Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.7% → 99.4%

Time: 23.2s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(t_0 \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 10^{+302}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* t_0 (* -2.0 J))
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0))))))
   (if (<= t_1 (- INFINITY)) (- U) (if (<= t_1 1e+302) t_1 U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 1e+302) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = (t_0 * (-2.0 * J)) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 1e+302) {
		tmp = t_1;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = (t_0 * (-2.0 * J)) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 1e+302:
		tmp = t_1
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(t_0 * Float64(-2.0 * J)) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 1e+302)
		tmp = t_1;
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = (t_0 * (-2.0 * J)) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 1e+302)
		tmp = t_1;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 1e+302], t$95$1, U]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.3%

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

   2. Taylor expanded in J around 0 52.7%

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

      1. neg-mul-1 52.7%

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

   4. Simplified 52.7%

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2)))) < 1.0000000000000001e302

   1. Initial program 99.7%

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

   if 1.0000000000000001e302 < (*.f64 (*.f64 (*.f64 -2 J) (cos.f64 (/.f64 K 2))) (sqrt.f64 (+.f64 1 (pow.f64 (/.f64 U (*.f64 (*.f64 2 J) (cos.f64 (/.f64 K 2)))) 2))))

   1. Initial program 5.9%

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

   2. Taylor expanded in U around -inf 32.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 10^{+302}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 2: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 6 \cdot 10^{+244}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U 6e+244)
     (* -2.0 (* t_0 (* J (hypot 1.0 (/ (/ U (* J 2.0)) t_0)))))
     U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U <= 6e+244) {
		tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * 2.0)) / t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U <= 6e+244) {
		tmp = -2.0 * (t_0 * (J * Math.hypot(1.0, ((U / (J * 2.0)) / t_0))));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= 6e+244:
		tmp = -2.0 * (t_0 * (J * math.hypot(1.0, ((U / (J * 2.0)) / t_0))))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= 6e+244)
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J * hypot(1.0, Float64(Float64(U / Float64(J * 2.0)) / t_0)))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U <= 6e+244)
		tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * 2.0)) / t_0))));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, 6e+244], N[(-2.0 * N[(t$95$0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if U < 5.9999999999999995e244

   1. Initial program 72.4%

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

   3. Simplified 87.9%

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

   if 5.9999999999999995e244 < U

   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. Taylor expanded in U around -inf 30.1%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 6 \cdot 10^{+244}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 3: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 4.3 \cdot 10^{+245}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot t_0\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{t_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= U 4.3e+245)
     (* -2.0 (* (* J t_0) (hypot 1.0 (/ (/ U (* J 2.0)) t_0))))
     U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U <= 4.3e+245) {
		tmp = -2.0 * ((J * t_0) * hypot(1.0, ((U / (J * 2.0)) / t_0)));
	} else {
		tmp = U;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (U <= 4.3e+245) {
		tmp = -2.0 * ((J * t_0) * Math.hypot(1.0, ((U / (J * 2.0)) / t_0)));
	} else {
		tmp = U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if U <= 4.3e+245:
		tmp = -2.0 * ((J * t_0) * math.hypot(1.0, ((U / (J * 2.0)) / t_0)))
	else:
		tmp = U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (U <= 4.3e+245)
		tmp = Float64(-2.0 * Float64(Float64(J * t_0) * hypot(1.0, Float64(Float64(U / Float64(J * 2.0)) / t_0))));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (U <= 4.3e+245)
		tmp = -2.0 * ((J * t_0) * hypot(1.0, ((U / (J * 2.0)) / t_0)));
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[U, 4.3e+245], N[(-2.0 * N[(N[(J * t$95$0), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 2 regimes

2. if U < 4.29999999999999979e245

   1. Initial program 72.4%

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

      1. associate-*l* 72.4%

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

      2. associate-*l* 72.4%

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

      3. unpow2 72.4%

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

      4. hypot-1-def 88.0%

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

      5. associate-/r* 87.9%

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

      6. cos-neg 87.9%

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

      7. distribute-frac-neg 87.9%

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

      8. associate-/r* 88.0%

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

      9. associate-/r* 87.9%

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

      10. distribute-frac-neg 87.9%

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

      11. cos-neg 87.9%

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

   3. Simplified 87.9%

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

   if 4.29999999999999979e245 < U

   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. Taylor expanded in U around -inf 30.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 4.3 \cdot 10^{+245}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{\frac{U}{J \cdot 2}}{\cos \left(\frac{K}{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 4: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ t_1 := J \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -4.1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{t_1}{\frac{U}{t_1}}\right)\\ \mathbf{elif}\;J \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;-2 \cdot \frac{{t_1}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (* J (cos (/ K 2.0))) (hypot 1.0 (/ (* U 0.5) J)))))
        (t_1 (* J (cos (* K 0.5)))))
   (if (<= J -4.1e-115)
     t_0
     (if (<= J -4e-310)
       (* -2.0 (- (* U -0.5) (/ t_1 (/ U t_1))))
       (if (<= J 7.8e-28) (- (* -2.0 (/ (pow t_1 2.0) U)) U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, ((U * 0.5) / J)));
	double t_1 = J * cos((K * 0.5));
	double tmp;
	if (J <= -4.1e-115) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * ((U * -0.5) - (t_1 / (U / t_1)));
	} else if (J <= 7.8e-28) {
		tmp = (-2.0 * (pow(t_1, 2.0) / U)) - U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U * 0.5) / J)));
	double t_1 = J * Math.cos((K * 0.5));
	double tmp;
	if (J <= -4.1e-115) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * ((U * -0.5) - (t_1 / (U / t_1)));
	} else if (J <= 7.8e-28) {
		tmp = (-2.0 * (Math.pow(t_1, 2.0) / U)) - U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * ((J * math.cos((K / 2.0))) * math.hypot(1.0, ((U * 0.5) / J)))
	t_1 = J * math.cos((K * 0.5))
	tmp = 0
	if J <= -4.1e-115:
		tmp = t_0
	elif J <= -4e-310:
		tmp = -2.0 * ((U * -0.5) - (t_1 / (U / t_1)))
	elif J <= 7.8e-28:
		tmp = (-2.0 * (math.pow(t_1, 2.0) / U)) - U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(Float64(J * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U * 0.5) / J))))
	t_1 = Float64(J * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J <= -4.1e-115)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(t_1 / Float64(U / t_1))));
	elseif (J <= 7.8e-28)
		tmp = Float64(Float64(-2.0 * Float64((t_1 ^ 2.0) / U)) - U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, ((U * 0.5) / J)));
	t_1 = J * cos((K * 0.5));
	tmp = 0.0;
	if (J <= -4.1e-115)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = -2.0 * ((U * -0.5) - (t_1 / (U / t_1)));
	elseif (J <= 7.8e-28)
		tmp = (-2.0 * ((t_1 ^ 2.0) / U)) - U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -4.1e-115], t$95$0, If[LessEqual[J, -4e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(t$95$1 / N[(U / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 7.8e-28], N[(N[(-2.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -4.0999999999999997e-115 or 7.79999999999999998e-28 < J

   1. Initial program 90.0%

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

      1. associate-*l* 90.0%

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

      2. associate-*l* 90.0%

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

      3. unpow2 90.0%

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

      4. hypot-1-def 96.9%

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

      5. associate-/r* 96.9%

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

      6. cos-neg 96.9%

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

      7. distribute-frac-neg 96.9%

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

      8. associate-/r* 96.9%

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

      9. associate-/r* 96.9%

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

      10. distribute-frac-neg 96.9%

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

      11. cos-neg 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in K around 0 88.5%

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

      1. associate-*r/ 88.5%

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

   6. Simplified 88.5%

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

   if -4.0999999999999997e-115 < J < -3.999999999999988e-310

   1. Initial program 34.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in U around -inf 34.3%

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

      1. +-commutative 34.3%

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

      2. mul-1-neg 34.3%

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

      3. unsub-neg 34.3%

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

      4. *-commutative 34.3%

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

      5. unpow2 34.3%

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

      6. *-commutative 34.3%

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

      7. unpow2 34.3%

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

      8. swap-sqr 34.3%

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

      9. unpow2 34.3%

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

      10. *-commutative 34.3%

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

   6. Simplified 34.3%

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

      1. *-commutative 34.3%

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

      2. pow2 34.3%

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

      3. *-un-lft-identity 34.3%

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

      4. times-frac 34.2%

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

   8. Applied egg-rr 34.2%

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

      1. clear-num 34.2%

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

      2. /-rgt-identity 34.2%

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

      3. un-div-inv 34.2%

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

      4. *-commutative 34.2%

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

      5. *-commutative 34.2%

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

   10. Applied egg-rr 34.2%

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

   if -3.999999999999988e-310 < J < 7.79999999999999998e-28

   1. Initial program 40.8%

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

   2. Taylor expanded in J around 0 42.8%

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

      1. neg-mul-1 42.8%

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

      2. unsub-neg 42.8%

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

      3. unpow2 42.8%

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

      4. *-commutative 42.8%

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

      5. unpow2 42.8%

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

      6. swap-sqr 42.8%

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

      7. unpow2 42.8%

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

      8. *-commutative 42.8%

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

   4. Simplified 42.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.1 \cdot 10^{-115}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot \cos \left(K \cdot 0.5\right)}{\frac{U}{J \cdot \cos \left(K \cdot 0.5\right)}}\right)\\ \mathbf{elif}\;J \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;-2 \cdot \frac{{\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \]

Alternative 5: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ t_1 := J \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -6.5 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - t_1 \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;-2 \cdot \frac{{t_1}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (* J (cos (/ K 2.0))) (hypot 1.0 (/ (* U 0.5) J)))))
        (t_1 (* J (cos (* K 0.5)))))
   (if (<= J -6.5e-115)
     t_0
     (if (<= J -4e-310)
       (* -2.0 (- (* U -0.5) (* t_1 (/ J U))))
       (if (<= J 7.8e-28) (- (* -2.0 (/ (pow t_1 2.0) U)) U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, ((U * 0.5) / J)));
	double t_1 = J * cos((K * 0.5));
	double tmp;
	if (J <= -6.5e-115) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * ((U * -0.5) - (t_1 * (J / U)));
	} else if (J <= 7.8e-28) {
		tmp = (-2.0 * (pow(t_1, 2.0) / U)) - U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * ((J * Math.cos((K / 2.0))) * Math.hypot(1.0, ((U * 0.5) / J)));
	double t_1 = J * Math.cos((K * 0.5));
	double tmp;
	if (J <= -6.5e-115) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = -2.0 * ((U * -0.5) - (t_1 * (J / U)));
	} else if (J <= 7.8e-28) {
		tmp = (-2.0 * (Math.pow(t_1, 2.0) / U)) - U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * ((J * math.cos((K / 2.0))) * math.hypot(1.0, ((U * 0.5) / J)))
	t_1 = J * math.cos((K * 0.5))
	tmp = 0
	if J <= -6.5e-115:
		tmp = t_0
	elif J <= -4e-310:
		tmp = -2.0 * ((U * -0.5) - (t_1 * (J / U)))
	elif J <= 7.8e-28:
		tmp = (-2.0 * (math.pow(t_1, 2.0) / U)) - U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(Float64(J * cos(Float64(K / 2.0))) * hypot(1.0, Float64(Float64(U * 0.5) / J))))
	t_1 = Float64(J * cos(Float64(K * 0.5)))
	tmp = 0.0
	if (J <= -6.5e-115)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(t_1 * Float64(J / U))));
	elseif (J <= 7.8e-28)
		tmp = Float64(Float64(-2.0 * Float64((t_1 ^ 2.0) / U)) - U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * ((J * cos((K / 2.0))) * hypot(1.0, ((U * 0.5) / J)));
	t_1 = J * cos((K * 0.5));
	tmp = 0.0;
	if (J <= -6.5e-115)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = -2.0 * ((U * -0.5) - (t_1 * (J / U)));
	elseif (J <= 7.8e-28)
		tmp = (-2.0 * ((t_1 ^ 2.0) / U)) - U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(N[(U * 0.5), $MachinePrecision] / J), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -6.5e-115], t$95$0, If[LessEqual[J, -4e-310], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(t$95$1 * N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 7.8e-28], N[(N[(-2.0 * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision] - U), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if J < -6.50000000000000033e-115 or 7.79999999999999998e-28 < J

   1. Initial program 90.0%

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

      1. associate-*l* 90.0%

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

      2. associate-*l* 90.0%

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

      3. unpow2 90.0%

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

      4. hypot-1-def 96.9%

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

      5. associate-/r* 96.9%

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

      6. cos-neg 96.9%

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

      7. distribute-frac-neg 96.9%

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

      8. associate-/r* 96.9%

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

      9. associate-/r* 96.9%

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

      10. distribute-frac-neg 96.9%

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

      11. cos-neg 96.9%

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

   3. Simplified 96.9%

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

   4. Taylor expanded in K around 0 88.5%

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

      1. associate-*r/ 88.5%

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

   6. Simplified 88.5%

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

   if -6.50000000000000033e-115 < J < -3.999999999999988e-310

   1. Initial program 34.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in U around -inf 34.3%

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

      1. +-commutative 34.3%

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

      2. mul-1-neg 34.3%

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

      3. unsub-neg 34.3%

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

      4. *-commutative 34.3%

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

      5. unpow2 34.3%

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

      6. *-commutative 34.3%

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

      7. unpow2 34.3%

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

      8. swap-sqr 34.3%

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

      9. unpow2 34.3%

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

      10. *-commutative 34.3%

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

   6. Simplified 34.3%

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

      1. *-commutative 34.3%

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

      2. pow2 34.3%

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

      3. *-un-lft-identity 34.3%

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

      4. times-frac 34.2%

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

   8. Applied egg-rr 34.2%

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

   9. Taylor expanded in K around 0 34.2%

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

   if -3.999999999999988e-310 < J < 7.79999999999999998e-28

   1. Initial program 40.8%

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

   2. Taylor expanded in J around 0 42.8%

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

      1. neg-mul-1 42.8%

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

      2. unsub-neg 42.8%

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

      3. unpow2 42.8%

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

      4. *-commutative 42.8%

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

      5. unpow2 42.8%

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

      6. swap-sqr 42.8%

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

      7. unpow2 42.8%

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

      8. *-commutative 42.8%

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

   4. Simplified 42.8%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -6.5 \cdot 10^{-115}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \frac{J}{U}\right)\\ \mathbf{elif}\;J \leq 7.8 \cdot 10^{-28}:\\ \;\;\;\;-2 \cdot \frac{{\left(J \cdot \cos \left(K \cdot 0.5\right)\right)}^{2}}{U} - U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U \cdot 0.5}{J}\right)\right)\\ \end{array} \]

Alternative 6: 61.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;K \leq 4.8 \cdot 10^{+58} \lor \neg \left(K \leq 3.1 \cdot 10^{+153}\right) \land K \leq 2.9 \cdot 10^{+172}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (or (<= K 4.8e+58) (and (not (<= K 3.1e+153)) (<= K 2.9e+172)))
   (* (* -2.0 J) (hypot 1.0 (* 0.5 (/ U J))))
   (* (cos (* K 0.5)) (* -2.0 J))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if ((K <= 4.8e+58) || (!(K <= 3.1e+153) && (K <= 2.9e+172))) {
		tmp = (-2.0 * J) * hypot(1.0, (0.5 * (U / J)));
	} else {
		tmp = cos((K * 0.5)) * (-2.0 * J);
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if ((K <= 4.8e+58) || (!(K <= 3.1e+153) && (K <= 2.9e+172))) {
		tmp = (-2.0 * J) * Math.hypot(1.0, (0.5 * (U / J)));
	} else {
		tmp = Math.cos((K * 0.5)) * (-2.0 * J);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if (K <= 4.8e+58) or (not (K <= 3.1e+153) and (K <= 2.9e+172)):
		tmp = (-2.0 * J) * math.hypot(1.0, (0.5 * (U / J)))
	else:
		tmp = math.cos((K * 0.5)) * (-2.0 * J)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if ((K <= 4.8e+58) || (!(K <= 3.1e+153) && (K <= 2.9e+172)))
		tmp = Float64(Float64(-2.0 * J) * hypot(1.0, Float64(0.5 * Float64(U / J))));
	else
		tmp = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if ((K <= 4.8e+58) || (~((K <= 3.1e+153)) && (K <= 2.9e+172)))
		tmp = (-2.0 * J) * hypot(1.0, (0.5 * (U / J)));
	else
		tmp = cos((K * 0.5)) * (-2.0 * J);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[Or[LessEqual[K, 4.8e+58], And[N[Not[LessEqual[K, 3.1e+153]], $MachinePrecision], LessEqual[K, 2.9e+172]]], N[(N[(-2.0 * J), $MachinePrecision] * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 4.8e58 or 3.1e153 < K < 2.8999999999999999e172

   1. Initial program 72.2%

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

   2. Taylor expanded in K around 0 38.7%

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

      1. associate-*r* 38.7%

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

      2. *-commutative 38.7%

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

      3. metadata-eval 38.7%

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

      4. unpow2 38.7%

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

      5. unpow2 38.7%

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

      6. times-frac 55.0%

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

      7. swap-sqr 55.5%

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

      8. associate-*r/ 55.5%

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

      9. *-commutative 55.5%

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

      10. associate-*r/ 55.4%

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

      11. associate-*r/ 55.4%

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

      12. *-commutative 55.4%

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

      13. associate-*r/ 55.4%

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

      14. unpow2 55.4%

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

      15. associate-*r/ 55.5%

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

      16. *-commutative 55.5%

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

      17. associate-*r/ 55.5%

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

   4. Simplified 55.5%

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

      1. expm1-log1p-u 54.1%

         \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\right)\right)} \]

      2. expm1-udef 54.1%

         \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{1 + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\right)} - 1\right)} \]

      3. metadata-eval 54.1%

         \[\leadsto \left(J \cdot -2\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{1 \cdot 1} + {\left(0.5 \cdot \frac{U}{J}\right)}^{2}}\right)} - 1\right) \]

      4. unpow2 54.1%

         \[\leadsto \left(J \cdot -2\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 \cdot 1 + \color{blue}{\left(0.5 \cdot \frac{U}{J}\right) \cdot \left(0.5 \cdot \frac{U}{J}\right)}}\right)} - 1\right) \]

      5. associate-*r/ 54.1%

         \[\leadsto \left(J \cdot -2\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 \cdot 1 + \color{blue}{\frac{0.5 \cdot U}{J}} \cdot \left(0.5 \cdot \frac{U}{J}\right)}\right)} - 1\right) \]

      6. associate-*r/ 54.1%

         \[\leadsto \left(J \cdot -2\right) \cdot \left(e^{\mathsf{log1p}\left(\sqrt{1 \cdot 1 + \frac{0.5 \cdot U}{J} \cdot \color{blue}{\frac{0.5 \cdot U}{J}}}\right)} - 1\right) \]

      7. hypot-udef 65.6%

         \[\leadsto \left(J \cdot -2\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(1, \frac{0.5 \cdot U}{J}\right)}\right)} - 1\right) \]

      8. associate-*r/ 65.6%

         \[\leadsto \left(J \cdot -2\right) \cdot \left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \color{blue}{0.5 \cdot \frac{U}{J}}\right)\right)} - 1\right) \]

   6. Applied egg-rr 65.6%

      \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 65.6%

         \[\leadsto \left(J \cdot -2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\right)} \]

      2. expm1-log1p 68.5%

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

   8. Simplified 68.5%

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

   if 4.8e58 < K < 3.1e153 or 2.8999999999999999e172 < K

   1. Initial program 66.6%

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

   2. Taylor expanded in J around inf 58.6%

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

      1. associate-*r* 58.6%

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

      2. *-commutative 58.6%

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

   4. Simplified 58.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 4.8 \cdot 10^{+58} \lor \neg \left(K \leq 3.1 \cdot 10^{+153}\right) \land K \leq 2.9 \cdot 10^{+172}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]

Alternative 7: 66.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;J \leq -2.05 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.4 \cdot 10^{-26}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (cos (* K 0.5)) (* -2.0 J))))
   (if (<= J -2.05e-71)
     t_0
     (if (<= J -4e-310) U (if (<= J 9.4e-26) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5)) * (-2.0 * J);
	double tmp;
	if (J <= -2.05e-71) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 9.4e-26) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0)) * ((-2.0d0) * j)
    if (j <= (-2.05d-71)) then
        tmp = t_0
    else if (j <= (-4d-310)) then
        tmp = u
    else if (j <= 9.4d-26) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K * 0.5)) * (-2.0 * J);
	double tmp;
	if (J <= -2.05e-71) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 9.4e-26) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K * 0.5)) * (-2.0 * J)
	tmp = 0
	if J <= -2.05e-71:
		tmp = t_0
	elif J <= -4e-310:
		tmp = U
	elif J <= 9.4e-26:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J))
	tmp = 0.0
	if (J <= -2.05e-71)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 9.4e-26)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K * 0.5)) * (-2.0 * J);
	tmp = 0.0;
	if (J <= -2.05e-71)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 9.4e-26)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -2.05e-71], t$95$0, If[LessEqual[J, -4e-310], U, If[LessEqual[J, 9.4e-26], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -2.04999999999999997e-71 or 9.39999999999999979e-26 < J

   1. Initial program 92.8%

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

   2. Taylor expanded in J around inf 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

   4. Simplified 74.1%

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

   if -2.04999999999999997e-71 < J < -3.999999999999988e-310

   1. Initial program 43.6%

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

   2. Taylor expanded in U around -inf 38.3%

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

   if -3.999999999999988e-310 < J < 9.39999999999999979e-26

   1. Initial program 41.1%

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

   2. Taylor expanded in J around 0 42.7%

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

      1. neg-mul-1 42.7%

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

   4. Simplified 42.7%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.05 \cdot 10^{-71}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.4 \cdot 10^{-26}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]

Alternative 8: 66.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{if}\;J \leq -1.22 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U + \frac{2 \cdot {J}^{2}}{U}\\ \mathbf{elif}\;J \leq 3.5 \cdot 10^{-26}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* (cos (* K 0.5)) (* -2.0 J))))
   (if (<= J -1.22e-73)
     t_0
     (if (<= J -4e-310)
       (+ U (/ (* 2.0 (pow J 2.0)) U))
       (if (<= J 3.5e-26) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K * 0.5)) * (-2.0 * J);
	double tmp;
	if (J <= -1.22e-73) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = U + ((2.0 * pow(J, 2.0)) / U);
	} else if (J <= 3.5e-26) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0)) * ((-2.0d0) * j)
    if (j <= (-1.22d-73)) then
        tmp = t_0
    else if (j <= (-4d-310)) then
        tmp = u + ((2.0d0 * (j ** 2.0d0)) / u)
    else if (j <= 3.5d-26) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K * 0.5)) * (-2.0 * J);
	double tmp;
	if (J <= -1.22e-73) {
		tmp = t_0;
	} else if (J <= -4e-310) {
		tmp = U + ((2.0 * Math.pow(J, 2.0)) / U);
	} else if (J <= 3.5e-26) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K * 0.5)) * (-2.0 * J)
	tmp = 0
	if J <= -1.22e-73:
		tmp = t_0
	elif J <= -4e-310:
		tmp = U + ((2.0 * math.pow(J, 2.0)) / U)
	elif J <= 3.5e-26:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(cos(Float64(K * 0.5)) * Float64(-2.0 * J))
	tmp = 0.0
	if (J <= -1.22e-73)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = Float64(U + Float64(Float64(2.0 * (J ^ 2.0)) / U));
	elseif (J <= 3.5e-26)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K * 0.5)) * (-2.0 * J);
	tmp = 0.0;
	if (J <= -1.22e-73)
		tmp = t_0;
	elseif (J <= -4e-310)
		tmp = U + ((2.0 * (J ^ 2.0)) / U);
	elseif (J <= 3.5e-26)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -1.22e-73], t$95$0, If[LessEqual[J, -4e-310], N[(U + N[(N[(2.0 * N[Power[J, 2.0], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 3.5e-26], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.22e-73 or 3.49999999999999985e-26 < J

   1. Initial program 92.8%

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

   2. Taylor expanded in J around inf 74.1%

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

      1. associate-*r* 74.1%

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

      2. *-commutative 74.1%

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

   4. Simplified 74.1%

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

   if -1.22e-73 < J < -3.999999999999988e-310

   1. Initial program 43.6%

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

   2. Taylor expanded in K around 0 17.6%

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

      1. associate-*r* 17.6%

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

      2. *-commutative 17.6%

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

      3. metadata-eval 17.6%

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

      4. unpow2 17.6%

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

      5. unpow2 17.6%

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

      6. times-frac 32.0%

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

      7. swap-sqr 32.0%

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

      8. associate-*r/ 32.0%

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

      9. *-commutative 32.0%

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

      10. associate-*r/ 31.9%

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

      11. associate-*r/ 31.9%

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

      12. *-commutative 31.9%

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

      13. associate-*r/ 31.9%

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

      14. unpow2 31.9%

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

      15. associate-*r/ 32.0%

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

      16. *-commutative 32.0%

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

      17. associate-*r/ 32.0%

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

   4. Simplified 32.0%

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

   5. Taylor expanded in U around -inf 38.6%

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

      1. associate-*r/ 38.6%

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

   7. Simplified 38.6%

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

   if -3.999999999999988e-310 < J < 3.49999999999999985e-26

   1. Initial program 41.1%

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

   2. Taylor expanded in J around 0 42.7%

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

      1. neg-mul-1 42.7%

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

   4. Simplified 42.7%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.22 \cdot 10^{-73}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U + \frac{2 \cdot {J}^{2}}{U}\\ \mathbf{elif}\;J \leq 3.5 \cdot 10^{-26}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot 0.5\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]

Alternative 9: 49.8% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -3.1 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{+82}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -3.1e-37)
   (* -2.0 J)
   (if (<= J -4e-310) U (if (<= J 9.2e+82) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -3.1e-37) {
		tmp = -2.0 * J;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 9.2e+82) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-3.1d-37)) then
        tmp = (-2.0d0) * j
    else if (j <= (-4d-310)) then
        tmp = u
    else if (j <= 9.2d+82) then
        tmp = -u
    else
        tmp = (-2.0d0) * j
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -3.1e-37) {
		tmp = -2.0 * J;
	} else if (J <= -4e-310) {
		tmp = U;
	} else if (J <= 9.2e+82) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -3.1e-37:
		tmp = -2.0 * J
	elif J <= -4e-310:
		tmp = U
	elif J <= 9.2e+82:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -3.1e-37)
		tmp = Float64(-2.0 * J);
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 9.2e+82)
		tmp = Float64(-U);
	else
		tmp = Float64(-2.0 * J);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -3.1e-37)
		tmp = -2.0 * J;
	elseif (J <= -4e-310)
		tmp = U;
	elseif (J <= 9.2e+82)
		tmp = -U;
	else
		tmp = -2.0 * J;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -3.1e-37], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, -4e-310], U, If[LessEqual[J, 9.2e+82], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -3.09999999999999993e-37 or 9.19999999999999953e82 < J

   1. Initial program 97.1%

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

   2. Taylor expanded in K around 0 45.9%

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

      1. associate-*r* 45.9%

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

      2. *-commutative 45.9%

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

      3. metadata-eval 45.9%

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

      4. unpow2 45.9%

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

      5. unpow2 45.9%

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

      6. times-frac 59.0%

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

      7. swap-sqr 59.0%

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

      8. associate-*r/ 59.0%

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

      9. *-commutative 59.0%

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

      10. associate-*r/ 59.0%

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

      11. associate-*r/ 59.0%

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

      12. *-commutative 59.0%

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

      13. associate-*r/ 59.0%

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

      14. unpow2 59.0%

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

      15. associate-*r/ 59.0%

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

      16. *-commutative 59.0%

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

      17. associate-*r/ 59.0%

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

   4. Simplified 59.0%

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

   5. Taylor expanded in J around inf 46.8%

      \[\leadsto \color{blue}{-2 \cdot J} \]
   6. Step-by-step derivation

      1. *-commutative 46.8%

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

   7. Simplified 46.8%

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

   if -3.09999999999999993e-37 < J < -3.999999999999988e-310

   1. Initial program 51.1%

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

   2. Taylor expanded in U around -inf 31.5%

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

   if -3.999999999999988e-310 < J < 9.19999999999999953e82

   1. Initial program 52.1%

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

   2. Taylor expanded in J around 0 40.9%

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

      1. neg-mul-1 40.9%

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

   4. Simplified 40.9%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.1 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{+82}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 10: 39.9% accurate, 103.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= J -4e-310) U (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -4e-310) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-4d-310)) then
        tmp = u
    else
        tmp = -u
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double tmp;
	if (J <= -4e-310) {
		tmp = U;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -4e-310:
		tmp = U
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -4e-310)
		tmp = U;
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (J <= -4e-310)
		tmp = U;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[J, -4e-310], U, (-U)]

Derivation

1. Split input into 2 regimes

2. if J < -3.999999999999988e-310

   1. Initial program 75.6%

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

   2. Taylor expanded in U around -inf 18.5%

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

   if -3.999999999999988e-310 < J

   1. Initial program 66.4%

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

   2. Taylor expanded in J around 0 30.9%

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

      1. neg-mul-1 30.9%

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

   4. Simplified 30.9%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4 \cdot 10^{-310}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 11: 27.1% accurate, 420.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.0%

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

2. Taylor expanded in U around -inf 23.4%

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

3. Final simplification 23.4%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023300 
(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)))))