Average Error: 17.6 → 7.8
Time: 46.6s
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{U}{J}}{\cos \left(\frac{K}{2}\right) \cdot 2}\right)^2}^*\right) \cdot \left(J \cdot -2\right)\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.6

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

  2. Simplified7.8

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

  4. Applied associate-*r*7.8

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

  6. Applied associate-/r*7.8

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

  7. Final simplification7.8

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

Reproduce

herbie shell --seed 2019100 +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)))))