Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 9.0

Time: 35.3s

Precision: binary64

Cost: 20748

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(J \cdot -2\right) \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\\ \mathbf{if}\;J \leq -2.1 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-272}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-262}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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)))
        (t_1 (* (* (* J -2.0) t_0) (hypot 1.0 (/ U (* 2.0 (* J t_0)))))))
   (if (<= J -2.1e-197)
     t_1
     (if (<= J -2.3e-272) U (if (<= J 6.2e-262) (- U) t_1)))))

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 t_1 = ((J * -2.0) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	double tmp;
	if (J <= -2.1e-197) {
		tmp = t_1;
	} else if (J <= -2.3e-272) {
		tmp = U;
	} else if (J <= 6.2e-262) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	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 t_1 = ((J * -2.0) * t_0) * Math.hypot(1.0, (U / (2.0 * (J * t_0))));
	double tmp;
	if (J <= -2.1e-197) {
		tmp = t_1;
	} else if (J <= -2.3e-272) {
		tmp = U;
	} else if (J <= 6.2e-262) {
		tmp = -U;
	} else {
		tmp = t_1;
	}
	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))
	t_1 = ((J * -2.0) * t_0) * math.hypot(1.0, (U / (2.0 * (J * t_0))))
	tmp = 0
	if J <= -2.1e-197:
		tmp = t_1
	elif J <= -2.3e-272:
		tmp = U
	elif J <= 6.2e-262:
		tmp = -U
	else:
		tmp = t_1
	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))
	t_1 = Float64(Float64(Float64(J * -2.0) * t_0) * hypot(1.0, Float64(U / Float64(2.0 * Float64(J * t_0)))))
	tmp = 0.0
	if (J <= -2.1e-197)
		tmp = t_1;
	elseif (J <= -2.3e-272)
		tmp = U;
	elseif (J <= 6.2e-262)
		tmp = Float64(-U);
	else
		tmp = t_1;
	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));
	t_1 = ((J * -2.0) * t_0) * hypot(1.0, (U / (2.0 * (J * t_0))));
	tmp = 0.0;
	if (J <= -2.1e-197)
		tmp = t_1;
	elseif (J <= -2.3e-272)
		tmp = U;
	elseif (J <= 6.2e-262)
		tmp = -U;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(J * -2.0), $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[J, -2.1e-197], t$95$1, If[LessEqual[J, -2.3e-272], U, If[LessEqual[J, 6.2e-262], (-U), t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.1e-197 or 6.1999999999999997e-262 < J

   1. Initial program 14.5

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

   2. Simplified 5.4

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

      [Start] 14.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      unpow2 [=>] 14.5	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}} \]
      hypot-1-def [=>] 5.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]
      associate-*l* [=>] 5.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}}\right) \]

   if -2.1e-197 < J < -2.29999999999999989e-272

   1. Initial program 40.6

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

   2. Simplified 25.3

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

      [Start] 40.6	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 40.6	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 40.6	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 25.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 25.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 25.3	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in U around -inf 35.4

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

   if -2.29999999999999989e-272 < J < 6.1999999999999997e-262

   1. Initial program 45.4

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

   2. Simplified 30.4

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

      [Start] 45.4	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      associate-*l* [=>] 45.4	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      unpow2 [=>] 45.4	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + \color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}}}\right) \]
      hypot-1-def [=>] 30.4	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{hypot}\left(1, \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}\right) \]
      *-commutative [=>] 30.4	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\color{blue}{\cos \left(\frac{K}{2}\right) \cdot \left(2 \cdot J\right)}}\right)\right) \]
      *-commutative [=>] 30.4	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\left(J \cdot 2\right)}}\right)\right) \]

   3. Taylor expanded in J around 0 32.7

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

   4. Simplified 32.7

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

      [Start] 32.7	\[ -1 \cdot U \]
      mul-1-neg [=>] 32.7	\[ \color{blue}{-U} \]

3. Recombined 3 regimes into one program.

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.1 \cdot 10^{-197}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-272}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 6.2 \cdot 10^{-262}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	9.0
Cost	20748

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := -2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \mathbf{if}\;J \leq -2.1 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.12 \cdot 10^{-271}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3.55 \cdot 10^{-261}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	9.0
Cost	20748

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -2 \cdot 10^{-197}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)\right)\\ \mathbf{elif}\;J \leq -1.3 \cdot 10^{-270}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{-257}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot t_0\right)}\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	16.9
Cost	14224

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U \leq -3.1 \cdot 10^{+194}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.12 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{+214}:\\ \;\;\;\;\frac{J}{U} \cdot \left(J \cdot \left(-1 - \cos K\right)\right) - U\\ \mathbf{elif}\;U \leq 7.2 \cdot 10^{+215}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(J \cdot \frac{{t_0}^{2}}{U}\right)\\ \end{array} \]

Alternative 4
Error	16.9
Cost	13960

\[\begin{array}{l} \mathbf{if}\;U \leq -2.4 \cdot 10^{+193}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\\ \mathbf{elif}\;U \leq 5.2 \cdot 10^{+214}:\\ \;\;\;\;\frac{J}{U} \cdot \left(J \cdot \left(-1 - \cos K\right)\right) - U\\ \mathbf{elif}\;U \leq 7.2 \cdot 10^{+215}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5
Error	26.6
Cost	8284

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \sqrt{1 + \frac{0.25 \cdot \left(U \cdot U\right)}{J \cdot J}}\\ t_1 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;U \leq -3 \cdot 10^{+196}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -2.8 \cdot 10^{+118}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.3 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3.2 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 8.5 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2.7 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq 8 \cdot 10^{+215}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 6
Error	27.4
Cost	7508

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -7 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.5 \cdot 10^{-172}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -1.32 \cdot 10^{-201}:\\ \;\;\;\;\frac{-2}{\frac{\frac{U}{J}}{J}} - U\\ \mathbf{elif}\;J \leq -2.3 \cdot 10^{-272}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 8.2 \cdot 10^{-143}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	27.4
Cost	7508

\[\begin{array}{l} t_0 := \left(J \cdot -2\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -6.5 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -9 \cdot 10^{-171}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -8.5 \cdot 10^{-202}:\\ \;\;\;\;\frac{J}{U} \cdot \left(J \cdot \left(-1 - \cos K\right)\right) - U\\ \mathbf{elif}\;J \leq -3.5 \cdot 10^{-272}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-142}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	46.8
Cost	920

\[\begin{array}{l} \mathbf{if}\;U \leq -1.65 \cdot 10^{+196}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -3.1 \cdot 10^{+63}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.6 \cdot 10^{-110}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 5.8 \cdot 10^{+91}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 9
Error	38.7
Cost	920

\[\begin{array}{l} \mathbf{if}\;U \leq -1.3 \cdot 10^{+194}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.6 \cdot 10^{+64}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -2.6 \cdot 10^{-90}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 1.35 \cdot 10^{+92}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.4 \cdot 10^{+131}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 10
Error	47.0
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023066 
(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)))))