Average Error: 17.7 → 16.5
Time: 15.7s
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -1.14149938228696511 \cdot 10^{-223} \lor \neg \left(J \le 2.5022959607105417 \cdot 10^{-175}\right):\\ \;\;\;\;\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sqrt{0.25} \cdot U\right)\\ \end{array}\]
\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 -1.14149938228696511 \cdot 10^{-223} \lor \neg \left(J \le 2.5022959607105417 \cdot 10^{-175}\right):\\
\;\;\;\;\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\\

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

\end{array}
double f(double J, double K, double U) {
        double r175336 = -2.0;
        double r175337 = J;
        double r175338 = r175336 * r175337;
        double r175339 = K;
        double r175340 = 2.0;
        double r175341 = r175339 / r175340;
        double r175342 = cos(r175341);
        double r175343 = r175338 * r175342;
        double r175344 = 1.0;
        double r175345 = U;
        double r175346 = r175340 * r175337;
        double r175347 = r175346 * r175342;
        double r175348 = r175345 / r175347;
        double r175349 = pow(r175348, r175340);
        double r175350 = r175344 + r175349;
        double r175351 = sqrt(r175350);
        double r175352 = r175343 * r175351;
        return r175352;
}
↓
double f(double J, double K, double U) {
        double r175353 = J;
        double r175354 = -1.1414993822869651e-223;
        bool r175355 = r175353 <= r175354;
        double r175356 = 2.5022959607105417e-175;
        bool r175357 = r175353 <= r175356;
        double r175358 = !r175357;
        bool r175359 = r175355 || r175358;
        double r175360 = -2.0;
        double r175361 = r175360 * r175353;
        double r175362 = K;
        double r175363 = 2.0;
        double r175364 = r175362 / r175363;
        double r175365 = cos(r175364);
        double r175366 = r175361 * r175365;
        double r175367 = 1.0;
        double r175368 = U;
        double r175369 = r175363 * r175353;
        double r175370 = r175369 * r175365;
        double r175371 = r175368 / r175370;
        double r175372 = pow(r175371, r175363);
        double r175373 = r175367 + r175372;
        double r175374 = sqrt(r175373);
        double r175375 = sqrt(r175374);
        double r175376 = r175366 * r175375;
        double r175377 = r175376 * r175375;
        double r175378 = 0.25;
        double r175379 = sqrt(r175378);
        double r175380 = r175379 * r175368;
        double r175381 = r175360 * r175380;
        double r175382 = r175359 ? r175377 : r175381;
        return r175382;
}

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 < -1.1414993822869651e-223 or 2.5022959607105417e-175 < J

    1. Initial program 12.7

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

    3. Applied add-sqr-sqrt12.7

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

    4. Applied sqrt-prod12.7

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

    5. Applied associate-*r*12.7

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

    if -1.1414993822869651e-223 < J < 2.5022959607105417e-175

    1. Initial program 42.1

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

    3. Applied associate-*l*42.1

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

    4. Simplified42.1

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

    5. Taylor expanded around 0 34.2

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -1.14149938228696511 \cdot 10^{-223} \lor \neg \left(J \le 2.5022959607105417 \cdot 10^{-175}\right):\\ \;\;\;\;\left(\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right) \cdot \sqrt{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sqrt{0.25} \cdot U\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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)))))