Toniolo and Linder, Equation (10+)

Average Error: 31.4 → 16.5

Time: 47.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le 6.763037352752422302077139099297667443997 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \le 1.491801365253996162354700266962756132707 \cdot 10^{71}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell}} \cdot \sin k\right)\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r120481 = 2.0;
        double r120482 = t;
        double r120483 = 3.0;
        double r120484 = pow(r120482, r120483);
        double r120485 = l;
        double r120486 = r120485 * r120485;
        double r120487 = r120484 / r120486;
        double r120488 = k;
        double r120489 = sin(r120488);
        double r120490 = r120487 * r120489;
        double r120491 = tan(r120488);
        double r120492 = r120490 * r120491;
        double r120493 = 1.0;
        double r120494 = r120488 / r120482;
        double r120495 = pow(r120494, r120481);
        double r120496 = r120493 + r120495;
        double r120497 = r120496 + r120493;
        double r120498 = r120492 * r120497;
        double r120499 = r120481 / r120498;
        return r120499;
}

↓

double f(double t, double l, double k) {
        double r120500 = l;
        double r120501 = 6.763037352752422e-16;
        bool r120502 = r120500 <= r120501;
        double r120503 = 2.0;
        double r120504 = t;
        double r120505 = cbrt(r120504);
        double r120506 = r120505 * r120505;
        double r120507 = 3.0;
        double r120508 = 2.0;
        double r120509 = r120507 / r120508;
        double r120510 = pow(r120506, r120509);
        double r120511 = r120510 / r120500;
        double r120512 = pow(r120505, r120507);
        double r120513 = r120512 / r120500;
        double r120514 = k;
        double r120515 = sin(r120514);
        double r120516 = r120513 * r120515;
        double r120517 = r120511 * r120516;
        double r120518 = tan(r120514);
        double r120519 = r120517 * r120518;
        double r120520 = r120510 * r120519;
        double r120521 = 1.0;
        double r120522 = r120514 / r120504;
        double r120523 = pow(r120522, r120503);
        double r120524 = r120521 + r120523;
        double r120525 = r120524 + r120521;
        double r120526 = r120520 * r120525;
        double r120527 = r120503 / r120526;
        double r120528 = 1.4918013652539962e+71;
        bool r120529 = r120500 <= r120528;
        double r120530 = 3.0;
        double r120531 = pow(r120504, r120530);
        double r120532 = pow(r120515, r120508);
        double r120533 = r120531 * r120532;
        double r120534 = cos(r120514);
        double r120535 = pow(r120500, r120508);
        double r120536 = r120534 * r120535;
        double r120537 = r120533 / r120536;
        double r120538 = r120503 * r120537;
        double r120539 = 1.0;
        double r120540 = -1.0;
        double r120541 = pow(r120540, r120507);
        double r120542 = r120539 / r120541;
        double r120543 = pow(r120542, r120521);
        double r120544 = pow(r120514, r120508);
        double r120545 = r120544 * r120532;
        double r120546 = r120504 * r120545;
        double r120547 = r120546 / r120536;
        double r120548 = r120543 * r120547;
        double r120549 = r120538 - r120548;
        double r120550 = r120503 / r120549;
        double r120551 = cbrt(r120512);
        double r120552 = r120551 * r120551;
        double r120553 = cbrt(r120500);
        double r120554 = r120553 * r120553;
        double r120555 = r120552 / r120554;
        double r120556 = r120551 / r120553;
        double r120557 = r120556 * r120515;
        double r120558 = r120555 * r120557;
        double r120559 = r120511 * r120558;
        double r120560 = r120510 * r120559;
        double r120561 = r120560 * r120518;
        double r120562 = r120561 * r120525;
        double r120563 = r120503 / r120562;
        double r120564 = r120529 ? r120550 : r120563;
        double r120565 = r120502 ? r120527 : r120564;
        return r120565;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if l < 6.763037352752422e-16

   1. Initial program 27.8

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

   3. Applied add-cube-cbrt 28.0

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

   4. Applied unpow-prod-down 28.0

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

   5. Applied times-frac 21.9

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

   6. Applied associate-*l* 19.9

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

   8. Applied *-un-lft-identity 19.9

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

   9. Applied sqr-pow 19.9

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

   10. Applied times-frac 15.4

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

   11. Simplified 15.4

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

   13. Applied associate-*l* 15.1

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

   15. Applied associate-*l* 14.6

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

   if 6.763037352752422e-16 < l < 1.4918013652539962e+71

   1. Initial program 27.8

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

   2. Taylor expanded around -inf 18.7

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

   if 1.4918013652539962e+71 < l

   1. Initial program 52.1

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

   3. Applied add-cube-cbrt 52.2

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

   4. Applied unpow-prod-down 52.2

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

   5. Applied times-frac 38.2

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

   6. Applied associate-*l* 38.2

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

   8. Applied *-un-lft-identity 38.2

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

   9. Applied sqr-pow 38.2

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

   10. Applied times-frac 25.8

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

   11. Simplified 25.8

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

   13. Applied associate-*l* 25.1

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

   15. Applied add-cube-cbrt 25.2

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

   16. Applied add-cube-cbrt 25.2

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

   17. Applied times-frac 25.2

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

   18. Applied associate-*l* 25.2

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

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 6.763037352752422302077139099297667443997 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \mathbf{elif}\;\ell \le 1.491801365253996162354700266962756132707 \cdot 10^{71}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell}} \cdot \sin k\right)\right)\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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))))