Maksimov and Kolovsky, Equation (3)

Average Error: 18.2 → 9.6

Time: 33.4s

Precision: binary64

Cost: 80776

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

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)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U) (if (<= t_1 1e+303) t_1 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 t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 1e+303) {
		tmp = t_1;
	} 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 t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 1e+303) {
		tmp = t_1;
	} 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))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 1e+303:
		tmp = t_1
	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))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 1e+303)
		tmp = t_1;
	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));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 1e+303)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 1e+303], t$95$1, U]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)}\right)}^{2}}\right)} \]
      Proof

      [Start] 64.0	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \]
      rational.json-simplify-43 [=>] 64.0	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      trig.json-simplify-24 [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\color{blue}{\cos \left(-\frac{K}{2}\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      trig.json-simplify-24 [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\color{blue}{\cos \left(-\left(-\frac{K}{2}\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-8 [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \color{blue}{\left(\left(-\frac{K}{2}\right) \cdot -1\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \color{blue}{\left(-1 \cdot \left(-\frac{K}{2}\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-10 [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(-1 \cdot \color{blue}{\frac{\frac{K}{2}}{-1}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-47 [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(-1 \cdot \color{blue}{\frac{K}{2 \cdot -1}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      metadata-eval [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(-1 \cdot \frac{K}{\color{blue}{-2}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-49 [<=] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \color{blue}{\left(\frac{K \cdot -1}{-2}\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-2 [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{\color{blue}{-1 \cdot K}}{-2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-49 [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot \frac{-1}{-2}\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      metadata-eval [=>] 64.0	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(K \cdot \color{blue}{0.5}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]

   3. Taylor expanded in J around 0 33.2

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

   4. Simplified 33.2

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

      [Start] 33.2	\[ -1 \cdot U \]
      rational.json-simplify-2 [=>] 33.2	\[ \color{blue}{U \cdot -1} \]
      rational.json-simplify-8 [<=] 33.2	\[ \color{blue}{-U} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 1e303

   1. Initial program 0.1

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

   if 1e303 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 61.2

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

   2. Simplified 61.2

      \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \sqrt{1 + {\left(\frac{U}{J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)}\right)}^{2}}\right)} \]
      Proof

      [Start] 61.2	\[ \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
      rational.json-simplify-2 [=>] 61.2	\[ \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \]
      rational.json-simplify-43 [=>] 61.2	\[ \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right)} \]
      trig.json-simplify-24 [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\color{blue}{\cos \left(-\frac{K}{2}\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      trig.json-simplify-24 [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\color{blue}{\cos \left(-\left(-\frac{K}{2}\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-8 [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \color{blue}{\left(\left(-\frac{K}{2}\right) \cdot -1\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-2 [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \color{blue}{\left(-1 \cdot \left(-\frac{K}{2}\right)\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-10 [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(-1 \cdot \color{blue}{\frac{\frac{K}{2}}{-1}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-47 [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(-1 \cdot \color{blue}{\frac{K}{2 \cdot -1}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      metadata-eval [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(-1 \cdot \frac{K}{\color{blue}{-2}}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-49 [<=] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \color{blue}{\left(\frac{K \cdot -1}{-2}\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-2 [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{\color{blue}{-1 \cdot K}}{-2}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      rational.json-simplify-49 [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot \frac{-1}{-2}\right)} \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]
      metadata-eval [=>] 61.2	\[ \left(-2 \cdot J\right) \cdot \left(\cos \left(K \cdot \color{blue}{0.5}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right) \]

   3. Taylor expanded in U around -inf 32.7

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \leq 10^{+303}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternatives

Alternative 1
Error	17.9
Cost	27804

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \left(-2 \cdot J\right) \cdot \left(t_0 \cdot \sqrt{1 + {\left(\frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)}^{2}}\right)\\ \mathbf{if}\;J \leq -3300:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -2.4 \cdot 10^{-10}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -1.2 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -1.15 \cdot 10^{-172}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 9.4 \cdot 10^{-297}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.75 \cdot 10^{-174}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	22.7
Cost	21148

\[\begin{array}{l} t_0 := \sqrt{1 + {\left(\frac{U}{J + J}\right)}^{2}}\\ t_1 := \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot t_0\\ \mathbf{if}\;J \leq -3300:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -5.8 \cdot 10^{-53}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -2 \cdot 10^{-76}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot t_0\\ \mathbf{elif}\;J \leq -1.7 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \frac{{J}^{2}}{U} + U\\ \mathbf{elif}\;J \leq -7.5 \cdot 10^{-231}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.5 \cdot 10^{-299}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.4 \cdot 10^{-175}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	26.7
Cost	13964

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{if}\;J \leq -3300:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -3.2 \cdot 10^{-76}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{1 + {\left(U \cdot \frac{0.5}{J}\right)}^{2}}\\ \mathbf{elif}\;J \leq -4.6 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \frac{{J}^{2}}{U} + U\\ \mathbf{elif}\;J \leq -1.45 \cdot 10^{-230}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.05 \cdot 10^{-302}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.3 \cdot 10^{-151}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	26.6
Cost	13964

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{if}\;J \leq -3300:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -6.2 \cdot 10^{-53}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -1.8 \cdot 10^{-76}:\\ \;\;\;\;\left(J \cdot -2\right) \cdot \sqrt{1 + {\left(\frac{U}{J + J}\right)}^{2}}\\ \mathbf{elif}\;J \leq -4.6 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \frac{{J}^{2}}{U} + U\\ \mathbf{elif}\;J \leq -8.5 \cdot 10^{-231}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.4 \cdot 10^{-297}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 7.5 \cdot 10^{-151}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	26.6
Cost	7772

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{if}\;J \leq -3300:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -5.6 \cdot 10^{-70}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -5.2 \cdot 10^{-76}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.6 \cdot 10^{-97}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq -5.8 \cdot 10^{-231}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.02 \cdot 10^{-296}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.5 \cdot 10^{-151}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	26.6
Cost	7772

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot -2\right)\\ \mathbf{if}\;J \leq -3400:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -1.3 \cdot 10^{-70}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -8 \cdot 10^{-76}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4.6 \cdot 10^{-97}:\\ \;\;\;\;2 \cdot \frac{{J}^{2}}{U} + U\\ \mathbf{elif}\;J \leq -9.5 \cdot 10^{-232}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 3.5 \cdot 10^{-296}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{-151}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	38.4
Cost	984

\[\begin{array}{l} \mathbf{if}\;J \leq -8500:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -1.1 \cdot 10^{-231}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.6 \cdot 10^{-292}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.42 \cdot 10^{-105}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 3 \cdot 10^{+109}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 8
Error	47.1
Cost	656

\[\begin{array}{l} \mathbf{if}\;J \leq -2.3 \cdot 10^{-230}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.38 \cdot 10^{-300}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.52 \cdot 10^{-107}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 9 \cdot 10^{-33}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 9
Error	46.7
Cost	64

\[U \]

Reproduce

herbie shell --seed 2023075 
(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)))))