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Derivation

1. Split input into 2 regimes

2. if (* l l) < 1.1997133799852223e+295

   1. Initial program 44.6

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

   2. Taylor expanded around -inf 17.2

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

   4. Applied associate-*r*15.7

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

   6. Applied unpow215.7

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

   7. Applied associate-*r*13.2

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

   9. Applied times-frac12.3

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

   if 1.1997133799852223e+295 < (* l l)

   1. Initial program 61.2

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

   3. Applied cube-mult61.2

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

   4. Applied times-frac51.5

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

4. Final simplification18.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.1997133799852223 \cdot 10^{+295}:\\ \;\;\;\;\frac{2}{\frac{{\left(\sin k\right)}^{2}}{\cos k} \cdot \frac{\left(t \cdot k\right) \cdot k}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(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))))

Details

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