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Derivation

1. Split input into 2 regimes

2. if J < -6.191074441095509e-260 or 2.0165152943700504e-256 < J

   1. Initial program 14.9

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

   2. Initial simplification6.0

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

   4. Applied associate-/l/5.9

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

   if -6.191074441095509e-260 < J < 2.0165152943700504e-256

   1. Initial program 42.3

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

   2. Initial simplification28.1

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

   4. Applied associate-/l/28.1

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

   6. Applied add-cube-cbrt28.6

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

   7. Applied associate-*l*28.6

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

   8. Taylor expanded around inf 30.2

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

   9. Simplified30.2

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

4. Final simplification8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -6.191074441095509 \cdot 10^{-260} \lor \neg \left(J \le 2.0165152943700504 \cdot 10^{-256}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{2}}{\cos \left(\frac{K}{2}\right) \cdot J}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Runtime

Time bar (total: 27.0s)Debug logProfile

herbie shell --seed 2018339 +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)))))