Toniolo and Linder, Equation (10-)

Average Error: 48.1 → 26.9

Time: 33.3s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.33963357783478596 \cdot 10^{-103}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3} \cdot {\left(\sin k\right)}^{2}}}{\frac{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}{\ell}} \cdot \frac{\ell}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}\right) \cdot \frac{\cos k}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2}}\\ \mathbf{elif}\;t \le -4.40605940276589298 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \left({\left(\frac{{\left({\left(e^{1}\right)}^{\left(\log \left(\frac{-1}{t}\right)\right)}\right)}^{1} \cdot {\left(e^{\log \left(\frac{-1}{k}\right) \cdot 2}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \le 5.5559757494791578 \cdot 10^{-175}:\\ \;\;\;\;\frac{\cos k}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2}} \cdot \left(\left(\frac{\ell \cdot \ell}{{\left(\sin k\right)}^{2}} \cdot e^{-\left(1.33333333333333326 \cdot \log k + 1.666666666666667 \cdot \log t\right)}\right) \cdot 2\right)\\ \mathbf{elif}\;t \le 5.21580220030066152 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3} \cdot {\left(\sin k\right)}^{2}}}{\frac{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}{\ell}} \cdot \frac{\ell}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}\right) \cdot \frac{\cos k}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if t < -3.33963357783478596e-103 or 5.21580220030066152e-99 < t

   1. Initial program 42.6

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

   2. Simplified 31.7

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

   4. Applied add-cube-cbrt 31.8

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

   5. Applied unpow-prod-down 31.8

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

   6. Applied tan-quot 31.8

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

   7. Applied associate-*r/ 31.8

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

   8. Applied associate-/r/ 31.8

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

   9. Applied times-frac 28.1

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

   10. Simplified 25.2

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

   12. Applied unpow-prod-down 25.2

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

   13. Applied associate-*l/ 24.6

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

   14. Applied associate-/r/ 24.6

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

   15. Applied times-frac 21.0

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

   17. Applied div-inv 20.9

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

   18. Applied cbrt-prod 20.9

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

   20. Applied associate-*l/ 20.8

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

   21. Applied associate-/r/ 20.9

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

   22. Applied associate-/l* 20.0

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

   if -3.33963357783478596e-103 < t < -4.40605940276589298e-275

   1. Initial program 63.1

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

   2. Simplified 63.1

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

   3. Taylor expanded around -inf 44.5

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

   4. Simplified 44.6

      \[\leadsto \color{blue}{\left({\left(\frac{{\left(e^{2 \cdot \left(0 + \log \left(\frac{-1}{k}\right)\right)}\right)}^{1} \cdot {\left({\left(e^{1}\right)}^{\left(0 + \log \left(\frac{-1}{t}\right)\right)}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(\sin k\right)}^{2}}\right) \cdot 2}\]

   if -4.40605940276589298e-275 < t < 5.5559757494791578e-175

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied add-cube-cbrt 64.0

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

   5. Applied unpow-prod-down 64.0

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

   6. Applied tan-quot 64.0

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

   7. Applied associate-*r/ 64.0

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

   8. Applied associate-/r/ 64.0

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

   9. Applied times-frac 64.0

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

   10. Simplified 64.0

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

   12. Applied unpow-prod-down 64.0

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

   13. Applied associate-*l/ 64.0

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

   14. Applied associate-/r/ 64.0

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

   15. Applied times-frac 64.0

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

   17. Applied div-inv 64.0

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

   18. Applied cbrt-prod 64.0

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

   19. Taylor expanded around inf 63.3

       \[\leadsto \color{blue}{\left(2 \cdot \frac{{\ell}^{2} \cdot \left(e^{1.33333333333333326 \cdot \left(\log 1 + \log \left(\frac{1}{k}\right)\right)} \cdot e^{1.666666666666667 \cdot \left(\log \left(\frac{1}{t}\right) + \log 1\right)}\right)}{{\left(\sin k\right)}^{2}}\right)} \cdot \frac{\cos k}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2}}\]

   20. Simplified 52.6

       \[\leadsto \color{blue}{\left(2 \cdot \left(\frac{\ell \cdot \ell}{{\left(\sin k\right)}^{2}} \cdot e^{1.33333333333333326 \cdot \left(-\log k\right) + 1.666666666666667 \cdot \left(-\log t\right)}\right)\right)} \cdot \frac{\cos k}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2}}\]

   if 5.5559757494791578e-175 < t < 5.21580220030066152e-99

   1. Initial program 60.7

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

   2. Simplified 60.7

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

   4. Applied sqr-pow 60.7

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

   5. Applied times-frac 34.6

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

4. Final simplification 26.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.33963357783478596 \cdot 10^{-103}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3} \cdot {\left(\sin k\right)}^{2}}}{\frac{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}{\ell}} \cdot \frac{\ell}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}\right) \cdot \frac{\cos k}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2}}\\ \mathbf{elif}\;t \le -4.40605940276589298 \cdot 10^{-275}:\\ \;\;\;\;2 \cdot \left({\left(\frac{{\left({\left(e^{1}\right)}^{\left(\log \left(\frac{-1}{t}\right)\right)}\right)}^{1} \cdot {\left(e^{\log \left(\frac{-1}{k}\right) \cdot 2}\right)}^{1}}{{-1}^{3}}\right)}^{1} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\left(\sin k\right)}^{2}}\right)\\ \mathbf{elif}\;t \le 5.5559757494791578 \cdot 10^{-175}:\\ \;\;\;\;\frac{\cos k}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2}} \cdot \left(\left(\frac{\ell \cdot \ell}{{\left(\sin k\right)}^{2}} \cdot e^{-\left(1.33333333333333326 \cdot \log k + 1.666666666666667 \cdot \log t\right)}\right) \cdot 2\right)\\ \mathbf{elif}\;t \le 5.21580220030066152 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3} \cdot {\left(\sin k\right)}^{2}}}{\frac{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}{\ell}} \cdot \frac{\ell}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}\right) \cdot \frac{\cos k}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{1}{t}}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))