Average Error: 18.7 → 9.3
Time: 30.3s
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -5.675372485901490572720424474717990752987 \cdot 10^{-306} \lor \neg \left(J \le 6.995469865281344460219534865255688195257 \cdot 10^{-177}\right):\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left({\left(\frac{\frac{U}{J}}{\cos \left(\frac{K}{2}\right) \cdot 2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot \sqrt{0.25}\right)\\ \end{array}\]
\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 -5.675372485901490572720424474717990752987 \cdot 10^{-306} \lor \neg \left(J \le 6.995469865281344460219534865255688195257 \cdot 10^{-177}\right):\\
\;\;\;\;J \cdot \left(\mathsf{hypot}\left({\left(\frac{\frac{U}{J}}{\cos \left(\frac{K}{2}\right) \cdot 2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(U \cdot \sqrt{0.25}\right)\\

\end{array}
double f(double J, double K, double U) {
        double r114652 = -2.0;
        double r114653 = J;
        double r114654 = r114652 * r114653;
        double r114655 = K;
        double r114656 = 2.0;
        double r114657 = r114655 / r114656;
        double r114658 = cos(r114657);
        double r114659 = r114654 * r114658;
        double r114660 = 1.0;
        double r114661 = U;
        double r114662 = r114656 * r114653;
        double r114663 = r114662 * r114658;
        double r114664 = r114661 / r114663;
        double r114665 = pow(r114664, r114656);
        double r114666 = r114660 + r114665;
        double r114667 = sqrt(r114666);
        double r114668 = r114659 * r114667;
        return r114668;
}
↓
double f(double J, double K, double U) {
        double r114669 = J;
        double r114670 = -5.6753724859014906e-306;
        bool r114671 = r114669 <= r114670;
        double r114672 = 6.9954698652813445e-177;
        bool r114673 = r114669 <= r114672;
        double r114674 = !r114673;
        bool r114675 = r114671 || r114674;
        double r114676 = U;
        double r114677 = r114676 / r114669;
        double r114678 = K;
        double r114679 = 2.0;
        double r114680 = r114678 / r114679;
        double r114681 = cos(r114680);
        double r114682 = r114681 * r114679;
        double r114683 = r114677 / r114682;
        double r114684 = 2.0;
        double r114685 = r114679 / r114684;
        double r114686 = pow(r114683, r114685);
        double r114687 = 1.0;
        double r114688 = sqrt(r114687);
        double r114689 = hypot(r114686, r114688);
        double r114690 = -2.0;
        double r114691 = r114690 * r114681;
        double r114692 = r114689 * r114691;
        double r114693 = r114669 * r114692;
        double r114694 = 0.25;
        double r114695 = sqrt(r114694);
        double r114696 = r114676 * r114695;
        double r114697 = r114690 * r114696;
        double r114698 = r114675 ? r114693 : r114697;
        return r114698;
}

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 < -5.6753724859014906e-306 or 6.9954698652813445e-177 < J

    1. Initial program 15.6

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

    2. Simplified15.6

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

    4. Applied add-sqr-sqrt15.6

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

    5. Applied sqr-pow15.6

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

    6. Applied hypot-def6.1

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

    8. Applied pow16.1

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

    9. Applied pow16.1

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

    10. Applied pow16.1

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

    11. Applied pow16.1

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

    12. Applied pow-prod-down6.1

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

    13. Applied pow-prod-down6.1

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

    14. Applied pow-prod-down6.1

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

    15. Simplified6.2

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

    17. Applied associate-*r*6.1

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

    18. Simplified6.1

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

    if -5.6753724859014906e-306 < J < 6.9954698652813445e-177

    1. Initial program 42.8

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

    2. Simplified42.8

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

    3. Taylor expanded around inf 34.4

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -5.675372485901490572720424474717990752987 \cdot 10^{-306} \lor \neg \left(J \le 6.995469865281344460219534865255688195257 \cdot 10^{-177}\right):\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left({\left(\frac{\frac{U}{J}}{\cos \left(\frac{K}{2}\right) \cdot 2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot \sqrt{0.25}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(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)))))