Toniolo and Linder, Equation (13)

Average Error: 34.3 → 31.5

Time: 1.4m

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}\;n \le -115.2035769670583960078147356398403644562:\\ \;\;\;\;\sqrt{e^{\log \left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2\right)}}\\ \mathbf{elif}\;n \le -1.928586768576104907029448118743871413968 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le -1.991223724932492166185779454526004556741 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\left(n \cdot n\right) \cdot U}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, \left(U \cdot n\right) \cdot t\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}\\ \mathbf{elif}\;n \le 1.190270096378231867728639520214394049373 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le 3.245686032965869293542566860037617070177 \cdot 10^{109}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\left(n \cdot n\right) \cdot U}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, \left(U \cdot n\right) \cdot t\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2\right) \cdot \sqrt{\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2}}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2976257 = 2.0;
        double r2976258 = n;
        double r2976259 = r2976257 * r2976258;
        double r2976260 = U;
        double r2976261 = r2976259 * r2976260;
        double r2976262 = t;
        double r2976263 = l;
        double r2976264 = r2976263 * r2976263;
        double r2976265 = Om;
        double r2976266 = r2976264 / r2976265;
        double r2976267 = r2976257 * r2976266;
        double r2976268 = r2976262 - r2976267;
        double r2976269 = r2976263 / r2976265;
        double r2976270 = pow(r2976269, r2976257);
        double r2976271 = r2976258 * r2976270;
        double r2976272 = U_;
        double r2976273 = r2976260 - r2976272;
        double r2976274 = r2976271 * r2976273;
        double r2976275 = r2976268 - r2976274;
        double r2976276 = r2976261 * r2976275;
        double r2976277 = sqrt(r2976276);
        return r2976277;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2976278 = n;
        double r2976279 = -115.2035769670584;
        bool r2976280 = r2976278 <= r2976279;
        double r2976281 = U;
        double r2976282 = r2976281 * r2976278;
        double r2976283 = t;
        double r2976284 = l;
        double r2976285 = Om;
        double r2976286 = r2976284 / r2976285;
        double r2976287 = 2.0;
        double r2976288 = 2.0;
        double r2976289 = r2976287 / r2976288;
        double r2976290 = pow(r2976286, r2976289);
        double r2976291 = U_;
        double r2976292 = r2976281 - r2976291;
        double r2976293 = r2976290 * r2976292;
        double r2976294 = r2976290 * r2976278;
        double r2976295 = r2976284 * r2976286;
        double r2976296 = r2976295 * r2976287;
        double r2976297 = fma(r2976293, r2976294, r2976296);
        double r2976298 = r2976283 - r2976297;
        double r2976299 = r2976282 * r2976298;
        double r2976300 = r2976299 * r2976287;
        double r2976301 = log(r2976300);
        double r2976302 = exp(r2976301);
        double r2976303 = sqrt(r2976302);
        double r2976304 = -1.928586768576105e-95;
        bool r2976305 = r2976278 <= r2976304;
        double r2976306 = sqrt(r2976281);
        double r2976307 = r2976287 * r2976284;
        double r2976308 = r2976294 * r2976290;
        double r2976309 = r2976292 * r2976308;
        double r2976310 = fma(r2976286, r2976307, r2976309);
        double r2976311 = r2976283 - r2976310;
        double r2976312 = r2976278 * r2976311;
        double r2976313 = r2976287 * r2976312;
        double r2976314 = sqrt(r2976313);
        double r2976315 = r2976306 * r2976314;
        double r2976316 = -1.9912237249324922e-147;
        bool r2976317 = r2976278 <= r2976316;
        double r2976318 = r2976284 * r2976284;
        double r2976319 = r2976291 * r2976318;
        double r2976320 = r2976319 / r2976285;
        double r2976321 = r2976278 * r2976278;
        double r2976322 = r2976321 * r2976281;
        double r2976323 = r2976322 / r2976285;
        double r2976324 = r2976320 * r2976323;
        double r2976325 = 1.0;
        double r2976326 = -1.0;
        double r2976327 = pow(r2976326, r2976287);
        double r2976328 = r2976325 / r2976327;
        double r2976329 = 1.0;
        double r2976330 = pow(r2976328, r2976329);
        double r2976331 = r2976282 * r2976283;
        double r2976332 = fma(r2976324, r2976330, r2976331);
        double r2976333 = r2976332 * r2976287;
        double r2976334 = 4.0;
        double r2976335 = r2976281 / r2976285;
        double r2976336 = r2976278 * r2976284;
        double r2976337 = r2976336 * r2976284;
        double r2976338 = r2976335 * r2976337;
        double r2976339 = r2976334 * r2976338;
        double r2976340 = r2976333 - r2976339;
        double r2976341 = sqrt(r2976340);
        double r2976342 = 1.1902700963782319e-56;
        bool r2976343 = r2976278 <= r2976342;
        double r2976344 = r2976281 * r2976313;
        double r2976345 = sqrt(r2976344);
        double r2976346 = 3.2456860329658693e+109;
        bool r2976347 = r2976278 <= r2976346;
        double r2976348 = sqrt(r2976300);
        double r2976349 = r2976300 * r2976348;
        double r2976350 = cbrt(r2976349);
        double r2976351 = r2976347 ? r2976341 : r2976350;
        double r2976352 = r2976343 ? r2976345 : r2976351;
        double r2976353 = r2976317 ? r2976341 : r2976352;
        double r2976354 = r2976305 ? r2976315 : r2976353;
        double r2976355 = r2976280 ? r2976303 : r2976354;
        return r2976355;
}

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 n < -115.2035769670584

   1. Initial program 32.2

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

   2. Simplified 36.2

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

   4. Applied sqr-pow 36.2

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

   5. Applied associate-*r* 35.1

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

   7. Applied add-exp-log 35.1

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

   8. Applied add-exp-log 50.8

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

   9. Applied add-exp-log 64.0

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

   10. Applied prod-exp 64.0

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

   11. Applied prod-exp 64.0

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

   12. Applied add-exp-log 64.0

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

   13. Applied prod-exp 64.0

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

   14. Simplified 30.1

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

   if -115.2035769670584 < n < -1.928586768576105e-95

   1. Initial program 32.3

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

   2. Simplified 26.7

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

   4. Applied sqr-pow 26.7

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

   5. Applied associate-*r* 26.5

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

   7. Applied sqrt-prod 38.5

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

   if -1.928586768576105e-95 < n < -1.9912237249324922e-147 or 1.1902700963782319e-56 < n < 3.2456860329658693e+109

   1. Initial program 29.9

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

   2. Simplified 27.0

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

   4. Applied sqr-pow 27.0

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

   5. Applied associate-*r* 26.5

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

   7. Applied *-un-lft-identity 26.5

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

   8. Applied add-sqr-sqrt 45.0

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

   9. Applied prod-diff 45.0

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

   10. Applied distribute-lft-in 45.0

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

   11. Simplified 26.1

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

   12. Taylor expanded around -inf 35.5

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \left(t \cdot \left(U \cdot n\right)\right) + 2 \cdot \left(\frac{U \cdot \left({n}^{2} \cdot \left(U* \cdot {\ell}^{2}\right)\right)}{{Om}^{2}} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right) - 4 \cdot \frac{U \cdot \left(n \cdot {\ell}^{2}\right)}{Om}}}\]

   13. Simplified 32.4

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U \cdot \left(n \cdot n\right)}{Om} \cdot \frac{\left(\ell \cdot \ell\right) \cdot U*}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, t \cdot \left(U \cdot n\right)\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}}\]

   if -1.9912237249324922e-147 < n < 1.1902700963782319e-56

   1. Initial program 37.3

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

   2. Simplified 29.8

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

   4. Applied sqr-pow 29.8

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

   5. Applied associate-*r* 28.3

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

   if 3.2456860329658693e+109 < n

   1. Initial program 35.2

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

   2. Simplified 40.6

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

   4. Applied sqr-pow 40.6

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

   5. Applied associate-*r* 39.8

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

   7. Applied add-cbrt-cube 44.9

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

   8. Simplified 40.1

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

4. Final simplification 31.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -115.2035769670583960078147356398403644562:\\ \;\;\;\;\sqrt{e^{\log \left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2\right)}}\\ \mathbf{elif}\;n \le -1.928586768576104907029448118743871413968 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{U} \cdot \sqrt{2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le -1.991223724932492166185779454526004556741 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\left(n \cdot n\right) \cdot U}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, \left(U \cdot n\right) \cdot t\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}\\ \mathbf{elif}\;n \le 1.190270096378231867728639520214394049373 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{U \cdot \left(2 \cdot \left(n \cdot \left(t - \mathsf{fma}\left(\frac{\ell}{Om}, 2 \cdot \ell, \left(U - U*\right) \cdot \left(\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right)\right)\right)}\\ \mathbf{elif}\;n \le 3.245686032965869293542566860037617070177 \cdot 10^{109}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{U* \cdot \left(\ell \cdot \ell\right)}{Om} \cdot \frac{\left(n \cdot n\right) \cdot U}{Om}, {\left(\frac{1}{{-1}^{2}}\right)}^{1}, \left(U \cdot n\right) \cdot t\right) \cdot 2 - 4 \cdot \left(\frac{U}{Om} \cdot \left(\left(n \cdot \ell\right) \cdot \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2\right) \cdot \sqrt{\left(\left(U \cdot n\right) \cdot \left(t - \mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right), {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot n, \left(\ell \cdot \frac{\ell}{Om}\right) \cdot 2\right)\right)\right) \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +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*))))))