 = r7992583 <= r7992624;
        double r7992626 = r7992586 * r7992587;
        double r7992627 = r7992603 * r7992626;
        double r7992628 = sqrt(r7992627);
        double r7992629 = r7992603 * r7992589;
        double r7992630 = -r7992610;
        double r7992631 = r7992630 * r7992594;
        double r7992632 = r7992629 - r7992631;
        double r7992633 = r7992609 * r7992632;
        double r7992634 = r7992583 - r7992633;
        double r7992635 = sqrt(r7992634);
        double r7992636 = r7992628 * r7992635;
        double r7992637 = 8.989646100831337e+233;
        bool r7992638 = r7992583 <= r7992637;
        double r7992639 = r7992586 * r7992583;
        double r7992640 = r7992639 * r7992616;
        double r7992641 = r7992640 + r7992615;
        double r7992642 = sqrt(r7992641);
        double r7992643 = sqrt(r7992642);
        double r7992644 = r7992643 * r7992643;
        double r7992645 = 2.3441672195036816e+280;
        bool r7992646 = r7992583 <= r7992645;
        double r7992647 = r7992646 ? r7992636 : r7992644;
        double r7992648 = r7992638 ? r7992644 : r7992647;
        double r7992649 = r7992625 ? r7992636 : r7992648;
        double r7992650 = r7992608 ? r7992623 : r7992649;
        double r7992651 = r7992585 ? r7992606 : r7992650;
        return r7992651;
}

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*

Derivation

1. Split input into 4 regimes

2. if t < -2.2929690545772815e+195

   1. Initial program 36.9

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

   3. Applied *-un-lft-identity 36.9

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

   4. Applied associate-*r* 36.9

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

   5. Simplified 33.5

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

   7. Applied sub-neg 33.5

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

   8. Applied distribute-rgt-in 33.5

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

   9. Simplified 29.7

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

   11. Applied pow1 29.7

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

   12. Applied pow1 29.7

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

   13. Applied pow-prod-down 29.7

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

   14. Applied pow1 29.7

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

   15. Applied pow-prod-down 29.7

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

   16. Simplified 29.7

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

   18. Applied pow1 29.7

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

   19. Applied sqrt-pow1 29.7

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

   20. Simplified 29.9

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

   if -2.2929690545772815e+195 < t < 4.7931236222139514e-300

   1. Initial program 32.1

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

   3. Applied *-un-lft-identity 32.1

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

   4. Applied associate-*r* 32.1

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

   5. Simplified 27.9

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

   7. Applied sub-neg 27.9

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

   8. Applied distribute-rgt-in 27.9

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

   9. Simplified 24.0

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

   11. Applied add-cube-cbrt 24.3

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

   if 4.7931236222139514e-300 < t < 9.503639693249915e-246 or 8.989646100831337e+233 < t < 2.3441672195036816e+280

   1. Initial program 36.6

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

   3. Applied *-un-lft-identity 36.6

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

   4. Applied associate-*r* 36.6

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

   5. Simplified 31.9

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

   7. Applied sqrt-prod 27.1

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

   8. Simplified 27.1

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

   if 9.503639693249915e-246 < t < 8.989646100831337e+233 or 2.3441672195036816e+280 < t

   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. Using strategy rm

   3. Applied *-un-lft-identity 32.7

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

   4. Applied associate-*r* 32.7

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

   5. Simplified 29.0

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

   7. Applied sub-neg 29.0

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

   8. Applied distribute-rgt-in 29.0

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

   9. Simplified 25.1

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

   11. Applied pow1 25.1

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

   12. Applied pow1 25.1

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

   13. Applied pow-prod-down 25.1

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

   14. Applied pow1 25.1

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

   15. Applied pow-prod-down 25.1

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

   16. Simplified 25.3

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

   18. Applied add-sqr-sqrt 25.5

       \[\leadsto \color{blue}{\sqrt{\sqrt{{\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{1} + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{{\left(\left(U \cdot t\right) \cdot \left(n \cdot 2\right)\right)}^{1} + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(\ell \cdot -2 - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.2929690545772815 \cdot 10^{+195}:\\ \;\;\;\;{\left(\left(U \cdot \left(n \cdot t + \frac{\left(\ell \cdot n\right) \cdot \left(-2 \cdot \ell - \frac{U - U*}{\frac{Om}{\ell \cdot n}}\right)}{Om}\right)\right) \cdot 2\right)}^{\frac{1}{2}}\\ \mathbf{elif}\;t \le 4.7931236222139514 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2 + \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \cdot \left(\sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t} \cdot \sqrt[3]{\left(\left(2 \cdot n\right) \cdot U\right) \cdot t}\right)}\\ \mathbf{elif}\;t \le 9.503639693249915 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(-\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{elif}\;t \le 8.989646100831337 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\\ \mathbf{elif}\;t \le 2.3441672195036816 \cdot 10^{+280}:\\ \;\;\;\;\sqrt{2 \cdot \left(U \cdot n\right)} \cdot \sqrt{t - \frac{\ell}{Om} \cdot \left(2 \cdot \ell - \left(-\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}} \cdot \sqrt{\sqrt{\left(U \cdot t\right) \cdot \left(2 \cdot n\right) + \left(\left(\left(\frac{\ell}{Om} \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \ell - \left(\frac{\ell}{Om} \cdot n\right) \cdot \left(U - U*\right)\right)\right) \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(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*))))))