Toniolo and Linder, Equation (13)

Average Error: 32.8 → 29.7

Time: 52.8s

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 -1.6760288117773952 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \frac{\left(\frac{\ell}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right) \cdot \left(U - U*\right)}{Om} + \left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)}\\ \mathbf{elif}\;t \le -1.908160256120522 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot U}\\ \mathbf{elif}\;t \le -6.039684361220293 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2 \cdot t, U \cdot n, \left(\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \frac{U}{Om}\right) \cdot -4\right) + \left(\left(\frac{\ell}{Om} \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{elif}\;t \le 3.0731526450190984 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot U}\\ \mathbf{elif}\;t \le 83934.69895366188:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 2.2801453703210546 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(\left(\sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n} \cdot U\right) \cdot \left(\sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n} \cdot \sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n}\right)\right) \cdot \frac{\ell}{Om}\right)\right) + \left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1836251 = 2.0;
        double r1836252 = n;
        double r1836253 = r1836251 * r1836252;
        double r1836254 = U;
        double r1836255 = r1836253 * r1836254;
        double r1836256 = t;
        double r1836257 = l;
        double r1836258 = r1836257 * r1836257;
        double r1836259 = Om;
        double r1836260 = r1836258 / r1836259;
        double r1836261 = r1836251 * r1836260;
        double r1836262 = r1836256 - r1836261;
        double r1836263 = r1836257 / r1836259;
        double r1836264 = pow(r1836263, r1836251);
        double r1836265 = r1836252 * r1836264;
        double r1836266 = U_;
        double r1836267 = r1836254 - r1836266;
        double r1836268 = r1836265 * r1836267;
        double r1836269 = r1836262 - r1836268;
        double r1836270 = r1836255 * r1836269;
        double r1836271 = sqrt(r1836270);
        return r1836271;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r1836272 = t;
        double r1836273 = -1.6760288117773952e+183;
        bool r1836274 = r1836272 <= r1836273;
        double r1836275 = n;
        double r1836276 = -2.0;
        double r1836277 = r1836275 * r1836276;
        double r1836278 = l;
        double r1836279 = Om;
        double r1836280 = r1836278 / r1836279;
        double r1836281 = r1836275 * r1836278;
        double r1836282 = U;
        double r1836283 = r1836281 * r1836282;
        double r1836284 = r1836280 * r1836283;
        double r1836285 = U_;
        double r1836286 = r1836282 - r1836285;
        double r1836287 = r1836284 * r1836286;
        double r1836288 = r1836287 / r1836279;
        double r1836289 = r1836277 * r1836288;
        double r1836290 = 2.0;
        double r1836291 = r1836290 * r1836275;
        double r1836292 = r1836278 * r1836280;
        double r1836293 = fma(r1836276, r1836292, r1836272);
        double r1836294 = r1836293 * r1836282;
        double r1836295 = r1836291 * r1836294;
        double r1836296 = r1836289 + r1836295;
        double r1836297 = sqrt(r1836296);
        double r1836298 = -1.908160256120522e+56;
        bool r1836299 = r1836272 <= r1836298;
        double r1836300 = r1836280 * r1836280;
        double r1836301 = r1836300 * r1836286;
        double r1836302 = r1836301 * r1836275;
        double r1836303 = fma(r1836292, r1836290, r1836302);
        double r1836304 = r1836272 - r1836303;
        double r1836305 = r1836291 * r1836304;
        double r1836306 = r1836305 * r1836282;
        double r1836307 = sqrt(r1836306);
        double r1836308 = -6.039684361220293e-180;
        bool r1836309 = r1836272 <= r1836308;
        double r1836310 = r1836290 * r1836272;
        double r1836311 = r1836282 * r1836275;
        double r1836312 = r1836278 * r1836281;
        double r1836313 = r1836282 / r1836279;
        double r1836314 = r1836312 * r1836313;
        double r1836315 = -4.0;
        double r1836316 = r1836314 * r1836315;
        double r1836317 = fma(r1836310, r1836311, r1836316);
        double r1836318 = r1836275 * r1836280;
        double r1836319 = r1836318 * r1836282;
        double r1836320 = r1836280 * r1836319;
        double r1836321 = r1836320 * r1836286;
        double r1836322 = r1836321 * r1836277;
        double r1836323 = r1836317 + r1836322;
        double r1836324 = sqrt(r1836323);
        double r1836325 = 3.0731526450190984e-256;
        bool r1836326 = r1836272 <= r1836325;
        double r1836327 = 83934.69895366188;
        bool r1836328 = r1836272 <= r1836327;
        double r1836329 = sqrt(r1836304);
        double r1836330 = r1836291 * r1836282;
        double r1836331 = sqrt(r1836330);
        double r1836332 = r1836329 * r1836331;
        double r1836333 = 2.2801453703210546e+171;
        bool r1836334 = r1836272 <= r1836333;
        double r1836335 = -r1836280;
        double r1836336 = r1836335 * r1836275;
        double r1836337 = cbrt(r1836336);
        double r1836338 = r1836337 * r1836282;
        double r1836339 = r1836337 * r1836337;
        double r1836340 = r1836338 * r1836339;
        double r1836341 = r1836340 * r1836280;
        double r1836342 = r1836286 * r1836341;
        double r1836343 = r1836291 * r1836342;
        double r1836344 = r1836343 + r1836295;
        double r1836345 = sqrt(r1836344);
        double r1836346 = r1836334 ? r1836345 : r1836332;
        double r1836347 = r1836328 ? r1836332 : r1836346;
        double r1836348 = r1836326 ? r1836307 : r1836347;
        double r1836349 = r1836309 ? r1836324 : r1836348;
        double r1836350 = r1836299 ? r1836307 : r1836349;
        double r1836351 = r1836274 ? r1836297 : r1836350;
        return r1836351;
}

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 t < -1.6760288117773952e+183

   1. Initial program 35.6

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

   3. Applied sub-neg 35.6

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

   4. Applied distribute-lft-in 35.7

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

   5. Simplified 34.5

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

   6. Simplified 32.4

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

   8. Applied *-un-lft-identity 32.4

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

   9. Applied associate-*r* 32.4

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

   10. Simplified 31.9

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

   12. Applied associate-*l/ 32.2

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

   13. Applied distribute-neg-frac 32.2

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

   14. Applied associate-*l/ 33.0

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

   15. Applied associate-*r/ 33.0

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

   16. Applied associate-*l/ 33.0

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

   if -1.6760288117773952e+183 < t < -1.908160256120522e+56 or -6.039684361220293e-180 < t < 3.0731526450190984e-256

   1. Initial program 34.1

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

   3. Applied pow1 34.1

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

   4. Applied pow1 34.1

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

   5. Applied pow1 34.1

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

   6. Applied pow1 34.1

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

   7. Applied pow-prod-down 34.1

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

   8. Applied pow-prod-down 34.1

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

   9. Applied pow-prod-down 34.1

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

   10. Simplified 32.8

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

   if -1.908160256120522e+56 < t < -6.039684361220293e-180

   1. Initial program 31.6

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

   3. Applied sub-neg 31.6

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

   4. Applied distribute-lft-in 31.6

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

   5. Simplified 29.7

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

   6. Simplified 28.1

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

   8. Applied *-un-lft-identity 28.1

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

   9. Applied associate-*r* 28.1

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

   10. Simplified 27.6

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

   11. Taylor expanded around inf 30.7

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

   12. Simplified 29.5

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

   if 3.0731526450190984e-256 < t < 83934.69895366188 or 2.2801453703210546e+171 < 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. Using strategy rm

   3. Applied sqrt-prod 30.2

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

   4. Simplified 28.6

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

   if 83934.69895366188 < t < 2.2801453703210546e+171

   1. Initial program 28.6

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

   3. Applied sub-neg 28.6

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

   4. Applied distribute-lft-in 28.6

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

   5. Simplified 26.1

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

   6. Simplified 24.9

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

   8. Applied *-un-lft-identity 24.9

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

   9. Applied associate-*r* 24.9

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

   10. Simplified 24.4

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

   12. Applied add-cube-cbrt 24.5

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

   13. Applied associate-*l* 24.5

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

4. Final simplification 29.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.6760288117773952 \cdot 10^{+183}:\\ \;\;\;\;\sqrt{\left(n \cdot -2\right) \cdot \frac{\left(\frac{\ell}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot U\right)\right) \cdot \left(U - U*\right)}{Om} + \left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)}\\ \mathbf{elif}\;t \le -1.908160256120522 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot U}\\ \mathbf{elif}\;t \le -6.039684361220293 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(2 \cdot t, U \cdot n, \left(\left(\ell \cdot \left(n \cdot \ell\right)\right) \cdot \frac{U}{Om}\right) \cdot -4\right) + \left(\left(\frac{\ell}{Om} \cdot \left(\left(n \cdot \frac{\ell}{Om}\right) \cdot U\right)\right) \cdot \left(U - U*\right)\right) \cdot \left(n \cdot -2\right)}\\ \mathbf{elif}\;t \le 3.0731526450190984 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot \left(t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)\right)\right) \cdot U}\\ \mathbf{elif}\;t \le 83934.69895366188:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \mathbf{elif}\;t \le 2.2801453703210546 \cdot 10^{+171}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \left(\left(U - U*\right) \cdot \left(\left(\left(\sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n} \cdot U\right) \cdot \left(\sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n} \cdot \sqrt[3]{\left(-\frac{\ell}{Om}\right) \cdot n}\right)\right) \cdot \frac{\ell}{Om}\right)\right) + \left(2 \cdot n\right) \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t - \mathsf{fma}\left(\ell \cdot \frac{\ell}{Om}, 2, \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right) \cdot n\right)} \cdot \sqrt{\left(2 \cdot n\right) \cdot U}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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*))))))