Average Error: 16.5 → 5.6
Time: 48.3s
Precision: 64
Internal Precision: 576
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\right) \cdot \left(J \cdot -2\right) \le 1.7696509624536364 \cdot 10^{+308}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))) (cbrt (cos (/ K 2)))) (* -2 J)) < 1.7696509624536364e+308

    1. Initial program 13.9

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

    2. Initial simplification4.1

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

    4. Applied associate-*r*4.1

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

    if 1.7696509624536364e+308 < (* (* (* (hypot 1 (/ (/ (/ U 2) J) (cos (/ K 2)))) (* (cbrt (cos (/ K 2))) (cbrt (cos (/ K 2))))) (cbrt (cos (/ K 2)))) (* -2 J))

    1. Initial program 59.9

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

    2. Initial simplification59.9

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

    3. Taylor expanded around inf 30.1

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

    4. Simplified30.1

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\right) \cdot \left(J \cdot -2\right) \le 1.7696509624536364 \cdot 10^{+308}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1^2 + \left(\frac{\frac{\frac{U}{2}}{J}}{\cos \left(\frac{K}{2}\right)}\right)^2}^*\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Runtime

Time bar (total: 48.3s)Debug logProfile

herbie shell --seed 2018225 +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)))))