Toniolo and Linder, Equation (10-)

Average Error: 48.0 → 12.3

Time: 26.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \le -2.484097675036736748520774305137126666949 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right)\right)\right)\\ \mathbf{elif}\;k \le -2.299675617169774612848622390750025820293 \cdot 10^{-136}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \ell\right) \cdot \cos k}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{elif}\;k \le 2.14912132342830487586681637498725524116 \cdot 10^{-85}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\left|\sin k\right|} \cdot \ell}{\frac{\left|\sin k\right|}{\ell}}\right)\\ \mathbf{elif}\;k \le 1.450138159937341979949839776479400714061 \cdot 10^{142}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\left|\sin k\right|}\right) \cdot \ell}{\frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right)\right)\right)\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r87586 = 2.0;
        double r87587 = t;
        double r87588 = 3.0;
        double r87589 = pow(r87587, r87588);
        double r87590 = l;
        double r87591 = r87590 * r87590;
        double r87592 = r87589 / r87591;
        double r87593 = k;
        double r87594 = sin(r87593);
        double r87595 = r87592 * r87594;
        double r87596 = tan(r87593);
        double r87597 = r87595 * r87596;
        double r87598 = 1.0;
        double r87599 = r87593 / r87587;
        double r87600 = pow(r87599, r87586);
        double r87601 = r87598 + r87600;
        double r87602 = r87601 - r87598;
        double r87603 = r87597 * r87602;
        double r87604 = r87586 / r87603;
        return r87604;
}

↓

double f(double t, double l, double k) {
        double r87605 = k;
        double r87606 = -2.4840976750367367e+154;
        bool r87607 = r87605 <= r87606;
        double r87608 = 2.0;
        double r87609 = 1.0;
        double r87610 = cbrt(r87609);
        double r87611 = r87610 * r87610;
        double r87612 = 2.0;
        double r87613 = r87608 / r87612;
        double r87614 = pow(r87605, r87613);
        double r87615 = r87611 / r87614;
        double r87616 = 1.0;
        double r87617 = pow(r87615, r87616);
        double r87618 = t;
        double r87619 = pow(r87618, r87616);
        double r87620 = r87614 * r87619;
        double r87621 = r87610 / r87620;
        double r87622 = pow(r87621, r87616);
        double r87623 = cos(r87605);
        double r87624 = sin(r87605);
        double r87625 = fabs(r87624);
        double r87626 = r87623 / r87625;
        double r87627 = l;
        double r87628 = r87625 / r87627;
        double r87629 = r87627 / r87628;
        double r87630 = r87626 * r87629;
        double r87631 = r87622 * r87630;
        double r87632 = r87617 * r87631;
        double r87633 = r87608 * r87632;
        double r87634 = -2.2996756171697746e-136;
        bool r87635 = r87605 <= r87634;
        double r87636 = pow(r87605, r87608);
        double r87637 = r87636 * r87619;
        double r87638 = r87609 / r87637;
        double r87639 = pow(r87638, r87616);
        double r87640 = r87639 * r87627;
        double r87641 = r87640 * r87623;
        double r87642 = r87625 * r87628;
        double r87643 = r87641 / r87642;
        double r87644 = r87608 * r87643;
        double r87645 = 2.149121323428305e-85;
        bool r87646 = r87605 <= r87645;
        double r87647 = r87614 * r87620;
        double r87648 = r87609 / r87647;
        double r87649 = pow(r87648, r87616);
        double r87650 = r87626 * r87627;
        double r87651 = r87650 / r87628;
        double r87652 = r87649 * r87651;
        double r87653 = r87608 * r87652;
        double r87654 = 1.450138159937342e+142;
        bool r87655 = r87605 <= r87654;
        double r87656 = r87639 * r87626;
        double r87657 = r87656 * r87627;
        double r87658 = r87657 / r87628;
        double r87659 = r87608 * r87658;
        double r87660 = r87655 ? r87659 : r87633;
        double r87661 = r87646 ? r87653 : r87660;
        double r87662 = r87635 ? r87644 : r87661;
        double r87663 = r87607 ? r87633 : r87662;
        return r87663;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if k < -2.4840976750367367e+154 or 1.450138159937342e+142 < k

   1. Initial program 38.8

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

   2. Simplified 33.6

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

   3. Taylor expanded around inf 23.6

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

   5. Applied sqr-pow 23.6

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

   6. Applied associate-*l* 19.3

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

   8. Applied add-sqr-sqrt 19.3

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

   9. Applied times-frac 19.3

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

   10. Simplified 19.3

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

   11. Simplified 19.3

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

   13. Applied add-cube-cbrt 19.3

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

   14. Applied times-frac 19.1

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

   15. Applied unpow-prod-down 19.1

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

   16. Applied associate-*l* 16.3

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

   if -2.4840976750367367e+154 < k < -2.2996756171697746e-136

   1. Initial program 53.6

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

   2. Simplified 43.9

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

   3. Taylor expanded around inf 17.9

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

   5. Applied sqr-pow 17.9

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

   6. Applied associate-*l* 17.9

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

   8. Applied add-sqr-sqrt 17.9

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

   9. Applied times-frac 17.9

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

   10. Simplified 17.9

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

   11. Simplified 16.3

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

   13. Applied frac-times 14.9

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

   14. Applied associate-*r/ 8.7

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

   15. Simplified 8.8

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

   if -2.2996756171697746e-136 < k < 2.149121323428305e-85

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   3. Taylor expanded around inf 50.2

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

   5. Applied sqr-pow 50.2

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

   6. Applied associate-*l* 50.2

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

   8. Applied add-sqr-sqrt 50.2

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

   9. Applied times-frac 50.2

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

   10. Simplified 50.2

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

   11. Simplified 29.0

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

   13. Applied associate-*r/ 15.9

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

   if 2.149121323428305e-85 < k < 1.450138159937342e+142

   1. Initial program 52.6

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

   2. Simplified 40.8

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

   3. Taylor expanded around inf 15.3

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

   5. Applied sqr-pow 15.3

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

   6. Applied associate-*l* 15.2

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

   8. Applied add-sqr-sqrt 15.2

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

   9. Applied times-frac 15.2

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

   10. Simplified 15.2

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

   11. Simplified 14.5

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

   13. Applied associate-*r/ 13.8

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

   14. Applied associate-*r/ 6.8

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

   15. Simplified 7.8

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

4. Final simplification 12.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -2.484097675036736748520774305137126666949 \cdot 10^{154}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right)\right)\right)\\ \mathbf{elif}\;k \le -2.299675617169774612848622390750025820293 \cdot 10^{-136}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \ell\right) \cdot \cos k}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{elif}\;k \le 2.14912132342830487586681637498725524116 \cdot 10^{-85}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\frac{\cos k}{\left|\sin k\right|} \cdot \ell}{\frac{\left|\sin k\right|}{\ell}}\right)\\ \mathbf{elif}\;k \le 1.450138159937341979949839776479400714061 \cdot 10^{142}:\\ \;\;\;\;2 \cdot \frac{\left({\left(\frac{1}{{k}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\left|\sin k\right|}\right) \cdot \ell}{\frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \frac{\ell}{\frac{\left|\sin k\right|}{\ell}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))))