Average Error: 16.8 → 16.0
Time: 2.2m
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -1.9478612451442132 \cdot 10^{-147}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(-2 \cdot \sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}\right)\\ \mathbf{elif}\;J \le 5.648842308741205 \cdot 10^{-186}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(-2 \cdot \sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}\right)\\ \end{array}\]
\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 -1.9478612451442132 \cdot 10^{-147}:\\
\;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(-2 \cdot \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)\\

\mathbf{elif}\;J \le 5.648842308741205 \cdot 10^{-186}:\\
\;\;\;\;-U\\

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

\end{array}
double f(double J, double K, double U) {
        double r44669679 = -2.0;
        double r44669680 = J;
        double r44669681 = r44669679 * r44669680;
        double r44669682 = K;
        double r44669683 = 2.0;
        double r44669684 = r44669682 / r44669683;
        double r44669685 = cos(r44669684);
        double r44669686 = r44669681 * r44669685;
        double r44669687 = 1.0;
        double r44669688 = U;
        double r44669689 = r44669683 * r44669680;
        double r44669690 = r44669689 * r44669685;
        double r44669691 = r44669688 / r44669690;
        double r44669692 = pow(r44669691, r44669683);
        double r44669693 = r44669687 + r44669692;
        double r44669694 = sqrt(r44669693);
        double r44669695 = r44669686 * r44669694;
        return r44669695;
}
↓
double f(double J, double K, double U) {
        double r44669696 = J;
        double r44669697 = -1.9478612451442132e-147;
        bool r44669698 = r44669696 <= r44669697;
        double r44669699 = K;
        double r44669700 = 2.0;
        double r44669701 = r44669699 / r44669700;
        double r44669702 = cos(r44669701);
        double r44669703 = r44669702 * r44669696;
        double r44669704 = -2.0;
        double r44669705 = 1.0;
        double r44669706 = U;
        double r44669707 = r44669700 * r44669703;
        double r44669708 = r44669706 / r44669707;
        double r44669709 = r44669708 * r44669708;
        double r44669710 = r44669705 + r44669709;
        double r44669711 = sqrt(r44669710);
        double r44669712 = r44669704 * r44669711;
        double r44669713 = r44669703 * r44669712;
        double r44669714 = 5.648842308741205e-186;
        bool r44669715 = r44669696 <= r44669714;
        double r44669716 = -r44669706;
        double r44669717 = r44669715 ? r44669716 : r44669713;
        double r44669718 = r44669698 ? r44669713 : r44669717;
        return r44669718;
}

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 < -1.9478612451442132e-147 or 5.648842308741205e-186 < J

    1. Initial program 10.3

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

    2. Simplified10.3

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

    if -1.9478612451442132e-147 < J < 5.648842308741205e-186

    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. Simplified37.7

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

    4. Applied add-cube-cbrt37.9

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

    5. Applied associate-*l*37.9

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

    7. Applied pow1/344.2

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

    8. Taylor expanded around -inf 34.1

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

    9. Simplified34.1

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -1.9478612451442132 \cdot 10^{-147}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(-2 \cdot \sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}\right)\\ \mathbf{elif}\;J \le 5.648842308741205 \cdot 10^{-186}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(-2 \cdot \sqrt{1 + \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)} \cdot \frac{U}{2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(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)))))