Toniolo and Linder, Equation (10-)

Average Error: 48.4 → 11.0

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -3.104316958624234 \cdot 10^{141}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le -7.585502794490059 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r300496 = 2.0;
        double r300497 = t;
        double r300498 = 3.0;
        double r300499 = pow(r300497, r300498);
        double r300500 = l;
        double r300501 = r300500 * r300500;
        double r300502 = r300499 / r300501;
        double r300503 = k;
        double r300504 = sin(r300503);
        double r300505 = r300502 * r300504;
        double r300506 = tan(r300503);
        double r300507 = r300505 * r300506;
        double r300508 = 1.0;
        double r300509 = r300503 / r300497;
        double r300510 = pow(r300509, r300496);
        double r300511 = r300508 + r300510;
        double r300512 = r300511 - r300508;
        double r300513 = r300507 * r300512;
        double r300514 = r300496 / r300513;
        return r300514;
}

↓

double f(double t, double l, double k) {
        double r300515 = l;
        double r300516 = -3.104316958624234e+141;
        bool r300517 = r300515 <= r300516;
        double r300518 = 2.0;
        double r300519 = 1.0;
        double r300520 = k;
        double r300521 = 2.0;
        double r300522 = r300518 / r300521;
        double r300523 = pow(r300520, r300522);
        double r300524 = t;
        double r300525 = 1.0;
        double r300526 = pow(r300524, r300525);
        double r300527 = r300523 * r300526;
        double r300528 = r300523 * r300527;
        double r300529 = r300519 / r300528;
        double r300530 = pow(r300529, r300525);
        double r300531 = cos(r300520);
        double r300532 = sin(r300520);
        double r300533 = cbrt(r300532);
        double r300534 = 4.0;
        double r300535 = pow(r300533, r300534);
        double r300536 = r300535 / r300515;
        double r300537 = r300531 / r300536;
        double r300538 = cbrt(r300533);
        double r300539 = r300538 * r300538;
        double r300540 = pow(r300539, r300521);
        double r300541 = r300537 / r300540;
        double r300542 = r300530 * r300541;
        double r300543 = pow(r300538, r300521);
        double r300544 = r300515 / r300543;
        double r300545 = r300542 * r300544;
        double r300546 = r300518 * r300545;
        double r300547 = -7.585502794490059e-156;
        bool r300548 = r300515 <= r300547;
        double r300549 = r300519 / r300523;
        double r300550 = pow(r300549, r300525);
        double r300551 = r300519 / r300527;
        double r300552 = pow(r300551, r300525);
        double r300553 = pow(r300515, r300521);
        double r300554 = r300531 * r300553;
        double r300555 = pow(r300532, r300521);
        double r300556 = r300554 / r300555;
        double r300557 = r300552 * r300556;
        double r300558 = r300550 * r300557;
        double r300559 = r300518 * r300558;
        double r300560 = cbrt(r300519);
        double r300561 = pow(r300560, r300521);
        double r300562 = r300537 / r300561;
        double r300563 = r300530 * r300562;
        double r300564 = pow(r300533, r300521);
        double r300565 = r300515 / r300564;
        double r300566 = r300563 * r300565;
        double r300567 = r300518 * r300566;
        double r300568 = r300548 ? r300559 : r300567;
        double r300569 = r300517 ? r300546 : r300568;
        return r300569;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < -3.104316958624234e+141

   1. Initial program 62.5

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

   2. Simplified 61.9

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

   3. Taylor expanded around inf 61.2

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

   5. Applied sqr-pow 61.2

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

   6. Applied associate-*l* 60.3

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

   8. Applied add-cube-cbrt 60.4

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

   9. Applied unpow-prod-down 60.4

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

   10. Applied associate-/r* 60.4

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

   11. Simplified 60.4

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

   13. Applied add-cube-cbrt 60.4

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

   14. Applied unpow-prod-down 60.4

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

   15. Applied associate-/r/ 60.4

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

   16. Applied times-frac 60.4

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

   17. Applied associate-*r* 38.5

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

   if -3.104316958624234e+141 < l < -7.585502794490059e-156

   1. Initial program 45.7

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

   2. Simplified 35.6

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

   3. Taylor expanded around inf 10.3

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

   5. Applied sqr-pow 10.3

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

   6. Applied associate-*l* 6.3

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

   8. Applied *-un-lft-identity 6.3

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

   9. Applied times-frac 6.0

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

   10. Applied unpow-prod-down 6.0

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

   11. Applied associate-*l* 3.1

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

   if -7.585502794490059e-156 < l

   1. Initial program 47.7

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

   2. Simplified 39.8

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

   3. Taylor expanded around inf 21.3

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

   5. Applied sqr-pow 21.3

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

   6. Applied associate-*l* 19.8

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

   8. Applied add-cube-cbrt 20.0

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

   9. Applied unpow-prod-down 20.0

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

   10. Applied associate-/r* 19.7

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

   11. Simplified 16.6

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

   13. Applied *-un-lft-identity 16.6

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

   14. Applied cbrt-prod 16.6

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

   15. Applied unpow-prod-down 16.6

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

   16. Applied associate-/r/ 16.3

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

   17. Applied times-frac 15.4

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

   18. Applied associate-*r* 10.6

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.104316958624234 \cdot 10^{141}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sqrt[3]{\sin k}}\right)}^{2}}\right)\\ \mathbf{elif}\;\ell \le -7.585502794490059 \cdot 10^{-156}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\frac{{\left(\sqrt[3]{\sin k}\right)}^{4}}{\ell}}}{{\left(\sqrt[3]{1}\right)}^{2}}\right) \cdot \frac{\ell}{{\left(\sqrt[3]{\sin k}\right)}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(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))))