Average Error: 18.0 → 17.3
Time: 1.5m
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\begin{array}{l} \mathbf{if}\;J \le -8.793677279707616483357998085298165644322 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\ \mathbf{elif}\;J \le -7.496155157232238258900223489995841630596 \cdot 10^{-180}:\\ \;\;\;\;\left(\sqrt{0.25} \cdot U\right) \cdot -2\\ \mathbf{elif}\;J \le -2.185755027344584049365129282325313290485 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\ \mathbf{elif}\;J \le 9.044065113033606576274056251320451028479 \cdot 10^{-176}:\\ \;\;\;\;\left(\sqrt{0.25} \cdot U\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\ \end{array}\]
\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 -8.793677279707616483357998085298165644322 \cdot 10^{-69}:\\
\;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\

\mathbf{elif}\;J \le -7.496155157232238258900223489995841630596 \cdot 10^{-180}:\\
\;\;\;\;\left(\sqrt{0.25} \cdot U\right) \cdot -2\\

\mathbf{elif}\;J \le -2.185755027344584049365129282325313290485 \cdot 10^{-254}:\\
\;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\

\mathbf{elif}\;J \le 9.044065113033606576274056251320451028479 \cdot 10^{-176}:\\
\;\;\;\;\left(\sqrt{0.25} \cdot U\right) \cdot -2\\

\mathbf{else}:\\
\;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\

\end{array}
double f(double J, double K, double U) {
        double r6729524 = -2.0;
        double r6729525 = J;
        double r6729526 = r6729524 * r6729525;
        double r6729527 = K;
        double r6729528 = 2.0;
        double r6729529 = r6729527 / r6729528;
        double r6729530 = cos(r6729529);
        double r6729531 = r6729526 * r6729530;
        double r6729532 = 1.0;
        double r6729533 = U;
        double r6729534 = r6729528 * r6729525;
        double r6729535 = r6729534 * r6729530;
        double r6729536 = r6729533 / r6729535;
        double r6729537 = pow(r6729536, r6729528);
        double r6729538 = r6729532 + r6729537;
        double r6729539 = sqrt(r6729538);
        double r6729540 = r6729531 * r6729539;
        return r6729540;
}
↓
double f(double J, double K, double U) {
        double r6729541 = J;
        double r6729542 = -8.793677279707616e-69;
        bool r6729543 = r6729541 <= r6729542;
        double r6729544 = 1.0;
        double r6729545 = U;
        double r6729546 = K;
        double r6729547 = 2.0;
        double r6729548 = r6729546 / r6729547;
        double r6729549 = cos(r6729548);
        double r6729550 = r6729549 * r6729541;
        double r6729551 = r6729550 * r6729547;
        double r6729552 = r6729545 / r6729551;
        double r6729553 = pow(r6729552, r6729547);
        double r6729554 = r6729544 + r6729553;
        double r6729555 = sqrt(r6729554);
        double r6729556 = -2.0;
        double r6729557 = r6729556 * r6729550;
        double r6729558 = r6729555 * r6729557;
        double r6729559 = -7.496155157232238e-180;
        bool r6729560 = r6729541 <= r6729559;
        double r6729561 = 0.25;
        double r6729562 = sqrt(r6729561);
        double r6729563 = r6729562 * r6729545;
        double r6729564 = r6729563 * r6729556;
        double r6729565 = -2.185755027344584e-254;
        bool r6729566 = r6729541 <= r6729565;
        double r6729567 = 9.044065113033607e-176;
        bool r6729568 = r6729541 <= r6729567;
        double r6729569 = r6729568 ? r6729564 : r6729558;
        double r6729570 = r6729566 ? r6729558 : r6729569;
        double r6729571 = r6729560 ? r6729564 : r6729570;
        double r6729572 = r6729543 ? r6729558 : r6729571;
        return r6729572;
}

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 < -8.793677279707616e-69 or -7.496155157232238e-180 < J < -2.185755027344584e-254 or 9.044065113033607e-176 < J

    1. Initial program 11.9

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

    2. Simplified11.9

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

    if -8.793677279707616e-69 < J < -7.496155157232238e-180 or -2.185755027344584e-254 < J < 9.044065113033607e-176

    1. Initial program 37.8

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

    2. Simplified37.8

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

    3. Taylor expanded around inf 35.1

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

  4. Final simplification17.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -8.793677279707616483357998085298165644322 \cdot 10^{-69}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\ \mathbf{elif}\;J \le -7.496155157232238258900223489995841630596 \cdot 10^{-180}:\\ \;\;\;\;\left(\sqrt{0.25} \cdot U\right) \cdot -2\\ \mathbf{elif}\;J \le -2.185755027344584049365129282325313290485 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\ \mathbf{elif}\;J \le 9.044065113033606576274056251320451028479 \cdot 10^{-176}:\\ \;\;\;\;\left(\sqrt{0.25} \cdot U\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + {\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{2}} \cdot \left(-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot J\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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)))))