Average Error: 18.4 → 8.3
Time: 13.3s
Precision: binary64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
↓
\[\begin{array}{l} \mathbf{if}\;U \leq 2.851168173361108 \cdot 10^{+219} \lor \neg \left(U \leq 5.629354276877763 \cdot 10^{+291}\right):\\ \;\;\;\;\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_0 \cdot \mathsf{hypot}\left(1, \frac{-U}{t_0}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
\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 \leq 2.851168173361108 \cdot 10^{+219} \lor \neg \left(U \leq 5.629354276877763 \cdot 10^{+291}\right):\\
\;\;\;\;\begin{array}{l}
t_0 := \left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\
t_0 \cdot \mathsf{hypot}\left(1, \frac{-U}{t_0}\right)
\end{array}\\

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


\end{array}

(FPCore (J K U)
 :precision binary64
 (*
  (* (* -2.0 J) (cos (/ K 2.0)))
  (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))
↓
(FPCore (J K U)
 :precision binary64
 (if (or (<= U 2.851168173361108e+219) (not (<= U 5.629354276877763e+291)))
   (let* ((t_0 (* (* -2.0 J) (cos (/ K 2.0)))))
     (* t_0 (hypot 1.0 (/ (- U) t_0))))
   (- U)))
double code(double J, double K, double U) {
	return ((-2.0 * J) * cos(K / 2.0)) * sqrt(1.0 + pow((U / ((2.0 * J) * cos(K / 2.0))), 2.0));
}
↓
double code(double J, double K, double U) {
	double tmp;
	if ((U <= 2.851168173361108e+219) || !(U <= 5.629354276877763e+291)) {
		double t_0_1 = (-2.0 * J) * cos(K / 2.0);
		tmp = t_0_1 * hypot(1.0, (-U / t_0_1));
	} else {
		tmp = -U;
	}
	return tmp;
}

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 < 2.8511681733611083e219 or 5.62935427687776278e291 < U

    1. Initial program 17.0

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

    2. Simplified6.8

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

    3. Applied frac-2neg_binary646.8

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

    if 2.8511681733611083e219 < U < 5.62935427687776278e291

    1. Initial program 41.7

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

    2. Simplified27.1

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

    3. Taylor expanded in J around 0 32.9

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

    4. Simplified32.9

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.851168173361108 \cdot 10^{+219} \lor \neg \left(U \leq 5.629354276877763 \cdot 10^{+291}\right):\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{-U}{\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Reproduce

herbie shell --seed 2022039 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))