Toniolo and Linder, Equation (10+)

Average Error: 33.1 → 11.7

Time: 57.9s

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.16994539344101880776243334960063175807 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\left(\frac{\sqrt[3]{\tan k}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}}}\\ \mathbf{elif}\;t \le 2.641659376196321363068322206147910460738 \cdot 10^{-80}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{{\ell}^{2}}{{\left(\sin k\right)}^{2}} \cdot \cos k}{{k}^{2}}\right) \cdot {\left(\frac{1}{{t}^{1} \cdot {-1}^{2}}\right)}^{1} - \left({\left(\frac{1}{{-1}^{4}}\right)}^{1} \cdot 4\right) \cdot \frac{t}{\frac{{\left(\sin k\right)}^{2}}{\cos k} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r131291 = 2.0;
        double r131292 = t;
        double r131293 = 3.0;
        double r131294 = pow(r131292, r131293);
        double r131295 = l;
        double r131296 = r131295 * r131295;
        double r131297 = r131294 / r131296;
        double r131298 = k;
        double r131299 = sin(r131298);
        double r131300 = r131297 * r131299;
        double r131301 = tan(r131298);
        double r131302 = r131300 * r131301;
        double r131303 = 1.0;
        double r131304 = r131298 / r131292;
        double r131305 = pow(r131304, r131291);
        double r131306 = r131303 + r131305;
        double r131307 = r131306 + r131303;
        double r131308 = r131302 * r131307;
        double r131309 = r131291 / r131308;
        return r131309;
}

↓

double f(double t, double l, double k) {
        double r131310 = t;
        double r131311 = -3.1699453934410188e-86;
        bool r131312 = r131310 <= r131311;
        double r131313 = 2.0;
        double r131314 = cbrt(r131313);
        double r131315 = r131314 * r131314;
        double r131316 = k;
        double r131317 = tan(r131316);
        double r131318 = cbrt(r131317);
        double r131319 = l;
        double r131320 = cbrt(r131319);
        double r131321 = r131318 / r131320;
        double r131322 = cbrt(r131310);
        double r131323 = r131322 * r131322;
        double r131324 = 3.0;
        double r131325 = 2.0;
        double r131326 = r131324 / r131325;
        double r131327 = pow(r131323, r131326);
        double r131328 = r131327 / r131320;
        double r131329 = r131321 * r131328;
        double r131330 = r131318 * r131318;
        double r131331 = r131320 * r131320;
        double r131332 = r131330 / r131331;
        double r131333 = r131329 * r131332;
        double r131334 = r131333 * r131328;
        double r131335 = r131315 / r131334;
        double r131336 = sin(r131316);
        double r131337 = pow(r131322, r131324);
        double r131338 = r131314 / r131337;
        double r131339 = r131338 * r131320;
        double r131340 = 1.0;
        double r131341 = r131316 / r131310;
        double r131342 = pow(r131341, r131313);
        double r131343 = r131342 + r131340;
        double r131344 = r131340 + r131343;
        double r131345 = r131339 / r131344;
        double r131346 = r131336 / r131345;
        double r131347 = r131335 / r131346;
        double r131348 = 2.6416593761963214e-80;
        bool r131349 = r131310 <= r131348;
        double r131350 = pow(r131319, r131325);
        double r131351 = pow(r131336, r131325);
        double r131352 = r131350 / r131351;
        double r131353 = cos(r131316);
        double r131354 = r131352 * r131353;
        double r131355 = pow(r131316, r131325);
        double r131356 = r131354 / r131355;
        double r131357 = r131313 * r131356;
        double r131358 = 1.0;
        double r131359 = pow(r131310, r131340);
        double r131360 = -1.0;
        double r131361 = pow(r131360, r131313);
        double r131362 = r131359 * r131361;
        double r131363 = r131358 / r131362;
        double r131364 = pow(r131363, r131340);
        double r131365 = r131357 * r131364;
        double r131366 = 4.0;
        double r131367 = pow(r131360, r131366);
        double r131368 = r131358 / r131367;
        double r131369 = pow(r131368, r131340);
        double r131370 = r131369 * r131366;
        double r131371 = r131351 / r131353;
        double r131372 = 4.0;
        double r131373 = pow(r131316, r131372);
        double r131374 = r131373 / r131350;
        double r131375 = r131371 * r131374;
        double r131376 = r131310 / r131375;
        double r131377 = r131370 * r131376;
        double r131378 = r131365 - r131377;
        double r131379 = pow(r131322, r131326);
        double r131380 = cbrt(r131331);
        double r131381 = r131379 / r131380;
        double r131382 = r131317 * r131381;
        double r131383 = r131382 / r131319;
        double r131384 = cbrt(r131320);
        double r131385 = r131379 / r131384;
        double r131386 = r131383 * r131385;
        double r131387 = r131386 * r131328;
        double r131388 = r131315 / r131387;
        double r131389 = r131388 / r131346;
        double r131390 = r131349 ? r131378 : r131389;
        double r131391 = r131312 ? r131347 : r131390;
        return r131391;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -3.1699453934410188e-86

   1. Initial program 24.1

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 18.7

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 18.8

      \[\leadsto \frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   5. Applied add-cube-cbrt 18.9

      \[\leadsto \frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   6. Applied unpow-prod-down 18.9

      \[\leadsto \frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   7. Applied times-frac 16.8

      \[\leadsto \frac{\frac{2}{\frac{\tan k}{\ell} \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right)}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   8. Applied associate-*r* 15.4

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]
   9. Using strategy rm

   10. Applied sqr-pow 15.4

       \[\leadsto \frac{\frac{2}{\left(\frac{\tan k}{\ell} \cdot \frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   11. Applied times-frac 12.5

       \[\leadsto \frac{\frac{2}{\left(\frac{\tan k}{\ell} \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   12. Applied associate-*r* 11.3

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 11.3

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{\left(\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   15. Applied times-frac 11.1

       \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   16. Applied associate-/l* 8.5

       \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}}\]

   17. Simplified 8.5

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\color{blue}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]
   18. Using strategy rm

   19. Applied add-cube-cbrt 8.6

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   20. Applied add-cube-cbrt 8.6

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\color{blue}{\left(\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}\right) \cdot \sqrt[3]{\tan k}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   21. Applied times-frac 8.6

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\color{blue}{\left(\frac{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{\tan k}}{\sqrt[3]{\ell}}\right)} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   22. Applied associate-*l* 6.0

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\color{blue}{\left(\frac{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{\tan k}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)\right)} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   if -3.1699453934410188e-86 < t < 2.6416593761963214e-80

   1. Initial program 60.4

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 59.5

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}}\]

   3. Taylor expanded around -inf 27.5

      \[\leadsto \color{blue}{2 \cdot \left({\left(\frac{1}{{t}^{1} \cdot {-1}^{2}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2} \cdot {k}^{2}}\right) - 4 \cdot \left(\frac{t \cdot \left({\ell}^{2} \cdot \cos k\right)}{{\left(\sin k\right)}^{2} \cdot {k}^{4}} \cdot {\left(\frac{1}{{-1}^{4}}\right)}^{1}\right)}\]

   4. Simplified 26.9

      \[\leadsto \color{blue}{{\left(\frac{1}{{t}^{1} \cdot {-1}^{2}}\right)}^{1} \cdot \left(\frac{\frac{{\ell}^{2}}{{\left(\sin k\right)}^{2}} \cdot \cos k}{{k}^{2}} \cdot 2\right) - \left(4 \cdot {\left(\frac{1}{{-1}^{4}}\right)}^{1}\right) \cdot \frac{t}{\frac{{\left(\sin k\right)}^{2}}{\cos k} \cdot \frac{{k}^{4}}{{\ell}^{2}}}}\]

   if 2.6416593761963214e-80 < t

   1. Initial program 23.1

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   2. Simplified 18.0

      \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\ell}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 18.1

      \[\leadsto \frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{{t}^{3}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   5. Applied add-cube-cbrt 18.3

      \[\leadsto \frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   6. Applied unpow-prod-down 18.3

      \[\leadsto \frac{\frac{2}{\frac{\tan k}{\ell} \cdot \frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   7. Applied times-frac 16.3

      \[\leadsto \frac{\frac{2}{\frac{\tan k}{\ell} \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right)}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   8. Applied associate-*r* 14.7

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]
   9. Using strategy rm

   10. Applied sqr-pow 14.7

       \[\leadsto \frac{\frac{2}{\left(\frac{\tan k}{\ell} \cdot \frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   11. Applied times-frac 12.2

       \[\leadsto \frac{\frac{2}{\left(\frac{\tan k}{\ell} \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   12. Applied associate-*r* 11.1

       \[\leadsto \frac{\frac{2}{\color{blue}{\left(\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]
   13. Using strategy rm

   14. Applied add-cube-cbrt 11.1

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{\left(\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   15. Applied times-frac 10.8

       \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}\]

   16. Applied associate-/l* 8.1

       \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 1\right) + 1\right)}{\frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}}\]

   17. Simplified 8.1

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\color{blue}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]
   18. Using strategy rm

   19. Applied add-cube-cbrt 8.1

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   20. Applied cbrt-prod 8.1

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   21. Applied unpow-prod-down 8.1

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   22. Applied times-frac 8.1

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k}{\ell} \cdot \color{blue}{\left(\frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   23. Applied associate-*r* 8.1

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\color{blue}{\left(\left(\frac{\tan k}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]

   24. Simplified 6.8

       \[\leadsto \frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\color{blue}{\frac{\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.16994539344101880776243334960063175807 \cdot 10^{-86}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\left(\frac{\sqrt[3]{\tan k}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}}}\\ \mathbf{elif}\;t \le 2.641659376196321363068322206147910460738 \cdot 10^{-80}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{{\ell}^{2}}{{\left(\sin k\right)}^{2}} \cdot \cos k}{{k}^{2}}\right) \cdot {\left(\frac{1}{{t}^{1} \cdot {-1}^{2}}\right)}^{1} - \left({\left(\frac{1}{{-1}^{4}}\right)}^{1} \cdot 4\right) \cdot \frac{t}{\frac{{\left(\sin k\right)}^{2}}{\cos k} \cdot \frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\left(\frac{\tan k \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}}}{\frac{\sin k}{\frac{\frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}}{1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))