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Derivation

1. Split input into 2 regimes

2. if U < -6.405075533169645e+191

   1. Initial program 40.2

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

   2. Initial simplification25.2

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

   4. Applied associate-/l/25.2

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

   6. Applied add-cube-cbrt25.5

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

   7. Applied associate-*l*25.5

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

   8. Taylor expanded around -inf 33.9

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

   9. Simplified33.9

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

   if -6.405075533169645e+191 < U

   1. Initial program 14.4

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

   2. Initial simplification5.8

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

   4. Applied associate-/l/5.8

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

   6. Applied add-sqr-sqrt6.0

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

   7. Applied associate-*l*6.0

      \[\leadsto \color{blue}{\sqrt{\sqrt{1^2 + \left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)^2}^*} \cdot \left(\sqrt{\sqrt{1^2 + \left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -6.405075533169645 \cdot 10^{+191}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{1^2 + \left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)^2}^*} \cdot \left(\sqrt{\sqrt{1^2 + \left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)^2}^*} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 22.8s)Debug logProfile

herbie shell --seed 2018340 +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)))))