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Derivation

1. Split input into 2 regimes

2. if J < -7.384923856485112e-264 or 7.035784431961942e-228 < J

   1. Initial program 14.1

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

   2. Initial simplification5.3

      \[\leadsto \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\]
   3. Using strategy rm

   4. Applied associate-/l/5.3

      \[\leadsto \sqrt{1^2 + \color{blue}{\left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)}^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\]

   if -7.384923856485112e-264 < J < 7.035784431961942e-228

   1. Initial program 40.9

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

   2. Initial simplification27.6

      \[\leadsto \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt27.8

      \[\leadsto \color{blue}{\left(\sqrt{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \sqrt{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*}\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\]

   5. Applied associate-*l*27.7

      \[\leadsto \color{blue}{\sqrt{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \left(\sqrt{\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right)}\]

   6. Taylor expanded around inf 33.5

      \[\leadsto \color{blue}{-1 \cdot U}\]

   7. Simplified33.5

      \[\leadsto \color{blue}{-U}\]
3. Recombined 2 regimes into one program.

4. Final simplification8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -7.384923856485112 \cdot 10^{-264} \lor \neg \left(J \le 7.035784431961942 \cdot 10^{-228}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))

Details

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