Toniolo and Linder, Equation (13)

Average Error: 34.6 → 29.3

Time: 1.0m

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}\;U \le -4.797165734624695899226646467454361223135 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\\ \mathbf{elif}\;U \le 4.114286479798257694747708807463659943796 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \le 1.055275531112388034015510787998742805358 \cdot 10^{79}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) + \left(\left(\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(U* - U\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right)}\\ \mathbf{elif}\;U \le 3.826937509623075222416630358699559428371 \cdot 10^{134}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \le 1.628240362856102545213093675189928586577 \cdot 10^{187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)} \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2} \cdot 2\right)} \cdot n, U* - U, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r184270 = 2.0;
        double r184271 = n;
        double r184272 = r184270 * r184271;
        double r184273 = U;
        double r184274 = r184272 * r184273;
        double r184275 = t;
        double r184276 = l;
        double r184277 = r184276 * r184276;
        double r184278 = Om;
        double r184279 = r184277 / r184278;
        double r184280 = r184270 * r184279;
        double r184281 = r184275 - r184280;
        double r184282 = r184276 / r184278;
        double r184283 = pow(r184282, r184270);
        double r184284 = r184271 * r184283;
        double r184285 = U_;
        double r184286 = r184273 - r184285;
        double r184287 = r184284 * r184286;
        double r184288 = r184281 - r184287;
        double r184289 = r184274 * r184288;
        double r184290 = sqrt(r184289);
        return r184290;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r184291 = U;
        double r184292 = -4.797165734624696e-88;
        bool r184293 = r184291 <= r184292;
        double r184294 = 2.0;
        double r184295 = n;
        double r184296 = r184294 * r184295;
        double r184297 = r184296 * r184291;
        double r184298 = U_;
        double r184299 = r184298 - r184291;
        double r184300 = l;
        double r184301 = Om;
        double r184302 = r184300 / r184301;
        double r184303 = 2.0;
        double r184304 = r184294 / r184303;
        double r184305 = pow(r184302, r184304);
        double r184306 = cbrt(r184295);
        double r184307 = r184306 * r184306;
        double r184308 = r184306 * r184305;
        double r184309 = r184307 * r184308;
        double r184310 = r184305 * r184309;
        double r184311 = r184300 * r184302;
        double r184312 = -r184294;
        double r184313 = t;
        double r184314 = fma(r184311, r184312, r184313);
        double r184315 = fma(r184299, r184310, r184314);
        double r184316 = r184297 * r184315;
        double r184317 = sqrt(r184316);
        double r184318 = 4.114286479798258e-24;
        bool r184319 = r184291 <= r184318;
        double r184320 = r184305 * r184295;
        double r184321 = r184299 * r184305;
        double r184322 = r184302 * r184312;
        double r184323 = fma(r184300, r184322, r184313);
        double r184324 = fma(r184320, r184321, r184323);
        double r184325 = r184291 * r184324;
        double r184326 = r184325 * r184296;
        double r184327 = sqrt(r184326);
        double r184328 = 1.055275531112388e+79;
        bool r184329 = r184291 <= r184328;
        double r184330 = r184323 * r184297;
        double r184331 = r184294 * r184291;
        double r184332 = r184295 * r184331;
        double r184333 = r184332 * r184299;
        double r184334 = r184333 * r184305;
        double r184335 = r184334 * r184320;
        double r184336 = r184330 + r184335;
        double r184337 = sqrt(r184336);
        double r184338 = 3.826937509623075e+134;
        bool r184339 = r184291 <= r184338;
        double r184340 = 1.6282403628561025e+187;
        bool r184341 = r184291 <= r184340;
        double r184342 = sqrt(r184324);
        double r184343 = sqrt(r184332);
        double r184344 = r184342 * r184343;
        double r184345 = r184304 * r184303;
        double r184346 = pow(r184302, r184345);
        double r184347 = r184346 * r184295;
        double r184348 = fma(r184347, r184299, r184323);
        double r184349 = r184291 * r184348;
        double r184350 = r184349 * r184296;
        double r184351 = sqrt(r184350);
        double r184352 = r184341 ? r184344 : r184351;
        double r184353 = r184339 ? r184327 : r184352;
        double r184354 = r184329 ? r184337 : r184353;
        double r184355 = r184319 ? r184327 : r184354;
        double r184356 = r184293 ? r184317 : r184355;
        return r184356;
}

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 5 regimes

2. if U < -4.797165734624696e-88

   1. Initial program 30.0

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

   2. Simplified 26.7

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

   4. Applied sqr-pow 26.7

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

   5. Applied associate-*r* 26.1

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

   7. Applied add-cube-cbrt 26.1

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

   8. Applied associate-*l* 26.1

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

   9. Simplified 26.1

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

   if -4.797165734624696e-88 < U < 4.114286479798258e-24 or 1.055275531112388e+79 < U < 3.826937509623075e+134

   1. Initial program 37.9

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

   2. Simplified 35.4

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

   4. Applied sqr-pow 35.4

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

   5. Applied associate-*r* 34.5

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

   7. Applied add-cube-cbrt 34.5

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

   8. Applied associate-*l* 34.5

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

   9. Simplified 34.5

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

   11. Applied pow1 34.5

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

   12. Applied pow1 34.5

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

   13. Applied pow1 34.5

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

   14. Applied pow1 34.5

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

   15. Applied pow-prod-down 34.5

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

   16. Applied pow-prod-down 34.5

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

   17. Applied pow-prod-down 34.5

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

   18. Simplified 30.1

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

   if 4.114286479798258e-24 < U < 1.055275531112388e+79

   1. Initial program 29.3

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

   2. Simplified 25.6

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

   4. Applied sqr-pow 25.6

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

   5. Applied associate-*r* 24.3

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

   7. Applied add-cube-cbrt 24.3

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

   8. Applied associate-*l* 24.3

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

   9. Simplified 24.3

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

   11. Applied fma-udef 24.3

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

   12. Applied distribute-lft-in 24.3

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

   13. Simplified 24.9

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

   14. Simplified 24.9

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

   if 3.826937509623075e+134 < U < 1.6282403628561025e+187

   1. Initial program 30.3

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

   2. Simplified 26.7

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

   4. Applied sqr-pow 26.7

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

   5. Applied associate-*r* 26.2

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

   7. Applied add-cube-cbrt 26.2

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

   8. Applied associate-*l* 26.2

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

   9. Simplified 26.2

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

   11. Applied sqrt-prod 35.9

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

   12. Simplified 35.9

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

   13. Simplified 35.9

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

   if 1.6282403628561025e+187 < U

   1. Initial program 34.3

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

   2. Simplified 31.8

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

   4. Applied sqr-pow 31.8

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

   5. Applied associate-*r* 31.8

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

   7. Applied associate-*l* 41.1

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

   8. Simplified 41.1

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \color{blue}{\left(U \cdot \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)} \cdot n, U* - U, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 29.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -4.797165734624695899226646467454361223135 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(U* - U, {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right), \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, -2, t\right)\right)}\\ \mathbf{elif}\;U \le 4.114286479798257694747708807463659943796 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \le 1.055275531112388034015510787998742805358 \cdot 10^{79}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right) \cdot \left(\left(2 \cdot n\right) \cdot U\right) + \left(\left(\left(n \cdot \left(2 \cdot U\right)\right) \cdot \left(U* - U\right)\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right)}\\ \mathbf{elif}\;U \le 3.826937509623075222416630358699559428371 \cdot 10^{134}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \mathbf{elif}\;U \le 1.628240362856102545213093675189928586577 \cdot 10^{187}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)} \cdot \sqrt{n \cdot \left(2 \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(U \cdot \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2} \cdot 2\right)} \cdot n, U* - U, \mathsf{fma}\left(\ell, \frac{\ell}{Om} \cdot \left(-2\right), t\right)\right)\right) \cdot \left(2 \cdot n\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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*))))))