Maksimov and Kolovsky, Equation (3)

Average Error: 18.3 → 8.3

Time: 23.9s

Precision: binary64

Cost: 20748

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot t_0}\right)\right) \cdot \left(-2 \cdot t_0\right)\\ \mathbf{if}\;J \leq 3.8717310147334127 \cdot 10^{-295}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.9813514828358625 \cdot 10^{-234}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 0.5)))
        (t_1 (* (* J (hypot 1.0 (* U (/ 0.5 (* J t_0))))) (* -2.0 t_0))))
   (if (<= J 3.8717310147334127e-295)
     t_1
     (if (<= J 4.896013935583698e-267)
       (- U)
       (if (<= J 2.9813514828358625e-234) 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 * 0.5));
	double t_1 = (J * hypot(1.0, (U * (0.5 / (J * t_0))))) * (-2.0 * t_0);
	double tmp;
	if (J <= 3.8717310147334127e-295) {
		tmp = t_1;
	} else if (J <= 4.896013935583698e-267) {
		tmp = -U;
	} else if (J <= 2.9813514828358625e-234) {
		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 * 0.5));
	double t_1 = (J * Math.hypot(1.0, (U * (0.5 / (J * t_0))))) * (-2.0 * t_0);
	double tmp;
	if (J <= 3.8717310147334127e-295) {
		tmp = t_1;
	} else if (J <= 4.896013935583698e-267) {
		tmp = -U;
	} else if (J <= 2.9813514828358625e-234) {
		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 * 0.5))
	t_1 = (J * math.hypot(1.0, (U * (0.5 / (J * t_0))))) * (-2.0 * t_0)
	tmp = 0
	if J <= 3.8717310147334127e-295:
		tmp = t_1
	elif J <= 4.896013935583698e-267:
		tmp = -U
	elif J <= 2.9813514828358625e-234:
		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 * 0.5))
	t_1 = Float64(Float64(J * hypot(1.0, Float64(U * Float64(0.5 / Float64(J * t_0))))) * Float64(-2.0 * t_0))
	tmp = 0.0
	if (J <= 3.8717310147334127e-295)
		tmp = t_1;
	elseif (J <= 4.896013935583698e-267)
		tmp = Float64(-U);
	elseif (J <= 2.9813514828358625e-234)
		tmp = 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 * 0.5));
	t_1 = (J * hypot(1.0, (U * (0.5 / (J * t_0))))) * (-2.0 * t_0);
	tmp = 0.0;
	if (J <= 3.8717310147334127e-295)
		tmp = t_1;
	elseif (J <= 4.896013935583698e-267)
		tmp = -U;
	elseif (J <= 2.9813514828358625e-234)
		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 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(J * N[Sqrt[1.0 ^ 2 + N[(U * N[(0.5 / N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, 3.8717310147334127e-295], t$95$1, If[LessEqual[J, 4.896013935583698e-267], (-U), If[LessEqual[J, 2.9813514828358625e-234], U, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if J < 3.8717310147334127e-295 or 2.98135148283586255e-234 < J

   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 7.1

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      Proof
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (*.f64 J 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (Rewrite<= *-commutative_binary64 (*.f64 2 J))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 38 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (sqrt.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 -2 (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J -2) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 J)) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.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))))): 2 points increase in error, 6 points decrease in error

   3. Applied egg-rr 35.0

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

   4. Applied egg-rr 7.2

      \[\leadsto \color{blue}{\left(\mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(K \cdot 0.5\right)}\right) \cdot J\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot -2\right)} \]

   if 3.8717310147334127e-295 < J < 4.8960139355837e-267

   1. Initial program 48.6

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

   2. Simplified 34.2

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      Proof
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (*.f64 J 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (Rewrite<= *-commutative_binary64 (*.f64 2 J))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 38 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (sqrt.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 -2 (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J -2) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 J)) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.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))))): 2 points increase in error, 6 points decrease in error

   3. Taylor expanded in J around 0 30.0

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

   4. Simplified 30.0

      \[\leadsto \color{blue}{-U} \]
      Proof
      (neg.f64 U): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 U)): 0 points increase in error, 0 points decrease in error

   if 4.8960139355837e-267 < J < 2.98135148283586255e-234

   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 30.4

      \[\leadsto \color{blue}{J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right) \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right)} \]
      Proof
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (*.f64 J 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (*.f64 (cos.f64 (/.f64 K 2)) (Rewrite<= *-commutative_binary64 (*.f64 2 J))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (hypot.f64 1 (/.f64 U (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))))))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 38 points increase in error, 0 points decrease in error
      (*.f64 J (*.f64 (sqrt.f64 (+.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) (*.f64 -2 (cos.f64 (/.f64 K 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 -2 (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 J (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 J -2) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 J)) (*.f64 (cos.f64 (/.f64 K 2)) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.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))))): 2 points increase in error, 6 points decrease in error

   3. Taylor expanded in U around -inf 28.3

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

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 3.8717310147334127 \cdot 10^{-295}:\\ \;\;\;\;\left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\right) \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.9813514828358625 \cdot 10^{-234}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J \cdot \cos \left(K \cdot 0.5\right)}\right)\right) \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.6
Cost	20616

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\mathsf{hypot}\left(1, \frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right) \cdot \left(t_0 \cdot -2\right)\right)\\ \mathbf{if}\;U \leq 1.6374273789213422 \cdot 10^{+230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;U \leq 1.1724937223613034 \cdot 10^{+258}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	17.8
Cost	20232

\[\begin{array}{l} t_0 := J \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot -2\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ t_1 := {\cos \left(K \cdot 0.5\right)}^{2}\\ \mathbf{if}\;J \leq -9.18130858673931 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{J}{\frac{U}{J}} \cdot t_1, U\right)\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;\frac{t_1}{\frac{\frac{\frac{U}{J}}{J}}{-2}} - U\\ \mathbf{elif}\;J \leq 2.9813514828358625 \cdot 10^{-234}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	26.6
Cost	14424

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := -2 \cdot \left(J \cdot t_0\right)\\ \mathbf{if}\;J \leq -1.1746261169672367 \cdot 10^{-110}:\\ \;\;\;\;\left(-2 \cdot t_0\right) \cdot \mathsf{fma}\left(0.125, U \cdot \frac{U}{J}, J\right)\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.052480368001563 \cdot 10^{-125}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.509320468873213 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq 2.1136848813166284 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{U}{J} \cdot \frac{U}{J}, 1\right)} \cdot \left(J \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.9
Cost	14224

\[\begin{array}{l} t_0 := \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -9.18130858673931 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \frac{J}{U}\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.9813514828358625 \cdot 10^{-234}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	17.9
Cost	14224

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \left(-2 \cdot t_0\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -9.18130858673931 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \frac{J}{U}\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;\frac{{t_0}^{2}}{\frac{\frac{\frac{U}{J}}{J}}{-2}} - U\\ \mathbf{elif}\;J \leq 2.9813514828358625 \cdot 10^{-234}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	17.8
Cost	14224

\[\begin{array}{l} t_0 := J \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot -2\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot 2}\right)\right)\\ \mathbf{if}\;J \leq -9.18130858673931 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \frac{J}{U}\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;\frac{{\cos \left(K \cdot 0.5\right)}^{2}}{\frac{\frac{\frac{U}{J}}{J}}{-2}} - U\\ \mathbf{elif}\;J \leq 2.9813514828358625 \cdot 10^{-234}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	26.1
Cost	13764

\[\begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -1.1746261169672367 \cdot 10^{-110}:\\ \;\;\;\;\left(-2 \cdot t_0\right) \cdot \mathsf{fma}\left(0.125, U \cdot \frac{U}{J}, J\right)\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.052480368001563 \cdot 10^{-125}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot t_0\right)\\ \end{array} \]

Alternative 8
Error	26.1
Cost	7376

\[\begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -1.40572722140906 \cdot 10^{-131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \frac{J}{U}\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 4.052480368001563 \cdot 10^{-125}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	38.4
Cost	984

\[\begin{array}{l} \mathbf{if}\;J \leq -1.8974517473907947 \cdot 10^{+28}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -2.9289562024025276 \cdot 10^{-49}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 7.511931529555997 \cdot 10^{-77}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 1.1782760615722013 \cdot 10^{+55}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 10
Error	46.7
Cost	656

\[\begin{array}{l} \mathbf{if}\;J \leq -2.9289562024025276 \cdot 10^{-49}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq -3.629981878847924 \cdot 10^{-241}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 4.896013935583698 \cdot 10^{-267}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 7.511931529555997 \cdot 10^{-77}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 11
Error	46.2
Cost	392

\[\begin{array}{l} \mathbf{if}\;U \leq -3.4093226826729622 \cdot 10^{-254}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.4344440419843935 \cdot 10^{-225}:\\ \;\;\;\;-J\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 12
Error	46.7
Cost	64

\[U \]

Reproduce

herbie shell --seed 2022302 
(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)))))