Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 18.4

Time: 15.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.0049859230952732 \cdot 10^{123}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)\right)}\\ \mathbf{elif}\;t \le -6.83674943102251321 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left(\sin k \cdot \left(\frac{{1}^{1} \cdot {\left(\frac{-1}{t}\right)}^{1}}{\cos k} \cdot \left(k \cdot \left(k \cdot \left(-\sin k\right)\right)\right) - 2 \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left({1}^{1} \cdot {\left(\frac{-1}{t}\right)}^{1}\right)}^{1}}\right)}^{1}\right)\right)\right)}\right)\\ \mathbf{elif}\;t \le 2.76746326083733335 \cdot 10^{108}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(2 \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left(\frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1}\right) + \sin k \cdot \frac{k \cdot k}{\cos k}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\left(\ell \cdot \left(\sqrt[3]{2} \cdot \frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)\right) \cdot \frac{\sqrt[3]{2}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}\right)\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if t < -1.0049859230952732e123

   1. Initial program 21.9

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

   2. Simplified 24.1

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

   4. Applied add-cube-cbrt 24.1

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

   5. Applied unpow-prod-down 24.1

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

   6. Applied associate-*l* 23.4

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

   7. Simplified 17.9

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

   9. Applied sqr-pow 17.9

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

   10. Applied associate-*l* 14.9

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

   11. Simplified 12.1

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

   13. Applied *-un-lft-identity 12.1

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

   14. Applied times-frac 11.9

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

   15. Applied associate-*l* 9.9

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

   16. Simplified 10.9

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

   18. Applied associate-*r/ 10.9

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

   19. Applied associate-*r/ 10.9

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

   20. Applied associate-*r/ 11.2

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

   21. Simplified 11.2

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

   if -1.0049859230952732e123 < t < -6.83674943102251321e-160

   1. Initial program 28.4

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

   2. Simplified 27.6

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

   4. Applied add-cube-cbrt 27.9

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

   5. Applied unpow-prod-down 27.9

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

   6. Applied associate-*l* 23.3

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

   7. Simplified 22.1

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

   8. Taylor expanded around -inf 21.3

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

   9. Simplified 20.7

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

   if -6.83674943102251321e-160 < t < 2.76746326083733335e108

   1. Initial program 45.2

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

   2. Simplified 43.9

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

   4. Applied add-cube-cbrt 44.1

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

   5. Applied unpow-prod-down 44.1

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

   6. Applied associate-*l* 40.7

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

   7. Simplified 40.1

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

   9. Applied sqr-pow 40.1

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

   10. Applied associate-*l* 30.7

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

   11. Simplified 30.2

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

   12. Taylor expanded around inf 30.3

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

   13. Simplified 25.3

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

   if 2.76746326083733335e108 < t

   1. Initial program 24.0

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

   2. Simplified 26.6

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

   4. Applied add-cube-cbrt 26.6

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

   5. Applied unpow-prod-down 26.6

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

   6. Applied associate-*l* 25.6

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

   7. Simplified 19.9

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

   9. Applied sqr-pow 20.0

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

   10. Applied associate-*l* 16.4

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

   11. Simplified 13.5

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

   13. Applied *-un-lft-identity 13.5

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

   14. Applied times-frac 13.5

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

   15. Applied associate-*l* 11.0

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

   16. Simplified 12.0

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

   18. Applied add-cube-cbrt 12.0

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

   19. Applied times-frac 12.0

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

   20. Applied associate-*r* 11.0

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

   21. Simplified 11.0

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

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.0049859230952732 \cdot 10^{123}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)\right)}\\ \mathbf{elif}\;t \le -6.83674943102251321 \cdot 10^{-160}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot \left(\sin k \cdot \left(\frac{{1}^{1} \cdot {\left(\frac{-1}{t}\right)}^{1}}{\cos k} \cdot \left(k \cdot \left(k \cdot \left(-\sin k\right)\right)\right) - 2 \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left({1}^{1} \cdot {\left(\frac{-1}{t}\right)}^{1}\right)}^{1}}\right)}^{1}\right)\right)\right)}\right)\\ \mathbf{elif}\;t \le 2.76746326083733335 \cdot 10^{108}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\sin k \cdot \left(2 \cdot \left(\frac{\sin k}{\cos k} \cdot {\left(\frac{1}{{\left(\frac{1}{{t}^{2}}\right)}^{1}}\right)}^{1}\right) + \sin k \cdot \frac{k \cdot k}{\cos k}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{1}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}} \cdot \left(\left(\ell \cdot \left(\sqrt[3]{2} \cdot \frac{\sqrt[3]{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}\right)\right) \cdot \frac{\sqrt[3]{2}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot {\left(\sqrt[3]{t}\right)}^{3}\right)\right)}\right)\right)\\ \end{array}\]

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))))