Average Error: 17.6 → 17.0
Time: 2.6m
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -8.959113311661415 \cdot 10^{-141}:\\ \;\;\;\;J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\left(\sqrt{\sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}} \cdot \sqrt{\sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}}\right) \cdot -2\right)\right)\\ \mathbf{elif}\;J \le 1.3828181672431056 \cdot 10^{-264}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\left(\sqrt{\sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}} \cdot \sqrt{\sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}}\right) \cdot -2\right)\right)\\ \end{array}\]

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if J < -8.959113311661415e-141 or 1.3828181672431056e-264 < J

    1. Initial program 12.9

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

    2. Simplified12.9

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

    4. Applied associate-*r*12.9

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

    6. Applied add-sqr-sqrt12.9

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

    7. Applied sqrt-prod13.0

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

    if -8.959113311661415e-141 < J < 1.3828181672431056e-264

    1. Initial program 38.8

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

    2. Simplified38.8

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

    4. Applied associate-*r*38.8

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

    6. Applied add-sqr-sqrt38.8

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

    7. Applied sqrt-prod38.9

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

    8. Taylor expanded around -inf 35.2

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

    9. Simplified35.2

      \[\leadsto \color{blue}{-U}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification17.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -8.959113311661415 \cdot 10^{-141}:\\ \;\;\;\;J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\left(\sqrt{\sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}} \cdot \sqrt{\sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}}\right) \cdot -2\right)\right)\\ \mathbf{elif}\;J \le 1.3828181672431056 \cdot 10^{-264}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(\left(\sqrt{\sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}} \cdot \sqrt{\sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}}\right) \cdot -2\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 
(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)))))