Toniolo and Linder, Equation (13)

Average Error: 34.6 → 29.3

Time: 1.3m

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 9.719006733297615868376551534099752919713 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\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)} \cdot \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 9.205201470071284026503566371060614934683 \cdot 10^{83}:\\ \;\;\;\;\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(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{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 2.124245349407411794952057275273122817895 \cdot 10^{137} \lor \neg \left(U \le 2.14798498375804656509468198073242974239 \cdot 10^{190}\right):\\ \;\;\;\;\sqrt{\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)} \cdot \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{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot 2} \cdot \sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), n, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r201177 = 2.0;
        double r201178 = n;
        double r201179 = r201177 * r201178;
        double r201180 = U;
        double r201181 = r201179 * r201180;
        double r201182 = t;
        double r201183 = l;
        double r201184 = r201183 * r201183;
        double r201185 = Om;
        double r201186 = r201184 / r201185;
        double r201187 = r201177 * r201186;
        double r201188 = r201182 - r201187;
        double r201189 = r201183 / r201185;
        double r201190 = pow(r201189, r201177);
        double r201191 = r201178 * r201190;
        double r201192 = U_;
        double r201193 = r201180 - r201192;
        double r201194 = r201191 * r201193;
        double r201195 = r201188 - r201194;
        double r201196 = r201181 * r201195;
        double r201197 = sqrt(r201196);
        return r201197;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r201198 = U;
        double r201199 = -4.797165734624696e-88;
        bool r201200 = r201198 <= r201199;
        double r201201 = 2.0;
        double r201202 = n;
        double r201203 = r201201 * r201202;
        double r201204 = r201203 * r201198;
        double r201205 = U_;
        double r201206 = r201205 - r201198;
        double r201207 = l;
        double r201208 = Om;
        double r201209 = r201207 / r201208;
        double r201210 = 2.0;
        double r201211 = r201201 / r201210;
        double r201212 = pow(r201209, r201211);
        double r201213 = cbrt(r201202);
        double r201214 = r201213 * r201213;
        double r201215 = r201213 * r201212;
        double r201216 = r201214 * r201215;
        double r201217 = r201212 * r201216;
        double r201218 = r201207 * r201209;
        double r201219 = -r201201;
        double r201220 = t;
        double r201221 = fma(r201218, r201219, r201220);
        double r201222 = fma(r201206, r201217, r201221);
        double r201223 = r201204 * r201222;
        double r201224 = sqrt(r201223);
        double r201225 = 9.719006733297616e-12;
        bool r201226 = r201198 <= r201225;
        double r201227 = r201212 * r201202;
        double r201228 = r201206 * r201212;
        double r201229 = r201209 * r201219;
        double r201230 = fma(r201207, r201229, r201220);
        double r201231 = fma(r201227, r201228, r201230);
        double r201232 = r201198 * r201231;
        double r201233 = r201232 * r201203;
        double r201234 = sqrt(r201233);
        double r201235 = r201234 * r201234;
        double r201236 = sqrt(r201235);
        double r201237 = 9.205201470071284e+83;
        bool r201238 = r201198 <= r201237;
        double r201239 = cbrt(r201208);
        double r201240 = r201207 / r201239;
        double r201241 = pow(r201240, r201211);
        double r201242 = 1.0;
        double r201243 = r201239 * r201239;
        double r201244 = r201242 / r201243;
        double r201245 = pow(r201244, r201211);
        double r201246 = r201202 * r201245;
        double r201247 = r201241 * r201246;
        double r201248 = r201212 * r201247;
        double r201249 = fma(r201206, r201248, r201221);
        double r201250 = r201204 * r201249;
        double r201251 = sqrt(r201250);
        double r201252 = 2.124245349407412e+137;
        bool r201253 = r201198 <= r201252;
        double r201254 = 2.1479849837580466e+190;
        bool r201255 = r201198 <= r201254;
        double r201256 = !r201255;
        bool r201257 = r201253 || r201256;
        double r201258 = r201202 * r201198;
        double r201259 = r201258 * r201201;
        double r201260 = sqrt(r201259);
        double r201261 = pow(r201209, r201201);
        double r201262 = r201261 * r201206;
        double r201263 = r201208 / r201207;
        double r201264 = r201207 / r201263;
        double r201265 = fma(r201219, r201264, r201220);
        double r201266 = fma(r201262, r201202, r201265);
        double r201267 = sqrt(r201266);
        double r201268 = r201260 * r201267;
        double r201269 = r201257 ? r201236 : r201268;
        double r201270 = r201238 ? r201251 : r201269;
        double r201271 = r201226 ? r201236 : r201270;
        double r201272 = r201200 ? r201224 : r201271;
        return r201272;
}

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 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 < 9.719006733297616e-12 or 9.205201470071284e+83 < U < 2.124245349407412e+137 or 2.1479849837580466e+190 < U

   1. Initial program 37.4

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

   2. Simplified 34.9

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

   4. Applied sqr-pow 34.9

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

   5. Applied associate-*r* 34.1

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

   7. Applied add-cube-cbrt 34.1

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

   8. Applied associate-*l* 34.1

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

   9. Simplified 34.1

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

   11. Applied add-sqr-sqrt 34.1

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

   12. Simplified 37.5

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

   13. Simplified 30.9

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

   if 9.719006733297616e-12 < U < 9.205201470071284e+83

   1. Initial program 29.5

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

   2. Simplified 25.9

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

   4. Applied sqr-pow 25.9

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

   5. Applied associate-*r* 24.5

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

   7. Applied add-cube-cbrt 24.5

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

   8. Applied *-un-lft-identity 24.5

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

   9. Applied times-frac 24.5

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

   10. Applied unpow-prod-down 24.5

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

   11. Applied associate-*r* 24.5

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

   if 2.124245349407412e+137 < U < 2.1479849837580466e+190

   1. Initial program 29.7

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

   2. Simplified 26.3

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

   4. Applied sqrt-prod 35.7

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

   5. Simplified 35.7

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

   6. Simplified 36.5

      \[\leadsto \sqrt{2 \cdot \left(n \cdot U\right)} \cdot \color{blue}{\sqrt{\mathsf{fma}\left(\left(U* - U\right) \cdot {\left(\frac{\ell}{Om}\right)}^{2}, n, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\right)}}\]
3. Recombined 4 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 9.719006733297615868376551534099752919713 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\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)} \cdot \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 9.205201470071284026503566371060614934683 \cdot 10^{83}:\\ \;\;\;\;\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(\frac{\ell}{\sqrt[3]{Om}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(n \cdot {\left(\frac{1}{\sqrt[3]{Om} \cdot \sqrt[3]{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 2.124245349407411794952057275273122817895 \cdot 10^{137} \lor \neg \left(U \le 2.14798498375804656509468198073242974239 \cdot 10^{190}\right):\\ \;\;\;\;\sqrt{\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)} \cdot \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{else}:\\ \;\;\;\;\sqrt{\left(n \cdot U\right) \cdot 2} \cdot \sqrt{\mathsf{fma}\left({\left(\frac{\ell}{Om}\right)}^{2} \cdot \left(U* - U\right), n, \mathsf{fma}\left(-2, \frac{\ell}{\frac{Om}{\ell}}, t\right)\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*))))))