Toniolo and Linder, Equation (13)

Average Error: 34.6 → 28.7

Time: 49.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -9.422947167583051961821239451488524021954 \cdot 10^{-260}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\left(-2\right) \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \cdot \sqrt[3]{U \cdot \left(2 \cdot n\right)}\right)}^{\frac{3}{2}}\\ \mathbf{elif}\;n \le 2.598149486260100212883263141948192542387 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t \cdot U, 2 \cdot n, \frac{\left(-\ell \cdot \ell\right) \cdot n}{\frac{Om}{U}} \cdot 4\right)}\\ \mathbf{elif}\;n \le 3.33490110259219064400249536435033242418 \cdot 10^{-187}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\left(-2\right) \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \cdot \sqrt[3]{U \cdot \left(2 \cdot n\right)}\right)}^{\frac{3}{2}}\\ \mathbf{elif}\;n \le 1.407519843109883897735864030283250356428 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t \cdot U, 2 \cdot n, \frac{\left(-\ell \cdot \ell\right) \cdot n}{\frac{Om}{U}} \cdot 4\right)}\\ \mathbf{elif}\;n \le 7.962768658598781395526822537989097838818 \cdot 10^{-7}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\left(-2\right) \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \cdot \sqrt[3]{U \cdot \left(2 \cdot n\right)}\right)}^{\frac{3}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r161374 = 2.0;
        double r161375 = n;
        double r161376 = r161374 * r161375;
        double r161377 = U;
        double r161378 = r161376 * r161377;
        double r161379 = t;
        double r161380 = l;
        double r161381 = r161380 * r161380;
        double r161382 = Om;
        double r161383 = r161381 / r161382;
        double r161384 = r161374 * r161383;
        double r161385 = r161379 - r161384;
        double r161386 = r161380 / r161382;
        double r161387 = pow(r161386, r161374);
        double r161388 = r161375 * r161387;
        double r161389 = U_;
        double r161390 = r161377 - r161389;
        double r161391 = r161388 * r161390;
        double r161392 = r161385 - r161391;
        double r161393 = r161378 * r161392;
        double r161394 = sqrt(r161393);
        return r161394;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r161395 = n;
        double r161396 = -9.422947167583052e-260;
        bool r161397 = r161395 <= r161396;
        double r161398 = l;
        double r161399 = Om;
        double r161400 = r161398 / r161399;
        double r161401 = 2.0;
        double r161402 = 2.0;
        double r161403 = r161402 / r161401;
        double r161404 = r161401 * r161403;
        double r161405 = pow(r161400, r161404);
        double r161406 = U_;
        double r161407 = U;
        double r161408 = r161406 - r161407;
        double r161409 = r161395 * r161408;
        double r161410 = -r161402;
        double r161411 = r161410 * r161398;
        double r161412 = t;
        double r161413 = fma(r161411, r161400, r161412);
        double r161414 = fma(r161405, r161409, r161413);
        double r161415 = cbrt(r161414);
        double r161416 = r161402 * r161395;
        double r161417 = r161407 * r161416;
        double r161418 = cbrt(r161417);
        double r161419 = r161415 * r161418;
        double r161420 = 1.5;
        double r161421 = pow(r161419, r161420);
        double r161422 = 2.5981494862601002e-273;
        bool r161423 = r161395 <= r161422;
        double r161424 = r161412 * r161407;
        double r161425 = r161398 * r161398;
        double r161426 = -r161425;
        double r161427 = r161426 * r161395;
        double r161428 = r161399 / r161407;
        double r161429 = r161427 / r161428;
        double r161430 = 4.0;
        double r161431 = r161429 * r161430;
        double r161432 = fma(r161424, r161416, r161431);
        double r161433 = sqrt(r161432);
        double r161434 = 3.3349011025921906e-187;
        bool r161435 = r161395 <= r161434;
        double r161436 = 1.407519843109884e-104;
        bool r161437 = r161395 <= r161436;
        double r161438 = 7.962768658598781e-07;
        bool r161439 = r161395 <= r161438;
        double r161440 = pow(r161400, r161403);
        double r161441 = r161395 * r161440;
        double r161442 = r161441 * r161440;
        double r161443 = r161398 * r161400;
        double r161444 = fma(r161443, r161410, r161412);
        double r161445 = fma(r161408, r161442, r161444);
        double r161446 = r161417 * r161445;
        double r161447 = sqrt(r161446);
        double r161448 = r161439 ? r161421 : r161447;
        double r161449 = r161437 ? r161433 : r161448;
        double r161450 = r161435 ? r161421 : r161449;
        double r161451 = r161423 ? r161433 : r161450;
        double r161452 = r161397 ? r161421 : r161451;
        return r161452;
}

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 3 regimes

2. if n < -9.422947167583052e-260 or 2.5981494862601002e-273 < n < 3.3349011025921906e-187 or 1.407519843109884e-104 < n < 7.962768658598781e-07

   1. Initial program 33.8

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

   2. Simplified 31.1

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

   4. Applied sqr-pow 31.1

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

   5. Applied associate-*r* 30.2

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

   7. Applied add-cube-cbrt 30.5

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

   8. Simplified 31.4

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

   9. Simplified 31.4

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

   11. Applied pow1 31.4

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

   12. Applied pow1 31.4

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

   13. Applied pow1 31.4

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

   14. Applied pow-prod-up 31.4

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

   15. Applied pow-prod-up 31.4

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

   16. Applied sqrt-pow1 31.4

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

   17. Simplified 31.4

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

   19. Applied cbrt-prod 24.1

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

   20. Simplified 24.1

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

   21. Simplified 26.1

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

   if -9.422947167583052e-260 < n < 2.5981494862601002e-273 or 3.3349011025921906e-187 < n < 1.407519843109884e-104

   1. Initial program 38.8

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

   2. Simplified 35.7

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

   4. Applied sqr-pow 35.7

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

   5. Applied associate-*r* 35.0

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

   6. Taylor expanded around inf 37.4

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(t \cdot \left(U \cdot n\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\]

   7. Simplified 36.6

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(t \cdot U, n \cdot 2, \left(-4\right) \cdot \frac{n \cdot \left(\ell \cdot \ell\right)}{\frac{Om}{U}}\right)}}\]

   if 7.962768658598781e-07 < n

   1. Initial program 33.4

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

   2. Simplified 31.2

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

   4. Applied sqr-pow 31.2

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

   5. Applied associate-*r* 30.2

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

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -9.422947167583051961821239451488524021954 \cdot 10^{-260}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\left(-2\right) \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \cdot \sqrt[3]{U \cdot \left(2 \cdot n\right)}\right)}^{\frac{3}{2}}\\ \mathbf{elif}\;n \le 2.598149486260100212883263141948192542387 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t \cdot U, 2 \cdot n, \frac{\left(-\ell \cdot \ell\right) \cdot n}{\frac{Om}{U}} \cdot 4\right)}\\ \mathbf{elif}\;n \le 3.33490110259219064400249536435033242418 \cdot 10^{-187}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\left(-2\right) \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \cdot \sqrt[3]{U \cdot \left(2 \cdot n\right)}\right)}^{\frac{3}{2}}\\ \mathbf{elif}\;n \le 1.407519843109883897735864030283250356428 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(t \cdot U, 2 \cdot n, \frac{\left(-\ell \cdot \ell\right) \cdot n}{\frac{Om}{U}} \cdot 4\right)}\\ \mathbf{elif}\;n \le 7.962768658598781395526822537989097838818 \cdot 10^{-7}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}, n \cdot \left(U* - U\right), \mathsf{fma}\left(\left(-2\right) \cdot \ell, \frac{\ell}{Om}, t\right)\right)} \cdot \sqrt[3]{U \cdot \left(2 \cdot n\right)}\right)}^{\frac{3}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \left(2 \cdot n\right)\right) \cdot \mathsf{fma}\left(U* - U, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))