Average Error: 17.1 → 7.4
Time: 32.2s
Precision: 64
Internal Precision: 320
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^* \cdot \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.1

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

  2. Initial simplification7.4

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

  4. Applied associate-*r*7.4

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

  5. Final simplification7.4

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

Runtime

Time bar (total: 32.2s)Debug logProfile

herbie shell --seed 2018219 +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)))))