Average Error: 17.9 → 8.2
Time: 10.6s
Precision: binary64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
↓
\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 3.616021255462154 \cdot 10^{+229}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]
\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}
t_0 := \cos \left(\frac{K}{2}\right)\\
\mathbf{if}\;U \leq 3.616021255462154 \cdot 10^{+229}:\\
\;\;\;\;\left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\\

\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
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U 3.616021255462154e+229)
     (* (* (* -2.0 J) 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 t_0 = cos((K / 2.0));
	double tmp;
	if (U <= 3.616021255462154e+229) {
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (t_0 * (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 U < 3.6160212554621541e229

    1. Initial program 16.0

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

    2. Simplified6.5

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

    3. Applied add-sqr-sqrt_binary646.6

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

    4. Applied associate-*r*_binary646.6

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

    5. Applied associate-*l*_binary646.6

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

    6. Simplified6.5

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

    if 3.6160212554621541e229 < U

    1. Initial program 44.3

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

    2. Simplified29.5

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

    3. Taylor expanded in J around 0 32.9

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

    4. Simplified32.9

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.616021255462154 \cdot 10^{+229}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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)))))