Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 11.2

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -7.529474516601856360117855515427412693526 \cdot 10^{87}:\\ \;\;\;\;\left(\frac{{\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\ell \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\ell \cdot \cos k\right)\right)\right)}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;\ell \le 1.09956338955371745401648431203451639418 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \cos k}}{\ell}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{\left(\sin k\right)}^{2}} \cdot \left(\left(\ell \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\ell \cdot \cos k\right)\right)\right) \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}\right)\right) \cdot 2\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r197525 = 2.0;
        double r197526 = t;
        double r197527 = 3.0;
        double r197528 = pow(r197526, r197527);
        double r197529 = l;
        double r197530 = r197529 * r197529;
        double r197531 = r197528 / r197530;
        double r197532 = k;
        double r197533 = sin(r197532);
        double r197534 = r197531 * r197533;
        double r197535 = tan(r197532);
        double r197536 = r197534 * r197535;
        double r197537 = 1.0;
        double r197538 = r197532 / r197526;
        double r197539 = pow(r197538, r197525);
        double r197540 = r197537 + r197539;
        double r197541 = r197540 - r197537;
        double r197542 = r197536 * r197541;
        double r197543 = r197525 / r197542;
        return r197543;
}

↓

double f(double t, double l, double k) {
        double r197544 = l;
        double r197545 = -7.529474516601856e+87;
        bool r197546 = r197544 <= r197545;
        double r197547 = 1.0;
        double r197548 = t;
        double r197549 = 1.0;
        double r197550 = pow(r197548, r197549);
        double r197551 = r197547 / r197550;
        double r197552 = pow(r197551, r197549);
        double r197553 = k;
        double r197554 = 2.0;
        double r197555 = 2.0;
        double r197556 = r197554 / r197555;
        double r197557 = pow(r197553, r197556);
        double r197558 = r197547 / r197557;
        double r197559 = pow(r197558, r197549);
        double r197560 = cos(r197553);
        double r197561 = r197544 * r197560;
        double r197562 = r197559 * r197561;
        double r197563 = r197544 * r197562;
        double r197564 = r197552 * r197563;
        double r197565 = sin(r197553);
        double r197566 = pow(r197565, r197555);
        double r197567 = r197564 / r197566;
        double r197568 = r197567 * r197559;
        double r197569 = r197568 * r197554;
        double r197570 = 1.0995633895537175e-10;
        bool r197571 = r197544 <= r197570;
        double r197572 = r197550 * r197557;
        double r197573 = r197547 / r197572;
        double r197574 = pow(r197573, r197549);
        double r197575 = r197566 / r197561;
        double r197576 = r197575 / r197544;
        double r197577 = r197559 / r197576;
        double r197578 = r197574 * r197577;
        double r197579 = r197578 * r197554;
        double r197580 = r197547 / r197566;
        double r197581 = r197558 / r197550;
        double r197582 = pow(r197581, r197549);
        double r197583 = r197563 * r197582;
        double r197584 = r197580 * r197583;
        double r197585 = r197584 * r197554;
        double r197586 = r197571 ? r197579 : r197585;
        double r197587 = r197546 ? r197569 : r197586;
        return r197587;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < -7.529474516601856e+87

   1. Initial program 60.1

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

   2. Simplified 57.9

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

   3. Taylor expanded around inf 52.4

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

   5. Applied sqr-pow 52.4

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

   6. Applied associate-*r* 49.2

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

   7. Simplified 49.2

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

   9. Applied *-un-lft-identity 49.2

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

   10. Applied times-frac 48.8

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

   11. Applied unpow-prod-down 48.8

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

   12. Applied associate-*l* 43.9

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

   13. Simplified 43.7

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

   15. Applied *-un-lft-identity 43.7

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

   16. Applied times-frac 43.4

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

   17. Applied unpow-prod-down 43.4

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

   18. Applied associate-*l* 42.9

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

   19. Simplified 24.3

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

   if -7.529474516601856e+87 < l < 1.0995633895537175e-10

   1. Initial program 44.5

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

   2. Simplified 35.7

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

   3. Taylor expanded around inf 13.3

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

   5. Applied sqr-pow 13.3

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

   6. Applied associate-*r* 12.0

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

   7. Simplified 12.0

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

   9. Applied *-un-lft-identity 12.0

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

   10. Applied times-frac 11.9

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

   11. Applied unpow-prod-down 11.9

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

   12. Applied associate-*l* 11.7

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

   13. Simplified 11.7

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

   15. Applied associate-/l* 11.7

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

   16. Simplified 7.9

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

   if 1.0995633895537175e-10 < l

   1. Initial program 53.1

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

   2. Simplified 48.4

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

   3. Taylor expanded around inf 35.3

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

   5. Applied sqr-pow 35.3

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

   6. Applied associate-*r* 31.1

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

   7. Simplified 31.1

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

   9. Applied *-un-lft-identity 31.1

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

   10. Applied times-frac 30.7

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

   11. Applied unpow-prod-down 30.7

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

   12. Applied associate-*l* 27.6

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

   13. Simplified 27.4

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

   15. Applied div-inv 27.4

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

   16. Applied associate-*r* 26.0

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

   17. Simplified 14.5

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -7.529474516601856360117855515427412693526 \cdot 10^{87}:\\ \;\;\;\;\left(\frac{{\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\ell \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\ell \cdot \cos k\right)\right)\right)}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{elif}\;\ell \le 1.09956338955371745401648431203451639418 \cdot 10^{-10}:\\ \;\;\;\;\left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}}{\frac{\frac{{\left(\sin k\right)}^{2}}{\ell \cdot \cos k}}{\ell}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{\left(\sin k\right)}^{2}} \cdot \left(\left(\ell \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\ell \cdot \cos k\right)\right)\right) \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1}\right)\right) \cdot 2\\ \end{array}\]

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))))