Average Error: 17.1 → 9.4
Time: 36.8s
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le 9.875491213433985 \cdot 10^{+171}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;U \le 9.166002875458719 \cdot 10^{+250}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)}\right)}\\ \end{array}\]
\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}\;U \le 9.875491213433985 \cdot 10^{+171}:\\
\;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\

\mathbf{elif}\;U \le 9.166002875458719 \cdot 10^{+250}:\\
\;\;\;\;-U\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)}\right)}\\

\end{array}
double f(double J, double K, double U) {
        double r19116752 = -2.0;
        double r19116753 = J;
        double r19116754 = r19116752 * r19116753;
        double r19116755 = K;
        double r19116756 = 2.0;
        double r19116757 = r19116755 / r19116756;
        double r19116758 = cos(r19116757);
        double r19116759 = r19116754 * r19116758;
        double r19116760 = 1.0;
        double r19116761 = U;
        double r19116762 = r19116756 * r19116753;
        double r19116763 = r19116762 * r19116758;
        double r19116764 = r19116761 / r19116763;
        double r19116765 = pow(r19116764, r19116756);
        double r19116766 = r19116760 + r19116765;
        double r19116767 = sqrt(r19116766);
        double r19116768 = r19116759 * r19116767;
        return r19116768;
}
↓
double f(double J, double K, double U) {
        double r19116769 = U;
        double r19116770 = 9.875491213433985e+171;
        bool r19116771 = r19116769 <= r19116770;
        double r19116772 = J;
        double r19116773 = -2.0;
        double r19116774 = r19116772 * r19116773;
        double r19116775 = K;
        double r19116776 = 2.0;
        double r19116777 = r19116775 / r19116776;
        double r19116778 = cos(r19116777);
        double r19116779 = r19116774 * r19116778;
        double r19116780 = 1.0;
        double r19116781 = r19116776 * r19116778;
        double r19116782 = r19116772 * r19116781;
        double r19116783 = r19116769 / r19116782;
        double r19116784 = hypot(r19116780, r19116783);
        double r19116785 = r19116779 * r19116784;
        double r19116786 = 9.166002875458719e+250;
        bool r19116787 = r19116769 <= r19116786;
        double r19116788 = -r19116769;
        double r19116789 = sqrt(r19116784);
        double r19116790 = r19116779 * r19116789;
        double r19116791 = r19116789 * r19116789;
        double r19116792 = r19116789 * r19116791;
        double r19116793 = cbrt(r19116792);
        double r19116794 = r19116790 * r19116793;
        double r19116795 = r19116787 ? r19116788 : r19116794;
        double r19116796 = r19116771 ? r19116785 : r19116795;
        return r19116796;
}

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 3 regimes

  2. if U < 9.875491213433985e+171

    1. Initial program 14.5

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

    2. Simplified6.0

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

    if 9.875491213433985e+171 < U < 9.166002875458719e+250

    1. Initial program 36.2

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

    2. Simplified18.7

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

    3. Taylor expanded around -inf 37.3

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

    4. Simplified37.3

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

    if 9.166002875458719e+250 < U

    1. Initial program 39.6

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

    2. Simplified26.9

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

    4. Applied add-sqr-sqrt27.1

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

    5. Applied associate-*l*27.1

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

    7. Applied add-cbrt-cube34.4

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 9.875491213433985 \cdot 10^{+171}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{elif}\;U \le 9.166002875458719 \cdot 10^{+250}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)}\right) \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \cdot \left(\sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)} \cdot \sqrt{\mathsf{hypot}\left(1, \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(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)))))