Maksimov and Kolovsky, Equation (3)

Average Error: 18.2 → 8.4

Time: 24.2s

Precision: binary64

Cost: 20484

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 1.95 \cdot 10^{+233}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{elif}\;U \leq 10^{+292}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

FPCore

(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 1.95e+233)
     (* (* (* -2.0 J) t_0) (hypot 1.0 (/ U (* 2.0 (* J t_0)))))
     (if (<= U 1e+292) (- U) U))))

C

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 <= 1.95e+233) {
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	} else if (U <= 1e+292) {
		tmp = -U;
	} else {
		tmp = U;
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	return ((-2.0 * J) * Math.cos((K / 2.0))) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * Math.cos((K / 2.0)))), 2.0)));
}

↓

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U <= 1.95e+233) {
		tmp = ((-2.0 * J) * t_0) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
	} else if (U <= 1e+292) {
		tmp = -U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, K, U):
	return ((-2.0 * J) * math.cos((K / 2.0))) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * math.cos((K / 2.0)))), 2.0)))

↓

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= 1.95e+233:
		tmp = ((-2.0 * J) * t_0) * math.hypot(1.0, (U / (2.0 * (J * t_0))))
	elif U <= 1e+292:
		tmp = -U
	else:
		tmp = U
	return tmp

Julia

function code(J, K, U)
	return Float64(Float64(Float64(-2.0 * J) * cos(Float64(K / 2.0))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * cos(Float64(K / 2.0)))) ^ 2.0))))
end

↓

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= 1.95e+233)
		tmp = Float64(Float64(Float64(-2.0 * J) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))));
	elseif (U <= 1e+292)
		tmp = Float64(-U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((-2.0 * J) * cos((K / 2.0))) * sqrt((1.0 + ((U / ((2.0 * J) * cos((K / 2.0)))) ^ 2.0)));
end

↓

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U <= 1.95e+233)
		tmp = ((-2.0 * J) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	elseif (U <= 1e+292)
		tmp = -U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(-2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, 1.95e+233], N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 1e+292], (-U), U]]]

Derivation

1. Split input into 3 regimes

2. if U < 1.9499999999999999e233

   1. Initial program 16.8

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

   2. Simplified 7.0

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

      [Start] 16.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 16.8	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 7.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 7.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if 1.9499999999999999e233 < U < 1e292

   1. Initial program 42.4

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

   2. Simplified 30.0

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

      [Start] 42.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 42.5	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 42.5	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 30.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 30.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 30.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in J around 0 33.3

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

   4. Simplified 33.3

      \[\leadsto \color{blue}{-U} \]
      Proof

      [Start] 33.3	\[ -1 \cdot U \]
      mul-1-neg [=>] 33.3	\[ \color{blue}{-U} \]

   if 1e292 < U

   1. Initial program 43.9

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

   2. Simplified 28.7

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

      [Start] 43.9	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 43.9	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 43.9	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 28.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 28.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 28.7	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in U around -inf 26.9

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

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.95 \cdot 10^{+233}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{elif}\;U \leq 10^{+292}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternatives

Alternative 1
Error	8.4
Cost	20484

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 1.32 \cdot 10^{+233}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{elif}\;U \leq 3.8 \cdot 10^{+291}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2
Error	8.5
Cost	20484

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 1.65 \cdot 10^{+233}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+292}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 3
Error	17.5
Cost	14476

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\\ \mathbf{if}\;J \leq -8.6 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -5.5 \cdot 10^{-199}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.9 \cdot 10^{-209}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 + K \cdot \left(K \cdot -0.25\right)\right)}\right)\right)\\ \mathbf{elif}\;J \leq -2.6 \cdot 10^{-258}:\\ \;\;\;\;\frac{-2}{\frac{\frac{U}{J}}{J \cdot {\cos \left(K \cdot 0.5\right)}^{2}}} - U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.6
Cost	14356

\[\begin{array}{l} t_0 := \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\\ \mathbf{if}\;J \leq -4.6 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.22 \cdot 10^{-197}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.15 \cdot 10^{-206}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;J \leq -4.1 \cdot 10^{-258}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	17.5
Cost	14356

\[\begin{array}{l} t_0 := \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{0.5}{\frac{J}{U}}\right)\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -6.2 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2.55 \cdot 10^{-197}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.15 \cdot 10^{-206}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot t_1\\ \mathbf{elif}\;J \leq -6 \cdot 10^{-257}:\\ \;\;\;\;\frac{-2}{\frac{\frac{U}{J}}{J \cdot {t_1}^{2}}} - U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	23.1
Cost	7832

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ t_1 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.08 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -8 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -7.4 \cdot 10^{-240}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -4.8 \cdot 10^{-257}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.4 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	26.8
Cost	7772

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.6 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -2.85 \cdot 10^{-236}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-257}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.7 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.35 \cdot 10^{-72}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 8.5 \cdot 10^{-33}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\frac{U}{J} \cdot -0.5 - \frac{J}{U}\right)\right)\\ \mathbf{elif}\;J \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\frac{J}{U} + 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	26.4
Cost	7768

\[\begin{array}{l} t_0 := \left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.6 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.6 \cdot 10^{-237}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -5.2 \cdot 10^{-257}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.4 \cdot 10^{-95}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.5 \cdot 10^{+22}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left|U \cdot \frac{-0.5}{J}\right|\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	38.1
Cost	1756

\[\begin{array}{l} \mathbf{if}\;J \leq -4.5 \cdot 10^{+30}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -2.9 \cdot 10^{-240}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-257}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.25 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7.2 \cdot 10^{-73}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 8.6 \cdot 10^{-33}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\frac{J}{U} + 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 10
Error	38.3
Cost	1756

\[\begin{array}{l} \mathbf{if}\;J \leq -3.6 \cdot 10^{+33}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -6.4 \cdot 10^{-237}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -2.4 \cdot 10^{-258}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.65 \cdot 10^{-74}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{-30}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\frac{U}{J} \cdot -0.5 - \frac{J}{U}\right)\right)\\ \mathbf{elif}\;J \leq 0.0022:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\frac{J}{U} + 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 11
Error	38.1
Cost	1116

\[\begin{array}{l} \mathbf{if}\;J \leq -3.8 \cdot 10^{+30}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -7.5 \cdot 10^{-236}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -4.8 \cdot 10^{-258}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2 \cdot 10^{-74}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-32}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 0.00095:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 12
Error	47.3
Cost	788

\[\begin{array}{l} \mathbf{if}\;J \leq -5.3 \cdot 10^{-238}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.15 \cdot 10^{-258}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.25 \cdot 10^{-167}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-73}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.2 \cdot 10^{-32}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 13
Error	46.7
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023027 
(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)))))