Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.4% → 88.1%

Time: 26.8s

Alternatives: 10

Speedup: 1.3×

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: 73.4% 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: 88.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-*l* 78.4%

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

   3. unpow2 78.4%

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

   4. hypot-1-def 90.4%

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

   5. *-commutative 90.4%

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

   6. associate-*l* 90.4%

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

3. Simplified 90.4%

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

4. Final simplification 90.4%

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

Alternative 2: 88.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-*l* 78.4%

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

   3. associate-*r* 78.4%

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

   4. *-commutative 78.4%

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

   5. associate-*l* 78.4%

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

   6. *-commutative 78.4%

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

   7. unpow2 78.4%

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

   8. hypot-1-def 90.3%

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

   9. *-commutative 90.3%

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

   10. associate-*l* 90.3%

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

3. Simplified 90.3%

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

4. Final simplification 90.3%

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

Alternative 3: 72.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -9.5 \cdot 10^{+236}:\\ \;\;\;\;U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{elif}\;U \leq -3.2 \cdot 10^{+170}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -4.2 \cdot 10^{+134}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\frac{J}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= U -9.5e+236)
   (+ U (* J (/ 2.0 (/ U J))))
   (if (<= U -3.2e+170)
     (- U)
     (if (<= U -4.2e+134)
       U
       (if (<= U 5e+274)
         (* (* J (* -2.0 (cos (/ K 2.0)))) (hypot 1.0 (/ U (/ J 0.5))))
         (- U))))))

C

double code(double J, double K, double U) {
	double tmp;
	if (U <= -9.5e+236) {
		tmp = U + (J * (2.0 / (U / J)));
	} else if (U <= -3.2e+170) {
		tmp = -U;
	} else if (U <= -4.2e+134) {
		tmp = U;
	} else if (U <= 5e+274) {
		tmp = (J * (-2.0 * cos((K / 2.0)))) * hypot(1.0, (U / (J / 0.5)));
	} else {
		tmp = -U;
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (U <= -9.5e+236) {
		tmp = U + (J * (2.0 / (U / J)));
	} else if (U <= -3.2e+170) {
		tmp = -U;
	} else if (U <= -4.2e+134) {
		tmp = U;
	} else if (U <= 5e+274) {
		tmp = (J * (-2.0 * Math.cos((K / 2.0)))) * Math.hypot(1.0, (U / (J / 0.5)));
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if U <= -9.5e+236:
		tmp = U + (J * (2.0 / (U / J)))
	elif U <= -3.2e+170:
		tmp = -U
	elif U <= -4.2e+134:
		tmp = U
	elif U <= 5e+274:
		tmp = (J * (-2.0 * math.cos((K / 2.0)))) * math.hypot(1.0, (U / (J / 0.5)))
	else:
		tmp = -U
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (U <= -9.5e+236)
		tmp = Float64(U + Float64(J * Float64(2.0 / Float64(U / J))));
	elseif (U <= -3.2e+170)
		tmp = Float64(-U);
	elseif (U <= -4.2e+134)
		tmp = U;
	elseif (U <= 5e+274)
		tmp = Float64(Float64(J * Float64(-2.0 * cos(Float64(K / 2.0)))) * hypot(1.0, Float64(U / Float64(J / 0.5))));
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= -9.5e+236)
		tmp = U + (J * (2.0 / (U / J)));
	elseif (U <= -3.2e+170)
		tmp = -U;
	elseif (U <= -4.2e+134)
		tmp = U;
	elseif (U <= 5e+274)
		tmp = (J * (-2.0 * cos((K / 2.0)))) * hypot(1.0, (U / (J / 0.5)));
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[U, -9.5e+236], N[(U + N[(J * N[(2.0 / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, -3.2e+170], (-U), If[LessEqual[U, -4.2e+134], U, If[LessEqual[U, 5e+274], N[(N[(J * N[(-2.0 * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(U / N[(J / 0.5), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], (-U)]]]]

Derivation

1. Split input into 4 regimes

2. if U < -9.4999999999999999e236

   1. Initial program 37.9%

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

      1. *-commutative 37.9%

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

      2. associate-*l* 37.9%

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

      3. associate-*r* 37.9%

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

      4. *-commutative 37.9%

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

      5. associate-*l* 37.9%

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

      6. *-commutative 37.9%

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

      7. unpow2 37.9%

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

      8. hypot-1-def 53.1%

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

      9. *-commutative 53.1%

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

      10. associate-*l* 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in U around -inf 29.5%

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

   5. Taylor expanded in K around 0 29.5%

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

      1. *-commutative 29.5%

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

      2. +-commutative 29.5%

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

      3. distribute-lft-in 29.5%

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

      4. associate-*r/ 52.4%

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

      5. associate-*l/ 60.6%

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

      6. *-inverses 60.6%

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

      7. *-lft-identity 60.6%

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

      8. associate-*r/ 60.6%

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

      9. associate-/l* 60.6%

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

   7. Simplified 60.6%

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

   if -9.4999999999999999e236 < U < -3.19999999999999979e170 or 4.9999999999999998e274 < U

   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. Step-by-step derivation

      1. *-commutative 48.6%

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

      2. associate-*l* 48.6%

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

      3. associate-*r* 48.6%

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

      4. *-commutative 48.6%

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

      5. associate-*l* 48.5%

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

      6. *-commutative 48.5%

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

      7. unpow2 48.5%

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

      8. hypot-1-def 59.6%

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

      9. *-commutative 59.6%

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

      10. associate-*l* 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in J around 0 72.7%

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

      1. neg-mul-1 72.7%

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

   6. Simplified 72.7%

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

   if -3.19999999999999979e170 < U < -4.2000000000000002e134

   1. Initial program 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*l* 61.6%

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

      3. associate-*r* 61.6%

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

      4. *-commutative 61.6%

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

      5. associate-*l* 61.6%

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

      6. *-commutative 61.6%

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

      7. unpow2 61.6%

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

      8. hypot-1-def 61.6%

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

      9. *-commutative 61.6%

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

      10. associate-*l* 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in U around -inf 100.0%

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

   if -4.2000000000000002e134 < U < 4.9999999999999998e274

   1. Initial program 84.6%

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

      1. *-commutative 84.6%

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

      2. associate-*l* 84.6%

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

      3. unpow2 84.6%

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

      4. hypot-1-def 96.7%

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

      5. *-commutative 96.7%

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

      6. associate-*l* 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in K around 0 81.6%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-/l* 81.6%

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

   6. Simplified 81.6%

      \[\leadsto \left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \color{blue}{\frac{U}{\frac{J}{0.5}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -9.5 \cdot 10^{+236}:\\ \;\;\;\;U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{elif}\;U \leq -3.2 \cdot 10^{+170}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -4.2 \cdot 10^{+134}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{\frac{J}{0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 4: 57.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{if}\;U \leq -1.2 \cdot 10^{+237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq -2.2 \cdot 10^{+169}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -6 \cdot 10^{-10}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.5 \cdot 10^{+55}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;U \leq 1.6 \cdot 10^{+208}:\\ \;\;\;\;-2 \cdot \frac{J}{\frac{U}{J}} - U\\ \mathbf{elif}\;U \leq 1.42 \cdot 10^{+267}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (/ 2.0 (/ U J))))))
   (if (<= U -1.2e+237)
     t_0
     (if (<= U -2.2e+169)
       (- U)
       (if (<= U -6e-10)
         U
         (if (<= U 2.5e+55)
           (* J (* -2.0 (cos (* K 0.5))))
           (if (<= U 1.6e+208)
             (- (* -2.0 (/ J (/ U J))) U)
             (if (<= U 1.42e+267) t_0 (- U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = U + (J * (2.0 / (U / J)));
	double tmp;
	if (U <= -1.2e+237) {
		tmp = t_0;
	} else if (U <= -2.2e+169) {
		tmp = -U;
	} else if (U <= -6e-10) {
		tmp = U;
	} else if (U <= 2.5e+55) {
		tmp = J * (-2.0 * cos((K * 0.5)));
	} else if (U <= 1.6e+208) {
		tmp = (-2.0 * (J / (U / J))) - U;
	} else if (U <= 1.42e+267) {
		tmp = t_0;
	} else {
		tmp = -U;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = u + (j * (2.0d0 / (u / j)))
    if (u <= (-1.2d+237)) then
        tmp = t_0
    else if (u <= (-2.2d+169)) then
        tmp = -u
    else if (u <= (-6d-10)) then
        tmp = u
    else if (u <= 2.5d+55) then
        tmp = j * ((-2.0d0) * cos((k * 0.5d0)))
    else if (u <= 1.6d+208) then
        tmp = ((-2.0d0) * (j / (u / j))) - u
    else if (u <= 1.42d+267) then
        tmp = t_0
    else
        tmp = -u
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = U + (J * (2.0 / (U / J)));
	double tmp;
	if (U <= -1.2e+237) {
		tmp = t_0;
	} else if (U <= -2.2e+169) {
		tmp = -U;
	} else if (U <= -6e-10) {
		tmp = U;
	} else if (U <= 2.5e+55) {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	} else if (U <= 1.6e+208) {
		tmp = (-2.0 * (J / (U / J))) - U;
	} else if (U <= 1.42e+267) {
		tmp = t_0;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = U + (J * (2.0 / (U / J)))
	tmp = 0
	if U <= -1.2e+237:
		tmp = t_0
	elif U <= -2.2e+169:
		tmp = -U
	elif U <= -6e-10:
		tmp = U
	elif U <= 2.5e+55:
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	elif U <= 1.6e+208:
		tmp = (-2.0 * (J / (U / J))) - U
	elif U <= 1.42e+267:
		tmp = t_0
	else:
		tmp = -U
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(U + Float64(J * Float64(2.0 / Float64(U / J))))
	tmp = 0.0
	if (U <= -1.2e+237)
		tmp = t_0;
	elseif (U <= -2.2e+169)
		tmp = Float64(-U);
	elseif (U <= -6e-10)
		tmp = U;
	elseif (U <= 2.5e+55)
		tmp = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))));
	elseif (U <= 1.6e+208)
		tmp = Float64(Float64(-2.0 * Float64(J / Float64(U / J))) - U);
	elseif (U <= 1.42e+267)
		tmp = t_0;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = U + (J * (2.0 / (U / J)));
	tmp = 0.0;
	if (U <= -1.2e+237)
		tmp = t_0;
	elseif (U <= -2.2e+169)
		tmp = -U;
	elseif (U <= -6e-10)
		tmp = U;
	elseif (U <= 2.5e+55)
		tmp = J * (-2.0 * cos((K * 0.5)));
	elseif (U <= 1.6e+208)
		tmp = (-2.0 * (J / (U / J))) - U;
	elseif (U <= 1.42e+267)
		tmp = t_0;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(2.0 / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1.2e+237], t$95$0, If[LessEqual[U, -2.2e+169], (-U), If[LessEqual[U, -6e-10], U, If[LessEqual[U, 2.5e+55], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 1.6e+208], N[(N[(-2.0 * N[(J / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], If[LessEqual[U, 1.42e+267], t$95$0, (-U)]]]]]]]

Derivation

1. Split input into 5 regimes

2. if U < -1.1999999999999999e237 or 1.6000000000000001e208 < U < 1.42000000000000003e267

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. associate-*r* 55.7%

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

      4. *-commutative 55.7%

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

      5. associate-*l* 55.7%

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

      6. *-commutative 55.7%

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

      7. unpow2 55.7%

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

      8. hypot-1-def 65.4%

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

      9. *-commutative 65.4%

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

      10. associate-*l* 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in U around -inf 37.8%

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

   5. Taylor expanded in K around 0 37.0%

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

      1. *-commutative 37.0%

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

      2. +-commutative 37.0%

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

      3. distribute-lft-in 37.0%

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

      4. associate-*r/ 40.6%

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

      5. associate-*l/ 61.7%

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

      6. *-inverses 61.7%

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

      7. *-lft-identity 61.7%

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

      8. associate-*r/ 61.7%

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

      9. associate-/l* 61.7%

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

   7. Simplified 61.7%

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

   if -1.1999999999999999e237 < U < -2.2e169 or 1.42000000000000003e267 < U

   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. Step-by-step derivation

      1. *-commutative 48.6%

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

      2. associate-*l* 48.6%

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

      3. associate-*r* 48.6%

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

      4. *-commutative 48.6%

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

      5. associate-*l* 48.5%

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

      6. *-commutative 48.5%

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

      7. unpow2 48.5%

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

      8. hypot-1-def 59.6%

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

      9. *-commutative 59.6%

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

      10. associate-*l* 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in J around 0 72.7%

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

      1. neg-mul-1 72.7%

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

   6. Simplified 72.7%

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

   if -2.2e169 < U < -6e-10

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-*l* 69.6%

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

      3. associate-*r* 69.6%

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

      4. *-commutative 69.6%

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

      5. associate-*l* 69.7%

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

      6. *-commutative 69.7%

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

      7. unpow2 69.7%

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

      8. hypot-1-def 91.7%

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

      9. *-commutative 91.7%

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

      10. associate-*l* 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in U around -inf 61.0%

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

   if -6e-10 < U < 2.50000000000000023e55

   1. Initial program 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-*l* 92.7%

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

      3. associate-*r* 92.7%

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

      4. *-commutative 92.7%

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

      5. associate-*l* 92.7%

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

      6. *-commutative 92.7%

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

      7. unpow2 92.7%

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

      8. hypot-1-def 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in U around 0 77.9%

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

   if 2.50000000000000023e55 < U < 1.6000000000000001e208

   1. Initial program 54.1%

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

      1. *-commutative 54.1%

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

      2. associate-*l* 54.1%

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

      3. unpow2 54.1%

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

      4. hypot-1-def 82.2%

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

      5. *-commutative 82.2%

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

      6. associate-*l* 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in K around 0 55.3%

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

      1. associate-*r/ 55.3%

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

      2. *-commutative 55.3%

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

      3. associate-/l* 55.3%

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

   6. Simplified 55.3%

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

   7. Taylor expanded in K around 0 67.0%

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

   8. Taylor expanded in J around 0 52.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   9. Step-by-step derivation

      1. neg-mul-1 52.9%

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

      2. unsub-neg 52.9%

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

      3. associate-*r/ 52.9%

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

      4. unpow2 52.9%

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

   10. Simplified 52.9%

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

   11. Taylor expanded in J around 0 52.9%

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U}} - U \]
   12. Step-by-step derivation

       1. unpow2 52.9%

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

       2. associate-/l* 56.5%

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

   13. Simplified 56.5%

       \[\leadsto \color{blue}{-2 \cdot \frac{J}{\frac{U}{J}}} - U \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.2 \cdot 10^{+237}:\\ \;\;\;\;U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{elif}\;U \leq -2.2 \cdot 10^{+169}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -6 \cdot 10^{-10}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 2.5 \cdot 10^{+55}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;U \leq 1.6 \cdot 10^{+208}:\\ \;\;\;\;-2 \cdot \frac{J}{\frac{U}{J}} - U\\ \mathbf{elif}\;U \leq 1.42 \cdot 10^{+267}:\\ \;\;\;\;U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 5: 69.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq -2.75 \cdot 10^{+31} \lor \neg \left(K \leq 0.095\right):\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{\frac{J}{0.5}}\right) \cdot \left(J \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (or (<= K -2.75e+31) (not (<= K 0.095)))
   (* J (* -2.0 (cos (* K 0.5))))
   (* (hypot 1.0 (/ U (/ J 0.5))) (* J -2.0))))

C

double code(double J, double K, double U) {
	double tmp;
	if ((K <= -2.75e+31) || !(K <= 0.095)) {
		tmp = J * (-2.0 * cos((K * 0.5)));
	} else {
		tmp = hypot(1.0, (U / (J / 0.5))) * (J * -2.0);
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double tmp;
	if ((K <= -2.75e+31) || !(K <= 0.095)) {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	} else {
		tmp = Math.hypot(1.0, (U / (J / 0.5))) * (J * -2.0);
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if (K <= -2.75e+31) or not (K <= 0.095):
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	else:
		tmp = math.hypot(1.0, (U / (J / 0.5))) * (J * -2.0)
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if ((K <= -2.75e+31) || !(K <= 0.095))
		tmp = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))));
	else
		tmp = Float64(hypot(1.0, Float64(U / Float64(J / 0.5))) * Float64(J * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if ((K <= -2.75e+31) || ~((K <= 0.095)))
		tmp = J * (-2.0 * cos((K * 0.5)));
	else
		tmp = hypot(1.0, (U / (J / 0.5))) * (J * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[Or[LessEqual[K, -2.75e+31], N[Not[LessEqual[K, 0.095]], $MachinePrecision]], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[1.0 ^ 2 + N[(U / N[(J / 0.5), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < -2.75000000000000001e31 or 0.095000000000000001 < K

   1. Initial program 79.5%

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

      1. *-commutative 79.5%

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

      2. associate-*l* 79.5%

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

      3. associate-*r* 79.5%

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

      4. *-commutative 79.5%

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

      5. associate-*l* 79.5%

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

      6. *-commutative 79.5%

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

      7. unpow2 79.5%

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

      8. hypot-1-def 88.5%

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

      9. *-commutative 88.5%

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

      10. associate-*l* 88.5%

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

   3. Simplified 88.5%

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

   4. Taylor expanded in U around 0 54.8%

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

   if -2.75000000000000001e31 < K < 0.095000000000000001

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*l* 77.4%

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

      3. unpow2 77.4%

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

      4. hypot-1-def 92.1%

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

      5. *-commutative 92.1%

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

      6. associate-*l* 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in K around 0 87.7%

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

      1. associate-*r/ 87.7%

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

      2. *-commutative 87.7%

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

      3. associate-/l* 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in K around 0 91.4%

      \[\leadsto \color{blue}{\left(-2 \cdot J\right)} \cdot \mathsf{hypot}\left(1, \frac{U}{\frac{J}{0.5}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq -2.75 \cdot 10^{+31} \lor \neg \left(K \leq 0.095\right):\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(1, \frac{U}{\frac{J}{0.5}}\right) \cdot \left(J \cdot -2\right)\\ \end{array} \]

Alternative 6: 39.3% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{if}\;U \leq -1.35 \cdot 10^{+237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq -1.65 \cdot 10^{+170}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -4.4 \cdot 10^{-17}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 6.6 \cdot 10^{-84}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 1.86 \cdot 10^{+208}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+266}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (/ 2.0 (/ U J))))))
   (if (<= U -1.35e+237)
     t_0
     (if (<= U -1.65e+170)
       (- U)
       (if (<= U -4.4e-17)
         U
         (if (<= U 6.6e-84)
           (* J -2.0)
           (if (<= U 1.86e+208) (- U) (if (<= U 5e+266) t_0 (- U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = U + (J * (2.0 / (U / J)));
	double tmp;
	if (U <= -1.35e+237) {
		tmp = t_0;
	} else if (U <= -1.65e+170) {
		tmp = -U;
	} else if (U <= -4.4e-17) {
		tmp = U;
	} else if (U <= 6.6e-84) {
		tmp = J * -2.0;
	} else if (U <= 1.86e+208) {
		tmp = -U;
	} else if (U <= 5e+266) {
		tmp = t_0;
	} else {
		tmp = -U;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = u + (j * (2.0d0 / (u / j)))
    if (u <= (-1.35d+237)) then
        tmp = t_0
    else if (u <= (-1.65d+170)) then
        tmp = -u
    else if (u <= (-4.4d-17)) then
        tmp = u
    else if (u <= 6.6d-84) then
        tmp = j * (-2.0d0)
    else if (u <= 1.86d+208) then
        tmp = -u
    else if (u <= 5d+266) then
        tmp = t_0
    else
        tmp = -u
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = U + (J * (2.0 / (U / J)));
	double tmp;
	if (U <= -1.35e+237) {
		tmp = t_0;
	} else if (U <= -1.65e+170) {
		tmp = -U;
	} else if (U <= -4.4e-17) {
		tmp = U;
	} else if (U <= 6.6e-84) {
		tmp = J * -2.0;
	} else if (U <= 1.86e+208) {
		tmp = -U;
	} else if (U <= 5e+266) {
		tmp = t_0;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = U + (J * (2.0 / (U / J)))
	tmp = 0
	if U <= -1.35e+237:
		tmp = t_0
	elif U <= -1.65e+170:
		tmp = -U
	elif U <= -4.4e-17:
		tmp = U
	elif U <= 6.6e-84:
		tmp = J * -2.0
	elif U <= 1.86e+208:
		tmp = -U
	elif U <= 5e+266:
		tmp = t_0
	else:
		tmp = -U
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(U + Float64(J * Float64(2.0 / Float64(U / J))))
	tmp = 0.0
	if (U <= -1.35e+237)
		tmp = t_0;
	elseif (U <= -1.65e+170)
		tmp = Float64(-U);
	elseif (U <= -4.4e-17)
		tmp = U;
	elseif (U <= 6.6e-84)
		tmp = Float64(J * -2.0);
	elseif (U <= 1.86e+208)
		tmp = Float64(-U);
	elseif (U <= 5e+266)
		tmp = t_0;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = U + (J * (2.0 / (U / J)));
	tmp = 0.0;
	if (U <= -1.35e+237)
		tmp = t_0;
	elseif (U <= -1.65e+170)
		tmp = -U;
	elseif (U <= -4.4e-17)
		tmp = U;
	elseif (U <= 6.6e-84)
		tmp = J * -2.0;
	elseif (U <= 1.86e+208)
		tmp = -U;
	elseif (U <= 5e+266)
		tmp = t_0;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(2.0 / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1.35e+237], t$95$0, If[LessEqual[U, -1.65e+170], (-U), If[LessEqual[U, -4.4e-17], U, If[LessEqual[U, 6.6e-84], N[(J * -2.0), $MachinePrecision], If[LessEqual[U, 1.86e+208], (-U), If[LessEqual[U, 5e+266], t$95$0, (-U)]]]]]]]

Derivation

1. Split input into 4 regimes

2. if U < -1.35e237 or 1.85999999999999993e208 < U < 4.9999999999999999e266

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. associate-*r* 55.7%

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

      4. *-commutative 55.7%

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

      5. associate-*l* 55.7%

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

      6. *-commutative 55.7%

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

      7. unpow2 55.7%

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

      8. hypot-1-def 65.4%

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

      9. *-commutative 65.4%

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

      10. associate-*l* 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in U around -inf 37.8%

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

   5. Taylor expanded in K around 0 37.0%

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

      1. *-commutative 37.0%

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

      2. +-commutative 37.0%

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

      3. distribute-lft-in 37.0%

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

      4. associate-*r/ 40.6%

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

      5. associate-*l/ 61.7%

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

      6. *-inverses 61.7%

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

      7. *-lft-identity 61.7%

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

      8. associate-*r/ 61.7%

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

      9. associate-/l* 61.7%

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

   7. Simplified 61.7%

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

   if -1.35e237 < U < -1.65000000000000012e170 or 6.59999999999999968e-84 < U < 1.85999999999999993e208 or 4.9999999999999999e266 < U

   1. Initial program 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-*l* 58.1%

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

      3. associate-*r* 58.1%

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

      4. *-commutative 58.1%

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

      5. associate-*l* 58.0%

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

      6. *-commutative 58.0%

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

      7. unpow2 58.0%

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

      8. hypot-1-def 81.5%

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

      9. *-commutative 81.5%

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

      10. associate-*l* 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in J around 0 51.8%

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

      1. neg-mul-1 51.8%

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

   6. Simplified 51.8%

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

   if -1.65000000000000012e170 < U < -4.4e-17

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-*l* 69.6%

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

      3. associate-*r* 69.6%

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

      4. *-commutative 69.6%

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

      5. associate-*l* 69.7%

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

      6. *-commutative 69.7%

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

      7. unpow2 69.7%

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

      8. hypot-1-def 91.7%

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

      9. *-commutative 91.7%

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

      10. associate-*l* 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in U around -inf 61.0%

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

   if -4.4e-17 < U < 6.59999999999999968e-84

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l* 98.2%

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

      3. associate-*r* 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*l* 98.1%

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

      6. *-commutative 98.1%

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

      7. unpow2 98.1%

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

      8. hypot-1-def 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in U around 0 84.4%

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

   5. Taylor expanded in K around 0 48.5%

      \[\leadsto \color{blue}{-2 \cdot J} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.35 \cdot 10^{+237}:\\ \;\;\;\;U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{elif}\;U \leq -1.65 \cdot 10^{+170}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -4.4 \cdot 10^{-17}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 6.6 \cdot 10^{-84}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 1.86 \cdot 10^{+208}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+266}:\\ \;\;\;\;U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 7: 39.5% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{if}\;U \leq -1 \cdot 10^{+237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;U \leq -2.3 \cdot 10^{+171}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.3 \cdot 10^{-82}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 4.2 \cdot 10^{+212}:\\ \;\;\;\;-2 \cdot \frac{J}{\frac{U}{J}} - U\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+266}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (/ 2.0 (/ U J))))))
   (if (<= U -1e+237)
     t_0
     (if (<= U -2.3e+171)
       (- U)
       (if (<= U -1.8e-16)
         U
         (if (<= U 3.3e-82)
           (* J -2.0)
           (if (<= U 4.2e+212)
             (- (* -2.0 (/ J (/ U J))) U)
             (if (<= U 5e+266) t_0 (- U)))))))))

C

double code(double J, double K, double U) {
	double t_0 = U + (J * (2.0 / (U / J)));
	double tmp;
	if (U <= -1e+237) {
		tmp = t_0;
	} else if (U <= -2.3e+171) {
		tmp = -U;
	} else if (U <= -1.8e-16) {
		tmp = U;
	} else if (U <= 3.3e-82) {
		tmp = J * -2.0;
	} else if (U <= 4.2e+212) {
		tmp = (-2.0 * (J / (U / J))) - U;
	} else if (U <= 5e+266) {
		tmp = t_0;
	} else {
		tmp = -U;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = u + (j * (2.0d0 / (u / j)))
    if (u <= (-1d+237)) then
        tmp = t_0
    else if (u <= (-2.3d+171)) then
        tmp = -u
    else if (u <= (-1.8d-16)) then
        tmp = u
    else if (u <= 3.3d-82) then
        tmp = j * (-2.0d0)
    else if (u <= 4.2d+212) then
        tmp = ((-2.0d0) * (j / (u / j))) - u
    else if (u <= 5d+266) then
        tmp = t_0
    else
        tmp = -u
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = U + (J * (2.0 / (U / J)));
	double tmp;
	if (U <= -1e+237) {
		tmp = t_0;
	} else if (U <= -2.3e+171) {
		tmp = -U;
	} else if (U <= -1.8e-16) {
		tmp = U;
	} else if (U <= 3.3e-82) {
		tmp = J * -2.0;
	} else if (U <= 4.2e+212) {
		tmp = (-2.0 * (J / (U / J))) - U;
	} else if (U <= 5e+266) {
		tmp = t_0;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = U + (J * (2.0 / (U / J)))
	tmp = 0
	if U <= -1e+237:
		tmp = t_0
	elif U <= -2.3e+171:
		tmp = -U
	elif U <= -1.8e-16:
		tmp = U
	elif U <= 3.3e-82:
		tmp = J * -2.0
	elif U <= 4.2e+212:
		tmp = (-2.0 * (J / (U / J))) - U
	elif U <= 5e+266:
		tmp = t_0
	else:
		tmp = -U
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(U + Float64(J * Float64(2.0 / Float64(U / J))))
	tmp = 0.0
	if (U <= -1e+237)
		tmp = t_0;
	elseif (U <= -2.3e+171)
		tmp = Float64(-U);
	elseif (U <= -1.8e-16)
		tmp = U;
	elseif (U <= 3.3e-82)
		tmp = Float64(J * -2.0);
	elseif (U <= 4.2e+212)
		tmp = Float64(Float64(-2.0 * Float64(J / Float64(U / J))) - U);
	elseif (U <= 5e+266)
		tmp = t_0;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = U + (J * (2.0 / (U / J)));
	tmp = 0.0;
	if (U <= -1e+237)
		tmp = t_0;
	elseif (U <= -2.3e+171)
		tmp = -U;
	elseif (U <= -1.8e-16)
		tmp = U;
	elseif (U <= 3.3e-82)
		tmp = J * -2.0;
	elseif (U <= 4.2e+212)
		tmp = (-2.0 * (J / (U / J))) - U;
	elseif (U <= 5e+266)
		tmp = t_0;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(2.0 / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[U, -1e+237], t$95$0, If[LessEqual[U, -2.3e+171], (-U), If[LessEqual[U, -1.8e-16], U, If[LessEqual[U, 3.3e-82], N[(J * -2.0), $MachinePrecision], If[LessEqual[U, 4.2e+212], N[(N[(-2.0 * N[(J / N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], If[LessEqual[U, 5e+266], t$95$0, (-U)]]]]]]]

Derivation

1. Split input into 5 regimes

2. if U < -9.9999999999999994e236 or 4.2e212 < U < 4.9999999999999999e266

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*l* 55.7%

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

      3. associate-*r* 55.7%

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

      4. *-commutative 55.7%

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

      5. associate-*l* 55.7%

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

      6. *-commutative 55.7%

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

      7. unpow2 55.7%

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

      8. hypot-1-def 65.4%

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

      9. *-commutative 65.4%

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

      10. associate-*l* 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in U around -inf 37.8%

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

   5. Taylor expanded in K around 0 37.0%

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

      1. *-commutative 37.0%

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

      2. +-commutative 37.0%

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

      3. distribute-lft-in 37.0%

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

      4. associate-*r/ 40.6%

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

      5. associate-*l/ 61.7%

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

      6. *-inverses 61.7%

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

      7. *-lft-identity 61.7%

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

      8. associate-*r/ 61.7%

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

      9. associate-/l* 61.7%

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

   7. Simplified 61.7%

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

   if -9.9999999999999994e236 < U < -2.30000000000000017e171 or 4.9999999999999999e266 < U

   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. Step-by-step derivation

      1. *-commutative 48.6%

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

      2. associate-*l* 48.6%

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

      3. associate-*r* 48.6%

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

      4. *-commutative 48.6%

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

      5. associate-*l* 48.5%

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

      6. *-commutative 48.5%

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

      7. unpow2 48.5%

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

      8. hypot-1-def 59.6%

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

      9. *-commutative 59.6%

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

      10. associate-*l* 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in J around 0 72.7%

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

      1. neg-mul-1 72.7%

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

   6. Simplified 72.7%

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

   if -2.30000000000000017e171 < U < -1.79999999999999991e-16

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-*l* 69.6%

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

      3. associate-*r* 69.6%

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

      4. *-commutative 69.6%

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

      5. associate-*l* 69.7%

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

      6. *-commutative 69.7%

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

      7. unpow2 69.7%

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

      8. hypot-1-def 91.7%

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

      9. *-commutative 91.7%

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

      10. associate-*l* 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in U around -inf 61.0%

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

   if -1.79999999999999991e-16 < U < 3.30000000000000022e-82

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l* 98.2%

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

      3. associate-*r* 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*l* 98.1%

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

      6. *-commutative 98.1%

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

      7. unpow2 98.1%

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

      8. hypot-1-def 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in U around 0 84.4%

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

   5. Taylor expanded in K around 0 48.5%

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

   if 3.30000000000000022e-82 < U < 4.2e212

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*l* 62.3%

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

      3. unpow2 62.3%

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

      4. hypot-1-def 91.3%

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

      5. *-commutative 91.3%

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

      6. associate-*l* 91.3%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in K around 0 64.3%

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

      1. associate-*r/ 64.3%

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

      2. *-commutative 64.3%

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

      3. associate-/l* 64.3%

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

   6. Simplified 64.3%

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

   7. Taylor expanded in K around 0 62.5%

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

   8. Taylor expanded in J around 0 40.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U} + -1 \cdot U} \]
   9. Step-by-step derivation

      1. neg-mul-1 40.9%

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

      2. unsub-neg 40.9%

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

      3. associate-*r/ 40.9%

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

      4. unpow2 40.9%

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

   10. Simplified 40.9%

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

   11. Taylor expanded in J around 0 40.9%

       \[\leadsto \color{blue}{-2 \cdot \frac{{J}^{2}}{U}} - U \]
   12. Step-by-step derivation

       1. unpow2 40.9%

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

       2. associate-/l* 42.7%

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

   13. Simplified 42.7%

       \[\leadsto \color{blue}{-2 \cdot \frac{J}{\frac{U}{J}}} - U \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1 \cdot 10^{+237}:\\ \;\;\;\;U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{elif}\;U \leq -2.3 \cdot 10^{+171}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -1.8 \cdot 10^{-16}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.3 \cdot 10^{-82}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 4.2 \cdot 10^{+212}:\\ \;\;\;\;-2 \cdot \frac{J}{\frac{U}{J}} - U\\ \mathbf{elif}\;U \leq 5 \cdot 10^{+266}:\\ \;\;\;\;U + J \cdot \frac{2}{\frac{U}{J}}\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 8: 26.6% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.2 \cdot 10^{+237}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.55 \cdot 10^{+169}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -9.8 \cdot 10^{-194}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 7800000000:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.95 \cdot 10^{+167}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.9 \cdot 10^{+210}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 6.7 \cdot 10^{+270}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= U -1.2e+237)
   U
   (if (<= U -1.55e+169)
     (- U)
     (if (<= U -9.8e-194)
       U
       (if (<= U 7800000000.0)
         (- U)
         (if (<= U 1.95e+167)
           U
           (if (<= U 3.9e+210) (- U) (if (<= U 6.7e+270) U (- U)))))))))

C

double code(double J, double K, double U) {
	double tmp;
	if (U <= -1.2e+237) {
		tmp = U;
	} else if (U <= -1.55e+169) {
		tmp = -U;
	} else if (U <= -9.8e-194) {
		tmp = U;
	} else if (U <= 7800000000.0) {
		tmp = -U;
	} else if (U <= 1.95e+167) {
		tmp = U;
	} else if (U <= 3.9e+210) {
		tmp = -U;
	} else if (U <= 6.7e+270) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

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) :: tmp
    if (u <= (-1.2d+237)) then
        tmp = u
    else if (u <= (-1.55d+169)) then
        tmp = -u
    else if (u <= (-9.8d-194)) then
        tmp = u
    else if (u <= 7800000000.0d0) then
        tmp = -u
    else if (u <= 1.95d+167) then
        tmp = u
    else if (u <= 3.9d+210) then
        tmp = -u
    else if (u <= 6.7d+270) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (U <= -1.2e+237) {
		tmp = U;
	} else if (U <= -1.55e+169) {
		tmp = -U;
	} else if (U <= -9.8e-194) {
		tmp = U;
	} else if (U <= 7800000000.0) {
		tmp = -U;
	} else if (U <= 1.95e+167) {
		tmp = U;
	} else if (U <= 3.9e+210) {
		tmp = -U;
	} else if (U <= 6.7e+270) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if U <= -1.2e+237:
		tmp = U
	elif U <= -1.55e+169:
		tmp = -U
	elif U <= -9.8e-194:
		tmp = U
	elif U <= 7800000000.0:
		tmp = -U
	elif U <= 1.95e+167:
		tmp = U
	elif U <= 3.9e+210:
		tmp = -U
	elif U <= 6.7e+270:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (U <= -1.2e+237)
		tmp = U;
	elseif (U <= -1.55e+169)
		tmp = Float64(-U);
	elseif (U <= -9.8e-194)
		tmp = U;
	elseif (U <= 7800000000.0)
		tmp = Float64(-U);
	elseif (U <= 1.95e+167)
		tmp = U;
	elseif (U <= 3.9e+210)
		tmp = Float64(-U);
	elseif (U <= 6.7e+270)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= -1.2e+237)
		tmp = U;
	elseif (U <= -1.55e+169)
		tmp = -U;
	elseif (U <= -9.8e-194)
		tmp = U;
	elseif (U <= 7800000000.0)
		tmp = -U;
	elseif (U <= 1.95e+167)
		tmp = U;
	elseif (U <= 3.9e+210)
		tmp = -U;
	elseif (U <= 6.7e+270)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[U, -1.2e+237], U, If[LessEqual[U, -1.55e+169], (-U), If[LessEqual[U, -9.8e-194], U, If[LessEqual[U, 7800000000.0], (-U), If[LessEqual[U, 1.95e+167], U, If[LessEqual[U, 3.9e+210], (-U), If[LessEqual[U, 6.7e+270], U, (-U)]]]]]]]

Derivation

1. Split input into 2 regimes

2. if U < -1.1999999999999999e237 or -1.55e169 < U < -9.80000000000000008e-194 or 7.8e9 < U < 1.9499999999999999e167 or 3.9e210 < U < 6.7e270

   1. Initial program 77.8%

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

      1. *-commutative 77.8%

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

      2. associate-*l* 77.8%

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

      3. associate-*r* 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-*l* 77.8%

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

      6. *-commutative 77.8%

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

      7. unpow2 77.8%

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

      8. hypot-1-def 89.6%

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

      9. *-commutative 89.6%

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

      10. associate-*l* 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in U around -inf 41.6%

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

   if -1.1999999999999999e237 < U < -1.55e169 or -9.80000000000000008e-194 < U < 7.8e9 or 1.9499999999999999e167 < U < 3.9e210 or 6.7e270 < U

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-*l* 79.0%

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

      3. associate-*r* 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-*l* 78.9%

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

      6. *-commutative 78.9%

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

      7. unpow2 78.9%

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

      8. hypot-1-def 90.9%

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

      9. *-commutative 90.9%

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

      10. associate-*l* 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in J around 0 33.6%

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

      1. neg-mul-1 33.6%

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

   6. Simplified 33.6%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.2 \cdot 10^{+237}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -1.55 \cdot 10^{+169}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -9.8 \cdot 10^{-194}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 7800000000:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 1.95 \cdot 10^{+167}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.9 \cdot 10^{+210}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 6.7 \cdot 10^{+270}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 9: 39.1% accurate, 29.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.1 \cdot 10^{+237}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -2.9 \cdot 10^{+162}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{-82}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{+212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2 \cdot 10^{+269}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= U -1.1e+237)
   U
   (if (<= U -2.9e+162)
     (- U)
     (if (<= U -4.5e-14)
       U
       (if (<= U 1.2e-82)
         (* J -2.0)
         (if (<= U 1.25e+212) (- U) (if (<= U 2e+269) U (- U))))))))

C

double code(double J, double K, double U) {
	double tmp;
	if (U <= -1.1e+237) {
		tmp = U;
	} else if (U <= -2.9e+162) {
		tmp = -U;
	} else if (U <= -4.5e-14) {
		tmp = U;
	} else if (U <= 1.2e-82) {
		tmp = J * -2.0;
	} else if (U <= 1.25e+212) {
		tmp = -U;
	} else if (U <= 2e+269) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

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) :: tmp
    if (u <= (-1.1d+237)) then
        tmp = u
    else if (u <= (-2.9d+162)) then
        tmp = -u
    else if (u <= (-4.5d-14)) then
        tmp = u
    else if (u <= 1.2d-82) then
        tmp = j * (-2.0d0)
    else if (u <= 1.25d+212) then
        tmp = -u
    else if (u <= 2d+269) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (U <= -1.1e+237) {
		tmp = U;
	} else if (U <= -2.9e+162) {
		tmp = -U;
	} else if (U <= -4.5e-14) {
		tmp = U;
	} else if (U <= 1.2e-82) {
		tmp = J * -2.0;
	} else if (U <= 1.25e+212) {
		tmp = -U;
	} else if (U <= 2e+269) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if U <= -1.1e+237:
		tmp = U
	elif U <= -2.9e+162:
		tmp = -U
	elif U <= -4.5e-14:
		tmp = U
	elif U <= 1.2e-82:
		tmp = J * -2.0
	elif U <= 1.25e+212:
		tmp = -U
	elif U <= 2e+269:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (U <= -1.1e+237)
		tmp = U;
	elseif (U <= -2.9e+162)
		tmp = Float64(-U);
	elseif (U <= -4.5e-14)
		tmp = U;
	elseif (U <= 1.2e-82)
		tmp = Float64(J * -2.0);
	elseif (U <= 1.25e+212)
		tmp = Float64(-U);
	elseif (U <= 2e+269)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= -1.1e+237)
		tmp = U;
	elseif (U <= -2.9e+162)
		tmp = -U;
	elseif (U <= -4.5e-14)
		tmp = U;
	elseif (U <= 1.2e-82)
		tmp = J * -2.0;
	elseif (U <= 1.25e+212)
		tmp = -U;
	elseif (U <= 2e+269)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[U, -1.1e+237], U, If[LessEqual[U, -2.9e+162], (-U), If[LessEqual[U, -4.5e-14], U, If[LessEqual[U, 1.2e-82], N[(J * -2.0), $MachinePrecision], If[LessEqual[U, 1.25e+212], (-U), If[LessEqual[U, 2e+269], U, (-U)]]]]]]

Derivation

1. Split input into 3 regimes

2. if U < -1.1e237 or -2.90000000000000006e162 < U < -4.4999999999999998e-14 or 1.24999999999999998e212 < U < 2.0000000000000001e269

   1. Initial program 64.7%

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

      1. *-commutative 64.7%

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

      2. associate-*l* 64.7%

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

      3. associate-*r* 64.7%

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

      4. *-commutative 64.7%

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

      5. associate-*l* 64.8%

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

      6. *-commutative 64.8%

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

      7. unpow2 64.8%

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

      8. hypot-1-def 82.4%

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

      9. *-commutative 82.4%

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

      10. associate-*l* 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in U around -inf 61.1%

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

   if -1.1e237 < U < -2.90000000000000006e162 or 1.20000000000000004e-82 < U < 1.24999999999999998e212 or 2.0000000000000001e269 < U

   1. Initial program 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-*l* 58.1%

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

      3. associate-*r* 58.1%

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

      4. *-commutative 58.1%

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

      5. associate-*l* 58.0%

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

      6. *-commutative 58.0%

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

      7. unpow2 58.0%

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

      8. hypot-1-def 81.5%

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

      9. *-commutative 81.5%

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

      10. associate-*l* 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in J around 0 51.8%

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

      1. neg-mul-1 51.8%

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

   6. Simplified 51.8%

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

   if -4.4999999999999998e-14 < U < 1.20000000000000004e-82

   1. Initial program 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l* 98.2%

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

      3. associate-*r* 98.2%

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

      4. *-commutative 98.2%

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

      5. associate-*l* 98.1%

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

      6. *-commutative 98.1%

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

      7. unpow2 98.1%

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

      8. hypot-1-def 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in U around 0 84.4%

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

   5. Taylor expanded in K around 0 48.5%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.1 \cdot 10^{+237}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq -2.9 \cdot 10^{+162}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 1.2 \cdot 10^{-82}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;U \leq 1.25 \cdot 10^{+212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;U \leq 2 \cdot 10^{+269}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 10: 26.1% accurate, 420.0× speedup

Math

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

FPCore

(FPCore (J K U) :precision binary64 U)

C

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

Fortran

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

public static double code(double J, double K, double U) {
	return U;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, K_, U_] := U

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. associate-*l* 78.4%

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

   3. associate-*r* 78.4%

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

   4. *-commutative 78.4%

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

   5. associate-*l* 78.4%

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

   6. *-commutative 78.4%

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

   7. unpow2 78.4%

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

   8. hypot-1-def 90.3%

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

   9. *-commutative 90.3%

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

   10. associate-*l* 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in U around -inf 25.7%

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

5. Final simplification 25.7%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023187 
(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)))))