Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 74.2% → 99.8%

Time: 21.8s

Alternatives: 9

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, 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]]

Initial Program: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t_0}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, 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]]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-2 \cdot \frac{{J}^{2}}{U} - U\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(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 (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY))
     (- (* -2.0 (/ (pow J 2.0) U)) U)
     (if (<= t_1 5e+307) t_1 U))))

C

U = abs(U);
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 / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-2.0 * (pow(J, 2.0) / U)) - U;
	} else if (t_1 <= 5e+307) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
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 / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (-2.0 * (Math.pow(J, 2.0) / U)) - U;
	} else if (t_1 <= 5e+307) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
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 / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (-2.0 * (math.pow(J, 2.0) / U)) - U
	elif t_1 <= 5e+307:
		tmp = t_1
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
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(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-2.0 * Float64((J ^ 2.0) / U)) - U);
	elseif (t_1 <= 5e+307)
		tmp = t_1;
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (-2.0 * ((J ^ 2.0) / U)) - U;
	elseif (t_1 <= 5e+307)
		tmp = t_1;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
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[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-2.0 * N[(N[Power[J, 2.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], If[LessEqual[t$95$1, 5e+307], 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 6.0%

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

   2. Taylor expanded in J around 0 34.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2} \cdot {\cos \left(0.5 \cdot K\right)}^{2}}{U} + -1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 34.8%

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

      2. unsub-neg 34.8%

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

      3. unpow2 34.8%

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

      4. *-commutative 34.8%

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

      5. unpow2 34.8%

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

      6. swap-sqr 34.8%

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

      7. unpow2 34.8%

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

      8. *-commutative 34.8%

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

   4. Simplified 34.8%

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

   5. Taylor expanded in K around 0 34.8%

      \[\leadsto -2 \cdot \color{blue}{\frac{{J}^{2}}{U}} - U \]

   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)))) < 5e307

   1. Initial program 99.8%

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

   if 5e307 < (*.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 5.0%

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

   2. Taylor expanded in U around -inf 49.3%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-2 \cdot \frac{{J}^{2}}{U} - U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;J \leq -1.8 \cdot 10^{-245} \lor \neg \left(J \leq 1.7 \cdot 10^{-292}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (or (<= J -1.8e-245) (not (<= J 1.7e-292)))
     (* -2.0 (* t_0 (* J (hypot 1.0 (/ (/ U (* J 2.0)) t_0)))))
     U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((J <= -1.8e-245) || !(J <= 1.7e-292)) {
		tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * 2.0)) / t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((J <= -1.8e-245) || !(J <= 1.7e-292)) {
		tmp = -2.0 * (t_0 * (J * Math.hypot(1.0, ((U / (J * 2.0)) / t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (J <= -1.8e-245) or not (J <= 1.7e-292):
		tmp = -2.0 * (t_0 * (J * math.hypot(1.0, ((U / (J * 2.0)) / t_0))))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if ((J <= -1.8e-245) || !(J <= 1.7e-292))
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J * hypot(1.0, Float64(Float64(U / Float64(J * 2.0)) / t_0)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((J <= -1.8e-245) || ~((J <= 1.7e-292)))
		tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * 2.0)) / t_0))));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[J, -1.8e-245], N[Not[LessEqual[J, 1.7e-292]], $MachinePrecision]], N[(-2.0 * N[(t$95$0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if J < -1.8e-245 or 1.70000000000000009e-292 < J

   1. Initial program 76.5%

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

   3. Simplified 90.6%

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

   if -1.8e-245 < J < 1.70000000000000009e-292

   1. Initial program 35.1%

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

   2. Taylor expanded in U around -inf 29.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.8 \cdot 10^{-245} \lor \neg \left(J \leq 1.7 \cdot 10^{-292}\right):\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 3: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\\ \mathbf{if}\;J \leq -1.8 \cdot 10^{-245}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot t_1\right)\right)\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t_1 \cdot \left(J \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (hypot 1.0 (/ (/ U (* J 2.0)) t_0))))
   (if (<= J -1.8e-245)
     (* -2.0 (* t_0 (* J t_1)))
     (if (<= J 2.6e-294) U (* -2.0 (* t_1 (* J t_0)))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = hypot(1.0, ((U / (J * 2.0)) / t_0));
	double tmp;
	if (J <= -1.8e-245) {
		tmp = -2.0 * (t_0 * (J * t_1));
	} else if (J <= 2.6e-294) {
		tmp = U;
	} else {
		tmp = -2.0 * (t_1 * (J * t_0));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.hypot(1.0, ((U / (J * 2.0)) / t_0));
	double tmp;
	if (J <= -1.8e-245) {
		tmp = -2.0 * (t_0 * (J * t_1));
	} else if (J <= 2.6e-294) {
		tmp = U;
	} else {
		tmp = -2.0 * (t_1 * (J * t_0));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.hypot(1.0, ((U / (J * 2.0)) / t_0))
	tmp = 0
	if J <= -1.8e-245:
		tmp = -2.0 * (t_0 * (J * t_1))
	elif J <= 2.6e-294:
		tmp = U
	else:
		tmp = -2.0 * (t_1 * (J * t_0))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = hypot(1.0, Float64(Float64(U / Float64(J * 2.0)) / t_0))
	tmp = 0.0
	if (J <= -1.8e-245)
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J * t_1)));
	elseif (J <= 2.6e-294)
		tmp = U;
	else
		tmp = Float64(-2.0 * Float64(t_1 * Float64(J * t_0)));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = hypot(1.0, ((U / (J * 2.0)) / t_0));
	tmp = 0.0;
	if (J <= -1.8e-245)
		tmp = -2.0 * (t_0 * (J * t_1));
	elseif (J <= 2.6e-294)
		tmp = U;
	else
		tmp = -2.0 * (t_1 * (J * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[J, -1.8e-245], N[(-2.0 * N[(t$95$0 * N[(J * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 2.6e-294], U, N[(-2.0 * N[(t$95$1 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.8e-245

   1. Initial program 77.0%

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

   3. Simplified 89.4%

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

   if -1.8e-245 < J < 2.5999999999999999e-294

   1. Initial program 35.1%

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

   2. Taylor expanded in U around -inf 29.1%

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

   if 2.5999999999999999e-294 < J

   1. Initial program 76.0%

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

      1. associate-*l* 76.7%

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

      2. associate-*l* 76.7%

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

      3. unpow2 76.7%

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

      4. hypot-1-def 91.9%

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

      5. associate-/r* 91.8%

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

      6. cos-neg 91.8%

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

      7. distribute-frac-neg 91.8%

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

      8. associate-/r* 91.9%

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

      9. associate-/r* 91.8%

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

      10. distribute-frac-neg 91.8%

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

      11. cos-neg 91.8%

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

   3. Simplified 91.8%

      \[\leadsto \color{blue}{-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.8 \cdot 10^{-245}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\right)\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right) \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)\\ \end{array} \]

Alternative 4: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 8.5 \cdot 10^{+197}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;U \leq 8.6 \cdot 10^{+250}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 8.5e+197)
   (* -2.0 (* (* J (cos (/ K 2.0))) (hypot 1.0 (/ (* U 0.5) J))))
   (if (<= U 8.6e+250) (- U) U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 8.5e+197) {
		tmp = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, ((U * 0.5) / J)));
	} else if (U <= 8.6e+250) {
		tmp = -U;
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (U <= 8.5e+197) {
		tmp = -2.0 * ((J * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U * 0.5) / J)));
	} else if (U <= 8.6e+250) {
		tmp = -U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 8.5e+197:
		tmp = -2.0 * ((J * math.cos((K / 2.0))) * math.hypot(1.0, ((U * 0.5) / J)))
	elif U <= 8.6e+250:
		tmp = -U
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 8.5e+197)
		tmp = Float64(-2.0 * Float64(Float64(J * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U * 0.5) / J))));
	elseif (U <= 8.6e+250)
		tmp = Float64(-U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 8.5e+197)
		tmp = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, ((U * 0.5) / J)));
	elseif (U <= 8.6e+250)
		tmp = -U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 8.5e+197], N[(-2.0 * N[(N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 8.6e+250], (-U), U]]

Derivation

1. Split input into 3 regimes

2. if U < 8.5000000000000001e197

   1. Initial program 76.3%

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

      1. associate-*l* 76.7%

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

      2. associate-*l* 76.7%

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

      3. unpow2 76.7%

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

      4. hypot-1-def 89.0%

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

      5. associate-/r* 88.9%

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

      6. cos-neg 88.9%

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

      7. distribute-frac-neg 88.9%

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

      8. associate-/r* 89.0%

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

      9. associate-/r* 88.9%

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

      10. distribute-frac-neg 88.9%

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

      11. cos-neg 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in K around 0 75.5%

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

      1. associate-*r/ 75.5%

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

   6. Simplified 75.5%

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

   if 8.5000000000000001e197 < U < 8.6e250

   1. Initial program 53.0%

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

   2. Taylor expanded in J around 0 63.5%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 63.5%

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

   4. Simplified 63.5%

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

   if 8.6e250 < U

   1. Initial program 27.6%

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

   2. Taylor expanded in U around -inf 72.1%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 8.5 \cdot 10^{+197}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;U \leq 8.6 \cdot 10^{+250}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 5: 63.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+32}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e+32)
   (* -2.0 (* J (hypot 1.0 (* 0.5 (/ U J)))))
   (* -2.0 (* J (cos (/ K 2.0))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e+32) {
		tmp = -2.0 * (J * hypot(1.0, (0.5 * (U / J))));
	} else {
		tmp = -2.0 * (J * cos((K / 2.0)));
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e+32) {
		tmp = -2.0 * (J * Math.hypot(1.0, (0.5 * (U / J))));
	} else {
		tmp = -2.0 * (J * Math.cos((K / 2.0)));
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if (K / 2.0) <= 1e+32:
		tmp = -2.0 * (J * math.hypot(1.0, (0.5 * (U / J))))
	else:
		tmp = -2.0 * (J * math.cos((K / 2.0)))
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e+32)
		tmp = Float64(-2.0 * Float64(J * hypot(1.0, Float64(0.5 * Float64(U / J)))));
	else
		tmp = Float64(-2.0 * Float64(J * cos(Float64(K / 2.0))));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 1e+32)
		tmp = -2.0 * (J * hypot(1.0, (0.5 * (U / J))));
	else
		tmp = -2.0 * (J * cos((K / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e+32], N[(-2.0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K 2) < 1.00000000000000005e32

   1. Initial program 71.9%

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

   3. Simplified 86.4%

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

      1. associate-*r* 86.4%

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

      2. *-commutative 86.4%

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

      3. add-cube-cbrt 84.9%

         \[\leadsto -2 \cdot \color{blue}{\left(\left(\sqrt[3]{\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)} \cdot \sqrt[3]{\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)}\right) \cdot \sqrt[3]{\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)}\right)} \]

      4. pow3 84.9%

         \[\leadsto -2 \cdot \color{blue}{{\left(\sqrt[3]{\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)}\right)}^{3}} \]

   5. Applied egg-rr 84.9%

      \[\leadsto -2 \cdot \color{blue}{{\left(\sqrt[3]{J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)\right)}\right)}^{3}} \]
   6. Step-by-step derivation

      1. rem-cube-cbrt 86.3%

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

      2. *-commutative 86.3%

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

      3. add-cube-cbrt 85.4%

         \[\leadsto -2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)} \cdot \sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)}\right) \cdot \sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)}\right)} \cdot J\right) \]

      4. associate-*l* 85.5%

         \[\leadsto -2 \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)} \cdot \sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)}\right) \cdot \left(\sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, U \cdot \frac{\frac{0.5}{J}}{\cos \left(K \cdot 0.5\right)}\right)} \cdot J\right)\right)} \]

   7. Applied egg-rr 85.5%

      \[\leadsto -2 \cdot \color{blue}{\left({\left(\sqrt[3]{\cos \left(0.5 \cdot K\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right)}\right)}\right)}^{2} \cdot \left(\sqrt[3]{\cos \left(0.5 \cdot K\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right)}\right)} \cdot J\right)\right)} \]

   8. Taylor expanded in K around 0 37.2%

      \[\leadsto -2 \cdot \color{blue}{\left(J \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]
   9. Step-by-step derivation

      1. metadata-eval 37.2%

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

      2. unpow2 37.2%

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

      3. unpow2 37.2%

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

      4. times-frac 53.7%

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

      5. swap-sqr 53.7%

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

      6. unpow2 53.7%

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

      7. unpow2 53.7%

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

      8. hypot-1-def 64.4%

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

   10. Simplified 64.4%

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

   if 1.00000000000000005e32 < (/.f64 K 2)

   1. Initial program 76.1%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in J around inf 60.1%

      \[\leadsto -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{J}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+32}:\\ \;\;\;\;-2 \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array} \]

Alternative 6: 66.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{if}\;J \leq -1.18 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (cos (/ K 2.0))))))
   (if (<= J -1.18e-7)
     t_0
     (if (<= J 2.6e-294) U (if (<= J 9.5e-22) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * cos((K / 2.0)));
	double tmp;
	if (J <= -1.18e-7) {
		tmp = t_0;
	} else if (J <= 2.6e-294) {
		tmp = U;
	} else if (J <= 9.5e-22) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * (j * cos((k / 2.0d0)))
    if (j <= (-1.18d-7)) then
        tmp = t_0
    else if (j <= 2.6d-294) then
        tmp = u
    else if (j <= 9.5d-22) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * Math.cos((K / 2.0)));
	double tmp;
	if (J <= -1.18e-7) {
		tmp = t_0;
	} else if (J <= 2.6e-294) {
		tmp = U;
	} else if (J <= 9.5e-22) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J * math.cos((K / 2.0)))
	tmp = 0
	if J <= -1.18e-7:
		tmp = t_0
	elif J <= 2.6e-294:
		tmp = U
	elif J <= 9.5e-22:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * cos(Float64(K / 2.0))))
	tmp = 0.0
	if (J <= -1.18e-7)
		tmp = t_0;
	elseif (J <= 2.6e-294)
		tmp = U;
	elseif (J <= 9.5e-22)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J * cos((K / 2.0)));
	tmp = 0.0;
	if (J <= -1.18e-7)
		tmp = t_0;
	elseif (J <= 2.6e-294)
		tmp = U;
	elseif (J <= 9.5e-22)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.18e-7], t$95$0, If[LessEqual[J, 2.6e-294], U, If[LessEqual[J, 9.5e-22], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.18e-7 or 9.4999999999999994e-22 < J

   1. Initial program 95.0%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in J around inf 83.5%

      \[\leadsto -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{J}\right) \]

   if -1.18e-7 < J < 2.5999999999999999e-294

   1. Initial program 45.9%

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

   2. Taylor expanded in U around -inf 41.8%

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

   if 2.5999999999999999e-294 < J < 9.4999999999999994e-22

   1. Initial program 54.8%

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

   2. Taylor expanded in J around 0 36.4%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 36.4%

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

   4. Simplified 36.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.18 \cdot 10^{-7}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array} \]

Alternative 7: 49.0% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -0.9:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 340000000:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -0.9)
   (* -2.0 J)
   (if (<= J 2.6e-294) U (if (<= J 340000000.0) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -0.9) {
		tmp = -2.0 * J;
	} else if (J <= 2.6e-294) {
		tmp = U;
	} else if (J <= 340000000.0) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-0.9d0)) then
        tmp = (-2.0d0) * j
    else if (j <= 2.6d-294) then
        tmp = u
    else if (j <= 340000000.0d0) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -0.9) {
		tmp = -2.0 * J;
	} else if (J <= 2.6e-294) {
		tmp = U;
	} else if (J <= 340000000.0) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -0.9:
		tmp = -2.0 * J
	elif J <= 2.6e-294:
		tmp = U
	elif J <= 340000000.0:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -0.9)
		tmp = Float64(-2.0 * J);
	elseif (J <= 2.6e-294)
		tmp = U;
	elseif (J <= 340000000.0)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -0.9)
		tmp = -2.0 * J;
	elseif (J <= 2.6e-294)
		tmp = U;
	elseif (J <= 340000000.0)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -0.9], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, 2.6e-294], U, If[LessEqual[J, 340000000.0], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -0.900000000000000022 or 3.4e8 < J

   1. Initial program 96.3%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in J around inf 84.4%

      \[\leadsto -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{J}\right) \]

   5. Taylor expanded in K around 0 47.0%

      \[\leadsto -2 \cdot \color{blue}{J} \]

   if -0.900000000000000022 < J < 2.5999999999999999e-294

   1. Initial program 45.9%

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

   2. Taylor expanded in U around -inf 41.8%

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

   if 2.5999999999999999e-294 < J < 3.4e8

   1. Initial program 55.4%

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

   2. Taylor expanded in J around 0 33.5%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 33.5%

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

   4. Simplified 33.5%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -0.9:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 340000000:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 8: 38.2% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J 2.6e-294) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= 2.6e-294) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= 2.6d-294) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= 2.6e-294) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= 2.6e-294:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= 2.6e-294)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= 2.6e-294)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, 2.6e-294], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < 2.5999999999999999e-294

   1. Initial program 70.2%

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

   2. Taylor expanded in U around -inf 26.8%

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

   if 2.5999999999999999e-294 < J

   1. Initial program 76.0%

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

   2. Taylor expanded in J around 0 21.9%

      \[\leadsto \color{blue}{-1 \cdot U} \]
   3. Step-by-step derivation

      1. neg-mul-1 21.9%

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

   4. Simplified 21.9%

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

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.6 \cdot 10^{-294}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 9: 26.4% accurate, 420.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ U \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 U)

C

U = abs(U);
double code(double J, double K, double U) {
	return U;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	return U;
}

Python

U = abs(U)
def code(J, K, U):
	return U

Julia

U = abs(U)
function code(J, K, U)
	return U
end

MATLAB

U = abs(U)
function tmp = code(J, K, U)
	tmp = U;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := U

Derivation

1. Initial program 72.9%

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

2. Taylor expanded in U around -inf 26.2%

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

3. Final simplification 26.2%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023306 
(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)))))