Average Error: 18.6 → 16.9
Time: 20.5s
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -1.7648872844499354 \cdot 10^{-122} \lor \neg \left(J \le 6.0013758387488983 \cdot 10^{-109}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}\\ \end{array}\]
\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 -1.7648872844499354 \cdot 10^{-122} \lor \neg \left(J \le 6.0013758387488983 \cdot 10^{-109}\right):\\
\;\;\;\;\left(\left(-2 \cdot J\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \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 \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}\\

\end{array}
double f(double J, double K, double U) {
        double r177792 = -2.0;
        double r177793 = J;
        double r177794 = r177792 * r177793;
        double r177795 = K;
        double r177796 = 2.0;
        double r177797 = r177795 / r177796;
        double r177798 = cos(r177797);
        double r177799 = r177794 * r177798;
        double r177800 = 1.0;
        double r177801 = U;
        double r177802 = r177796 * r177793;
        double r177803 = r177802 * r177798;
        double r177804 = r177801 / r177803;
        double r177805 = pow(r177804, r177796);
        double r177806 = r177800 + r177805;
        double r177807 = sqrt(r177806);
        double r177808 = r177799 * r177807;
        return r177808;
}
↓
double f(double J, double K, double U) {
        double r177809 = J;
        double r177810 = -1.7648872844499354e-122;
        bool r177811 = r177809 <= r177810;
        double r177812 = 6.001375838748898e-109;
        bool r177813 = r177809 <= r177812;
        double r177814 = !r177813;
        bool r177815 = r177811 || r177814;
        double r177816 = -2.0;
        double r177817 = r177816 * r177809;
        double r177818 = K;
        double r177819 = 2.0;
        double r177820 = r177818 / r177819;
        double r177821 = cos(r177820);
        double r177822 = cbrt(r177821);
        double r177823 = r177822 * r177822;
        double r177824 = cbrt(r177823);
        double r177825 = cbrt(r177822);
        double r177826 = r177824 * r177825;
        double r177827 = r177822 * r177826;
        double r177828 = r177817 * r177827;
        double r177829 = 1.0;
        double r177830 = U;
        double r177831 = r177819 * r177809;
        double r177832 = r177831 * r177821;
        double r177833 = r177830 / r177832;
        double r177834 = pow(r177833, r177819);
        double r177835 = r177829 + r177834;
        double r177836 = sqrt(r177835);
        double r177837 = r177822 * r177836;
        double r177838 = r177828 * r177837;
        double r177839 = r177817 * r177821;
        double r177840 = 0.25;
        double r177841 = sqrt(r177840);
        double r177842 = r177841 * r177830;
        double r177843 = 0.5;
        double r177844 = r177843 * r177818;
        double r177845 = cos(r177844);
        double r177846 = r177809 * r177845;
        double r177847 = r177842 / r177846;
        double r177848 = r177839 * r177847;
        double r177849 = r177815 ? r177838 : r177848;
        return r177849;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if J < -1.7648872844499354e-122 or 6.001375838748898e-109 < J

    1. Initial program 9.2

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

    3. Applied add-cube-cbrt9.6

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

    4. Applied associate-*r*9.6

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

    6. Applied associate-*l*9.6

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

    8. Applied add-cube-cbrt9.7

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

    9. Applied cbrt-prod9.7

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

    if -1.7648872844499354e-122 < J < 6.001375838748898e-109

    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. Taylor expanded around inf 32.7

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -1.7648872844499354 \cdot 10^{-122} \lor \neg \left(J \le 6.0013758387488983 \cdot 10^{-109}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}} \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{K}{2}\right)}}\right)\right)\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(0.5 \cdot K\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))