Error

Try it out

Derivation

1. Split input into 2 regimes

2. if J < -8.337566087302848e-234 or 2.216732952222694e-301 < J

   1. Initial program 15.4

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

   2. Initial simplification6.4

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

   3. Taylor expanded around -inf 6.3

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

   if -8.337566087302848e-234 < J < 2.216732952222694e-301

   1. Initial program 40.5

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

   2. Initial simplification26.4

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

   3. Taylor expanded around -inf 26.4

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

   5. Applied div-inv26.6

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

   6. Taylor expanded around inf 30.4

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

   7. Simplified30.4

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

4. Final simplification8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -8.337566087302848 \cdot 10^{-234} \lor \neg \left(J \le 2.216732952222694 \cdot 10^{-301}\right):\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \cdot \sqrt{1^2 + \left(\frac{1}{2} \cdot \frac{U}{J \cdot \cos \left(K \cdot \frac{1}{2}\right)}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 +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

Time bar (total: 24.5s)Debug log