Average Error: 17.5 → 16.8
Time: 49.8s
Precision: 64
Internal Precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -6.266434501174935 \cdot 10^{+197}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le 3.213415482989335 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}} \cdot \left(\left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right) \cdot \sqrt{\sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if U < -6.266434501174935e+197 or 3.213415482989335e+218 < U

    1. Initial program 39.2

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

    2. Simplified39.2

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

    3. Taylor expanded around inf 34.5

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

    4. Simplified34.5

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

    if -6.266434501174935e+197 < U < 3.213415482989335e+218

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied add-sqr-sqrt13.2

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

    5. Applied sqrt-prod13.3

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

    6. Applied associate-*l*13.3

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -6.266434501174935 \cdot 10^{+197}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le 3.213415482989335 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}} \cdot \left(\left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right) \cdot \sqrt{\sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(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)))))