Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.9% → 88.1%

Time: 16.6s

Alternatives: 9

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: 73.9% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -4 \cdot 10^{+270}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot t_0\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U -4e+270)
     U
     (* (* 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));
	double tmp;
	if (U <= -4e+270) {
		tmp = U;
	} else {
		tmp = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U <= -4e+270) {
		tmp = U;
	} else {
		tmp = (J * (-2.0 * t_0)) * Math.hypot(1.0, (U / (J * (2.0 * t_0))));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= -4e+270:
		tmp = U
	else:
		tmp = (J * (-2.0 * t_0)) * math.hypot(1.0, (U / (J * (2.0 * t_0))))
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= -4e+270)
		tmp = U;
	else
		tmp = Float64(Float64(J * Float64(-2.0 * t_0)) * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U <= -4e+270)
		tmp = U;
	else
		tmp = (J * (-2.0 * t_0)) * hypot(1.0, (U / (J * (2.0 * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -4e+270], U, 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. Split input into 2 regimes

2. if U < -4.0000000000000002e270

   1. Initial program 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. associate-*r* 25.0%

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

      4. *-commutative 25.0%

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

      5. associate-*l* 25.0%

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

      6. *-commutative 25.0%

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

      7. unpow2 25.0%

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

      8. hypot-1-def 32.7%

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

      9. *-commutative 32.7%

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

      10. associate-*l* 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in U around -inf 66.7%

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

   if -4.0000000000000002e270 < U

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 74.1%

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

      3. unpow2 74.1%

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

      4. hypot-1-def 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%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -4 \cdot 10^{+270}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(-2 \cdot \cos \left(\frac{K}{2}\right)\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\\ \end{array} \]

Alternative 2: 88.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq -5.1 \cdot 10^{+269}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(t_0 \cdot \left(-2 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U -5.1e+269)
     U
     (* 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));
	double tmp;
	if (U <= -5.1e+269) {
		tmp = U;
	} else {
		tmp = J * (t_0 * (-2.0 * hypot(1.0, (U / (J * (2.0 * t_0))))));
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U <= -5.1e+269) {
		tmp = U;
	} else {
		tmp = J * (t_0 * (-2.0 * Math.hypot(1.0, (U / (J * (2.0 * t_0))))));
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= -5.1e+269:
		tmp = U
	else:
		tmp = J * (t_0 * (-2.0 * math.hypot(1.0, (U / (J * (2.0 * t_0))))))
	return tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= -5.1e+269)
		tmp = U;
	else
		tmp = Float64(J * Float64(t_0 * Float64(-2.0 * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U <= -5.1e+269)
		tmp = U;
	else
		tmp = J * (t_0 * (-2.0 * hypot(1.0, (U / (J * (2.0 * t_0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, -5.1e+269], U, 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. Split input into 2 regimes

2. if U < -5.10000000000000012e269

   1. Initial program 25.0%

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

      1. *-commutative 25.0%

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

      2. associate-*l* 25.0%

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

      3. associate-*r* 25.0%

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

      4. *-commutative 25.0%

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

      5. associate-*l* 25.0%

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

      6. *-commutative 25.0%

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

      7. unpow2 25.0%

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

      8. hypot-1-def 32.7%

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

      9. *-commutative 32.7%

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

      10. associate-*l* 32.7%

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

   3. Simplified 32.7%

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

   4. Taylor expanded in U around -inf 66.7%

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

   if -5.10000000000000012e269 < U

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 74.1%

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

      3. associate-*r* 74.1%

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

      4. *-commutative 74.1%

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

      5. associate-*l* 74.0%

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

      6. *-commutative 74.0%

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

      7. unpow2 74.0%

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

      8. hypot-1-def 92.0%

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

      9. *-commutative 92.0%

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

      10. associate-*l* 92.0%

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

   3. Simplified 92.0%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -5.1 \cdot 10^{+269}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\right)\\ \end{array} \]

Alternative 3: 73.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq -1.45 \cdot 10^{+178}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.8 \cdot 10^{+185}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= U -1.45e+178)
   U
   (if (<= U 3.8e+185)
     (* (* (cos (/ K 2.0)) (* J -2.0)) (hypot 1.0 (* 0.5 (/ U J))))
     U)))

C

double code(double J, double K, double U) {
	double tmp;
	if (U <= -1.45e+178) {
		tmp = U;
	} else if (U <= 3.8e+185) {
		tmp = (cos((K / 2.0)) * (J * -2.0)) * hypot(1.0, (0.5 * (U / J)));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (U <= -1.45e+178) {
		tmp = U;
	} else if (U <= 3.8e+185) {
		tmp = (Math.cos((K / 2.0)) * (J * -2.0)) * Math.hypot(1.0, (0.5 * (U / J)));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if U <= -1.45e+178:
		tmp = U
	elif U <= 3.8e+185:
		tmp = (math.cos((K / 2.0)) * (J * -2.0)) * math.hypot(1.0, (0.5 * (U / J)))
	else:
		tmp = U
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (U <= -1.45e+178)
		tmp = U;
	elseif (U <= 3.8e+185)
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(J * -2.0)) * hypot(1.0, Float64(0.5 * Float64(U / J))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= -1.45e+178)
		tmp = U;
	elseif (U <= 3.8e+185)
		tmp = (cos((K / 2.0)) * (J * -2.0)) * hypot(1.0, (0.5 * (U / J)));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[U, -1.45e+178], U, If[LessEqual[U, 3.8e+185], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * -2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if U < -1.45e178 or 3.7999999999999998e185 < U

   1. Initial program 41.0%

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

      1. *-commutative 41.0%

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

      2. associate-*l* 41.0%

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

      3. associate-*r* 41.0%

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

      4. *-commutative 41.0%

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

      5. associate-*l* 40.9%

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

      6. *-commutative 40.9%

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

      7. unpow2 40.9%

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

      8. hypot-1-def 59.5%

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

      9. *-commutative 59.5%

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

      10. associate-*l* 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in U around -inf 60.4%

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

   if -1.45e178 < U < 3.7999999999999998e185

   1. Initial program 79.3%

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

   2. Taylor expanded in K around 0 61.6%

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

      1. unpow2 61.6%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{\color{blue}{J \cdot J}}} \]

      2. associate-/r* 65.7%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + 0.25 \cdot \color{blue}{\frac{\frac{{U}^{2}}{J}}{J}}} \]

      3. metadata-eval 65.7%

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

      4. unpow2 65.7%

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

      5. associate-*l/ 69.6%

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

      6. associate-*r/ 69.6%

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

      7. swap-sqr 69.7%

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

      8. hypot-1-def 80.9%

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

   4. Simplified 80.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq -1.45 \cdot 10^{+178}:\\ \;\;\;\;U\\ \mathbf{elif}\;U \leq 3.8 \cdot 10^{+185}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot -2\right)\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 4: 69.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq -8.2 \cdot 10^{+34} \lor \neg \left(K \leq 2.4 \cdot 10^{-23}\right):\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (or (<= K -8.2e+34) (not (<= K 2.4e-23)))
   (* J (* -2.0 (cos (* K 0.5))))
   (* J (* -2.0 (hypot 1.0 (* 0.5 (/ U J)))))))

C

double code(double J, double K, double U) {
	double tmp;
	if ((K <= -8.2e+34) || !(K <= 2.4e-23)) {
		tmp = J * (-2.0 * cos((K * 0.5)));
	} else {
		tmp = J * (-2.0 * hypot(1.0, (0.5 * (U / J))));
	}
	return tmp;
}

Java

public static double code(double J, double K, double U) {
	double tmp;
	if ((K <= -8.2e+34) || !(K <= 2.4e-23)) {
		tmp = J * (-2.0 * Math.cos((K * 0.5)));
	} else {
		tmp = J * (-2.0 * Math.hypot(1.0, (0.5 * (U / J))));
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if (K <= -8.2e+34) or not (K <= 2.4e-23):
		tmp = J * (-2.0 * math.cos((K * 0.5)))
	else:
		tmp = J * (-2.0 * math.hypot(1.0, (0.5 * (U / J))))
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if ((K <= -8.2e+34) || !(K <= 2.4e-23))
		tmp = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))));
	else
		tmp = Float64(J * Float64(-2.0 * hypot(1.0, Float64(0.5 * Float64(U / J)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if ((K <= -8.2e+34) || ~((K <= 2.4e-23)))
		tmp = J * (-2.0 * cos((K * 0.5)));
	else
		tmp = J * (-2.0 * hypot(1.0, (0.5 * (U / J))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[Or[LessEqual[K, -8.2e+34], N[Not[LessEqual[K, 2.4e-23]], $MachinePrecision]], N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(-2.0 * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < -8.1999999999999997e34 or 2.39999999999999996e-23 < 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 92.4%

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

      9. *-commutative 92.4%

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

      10. associate-*l* 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in U around 0 56.1%

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

   if -8.1999999999999997e34 < K < 2.39999999999999996e-23

   1. Initial program 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*l* 63.7%

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

      3. associate-*r* 63.7%

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

      4. *-commutative 63.7%

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

      5. associate-*l* 63.7%

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

      6. *-commutative 63.7%

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

      7. unpow2 63.7%

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

      8. hypot-1-def 85.8%

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

      9. *-commutative 85.8%

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

      10. associate-*l* 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in K around 0 49.2%

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

      1. associate-*r* 49.2%

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \cdot J} \]

      2. *-commutative 49.2%

         \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      3. unpow2 49.2%

         \[\leadsto J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{\color{blue}{J \cdot J}}}\right) \]

      4. associate-/r* 53.4%

         \[\leadsto J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \color{blue}{\frac{\frac{{U}^{2}}{J}}{J}}}\right) \]

      5. metadata-eval 53.4%

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

      6. unpow2 53.4%

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

      7. associate-*l/ 58.8%

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

      8. associate-*r/ 61.8%

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

      9. swap-sqr 61.8%

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

      10. hypot-1-def 83.9%

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

   6. Simplified 83.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq -8.2 \cdot 10^{+34} \lor \neg \left(K \leq 2.4 \cdot 10^{-23}\right):\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\\ \end{array} \]

Alternative 5: 58.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;J \leq -2.3 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -5.4 \cdot 10^{-212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{-60}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (* -2.0 (cos (* K 0.5))))))
   (if (<= J -2.3e-58)
     t_0
     (if (<= J -5.4e-212) (- U) (if (<= J 1.1e-60) U t_0)))))

C

double code(double J, double K, double U) {
	double t_0 = J * (-2.0 * cos((K * 0.5)));
	double tmp;
	if (J <= -2.3e-58) {
		tmp = t_0;
	} else if (J <= -5.4e-212) {
		tmp = -U;
	} else if (J <= 1.1e-60) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	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 = j * ((-2.0d0) * cos((k * 0.5d0)))
    if (j <= (-2.3d-58)) then
        tmp = t_0
    else if (j <= (-5.4d-212)) then
        tmp = -u
    else if (j <= 1.1d-60) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = J * (-2.0 * Math.cos((K * 0.5)));
	double tmp;
	if (J <= -2.3e-58) {
		tmp = t_0;
	} else if (J <= -5.4e-212) {
		tmp = -U;
	} else if (J <= 1.1e-60) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = J * (-2.0 * math.cos((K * 0.5)))
	tmp = 0
	if J <= -2.3e-58:
		tmp = t_0
	elif J <= -5.4e-212:
		tmp = -U
	elif J <= 1.1e-60:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(J * Float64(-2.0 * cos(Float64(K * 0.5))))
	tmp = 0.0
	if (J <= -2.3e-58)
		tmp = t_0;
	elseif (J <= -5.4e-212)
		tmp = Float64(-U);
	elseif (J <= 1.1e-60)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = J * (-2.0 * cos((K * 0.5)));
	tmp = 0.0;
	if (J <= -2.3e-58)
		tmp = t_0;
	elseif (J <= -5.4e-212)
		tmp = -U;
	elseif (J <= 1.1e-60)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[(-2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.3e-58], t$95$0, If[LessEqual[J, -5.4e-212], (-U), If[LessEqual[J, 1.1e-60], U, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.2999999999999999e-58 or 1.0999999999999999e-60 < J

   1. Initial program 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-*l* 93.2%

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

      3. associate-*r* 93.2%

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

      4. *-commutative 93.2%

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

      5. associate-*l* 93.2%

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

      6. *-commutative 93.2%

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

      7. unpow2 93.2%

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

      8. hypot-1-def 98.6%

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

      9. *-commutative 98.6%

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

      10. associate-*l* 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in U around 0 76.0%

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

   if -2.2999999999999999e-58 < J < -5.39999999999999962e-212

   1. Initial program 44.3%

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

      1. *-commutative 44.3%

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

      2. associate-*l* 44.3%

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

      3. associate-*r* 44.3%

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

      4. *-commutative 44.3%

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

      5. associate-*l* 44.2%

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

      6. *-commutative 44.2%

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

      7. unpow2 44.2%

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

      8. hypot-1-def 83.2%

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

      9. *-commutative 83.2%

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

      10. associate-*l* 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in J around 0 54.9%

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

      1. neg-mul-1 54.9%

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

   6. Simplified 54.9%

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

   if -5.39999999999999962e-212 < J < 1.0999999999999999e-60

   1. Initial program 36.5%

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

      1. *-commutative 36.5%

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

      2. associate-*l* 36.5%

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

      3. associate-*r* 36.5%

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

      4. *-commutative 36.5%

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

      5. associate-*l* 36.5%

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

      6. *-commutative 36.5%

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

      7. unpow2 36.5%

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

      8. hypot-1-def 70.0%

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

      9. *-commutative 70.0%

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

      10. associate-*l* 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in U around -inf 58.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.3 \cdot 10^{-58}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;J \leq -5.4 \cdot 10^{-212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.1 \cdot 10^{-60}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 40.6% accurate, 19.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(-2 + \frac{0.125}{J} \cdot \left(-2 \cdot \frac{U}{\frac{J}{U}}\right)\right)\\ \mathbf{if}\;J \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -6.8 \cdot 10^{-212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* J (+ -2.0 (* (/ 0.125 J) (* -2.0 (/ U (/ J U))))))))
   (if (<= J -1.6e-18)
     t_0
     (if (<= J -6.8e-212) (- U) (if (<= J 2.1e-16) U t_0)))))

C

double code(double J, double K, double U) {
	double t_0 = J * (-2.0 + ((0.125 / J) * (-2.0 * (U / (J / U)))));
	double tmp;
	if (J <= -1.6e-18) {
		tmp = t_0;
	} else if (J <= -6.8e-212) {
		tmp = -U;
	} else if (J <= 2.1e-16) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	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 = j * ((-2.0d0) + ((0.125d0 / j) * ((-2.0d0) * (u / (j / u)))))
    if (j <= (-1.6d-18)) then
        tmp = t_0
    else if (j <= (-6.8d-212)) then
        tmp = -u
    else if (j <= 2.1d-16) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = J * (-2.0 + ((0.125 / J) * (-2.0 * (U / (J / U)))));
	double tmp;
	if (J <= -1.6e-18) {
		tmp = t_0;
	} else if (J <= -6.8e-212) {
		tmp = -U;
	} else if (J <= 2.1e-16) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, K, U):
	t_0 = J * (-2.0 + ((0.125 / J) * (-2.0 * (U / (J / U)))))
	tmp = 0
	if J <= -1.6e-18:
		tmp = t_0
	elif J <= -6.8e-212:
		tmp = -U
	elif J <= 2.1e-16:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, K, U)
	t_0 = Float64(J * Float64(-2.0 + Float64(Float64(0.125 / J) * Float64(-2.0 * Float64(U / Float64(J / U))))))
	tmp = 0.0
	if (J <= -1.6e-18)
		tmp = t_0;
	elseif (J <= -6.8e-212)
		tmp = Float64(-U);
	elseif (J <= 2.1e-16)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = J * (-2.0 + ((0.125 / J) * (-2.0 * (U / (J / U)))));
	tmp = 0.0;
	if (J <= -1.6e-18)
		tmp = t_0;
	elseif (J <= -6.8e-212)
		tmp = -U;
	elseif (J <= 2.1e-16)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(J * N[(-2.0 + N[(N[(0.125 / J), $MachinePrecision] * N[(-2.0 * N[(U / N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.6e-18], t$95$0, If[LessEqual[J, -6.8e-212], (-U), If[LessEqual[J, 2.1e-16], U, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.6e-18 or 2.1000000000000001e-16 < J

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

      2. associate-*l* 96.3%

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

      3. associate-*r* 96.3%

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

      4. *-commutative 96.3%

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

      5. associate-*l* 96.2%

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

      6. *-commutative 96.2%

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

      7. unpow2 96.2%

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

      8. hypot-1-def 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in K around 0 41.4%

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

      1. associate-*r* 41.4%

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \cdot J} \]

      2. *-commutative 41.4%

         \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      3. unpow2 41.4%

         \[\leadsto J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{\color{blue}{J \cdot J}}}\right) \]

      4. associate-/r* 42.1%

         \[\leadsto J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \color{blue}{\frac{\frac{{U}^{2}}{J}}{J}}}\right) \]

      5. metadata-eval 42.1%

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

      6. unpow2 42.1%

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

      7. associate-*l/ 46.0%

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

      8. associate-*r/ 50.2%

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

      9. swap-sqr 50.2%

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

      10. hypot-1-def 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in U around 0 39.3%

      \[\leadsto J \cdot \left(-2 \cdot \color{blue}{\left(1 + 0.125 \cdot \frac{{U}^{2}}{{J}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. associate-*r/ 39.3%

         \[\leadsto J \cdot \left(-2 \cdot \left(1 + \color{blue}{\frac{0.125 \cdot {U}^{2}}{{J}^{2}}}\right)\right) \]

      2. unpow2 39.3%

         \[\leadsto J \cdot \left(-2 \cdot \left(1 + \frac{0.125 \cdot \color{blue}{\left(U \cdot U\right)}}{{J}^{2}}\right)\right) \]

      3. unpow2 39.3%

         \[\leadsto J \cdot \left(-2 \cdot \left(1 + \frac{0.125 \cdot \left(U \cdot U\right)}{\color{blue}{J \cdot J}}\right)\right) \]

   9. Simplified 39.3%

      \[\leadsto J \cdot \left(-2 \cdot \color{blue}{\left(1 + \frac{0.125 \cdot \left(U \cdot U\right)}{J \cdot J}\right)}\right) \]
   10. Step-by-step derivation

       1. expm1-log1p-u 20.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(J \cdot \left(-2 \cdot \left(1 + \frac{0.125 \cdot \left(U \cdot U\right)}{J \cdot J}\right)\right)\right)\right)} \]

       2. expm1-udef 20.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(J \cdot \left(-2 \cdot \left(1 + \frac{0.125 \cdot \left(U \cdot U\right)}{J \cdot J}\right)\right)\right)} - 1} \]

       3. distribute-rgt-in 20.0%

          \[\leadsto e^{\mathsf{log1p}\left(J \cdot \color{blue}{\left(1 \cdot -2 + \frac{0.125 \cdot \left(U \cdot U\right)}{J \cdot J} \cdot -2\right)}\right)} - 1 \]

       4. metadata-eval 20.0%

          \[\leadsto e^{\mathsf{log1p}\left(J \cdot \left(\color{blue}{-2} + \frac{0.125 \cdot \left(U \cdot U\right)}{J \cdot J} \cdot -2\right)\right)} - 1 \]

       5. times-frac 20.5%

          \[\leadsto e^{\mathsf{log1p}\left(J \cdot \left(-2 + \color{blue}{\left(\frac{0.125}{J} \cdot \frac{U \cdot U}{J}\right)} \cdot -2\right)\right)} - 1 \]

   11. Applied egg-rr 20.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(J \cdot \left(-2 + \left(\frac{0.125}{J} \cdot \frac{U \cdot U}{J}\right) \cdot -2\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 20.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(J \cdot \left(-2 + \left(\frac{0.125}{J} \cdot \frac{U \cdot U}{J}\right) \cdot -2\right)\right)\right)} \]

       2. expm1-log1p 40.0%

          \[\leadsto \color{blue}{J \cdot \left(-2 + \left(\frac{0.125}{J} \cdot \frac{U \cdot U}{J}\right) \cdot -2\right)} \]

       3. associate-*l* 40.0%

          \[\leadsto J \cdot \left(-2 + \color{blue}{\frac{0.125}{J} \cdot \left(\frac{U \cdot U}{J} \cdot -2\right)}\right) \]

       4. associate-/l* 42.5%

          \[\leadsto J \cdot \left(-2 + \frac{0.125}{J} \cdot \left(\color{blue}{\frac{U}{\frac{J}{U}}} \cdot -2\right)\right) \]

   13. Simplified 42.5%

       \[\leadsto \color{blue}{J \cdot \left(-2 + \frac{0.125}{J} \cdot \left(\frac{U}{\frac{J}{U}} \cdot -2\right)\right)} \]

   if -1.6e-18 < J < -6.79999999999999995e-212

   1. Initial program 53.7%

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

      1. *-commutative 53.7%

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

      2. associate-*l* 53.7%

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

      3. associate-*r* 53.7%

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

      4. *-commutative 53.7%

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

      5. associate-*l* 53.6%

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

      6. *-commutative 53.6%

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

      7. unpow2 53.6%

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

      8. hypot-1-def 85.8%

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

      9. *-commutative 85.8%

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

      10. associate-*l* 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in J around 0 48.5%

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

      1. neg-mul-1 48.5%

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

   6. Simplified 48.5%

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

   if -6.79999999999999995e-212 < J < 2.1000000000000001e-16

   1. Initial program 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-*l* 40.6%

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

      3. associate-*r* 40.6%

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

      4. *-commutative 40.6%

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

      5. associate-*l* 40.6%

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

      6. *-commutative 40.6%

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

      7. unpow2 40.6%

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

      8. hypot-1-def 72.5%

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

      9. *-commutative 72.5%

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

      10. associate-*l* 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in U around -inf 56.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;J \cdot \left(-2 + \frac{0.125}{J} \cdot \left(-2 \cdot \frac{U}{\frac{J}{U}}\right)\right)\\ \mathbf{elif}\;J \leq -6.8 \cdot 10^{-212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 2.1 \cdot 10^{-16}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(-2 + \frac{0.125}{J} \cdot \left(-2 \cdot \frac{U}{\frac{J}{U}}\right)\right)\\ \end{array} \]

Alternative 7: 40.4% accurate, 45.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -7.2 \cdot 10^{-212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (if (<= J -1.6e-18)
   (* J -2.0)
   (if (<= J -7.2e-212) (- U) (if (<= J 1.15e-20) U (* J -2.0)))))

C

double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.6e-18) {
		tmp = J * -2.0;
	} else if (J <= -7.2e-212) {
		tmp = -U;
	} else if (J <= 1.15e-20) {
		tmp = U;
	} else {
		tmp = J * -2.0;
	}
	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 (j <= (-1.6d-18)) then
        tmp = j * (-2.0d0)
    else if (j <= (-7.2d-212)) then
        tmp = -u
    else if (j <= 1.15d-20) then
        tmp = u
    else
        tmp = j * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.6e-18) {
		tmp = J * -2.0;
	} else if (J <= -7.2e-212) {
		tmp = -U;
	} else if (J <= 1.15e-20) {
		tmp = U;
	} else {
		tmp = J * -2.0;
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if J <= -1.6e-18:
		tmp = J * -2.0
	elif J <= -7.2e-212:
		tmp = -U
	elif J <= 1.15e-20:
		tmp = U
	else:
		tmp = J * -2.0
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (J <= -1.6e-18)
		tmp = Float64(J * -2.0);
	elseif (J <= -7.2e-212)
		tmp = Float64(-U);
	elseif (J <= 1.15e-20)
		tmp = U;
	else
		tmp = Float64(J * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -1.6e-18)
		tmp = J * -2.0;
	elseif (J <= -7.2e-212)
		tmp = -U;
	elseif (J <= 1.15e-20)
		tmp = U;
	else
		tmp = J * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[J, -1.6e-18], N[(J * -2.0), $MachinePrecision], If[LessEqual[J, -7.2e-212], (-U), If[LessEqual[J, 1.15e-20], U, N[(J * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.6e-18 or 1.15e-20 < J

   1. Initial program 96.3%

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

      1. *-commutative 96.3%

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

      2. associate-*l* 96.3%

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

      3. associate-*r* 96.3%

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

      4. *-commutative 96.3%

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

      5. associate-*l* 96.2%

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

      6. *-commutative 96.2%

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

      7. unpow2 96.2%

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

      8. hypot-1-def 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in K around 0 41.4%

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

      1. associate-*r* 41.4%

         \[\leadsto \color{blue}{\left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right) \cdot J} \]

      2. *-commutative 41.4%

         \[\leadsto \color{blue}{J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{{J}^{2}}}\right)} \]

      3. unpow2 41.4%

         \[\leadsto J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \frac{{U}^{2}}{\color{blue}{J \cdot J}}}\right) \]

      4. associate-/r* 42.1%

         \[\leadsto J \cdot \left(-2 \cdot \sqrt{1 + 0.25 \cdot \color{blue}{\frac{\frac{{U}^{2}}{J}}{J}}}\right) \]

      5. metadata-eval 42.1%

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

      6. unpow2 42.1%

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

      7. associate-*l/ 46.0%

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

      8. associate-*r/ 50.2%

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

      9. swap-sqr 50.2%

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

      10. hypot-1-def 53.1%

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

   6. Simplified 53.1%

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

   7. Taylor expanded in J around inf 42.0%

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

   if -1.6e-18 < J < -7.2000000000000002e-212

   1. Initial program 53.7%

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

      1. *-commutative 53.7%

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

      2. associate-*l* 53.7%

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

      3. associate-*r* 53.7%

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

      4. *-commutative 53.7%

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

      5. associate-*l* 53.6%

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

      6. *-commutative 53.6%

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

      7. unpow2 53.6%

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

      8. hypot-1-def 85.8%

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

      9. *-commutative 85.8%

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

      10. associate-*l* 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in J around 0 48.5%

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

      1. neg-mul-1 48.5%

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

   6. Simplified 48.5%

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

   if -7.2000000000000002e-212 < J < 1.15e-20

   1. Initial program 40.6%

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

      1. *-commutative 40.6%

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

      2. associate-*l* 40.6%

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

      3. associate-*r* 40.6%

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

      4. *-commutative 40.6%

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

      5. associate-*l* 40.6%

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

      6. *-commutative 40.6%

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

      7. unpow2 40.6%

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

      8. hypot-1-def 72.5%

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

      9. *-commutative 72.5%

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

      10. associate-*l* 72.5%

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

   3. Simplified 72.5%

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

   4. Taylor expanded in U around -inf 56.9%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;J \cdot -2\\ \mathbf{elif}\;J \leq -7.2 \cdot 10^{-212}:\\ \;\;\;\;-U\\ \mathbf{elif}\;J \leq 1.15 \cdot 10^{-20}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot -2\\ \end{array} \]

Alternative 8: 26.2% accurate, 103.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.2 \cdot 10^{-119}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U) :precision binary64 (if (<= K 1.2e-119) U (- U)))

C

double code(double J, double K, double U) {
	double tmp;
	if (K <= 1.2e-119) {
		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 (k <= 1.2d-119) 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 (K <= 1.2e-119) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

def code(J, K, U):
	tmp = 0
	if K <= 1.2e-119:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

function code(J, K, U)
	tmp = 0.0
	if (K <= 1.2e-119)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (K <= 1.2e-119)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, K_, U_] := If[LessEqual[K, 1.2e-119], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if K < 1.20000000000000004e-119

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-*l* 70.1%

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

      3. associate-*r* 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-*l* 70.0%

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

      6. *-commutative 70.0%

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

      7. unpow2 70.0%

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

      8. hypot-1-def 90.2%

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

      9. *-commutative 90.2%

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

      10. associate-*l* 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in U around -inf 34.1%

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

   if 1.20000000000000004e-119 < K

   1. Initial program 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*l* 74.9%

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

      3. associate-*r* 74.9%

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

      4. *-commutative 74.9%

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

      5. associate-*l* 74.9%

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

      6. *-commutative 74.9%

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

      7. unpow2 74.9%

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

      8. hypot-1-def 87.3%

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

      9. *-commutative 87.3%

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

      10. associate-*l* 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in J around 0 29.1%

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

      1. neg-mul-1 29.1%

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

   6. Simplified 29.1%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1.2 \cdot 10^{-119}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 9: 25.6% 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 71.8%

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

   1. *-commutative 71.8%

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

   2. associate-*l* 71.8%

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

   3. associate-*r* 71.8%

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

   4. *-commutative 71.8%

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

   5. associate-*l* 71.7%

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

   6. *-commutative 71.7%

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

   7. unpow2 71.7%

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

   8. hypot-1-def 89.2%

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

   9. *-commutative 89.2%

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

   10. associate-*l* 89.2%

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

3. Simplified 89.2%

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

4. Taylor expanded in U around -inf 29.7%

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

5. Final simplification 29.7%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023178 
(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)))))