Toniolo and Linder, Equation (13)

Average Error: 34.9 → 27.8

Time: 40.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1.3247266730184721 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;n \le -4.44897802822854103 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U + 2 \cdot \left(\left(U \cdot n\right) \cdot \left(\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right) \cdot 0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r157387 = 2.0;
        double r157388 = n;
        double r157389 = r157387 * r157388;
        double r157390 = U;
        double r157391 = r157389 * r157390;
        double r157392 = t;
        double r157393 = l;
        double r157394 = r157393 * r157393;
        double r157395 = Om;
        double r157396 = r157394 / r157395;
        double r157397 = r157387 * r157396;
        double r157398 = r157392 - r157397;
        double r157399 = r157393 / r157395;
        double r157400 = pow(r157399, r157387);
        double r157401 = r157388 * r157400;
        double r157402 = U_;
        double r157403 = r157390 - r157402;
        double r157404 = r157401 * r157403;
        double r157405 = r157398 - r157404;
        double r157406 = r157391 * r157405;
        double r157407 = sqrt(r157406);
        return r157407;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r157408 = n;
        double r157409 = -1.324726673018472e-161;
        bool r157410 = r157408 <= r157409;
        double r157411 = 2.0;
        double r157412 = r157411 * r157408;
        double r157413 = U;
        double r157414 = r157412 * r157413;
        double r157415 = t;
        double r157416 = l;
        double r157417 = Om;
        double r157418 = r157417 / r157416;
        double r157419 = r157416 / r157418;
        double r157420 = r157408 * r157416;
        double r157421 = 1.0;
        double r157422 = 1.0;
        double r157423 = pow(r157417, r157422);
        double r157424 = r157421 / r157423;
        double r157425 = pow(r157424, r157422);
        double r157426 = r157420 * r157425;
        double r157427 = r157416 / r157417;
        double r157428 = 2.0;
        double r157429 = r157411 / r157428;
        double r157430 = pow(r157427, r157429);
        double r157431 = r157426 * r157430;
        double r157432 = U_;
        double r157433 = r157413 - r157432;
        double r157434 = r157431 * r157433;
        double r157435 = fma(r157411, r157419, r157434);
        double r157436 = r157415 - r157435;
        double r157437 = r157414 * r157436;
        double r157438 = sqrt(r157437);
        double r157439 = -4.448978028228541e-305;
        bool r157440 = r157408 <= r157439;
        double r157441 = r157428 * r157429;
        double r157442 = pow(r157427, r157441);
        double r157443 = r157408 * r157442;
        double r157444 = r157433 * r157443;
        double r157445 = fma(r157411, r157419, r157444);
        double r157446 = r157415 - r157445;
        double r157447 = r157446 * r157412;
        double r157448 = r157447 * r157413;
        double r157449 = r157413 * r157408;
        double r157450 = 0.0;
        double r157451 = r157445 * r157450;
        double r157452 = r157449 * r157451;
        double r157453 = r157411 * r157452;
        double r157454 = r157448 + r157453;
        double r157455 = sqrt(r157454);
        double r157456 = sqrt(r157412);
        double r157457 = pow(r157427, r157411);
        double r157458 = r157408 * r157457;
        double r157459 = r157458 * r157433;
        double r157460 = fma(r157411, r157419, r157459);
        double r157461 = r157415 - r157460;
        double r157462 = r157413 * r157461;
        double r157463 = sqrt(r157462);
        double r157464 = r157456 * r157463;
        double r157465 = r157440 ? r157455 : r157464;
        double r157466 = r157410 ? r157438 : r157465;
        return r157466;
}

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 < -1.324726673018472e-161

   1. Initial program 33.2

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

   2. Simplified 33.3

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

   4. Applied associate-/l* 30.9

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

   6. Applied associate-*r* 30.6

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

   8. Applied sqr-pow 30.6

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

   9. Applied associate-*r* 29.5

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

   10. Taylor expanded around 0 30.7

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

   if -1.324726673018472e-161 < n < -4.448978028228541e-305

   1. Initial program 39.9

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

   2. Simplified 39.4

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

   4. Applied associate-/l* 36.4

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

   6. Applied associate-*r* 36.8

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

   8. Applied sqr-pow 36.8

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

   9. Applied associate-*r* 35.9

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

   11. Applied add-cube-cbrt 36.0

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

   12. Applied add-sqr-sqrt 50.7

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

   13. Applied prod-diff 50.7

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

   14. Applied distribute-lft-in 50.7

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

   15. Simplified 31.8

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

   16. Simplified 31.8

       \[\leadsto \sqrt{\left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U + \color{blue}{2 \cdot \left(\left(U \cdot n\right) \cdot \left(\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right) \cdot 0\right)\right)}}\]

   if -4.448978028228541e-305 < n

   1. Initial program 34.6

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

   2. Simplified 34.4

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

   4. Applied associate-/l* 31.6

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

   6. Applied sqrt-prod 24.6

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

4. Final simplification 27.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.3247266730184721 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(\left(\left(n \cdot \ell\right) \cdot {\left(\frac{1}{{Om}^{1}}\right)}^{1}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(U - U*\right)\right)\right)}\\ \mathbf{elif}\;n \le -4.44897802822854103 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{\left(\left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right)\right) \cdot \left(2 \cdot n\right)\right) \cdot U + 2 \cdot \left(\left(U \cdot n\right) \cdot \left(\mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(U - U*\right) \cdot \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right)\right) \cdot 0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{U \cdot \left(t - \mathsf{fma}\left(2, \frac{\ell}{\frac{Om}{\ell}}, \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))