Average Error: 16.9 → 8.6
Time: 1.0m
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;U \le 2.6727218722636747 \cdot 10^{+230}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1^2 + \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]
\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 2.6727218722636747 \cdot 10^{+230}:\\
\;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1^2 + \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)^2}^*\\

\mathbf{else}:\\
\;\;\;\;-U\\

\end{array}
double f(double J, double K, double U) {
        double r21785544 = -2.0;
        double r21785545 = J;
        double r21785546 = r21785544 * r21785545;
        double r21785547 = K;
        double r21785548 = 2.0;
        double r21785549 = r21785547 / r21785548;
        double r21785550 = cos(r21785549);
        double r21785551 = r21785546 * r21785550;
        double r21785552 = 1.0;
        double r21785553 = U;
        double r21785554 = r21785548 * r21785545;
        double r21785555 = r21785554 * r21785550;
        double r21785556 = r21785553 / r21785555;
        double r21785557 = pow(r21785556, r21785548);
        double r21785558 = r21785552 + r21785557;
        double r21785559 = sqrt(r21785558);
        double r21785560 = r21785551 * r21785559;
        return r21785560;
}
↓
double f(double J, double K, double U) {
        double r21785561 = U;
        double r21785562 = 2.6727218722636747e+230;
        bool r21785563 = r21785561 <= r21785562;
        double r21785564 = J;
        double r21785565 = -2.0;
        double r21785566 = r21785564 * r21785565;
        double r21785567 = K;
        double r21785568 = 2.0;
        double r21785569 = r21785567 / r21785568;
        double r21785570 = cos(r21785569);
        double r21785571 = r21785566 * r21785570;
        double r21785572 = 1.0;
        double r21785573 = r21785568 * r21785570;
        double r21785574 = r21785564 * r21785573;
        double r21785575 = r21785561 / r21785574;
        double r21785576 = hypot(r21785572, r21785575);
        double r21785577 = r21785571 * r21785576;
        double r21785578 = -r21785561;
        double r21785579 = r21785563 ? r21785577 : r21785578;
        return r21785579;
}

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 U < 2.6727218722636747e+230

    1. Initial program 15.3

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

    2. Simplified6.7

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

    if 2.6727218722636747e+230 < U

    1. Initial program 41.5

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

    2. Simplified26.4

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

    4. Applied add-cube-cbrt26.7

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

    5. Applied associate-*l*26.7

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

    7. Applied add-cbrt-cube26.7

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

    8. Taylor expanded around -inf 36.1

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

    9. Simplified36.1

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 2.6727218722636747 \cdot 10^{+230}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1^2 + \left(\frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)^2}^*\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +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)))))