Error

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Derivation

1. Split input into 3 regimes

2. if l < -1.5498810850203252e-197

   1. Initial program 47.7

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

   2. Taylor expanded around -inf 25.2

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

   4. Applied associate-*r*23.9

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

   6. Applied unpow223.9

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

   7. Applied associate-*r*20.9

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

   9. Applied times-frac20.0

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

   if -1.5498810850203252e-197 < l < 2.933507809807463e-177

   1. Initial program 46.2

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

   2. Taylor expanded around -inf 21.0

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

   4. Applied associate-*r*20.2

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

   6. Applied unpow220.2

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

   7. Applied associate-*r*20.2

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

   9. Applied add-cbrt-cube20.2

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

   10. Applied add-cbrt-cube23.0

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

   11. Applied cbrt-undiv23.0

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

   12. Simplified15.6

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

   if 2.933507809807463e-177 < l

   1. Initial program 47.3

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

   2. Taylor expanded around -inf 25.0

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

   4. Applied associate-*r*23.6

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

   6. Applied unpow223.6

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

   7. Applied associate-*r*20.9

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

   9. Applied times-frac19.9

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

   10. Applied associate-/r*19.8

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

4. Final simplification18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.5498810850203252 \cdot 10^{-197}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{{\ell}^{2}} \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;\ell \le 2.933507809807463 \cdot 10^{-177}:\\ \;\;\;\;\frac{2}{\sqrt[3]{{\left(\left(\frac{\sin k}{\ell} \cdot \frac{\sin k}{\ell}\right) \cdot \frac{k \cdot \left(t \cdot k\right)}{\cos k}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{k \cdot \left(t \cdot k\right)}{{\ell}^{2}}}}{\frac{{\left(\sin k\right)}^{2}}{\cos k}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019021 
(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))))