Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 10.7

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.6704179460034538 \cdot 10^{-223}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{1}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\sqrt[3]{\ell}}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 3.08766200373374408 \cdot 10^{216}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{1}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{1}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r366320 = 2.0;
        double r366321 = t;
        double r366322 = 3.0;
        double r366323 = pow(r366321, r366322);
        double r366324 = l;
        double r366325 = r366324 * r366324;
        double r366326 = r366323 / r366325;
        double r366327 = k;
        double r366328 = sin(r366327);
        double r366329 = r366326 * r366328;
        double r366330 = tan(r366327);
        double r366331 = r366329 * r366330;
        double r366332 = 1.0;
        double r366333 = r366327 / r366321;
        double r366334 = pow(r366333, r366320);
        double r366335 = r366332 + r366334;
        double r366336 = r366335 - r366332;
        double r366337 = r366331 * r366336;
        double r366338 = r366320 / r366337;
        return r366338;
}

↓

double f(double t, double l, double k) {
        double r366339 = l;
        double r366340 = r366339 * r366339;
        double r366341 = 4.670417946003454e-223;
        bool r366342 = r366340 <= r366341;
        double r366343 = 2.0;
        double r366344 = 1.0;
        double r366345 = k;
        double r366346 = 2.0;
        double r366347 = r366343 / r366346;
        double r366348 = pow(r366345, r366347);
        double r366349 = t;
        double r366350 = 1.0;
        double r366351 = pow(r366349, r366350);
        double r366352 = r366348 * r366351;
        double r366353 = r366348 * r366352;
        double r366354 = r366344 / r366353;
        double r366355 = pow(r366354, r366350);
        double r366356 = cos(r366345);
        double r366357 = cbrt(r366356);
        double r366358 = r366357 * r366357;
        double r366359 = sin(r366345);
        double r366360 = cbrt(r366359);
        double r366361 = pow(r366360, r366346);
        double r366362 = cbrt(r366339);
        double r366363 = r366362 * r366362;
        double r366364 = r366361 / r366363;
        double r366365 = r366364 / r366344;
        double r366366 = r366358 / r366365;
        double r366367 = r366366 / r366344;
        double r366368 = r366355 * r366367;
        double r366369 = r366361 / r366362;
        double r366370 = r366369 / r366339;
        double r366371 = r366357 / r366370;
        double r366372 = r366371 / r366361;
        double r366373 = r366368 * r366372;
        double r366374 = r366343 * r366373;
        double r366375 = 3.087662003733744e+216;
        bool r366376 = r366340 <= r366375;
        double r366377 = sqrt(r366344);
        double r366378 = r366377 / r366348;
        double r366379 = pow(r366378, r366350);
        double r366380 = r366344 / r366352;
        double r366381 = pow(r366380, r366350);
        double r366382 = pow(r366339, r366346);
        double r366383 = r366356 * r366382;
        double r366384 = pow(r366359, r366346);
        double r366385 = r366383 / r366384;
        double r366386 = r366381 * r366385;
        double r366387 = r366379 * r366386;
        double r366388 = r366343 * r366387;
        double r366389 = 4.0;
        double r366390 = pow(r366360, r366389);
        double r366391 = r366390 / r366339;
        double r366392 = r366358 / r366391;
        double r366393 = pow(r366360, r366344);
        double r366394 = r366392 / r366393;
        double r366395 = r366355 * r366394;
        double r366396 = r366344 / r366339;
        double r366397 = r366357 / r366396;
        double r366398 = r366397 / r366393;
        double r366399 = r366395 * r366398;
        double r366400 = r366343 * r366399;
        double r366401 = r366376 ? r366388 : r366400;
        double r366402 = r366342 ? r366374 : r366401;
        return r366402;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if (* l l) < 4.670417946003454e-223

   1. Initial program 45.6

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

   2. Simplified 36.2

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

   3. Taylor expanded around inf 16.3

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

   5. Applied sqr-pow 16.3

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

   6. Applied associate-*l* 16.2

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

   8. Applied add-cube-cbrt 16.3

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

   9. Applied unpow-prod-down 16.3

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

   10. Applied associate-/r* 16.0

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

   11. Simplified 11.4

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
   12. Using strategy rm

   13. Applied *-un-lft-identity 11.4

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{\color{blue}{1 \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}}\right)\]

   14. Applied *-un-lft-identity 11.4

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\color{blue}{1 \cdot \ell}}}}{1 \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   15. Applied add-cube-cbrt 11.5

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{1 \cdot \ell}}}{1 \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   16. Applied sqr-pow 11.5

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{\color{blue}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)} \cdot {\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{1 \cdot \ell}}}{1 \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   17. Applied times-frac 11.0

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\color{blue}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell}}}}{1 \cdot \ell}}}{1 \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   18. Applied times-frac 11.0

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\color{blue}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1} \cdot \frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell}}}{\ell}}}}{1 \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   19. Applied add-cube-cbrt 11.0

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}\right) \cdot \sqrt[3]{\cos k}}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1} \cdot \frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell}}}{\ell}}}{1 \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   20. Applied times-frac 10.4

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell}}}{\ell}}}}{1 \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]

   21. Applied times-frac 8.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{1} \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell}}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)}\right)\]

   22. Applied associate-*r* 6.2

       \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{1}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{4}{2}\right)}}{\sqrt[3]{\ell}}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)}\]

   if 4.670417946003454e-223 < (* l l) < 3.087662003733744e+216

   1. Initial program 44.2

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

   2. Simplified 34.3

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

   3. Taylor expanded around inf 9.7

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

   5. Applied sqr-pow 9.7

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

   6. Applied associate-*l* 5.1

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

   8. Applied add-sqr-sqrt 5.1

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

   9. Applied times-frac 4.8

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

   10. Applied unpow-prod-down 4.8

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

   11. Applied associate-*l* 2.5

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

   12. Simplified 2.5

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

   if 3.087662003733744e+216 < (* l l)

   1. Initial program 59.9

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

   2. Simplified 58.0

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

   3. Taylor expanded around inf 53.8

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

   5. Applied sqr-pow 53.8

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

   6. Applied associate-*l* 51.6

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

   8. Applied add-cube-cbrt 51.8

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

   9. Applied unpow-prod-down 51.8

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

   10. Applied associate-/r* 51.8

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

   11. Simplified 51.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\]
   12. Using strategy rm

   13. Applied sqr-pow 51.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}{\ell}}}{\color{blue}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)}}}\right)\]

   14. Applied div-inv 51.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\color{blue}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell} \cdot \frac{1}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)}}\right)\]

   15. Applied add-cube-cbrt 51.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}\right) \cdot \sqrt[3]{\cos k}}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell} \cdot \frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)}}\right)\]

   16. Applied times-frac 51.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\color{blue}{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)}}\right)\]

   17. Applied times-frac 51.8

       \[\leadsto 2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)}}\right)}\right)\]

   18. Applied associate-*r* 34.0

       \[\leadsto 2 \cdot \color{blue}{\left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{\left(\frac{2}{2}\right)}}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.6704179460034538 \cdot 10^{-223}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{1}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{\frac{{\left(\sqrt[3]{\sin k}\right)}^{2}}{\sqrt[3]{\ell}}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \cdot \ell \le 3.08766200373374408 \cdot 10^{216}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\cos k}}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{1}}\right) \cdot \frac{\frac{\sqrt[3]{\cos k}}{\frac{1}{\ell}}}{{\left(\sqrt[3]{\sin k}\right)}^{1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))