Average Error: 16.6 → 15.9
Time: 1.6m
Precision: 64
Internal Precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -3.054615077325277 \cdot 10^{+246}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le 1.768682694554876 \cdot 10^{+230}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\\ \mathbf{elif}\;U \le 9.90949087873545 \cdot 10^{+300}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if U < -3.054615077325277e+246 or 1.768682694554876e+230 < U < 9.90949087873545e+300

    1. Initial program 40.4

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

    2. Simplified40.4

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

    3. Taylor expanded around inf 34.2

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

    4. Simplified34.2

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

    if -3.054615077325277e+246 < U < 1.768682694554876e+230 or 9.90949087873545e+300 < U

    1. Initial program 13.8

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

    2. Simplified13.8

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

  4. Final simplification15.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -3.054615077325277 \cdot 10^{+246}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le 1.768682694554876 \cdot 10^{+230}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\\ \mathbf{elif}\;U \le 9.90949087873545 \cdot 10^{+300}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(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)))))