Toniolo and Linder, Equation (13)

Average Error: 33.7 → 23.9

Time: 45.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le 1.0022495870274159 \cdot 10^{-154}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot 2} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}}\right)}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right)}^{\frac{3}{2}}\\ \mathbf{elif}\;t \le 2.6158530040231545 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot 2} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}}\right)}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right)}^{\frac{3}{2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3141366 = 2.0;
        double r3141367 = n;
        double r3141368 = r3141366 * r3141367;
        double r3141369 = U;
        double r3141370 = r3141368 * r3141369;
        double r3141371 = t;
        double r3141372 = l;
        double r3141373 = r3141372 * r3141372;
        double r3141374 = Om;
        double r3141375 = r3141373 / r3141374;
        double r3141376 = r3141366 * r3141375;
        double r3141377 = r3141371 - r3141376;
        double r3141378 = r3141372 / r3141374;
        double r3141379 = pow(r3141378, r3141366);
        double r3141380 = r3141367 * r3141379;
        double r3141381 = U_;
        double r3141382 = r3141369 - r3141381;
        double r3141383 = r3141380 * r3141382;
        double r3141384 = r3141377 - r3141383;
        double r3141385 = r3141370 * r3141384;
        double r3141386 = sqrt(r3141385);
        return r3141386;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r3141387 = t;
        double r3141388 = 1.0022495870274159e-154;
        bool r3141389 = r3141387 <= r3141388;
        double r3141390 = U;
        double r3141391 = 2.0;
        double r3141392 = r3141390 * r3141391;
        double r3141393 = cbrt(r3141392);
        double r3141394 = U_;
        double r3141395 = r3141394 - r3141390;
        double r3141396 = l;
        double r3141397 = cbrt(r3141396);
        double r3141398 = n;
        double r3141399 = cbrt(r3141398);
        double r3141400 = r3141397 * r3141399;
        double r3141401 = Om;
        double r3141402 = cbrt(r3141401);
        double r3141403 = r3141400 / r3141402;
        double r3141404 = r3141403 * r3141403;
        double r3141405 = r3141403 * r3141404;
        double r3141406 = r3141401 / r3141396;
        double r3141407 = r3141405 / r3141406;
        double r3141408 = r3141396 / r3141406;
        double r3141409 = -2.0;
        double r3141410 = fma(r3141408, r3141409, r3141387);
        double r3141411 = fma(r3141395, r3141407, r3141410);
        double r3141412 = r3141411 * r3141398;
        double r3141413 = cbrt(r3141412);
        double r3141414 = r3141393 * r3141413;
        double r3141415 = 1.5;
        double r3141416 = pow(r3141414, r3141415);
        double r3141417 = 2.6158530040231545e-73;
        bool r3141418 = r3141387 <= r3141417;
        double r3141419 = r3141398 / r3141406;
        double r3141420 = r3141419 / r3141406;
        double r3141421 = fma(r3141395, r3141420, r3141410);
        double r3141422 = r3141398 * r3141392;
        double r3141423 = r3141421 * r3141422;
        double r3141424 = sqrt(r3141423);
        double r3141425 = r3141418 ? r3141424 : r3141416;
        double r3141426 = r3141389 ? r3141416 : r3141425;
        return r3141426;
}

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 2 regimes

2. if t < 1.0022495870274159e-154 or 2.6158530040231545e-73 < t

   1. Initial program 33.6

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

   2. Simplified 29.8

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

   4. Applied add-cube-cbrt 29.9

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

   5. Applied add-cube-cbrt 29.9

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

   6. Applied times-frac 29.9

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

   7. Applied add-cube-cbrt 29.9

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

   8. Applied times-frac 29.9

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

   9. Simplified 29.9

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

   10. Simplified 29.9

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

   12. Applied add-cube-cbrt 30.2

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

   14. Applied pow1 30.2

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

   15. Applied pow1 30.2

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

   16. Applied pow1 30.2

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

   17. Applied pow-prod-up 30.2

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

   18. Applied pow-prod-up 30.2

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

   19. Applied sqrt-pow1 30.2

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

   20. Simplified 30.2

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

   22. Applied cbrt-prod 23.5

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

   if 1.0022495870274159e-154 < t < 2.6158530040231545e-73

   1. Initial program 34.4

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

   2. Simplified 31.0

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

   4. Applied associate-*r* 30.4

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

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.0022495870274159 \cdot 10^{-154}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot 2} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}}\right)}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right)}^{\frac{3}{2}}\\ \mathbf{elif}\;t \le 2.6158530040231545 \cdot 10^{-73}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(U* - U, \frac{\frac{n}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot \left(n \cdot \left(U \cdot 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{U \cdot 2} \cdot \sqrt[3]{\mathsf{fma}\left(U* - U, \frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \left(\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}} \cdot \frac{\sqrt[3]{\ell} \cdot \sqrt[3]{n}}{\sqrt[3]{Om}}\right)}{\frac{Om}{\ell}}, \mathsf{fma}\left(\frac{\ell}{\frac{Om}{\ell}}, -2, t\right)\right) \cdot n}\right)}^{\frac{3}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(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*))))))