Toniolo and Linder, Equation (10+)

Average Error: 31.5 → 12.4

Time: 38.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le -2.5994779280181517 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(t \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;\ell \le -1.0539765897933676 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{\frac{\frac{2}{\ell} \cdot \frac{\left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right) \cdot t}{\ell} + \frac{t}{\frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}}{k \cdot k}}}{\cos k}}\\ \mathbf{elif}\;\ell \le 7.328736741637738 \cdot 10^{-217}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + \frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)}{\cos k}}\\ \mathbf{elif}\;\ell \le 8.785675515272592 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\frac{\frac{2}{\ell} \cdot \frac{\left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right) \cdot t}{\ell} + \frac{t}{\frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}}{k \cdot k}}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot t\right)\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r3327541 = 2.0;
        double r3327542 = t;
        double r3327543 = 3.0;
        double r3327544 = pow(r3327542, r3327543);
        double r3327545 = l;
        double r3327546 = r3327545 * r3327545;
        double r3327547 = r3327544 / r3327546;
        double r3327548 = k;
        double r3327549 = sin(r3327548);
        double r3327550 = r3327547 * r3327549;
        double r3327551 = tan(r3327548);
        double r3327552 = r3327550 * r3327551;
        double r3327553 = 1.0;
        double r3327554 = r3327548 / r3327542;
        double r3327555 = pow(r3327554, r3327541);
        double r3327556 = r3327553 + r3327555;
        double r3327557 = r3327556 + r3327553;
        double r3327558 = r3327552 * r3327557;
        double r3327559 = r3327541 / r3327558;
        return r3327559;
}

↓

double f(double t, double l, double k) {
        double r3327560 = l;
        double r3327561 = -2.5994779280181517e+141;
        bool r3327562 = r3327560 <= r3327561;
        double r3327563 = 2.0;
        double r3327564 = k;
        double r3327565 = sin(r3327564);
        double r3327566 = t;
        double r3327567 = tan(r3327564);
        double r3327568 = r3327564 / r3327566;
        double r3327569 = r3327568 * r3327568;
        double r3327570 = r3327563 + r3327569;
        double r3327571 = r3327567 * r3327570;
        double r3327572 = r3327566 * r3327571;
        double r3327573 = r3327566 / r3327560;
        double r3327574 = r3327573 * r3327573;
        double r3327575 = r3327572 * r3327574;
        double r3327576 = r3327565 * r3327575;
        double r3327577 = r3327563 / r3327576;
        double r3327578 = -1.0539765897933676e-121;
        bool r3327579 = r3327560 <= r3327578;
        double r3327580 = r3327563 / r3327560;
        double r3327581 = r3327565 * r3327566;
        double r3327582 = r3327581 * r3327581;
        double r3327583 = r3327582 * r3327566;
        double r3327584 = r3327583 / r3327560;
        double r3327585 = r3327580 * r3327584;
        double r3327586 = r3327560 / r3327565;
        double r3327587 = r3327586 * r3327586;
        double r3327588 = r3327564 * r3327564;
        double r3327589 = r3327587 / r3327588;
        double r3327590 = r3327566 / r3327589;
        double r3327591 = r3327585 + r3327590;
        double r3327592 = cos(r3327564);
        double r3327593 = r3327591 / r3327592;
        double r3327594 = r3327563 / r3327593;
        double r3327595 = 7.328736741637738e-217;
        bool r3327596 = r3327560 <= r3327595;
        double r3327597 = r3327565 * r3327573;
        double r3327598 = r3327597 * r3327597;
        double r3327599 = r3327566 * r3327598;
        double r3327600 = r3327570 * r3327599;
        double r3327601 = r3327600 / r3327592;
        double r3327602 = r3327563 / r3327601;
        double r3327603 = 8.785675515272592e+137;
        bool r3327604 = r3327560 <= r3327603;
        double r3327605 = r3327597 * r3327573;
        double r3327606 = r3327567 * r3327566;
        double r3327607 = r3327605 * r3327606;
        double r3327608 = r3327607 * r3327570;
        double r3327609 = r3327563 / r3327608;
        double r3327610 = r3327604 ? r3327594 : r3327609;
        double r3327611 = r3327596 ? r3327602 : r3327610;
        double r3327612 = r3327579 ? r3327594 : r3327611;
        double r3327613 = r3327562 ? r3327577 : r3327612;
        return r3327613;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if l < -2.5994779280181517e+141

   1. Initial program 60.8

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

   2. Simplified 28.8

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

   4. Applied div-inv 28.8

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

   5. Applied times-frac 28.8

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

   6. Applied associate-/l* 29.1

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

   7. Simplified 28.1

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

   9. Applied associate-/l/ 28.1

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

   if -2.5994779280181517e+141 < l < -1.0539765897933676e-121 or 7.328736741637738e-217 < l < 8.785675515272592e+137

   1. Initial program 26.2

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

   2. Simplified 21.9

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

   4. Applied div-inv 21.9

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

   5. Applied times-frac 21.9

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

   6. Applied associate-/l* 21.8

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

   7. Simplified 17.2

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

   9. Applied div-inv 17.2

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

   10. Applied associate-/l* 17.2

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

   11. Simplified 15.9

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

   13. Applied tan-quot 15.9

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

   14. Applied associate-*l/ 15.9

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

   15. Applied associate-*l/ 15.9

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

   16. Applied associate-*r/ 15.9

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

   17. Simplified 16.3

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

   18. Taylor expanded around inf 19.3

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

   19. Simplified 10.7

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

   if -1.0539765897933676e-121 < l < 7.328736741637738e-217

   1. Initial program 23.8

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

   2. Simplified 13.0

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

   4. Applied div-inv 13.0

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

   5. Applied times-frac 13.0

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

   6. Applied associate-/l* 12.5

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

   7. Simplified 10.1

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

   9. Applied div-inv 10.1

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

   10. Applied associate-/l* 10.1

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

   11. Simplified 9.4

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

   13. Applied tan-quot 9.4

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

   14. Applied associate-*l/ 9.4

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

   15. Applied associate-*l/ 9.4

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

   16. Applied associate-*r/ 9.4

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

   17. Simplified 7.3

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

   if 8.785675515272592e+137 < 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 28.8

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

   4. Applied div-inv 28.8

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

   5. Applied times-frac 28.8

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

   6. Applied associate-/l* 28.9

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

   7. Simplified 26.7

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

   9. Applied div-inv 26.7

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

   10. Applied associate-/l* 26.7

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

   11. Simplified 23.7

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -2.5994779280181517 \cdot 10^{+141}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(t \cdot \left(\tan k \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;\ell \le -1.0539765897933676 \cdot 10^{-121}:\\ \;\;\;\;\frac{2}{\frac{\frac{2}{\ell} \cdot \frac{\left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right) \cdot t}{\ell} + \frac{t}{\frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}}{k \cdot k}}}{\cos k}}\\ \mathbf{elif}\;\ell \le 7.328736741637738 \cdot 10^{-217}:\\ \;\;\;\;\frac{2}{\frac{\left(2 + \frac{k}{t} \cdot \frac{k}{t}\right) \cdot \left(t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \frac{t}{\ell}\right)\right)\right)}{\cos k}}\\ \mathbf{elif}\;\ell \le 8.785675515272592 \cdot 10^{+137}:\\ \;\;\;\;\frac{2}{\frac{\frac{2}{\ell} \cdot \frac{\left(\left(\sin k \cdot t\right) \cdot \left(\sin k \cdot t\right)\right) \cdot t}{\ell} + \frac{t}{\frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}}{k \cdot k}}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot t\right)\right) \cdot \left(2 + \frac{k}{t} \cdot \frac{k}{t}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))