Error

Derivation

1. Split input into 5 regimes

2. if k < -1.926020170106595e+67

   1. Initial program 40.9

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

   2. Simplified23.4

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

   3. Taylor expanded around -inf 20.3

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

   4. Simplified20.4

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

   6. Applied *-un-lft-identity20.4

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

   7. Applied associate-/r*20.4

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

   8. Simplified5.0

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

   10. Applied div-inv5.0

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

   11. Taylor expanded around -inf 20.3

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

   12. Simplified4.8

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

   if -1.926020170106595e+67 < k < -4.932748447749146e-95

   1. Initial program 55.0

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

   2. Simplified37.0

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

   3. Taylor expanded around -inf 18.6

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

   4. Simplified13.4

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

   6. Applied times-frac1.5

      \[\leadsto \frac{\cos k}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\sin k}}}} \cdot 2\]

   if -4.932748447749146e-95 < k < 6.863227758783006e-135

   1. Initial program 62.3

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

   2. Simplified57.0

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

   3. Taylor expanded around -inf 61.8

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

   4. Simplified35.8

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

   6. Applied associate-*l*14.9

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

   if 6.863227758783006e-135 < k < 1.6229494989538582e+125

   1. Initial program 53.4

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

   2. Simplified34.5

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

   3. Taylor expanded around -inf 21.0

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

   4. Simplified14.5

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

   6. Applied times-frac3.2

      \[\leadsto \frac{\cos k}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\sin k}}}} \cdot 2\]

   7. Applied add-cube-cbrt3.4

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

   8. Applied times-frac3.3

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

   9. Simplified3.3

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

   if 1.6229494989538582e+125 < k

   1. Initial program 39.1

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

   2. Simplified23.3

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

   3. Taylor expanded around -inf 23.2

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

   4. Simplified23.2

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

   6. Applied *-un-lft-identity23.2

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

   7. Applied associate-/r*23.2

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

   8. Simplified3.7

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

   10. Applied div-inv3.7

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

   11. Applied times-frac3.6

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

   12. Simplified3.5

       \[\leadsto \left(\frac{\cos k}{t} \cdot \color{blue}{\left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell}{k \cdot \sin k}\right)}\right) \cdot 2\]
3. Recombined 5 regimes into one program.

4. Final simplification4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.926020170106595 \cdot 10^{+67}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\left(\sin k \cdot \sin k\right) \cdot t}\right)\\ \mathbf{elif}\;k \le -4.932748447749146 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k \cdot k}{\frac{\ell}{\sin k}} \cdot \frac{t}{\frac{\ell}{\sin k}}}\\ \mathbf{elif}\;k \le 6.863227758783006 \cdot 10^{-135}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\ell}{\sin k} \cdot \frac{\ell}{\sin k}}}\\ \mathbf{elif}\;k \le 1.6229494989538582 \cdot 10^{+125}:\\ \;\;\;\;\left(\frac{\frac{\ell}{\sin k}}{\frac{k}{\sqrt[3]{\cos k}} \cdot \frac{k}{\sqrt[3]{\cos k}}} \cdot \frac{\sqrt[3]{\cos k}}{\frac{t}{\frac{\ell}{\sin k}}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell}{k \cdot \sin k}\right) \cdot \frac{\cos k}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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))))