Average Error: 17.8 → 8.6
Time: 11.9s
Precision: binary64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
↓
\[\begin{array}{l} \mathbf{if}\;J \leq -1.6525644811133454 \cdot 10^{-209} \lor \neg \left(J \leq 5.019231347031117 \cdot 10^{-278}\right):\\ \;\;\;\;\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
\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}\;J \leq -1.6525644811133454 \cdot 10^{-209} \lor \neg \left(J \leq 5.019231347031117 \cdot 10^{-278}\right):\\
\;\;\;\;\begin{array}{l}
t_0 := \cos \left(\frac{K}{2}\right)\\
\left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\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 (<= J -1.6525644811133454e-209) (not (<= J 5.019231347031117e-278)))
   (let* ((t_0 (cos (/ K 2.0))))
     (* (* (* J -2.0) t_0) (hypot 1.0 (/ U (* t_0 (* J 2.0))))))
   (- 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 ((J <= -1.6525644811133454e-209) || !(J <= 5.019231347031117e-278)) {
		double t_0_1 = cos(K / 2.0);
		tmp = ((J * -2.0) * t_0_1) * hypot(1.0, (U / (t_0_1 * (J * 2.0))));
	} 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 J < -1.65256448111334541e-209 or 5.01923134703111726e-278 < J

    1. Initial program 14.9

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

    2. Simplified5.7

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

    3. Applied pow1_binary645.7

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

    4. Applied pow1_binary645.7

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

    5. Applied pow1_binary645.7

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

    6. Applied pow1_binary645.7

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

    7. Applied pow-prod-down_binary645.7

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

    8. Applied pow-prod-down_binary645.7

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

    9. Applied pow-prod-down_binary645.7

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

    if -1.65256448111334541e-209 < J < 5.01923134703111726e-278

    1. Initial program 42.2

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

    2. Simplified26.4

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

    3. Taylor expanded in J around 0 32.4

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

    4. Simplified32.4

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.6525644811133454 \cdot 10^{-209} \lor \neg \left(J \leq 5.019231347031117 \cdot 10^{-278}\right):\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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)))))