Derivation

Split input into 3 regimes
if (* l l) < 2.5983975435393225e-288
1. Initial program 45.7
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied associate-*l*45.7
  \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}\]
4. Applied simplify37.8
  \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\tan k \cdot \frac{k}{t}\right) \cdot \frac{k}{t}\right)}}\]
5. Taylor expanded around 0 20.5
  \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} - \frac{1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}\]
6. Applied simplify18.1
  \[\leadsto \color{blue}{\frac{\ell}{\frac{t}{\ell}} \cdot \left(\frac{2}{{k}^{4}} - \frac{\frac{\frac{1}{3}}{k}}{k}\right)}\]
if 2.5983975435393225e-288 < (* l l) < 5.228365117095298e+283
1. Initial program 42.9
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied associate-*l*42.9
  \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}\]
4. Applied simplify33.5
  \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\tan k \cdot \frac{k}{t}\right) \cdot \frac{k}{t}\right)}}\]
5. Taylor expanded around inf 11.8
  \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}}\]
6. Using strategy rm
7. Applied unpow211.8
  \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}\]
8. Applied associate-*l*7.7
  \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)}}\]
9. Using strategy rm
10. Applied times-frac3.8
  \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{k} \cdot \frac{\cos k}{k \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}\right)}\]
if 5.228365117095298e+283 < (* l l)
1. Initial program 61.0
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Using strategy rm
3. Applied associate-*l*61.1
  \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}}\]
4. Applied simplify60.6
  \[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\tan k \cdot \frac{k}{t}\right) \cdot \frac{k}{t}\right)}}\]
5. Taylor expanded around inf 59.1
  \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}}\]
6. Using strategy rm
7. Applied add-cbrt-cube60.7
  \[\leadsto 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\sqrt[3]{\left(\left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right) \cdot \left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)\right) \cdot \left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)}}}\]
8. Applied add-cbrt-cube62.4
  \[\leadsto 2 \cdot \frac{\color{blue}{\sqrt[3]{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \left({\ell}^{2} \cdot \cos k\right)\right) \cdot \left({\ell}^{2} \cdot \cos k\right)}}}{\sqrt[3]{\left(\left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right) \cdot \left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)\right) \cdot \left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)}}\]
9. Applied cbrt-undiv62.4
  \[\leadsto 2 \cdot \color{blue}{\sqrt[3]{\frac{\left(\left({\ell}^{2} \cdot \cos k\right) \cdot \left({\ell}^{2} \cdot \cos k\right)\right) \cdot \left({\ell}^{2} \cdot \cos k\right)}{\left(\left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right) \cdot \left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)\right) \cdot \left({k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)\right)}}}\]
10. Applied simplify35.2
  \[\leadsto 2 \cdot \sqrt[3]{\color{blue}{{\left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k}{\sin k \cdot \sin k}\right)}^{3}}}\]
Recombined 3 regimes into one program.

Error

Derivation

`if (* l l) < 2.5983975435393225e-288`

`if 2.5983975435393225e-288 < (* l l) < 5.228365117095298e+283`

`if 5.228365117095298e+283 < (* l l)`

Runtime