Average Error: 16.8 → 7.9
Time: 24.9s
Precision: 64
Internal Precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -1.1026116457842678 \cdot 10^{+247}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1^2 + \left(\frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)^2}^* \cdot \left(-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if U < -1.1026116457842678e+247

    1. Initial program 38.9

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

    2. Simplified25.2

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

    4. Applied add-cube-cbrt25.5

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

    5. Applied associate-*l*25.5

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

    6. Taylor expanded around inf 34.4

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

    7. Simplified34.4

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

    if -1.1026116457842678e+247 < U

    1. Initial program 15.8

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

    2. Simplified6.6

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

  4. Final simplification7.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -1.1026116457842678 \cdot 10^{+247}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1^2 + \left(\frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)^2}^* \cdot \left(-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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)))))