Average Error: 17.1 → 16.6
Time: 2.4m
Precision: 64
Internal Precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -8.197015198459638 \cdot 10^{+220}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le -1.769594101284306 \cdot 10^{+172}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}\right) \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\\ \mathbf{elif}\;U \le -3.411237592812121 \cdot 10^{+125}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le 2.9320804634238083 \cdot 10^{+230}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}\right) \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if U < -8.197015198459638e+220 or -1.769594101284306e+172 < U < -3.411237592812121e+125 or 2.9320804634238083e+230 < U

    1. Initial program 37.6

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

    2. Simplified37.6

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

    3. Taylor expanded around inf 34.6

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

    4. Simplified34.6

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

    if -8.197015198459638e+220 < U < -1.769594101284306e+172 or -3.411237592812121e+125 < U < 2.9320804634238083e+230

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around inf 12.8

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

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -8.197015198459638 \cdot 10^{+220}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le -1.769594101284306 \cdot 10^{+172}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}\right) \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\\ \mathbf{elif}\;U \le -3.411237592812121 \cdot 10^{+125}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le 2.9320804634238083 \cdot 10^{+230}:\\ \;\;\;\;\left(-2 \cdot \sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1}\right) \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 
(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)))))