Toniolo and Linder, Equation (13)

Average Error: 33.2 → 28.7

Time: 46.1s

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 -9.628340007082016 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt[3]{\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(\sqrt[3]{\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 \sqrt[3]{\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)} \cdot \left(\left(\sqrt[3]{\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 \sqrt[3]{\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) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}\\ \mathbf{elif}\;t \le -3.2377717899062056 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\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 n} \cdot \sqrt[3]{\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 n}\right) \cdot \sqrt[3]{\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 n}\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;t \le 7.3820940872889895 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt[3]{\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(\sqrt[3]{\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 \sqrt[3]{\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)} \cdot \left(\left(\sqrt[3]{\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 \sqrt[3]{\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) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}\\ \mathbf{elif}\;t \le 1.412213387362571 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot 2\right) \cdot \left(\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 n\right)}} \cdot \sqrt{\sqrt{\left(U \cdot 2\right) \cdot \left(\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 n\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \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)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1028308 = 2.0;
        double r1028309 = n;
        double r1028310 = r1028308 * r1028309;
        double r1028311 = U;
        double r1028312 = r1028310 * r1028311;
        double r1028313 = t;
        double r1028314 = l;
        double r1028315 = r1028314 * r1028314;
        double r1028316 = Om;
        double r1028317 = r1028315 / r1028316;
        double r1028318 = r1028308 * r1028317;
        double r1028319 = r1028313 - r1028318;
        double r1028320 = r1028314 / r1028316;
        double r1028321 = pow(r1028320, r1028308);
        double r1028322 = r1028309 * r1028321;
        double r1028323 = U_;
        double r1028324 = r1028311 - r1028323;
        double r1028325 = r1028322 * r1028324;
        double r1028326 = r1028319 - r1028325;
        double r1028327 = r1028312 * r1028326;
        double r1028328 = sqrt(r1028327);
        return r1028328;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1028329 = t;
        double r1028330 = -9.628340007082016e+98;
        bool r1028331 = r1028329 <= r1028330;
        double r1028332 = U_;
        double r1028333 = U;
        double r1028334 = r1028332 - r1028333;
        double r1028335 = n;
        double r1028336 = Om;
        double r1028337 = l;
        double r1028338 = r1028336 / r1028337;
        double r1028339 = r1028335 / r1028338;
        double r1028340 = r1028339 / r1028338;
        double r1028341 = r1028337 / r1028338;
        double r1028342 = -2.0;
        double r1028343 = fma(r1028341, r1028342, r1028329);
        double r1028344 = fma(r1028334, r1028340, r1028343);
        double r1028345 = cbrt(r1028344);
        double r1028346 = r1028345 * r1028345;
        double r1028347 = r1028345 * r1028346;
        double r1028348 = cbrt(r1028347);
        double r1028349 = 2.0;
        double r1028350 = r1028333 * r1028349;
        double r1028351 = r1028350 * r1028335;
        double r1028352 = r1028346 * r1028351;
        double r1028353 = r1028348 * r1028352;
        double r1028354 = sqrt(r1028353);
        double r1028355 = -3.2377717899062056e-141;
        bool r1028356 = r1028329 <= r1028355;
        double r1028357 = r1028344 * r1028335;
        double r1028358 = cbrt(r1028357);
        double r1028359 = r1028358 * r1028358;
        double r1028360 = r1028359 * r1028358;
        double r1028361 = r1028360 * r1028350;
        double r1028362 = sqrt(r1028361);
        double r1028363 = 7.3820940872889895e-298;
        bool r1028364 = r1028329 <= r1028363;
        double r1028365 = 1.412213387362571e+29;
        bool r1028366 = r1028329 <= r1028365;
        double r1028367 = r1028350 * r1028357;
        double r1028368 = sqrt(r1028367);
        double r1028369 = sqrt(r1028368);
        double r1028370 = r1028369 * r1028369;
        double r1028371 = sqrt(r1028351);
        double r1028372 = sqrt(r1028344);
        double r1028373 = r1028371 * r1028372;
        double r1028374 = r1028366 ? r1028370 : r1028373;
        double r1028375 = r1028364 ? r1028354 : r1028374;
        double r1028376 = r1028356 ? r1028362 : r1028375;
        double r1028377 = r1028331 ? r1028354 : r1028376;
        return r1028377;
}

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 < -9.628340007082016e+98 or -3.2377717899062056e-141 < t < 7.3820940872889895e-298

   1. Initial program 35.0

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

   2. Simplified 31.4

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

   4. Applied associate-*r* 31.5

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

   6. Applied add-cube-cbrt 31.7

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

   7. Applied associate-*r* 31.7

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

   9. Applied add-cbrt-cube 31.8

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

   if -9.628340007082016e+98 < t < -3.2377717899062056e-141

   1. Initial program 30.8

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

   2. Simplified 27.1

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

   4. Applied add-cube-cbrt 27.5

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

   if 7.3820940872889895e-298 < t < 1.412213387362571e+29

   1. Initial program 32.8

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

   2. Simplified 29.5

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

   4. Applied add-sqr-sqrt 29.7

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

   if 1.412213387362571e+29 < t

   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 29.0

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

   4. Applied associate-*r* 30.4

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

   6. Applied sqrt-prod 24.3

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

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -9.628340007082016 \cdot 10^{+98}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt[3]{\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(\sqrt[3]{\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 \sqrt[3]{\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)} \cdot \left(\left(\sqrt[3]{\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 \sqrt[3]{\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) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}\\ \mathbf{elif}\;t \le -3.2377717899062056 \cdot 10^{-141}:\\ \;\;\;\;\sqrt{\left(\left(\sqrt[3]{\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 n} \cdot \sqrt[3]{\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 n}\right) \cdot \sqrt[3]{\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 n}\right) \cdot \left(U \cdot 2\right)}\\ \mathbf{elif}\;t \le 7.3820940872889895 \cdot 10^{-298}:\\ \;\;\;\;\sqrt{\sqrt[3]{\sqrt[3]{\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(\sqrt[3]{\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 \sqrt[3]{\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)} \cdot \left(\left(\sqrt[3]{\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 \sqrt[3]{\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) \cdot \left(\left(U \cdot 2\right) \cdot n\right)\right)}\\ \mathbf{elif}\;t \le 1.412213387362571 \cdot 10^{+29}:\\ \;\;\;\;\sqrt{\sqrt{\left(U \cdot 2\right) \cdot \left(\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 n\right)}} \cdot \sqrt{\sqrt{\left(U \cdot 2\right) \cdot \left(\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 n\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot 2\right) \cdot n} \cdot \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)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +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*))))))