Maksimov and Kolovsky, Equation (3)

Average Error: 18.0 → 18.1

Time: 35.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;J \le -9.697315888830236817320599117892605157675 \cdot 10^{-63}:\\ \;\;\;\;\left(\sqrt{\sqrt{\sqrt[3]{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \sqrt[3]{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}}} \cdot \sqrt{\sqrt{\sqrt[3]{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}}}\right) \cdot \left(\left(-2 \cdot \left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot J\right)\right)\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}}\right)\\ \mathbf{elif}\;J \le 5.176467849961618590484301862929324273807 \cdot 10^{-152}:\\ \;\;\;\;-2 \cdot \left(U \cdot \sqrt{0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}} \cdot \left(\left(-2 \cdot \left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot J\right)\right)\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}}\right)\\ \end{array}\]

C

double f(double J, double K, double U) {
        double r5450801 = -2.0;
        double r5450802 = J;
        double r5450803 = r5450801 * r5450802;
        double r5450804 = K;
        double r5450805 = 2.0;
        double r5450806 = r5450804 / r5450805;
        double r5450807 = cos(r5450806);
        double r5450808 = r5450803 * r5450807;
        double r5450809 = 1.0;
        double r5450810 = U;
        double r5450811 = r5450805 * r5450802;
        double r5450812 = r5450811 * r5450807;
        double r5450813 = r5450810 / r5450812;
        double r5450814 = pow(r5450813, r5450805);
        double r5450815 = r5450809 + r5450814;
        double r5450816 = sqrt(r5450815);
        double r5450817 = r5450808 * r5450816;
        return r5450817;
}

↓

double f(double J, double K, double U) {
        double r5450818 = J;
        double r5450819 = -9.697315888830237e-63;
        bool r5450820 = r5450818 <= r5450819;
        double r5450821 = 1.0;
        double r5450822 = U;
        double r5450823 = K;
        double r5450824 = 2.0;
        double r5450825 = r5450823 / r5450824;
        double r5450826 = cos(r5450825);
        double r5450827 = r5450826 * r5450818;
        double r5450828 = r5450827 * r5450824;
        double r5450829 = r5450822 / r5450828;
        double r5450830 = pow(r5450829, r5450824);
        double r5450831 = r5450821 + r5450830;
        double r5450832 = cbrt(r5450831);
        double r5450833 = r5450832 * r5450832;
        double r5450834 = sqrt(r5450833);
        double r5450835 = sqrt(r5450834);
        double r5450836 = sqrt(r5450832);
        double r5450837 = sqrt(r5450836);
        double r5450838 = r5450835 * r5450837;
        double r5450839 = -2.0;
        double r5450840 = cbrt(r5450826);
        double r5450841 = cbrt(r5450840);
        double r5450842 = r5450841 * r5450841;
        double r5450843 = r5450841 * r5450842;
        double r5450844 = r5450840 * r5450843;
        double r5450845 = r5450840 * r5450818;
        double r5450846 = r5450844 * r5450845;
        double r5450847 = r5450839 * r5450846;
        double r5450848 = sqrt(r5450831);
        double r5450849 = sqrt(r5450848);
        double r5450850 = r5450847 * r5450849;
        double r5450851 = r5450838 * r5450850;
        double r5450852 = 5.176467849961619e-152;
        bool r5450853 = r5450818 <= r5450852;
        double r5450854 = 0.25;
        double r5450855 = sqrt(r5450854);
        double r5450856 = r5450822 * r5450855;
        double r5450857 = r5450839 * r5450856;
        double r5450858 = r5450849 * r5450850;
        double r5450859 = r5450853 ? r5450857 : r5450858;
        double r5450860 = r5450820 ? r5450851 : r5450859;
        return r5450860;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 3 regimes

2. if J < -9.697315888830237e-63

   1. Initial program 6.2

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

   2. Simplified 6.2

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

   4. Applied add-cube-cbrt 6.7

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

   5. Applied associate-*l* 6.7

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

   7. Applied add-cube-cbrt 6.8

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

   9. Applied add-sqr-sqrt 6.8

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

   10. Applied sqrt-prod 6.8

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

   11. Applied associate-*l* 6.8

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

   13. Applied add-cube-cbrt 6.8

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

   14. Applied sqrt-prod 6.8

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

   15. Applied sqrt-prod 6.9

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

   if -9.697315888830237e-63 < J < 5.176467849961619e-152

   1. Initial program 37.7

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

   2. Simplified 37.7

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

   3. Taylor expanded around inf 36.8

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{0.25} \cdot U\right)}\]

   if 5.176467849961619e-152 < J

   1. Initial program 10.4

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

   2. Simplified 10.4

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

   4. Applied add-cube-cbrt 10.8

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

   5. Applied associate-*l* 10.8

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

   7. Applied add-cube-cbrt 10.9

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

   9. Applied add-sqr-sqrt 10.9

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

   10. Applied sqrt-prod 11.0

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

   11. Applied associate-*l* 11.0

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -9.697315888830236817320599117892605157675 \cdot 10^{-63}:\\ \;\;\;\;\left(\sqrt{\sqrt{\sqrt[3]{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \sqrt[3]{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}}} \cdot \sqrt{\sqrt{\sqrt[3]{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}}}\right) \cdot \left(\left(-2 \cdot \left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot J\right)\right)\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}}\right)\\ \mathbf{elif}\;J \le 5.176467849961618590484301862929324273807 \cdot 10^{-152}:\\ \;\;\;\;-2 \cdot \left(U \cdot \sqrt{0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}} \cdot \left(\left(-2 \cdot \left(\left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot J\right)\right)\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))