Average Error: 17.2 → 16.5
Time: 2.0m
Precision: 64
Internal Precision: 128
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le -2.4456710616777348 \cdot 10^{+243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le 3.130616183306899 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1} \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if U < -2.4456710616777348e+243 or 3.130616183306899e+165 < U

    1. Initial program 39.4

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

    2. Simplified39.4

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

    3. Taylor expanded around inf 35.4

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

    4. Simplified35.4

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

    if -2.4456710616777348e+243 < U < 3.130616183306899e+165

    1. Initial program 12.7

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

    2. Simplified12.7

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le -2.4456710616777348 \cdot 10^{+243}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \le 3.130616183306899 \cdot 10^{+165}:\\ \;\;\;\;\sqrt{\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} \cdot \frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2} + 1} \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Reproduce

herbie shell --seed 2019051 
(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)))))