Average Error: 17.8 → 8.5
Time: 11.4s
Precision: binary64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
↓
\[\begin{array}{l} \mathbf{if}\;J \leq -9.456952117734688 \cdot 10^{-234} \lor \neg \left(J \leq 3.848224094713413 \cdot 10^{-273}\right):\\ \;\;\;\;\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\\ t_0 \cdot \mathsf{hypot}\left(1, \frac{-U}{t_0}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(2, \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U}, U\right)\\ \end{array} \]
\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 \leq -9.456952117734688 \cdot 10^{-234} \lor \neg \left(J \leq 3.848224094713413 \cdot 10^{-273}\right):\\
\;\;\;\;\begin{array}{l}
t_0 := \left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\\
t_0 \cdot \mathsf{hypot}\left(1, \frac{-U}{t_0}\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;-\mathsf{fma}\left(2, \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U}, U\right)\\


\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 (<= J -9.456952117734688e-234) (not (<= J 3.848224094713413e-273)))
   (let* ((t_0 (* (* J -2.0) (cos (/ K 2.0)))))
     (* t_0 (hypot 1.0 (/ (- U) t_0))))
   (- (fma 2.0 (/ (* (* J J) (pow (cos (* K 0.5)) 2.0)) U) 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 ((J <= -9.456952117734688e-234) || !(J <= 3.848224094713413e-273)) {
		double t_0_1 = (J * -2.0) * cos(K / 2.0);
		tmp = t_0_1 * hypot(1.0, (-U / t_0_1));
	} else {
		tmp = -fma(2.0, (((J * J) * pow(cos(K * 0.5), 2.0)) / U), U);
	}
	return tmp;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

  1. Split input into 2 regimes

  2. if J < -9.45695211773468832e-234 or 3.84822409471341332e-273 < J

    1. Initial program 15.4

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

    2. Simplified6.0

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

    3. Applied frac-2neg_binary646.0

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

    4. Simplified6.0

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

    if -9.45695211773468832e-234 < J < 3.84822409471341332e-273

    1. Initial program 41.6

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

    2. Simplified26.5

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

    3. Taylor expanded in J around 0 34.0

      \[\leadsto \color{blue}{-\left(2 \cdot \frac{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}{U} + U\right)} \]

    4. Simplified34.0

      \[\leadsto \color{blue}{-\mathsf{fma}\left(2, \frac{\left(J \cdot J\right) \cdot {\cos \left(0.5 \cdot K\right)}^{2}}{U}, U\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9.456952117734688 \cdot 10^{-234} \lor \neg \left(J \leq 3.848224094713413 \cdot 10^{-273}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{-U}{\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(2, \frac{\left(J \cdot J\right) \cdot {\cos \left(K \cdot 0.5\right)}^{2}}{U}, U\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(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)))))