Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.3% → 98.7%

Time: 19.5s

Alternatives: 12

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: 73.3% 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: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := \left(\left(-2 \cdot J\right) \cdot t_0\right) \cdot \sqrt{1 + {\left(\frac{U}{t_0 \cdot \left(J \cdot 2\right)}\right)}^{2}}\\ t_2 := J \cdot t_0\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+294}:\\ \;\;\;\;-2 \cdot \left(t_2 \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1
         (*
          (* (* -2.0 J) t_0)
          (sqrt (+ 1.0 (pow (/ U (* t_0 (* J 2.0))) 2.0)))))
        (t_2 (* J t_0)))
   (if (<= t_1 (- INFINITY))
     (- U)
     (if (<= t_1 5e+294)
       (* -2.0 (* t_2 (hypot 1.0 (/ U (* 2.0 t_2)))))
       (* -2.0 (* U -0.5))))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = J * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -U;
	} else if (t_1 <= 5e+294) {
		tmp = -2.0 * (t_2 * hypot(1.0, (U / (2.0 * t_2))));
	} else {
		tmp = -2.0 * (U * -0.5);
	}
	return tmp;
}

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / (t_0 * (J * 2.0))), 2.0)));
	double t_2 = J * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -U;
	} else if (t_1 <= 5e+294) {
		tmp = -2.0 * (t_2 * Math.hypot(1.0, (U / (2.0 * t_2))));
	} else {
		tmp = -2.0 * (U * -0.5);
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / (t_0 * (J * 2.0))), 2.0)))
	t_2 = J * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -U
	elif t_1 <= 5e+294:
		tmp = -2.0 * (t_2 * math.hypot(1.0, (U / (2.0 * t_2))))
	else:
		tmp = -2.0 * (U * -0.5)
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(t_0 * Float64(J * 2.0))) ^ 2.0))))
	t_2 = Float64(J * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-U);
	elseif (t_1 <= 5e+294)
		tmp = Float64(-2.0 * Float64(t_2 * hypot(1.0, Float64(U / Float64(2.0 * t_2)))));
	else
		tmp = Float64(-2.0 * Float64(U * -0.5));
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = cos((K / 2.0));
	t_1 = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / (t_0 * (J * 2.0))) ^ 2.0)));
	t_2 = J * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -U;
	elseif (t_1 <= 5e+294)
		tmp = -2.0 * (t_2 * hypot(1.0, (U / (2.0 * t_2))));
	else
		tmp = -2.0 * (U * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(J * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-U), If[LessEqual[t$95$1, 5e+294], N[(-2.0 * N[(t$95$2 * N[Sqrt[1.0 ^ 2 + N[(U / N[(2.0 * t$95$2), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.4%

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

   3. Simplified 52.3%

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

   4. Taylor expanded in J around 0 56.7%

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

   5. Taylor expanded in U around 0 56.7%

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

      1. mul-1-neg 56.7%

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

   7. Simplified 56.7%

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

   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)))) < 4.9999999999999999e294

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. associate-*l* 99.8%

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

      3. unpow2 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. unpow2 99.8%

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

   3. Simplified 99.8%

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

   if 4.9999999999999999e294 < (*.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 5.6%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in U around -inf 55.9%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 55.9%

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

   6. Simplified 55.9%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot 2\right)}\right)}^{2}} \leq 5 \cdot 10^{+294}:\\ \;\;\;\;-2 \cdot \left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \mathsf{hypot}\left(1, \frac{U}{2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \end{array} \]

Alternative 2: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 1.4 \cdot 10^{+293}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(t_0 \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot t_0\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

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 1.4e+293)
     (* -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 <= 1.4e+293) {
		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 <= 1.4e+293) {
		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 <= 1.4e+293:
		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 <= 1.4e+293)
		tmp = Float64(-2.0 * Float64(J * Float64(t_0 * hypot(1.0, Float64(U / Float64(J * Float64(2.0 * t_0)))))));
	else
		tmp = Float64(-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 <= 1.4e+293)
		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, 1.4e+293], N[(-2.0 * N[(J * N[(t$95$0 * N[Sqrt[1.0 ^ 2 + N[(U / N[(J * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U)]]

Derivation

1. Split input into 2 regimes

2. if U < 1.39999999999999993e293

   1. Initial program 77.2%

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

   3. Simplified 88.8%

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

   if 1.39999999999999993e293 < U

   1. Initial program 11.4%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in J around 0 83.3%

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

   5. Taylor expanded in U around 0 83.3%

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

      1. mul-1-neg 83.3%

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

   7. Simplified 83.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.4 \cdot 10^{+293}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 3: 88.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;U \leq 3.1 \cdot 10^{+293}:\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot t_0} \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

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 3.1e+293)
     (* -2.0 (* t_0 (* J (hypot 1.0 (* (/ U (* J t_0)) 0.5)))))
     (- U))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (U <= 3.1e+293) {
		tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * t_0)) * 0.5))));
	} 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 <= 3.1e+293) {
		tmp = -2.0 * (t_0 * (J * Math.hypot(1.0, ((U / (J * t_0)) * 0.5))));
	} 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 <= 3.1e+293:
		tmp = -2.0 * (t_0 * (J * math.hypot(1.0, ((U / (J * t_0)) * 0.5))))
	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 <= 3.1e+293)
		tmp = Float64(-2.0 * Float64(t_0 * Float64(J * hypot(1.0, Float64(Float64(U / Float64(J * t_0)) * 0.5)))));
	else
		tmp = Float64(-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 <= 3.1e+293)
		tmp = -2.0 * (t_0 * (J * hypot(1.0, ((U / (J * t_0)) * 0.5))));
	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, 3.1e+293], N[(-2.0 * N[(t$95$0 * N[(J * N[Sqrt[1.0 ^ 2 + N[(N[(U / N[(J * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-U)]]

Derivation

1. Split input into 2 regimes

2. if U < 3.0999999999999999e293

   1. Initial program 77.2%

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

      1. associate-*l* 77.2%

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

      2. associate-*l* 77.2%

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

      3. *-commutative 77.2%

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

      4. unpow2 77.2%

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

      5. sqr-neg 77.2%

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

      6. distribute-frac-neg 77.2%

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

      7. distribute-frac-neg 77.2%

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

      8. unpow2 77.2%

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

   3. Simplified 88.8%

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

   if 3.0999999999999999e293 < U

   1. Initial program 11.4%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in J around 0 83.3%

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

   5. Taylor expanded in U around 0 83.3%

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

      1. mul-1-neg 83.3%

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

   7. Simplified 83.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 3.1 \cdot 10^{+293}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, \frac{U}{J \cdot \cos \left(\frac{K}{2}\right)} \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 4: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\right)\\ \mathbf{if}\;J \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* (cos (/ K 2.0)) (* J (hypot 1.0 (* 0.5 (/ U J))))))))
   (if (<= J -3.2e-186)
     t_0
     (if (<= J 7.6e-297) (* -2.0 (* U -0.5)) (if (<= J 3.5e-57) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, (0.5 * (U / J)))));
	double tmp;
	if (J <= -3.2e-186) {
		tmp = t_0;
	} else if (J <= 7.6e-297) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 3.5e-57) {
		tmp = -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 * (Math.cos((K / 2.0)) * (J * Math.hypot(1.0, (0.5 * (U / J)))));
	double tmp;
	if (J <= -3.2e-186) {
		tmp = t_0;
	} else if (J <= 7.6e-297) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 3.5e-57) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (math.cos((K / 2.0)) * (J * math.hypot(1.0, (0.5 * (U / J)))))
	tmp = 0
	if J <= -3.2e-186:
		tmp = t_0
	elif J <= 7.6e-297:
		tmp = -2.0 * (U * -0.5)
	elif J <= 3.5e-57:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(cos(Float64(K / 2.0)) * Float64(J * hypot(1.0, Float64(0.5 * Float64(U / J))))))
	tmp = 0.0
	if (J <= -3.2e-186)
		tmp = t_0;
	elseif (J <= 7.6e-297)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 3.5e-57)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (cos((K / 2.0)) * (J * hypot(1.0, (0.5 * (U / J)))));
	tmp = 0.0;
	if (J <= -3.2e-186)
		tmp = t_0;
	elseif (J <= 7.6e-297)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 3.5e-57)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[Sqrt[1.0 ^ 2 + N[(0.5 * N[(U / J), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -3.2e-186], t$95$0, If[LessEqual[J, 7.6e-297], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 3.5e-57], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -3.2e-186 or 3.49999999999999991e-57 < J

   1. Initial program 88.4%

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

      1. associate-*l* 88.4%

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

      2. associate-*l* 88.4%

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

      3. *-commutative 88.4%

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

      4. unpow2 88.4%

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

      5. sqr-neg 88.4%

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

      6. distribute-frac-neg 88.4%

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

      7. distribute-frac-neg 88.4%

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

      8. unpow2 88.4%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in K around 0 85.1%

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

   if -3.2e-186 < J < 7.6000000000000001e-297

   1. Initial program 40.2%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in U around -inf 60.5%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 60.5%

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

   6. Simplified 60.5%

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

   if 7.6000000000000001e-297 < J < 3.49999999999999991e-57

   1. Initial program 49.0%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in J around 0 47.2%

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

   5. Taylor expanded in U around 0 47.2%

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

      1. mul-1-neg 47.2%

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

   7. Simplified 47.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\right)\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \mathsf{hypot}\left(1, 0.5 \cdot \frac{U}{J}\right)\right)\right)\\ \end{array} \]

Alternative 5: 57.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;U \leq 8 \cdot 10^{+205}:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \left(\frac{U}{J} \cdot \frac{U}{J}\right)}\right)\\ \mathbf{elif}\;U \leq 3 \cdot 10^{+293}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 1.2e-29)
   (* -2.0 (* J (cos (/ K 2.0))))
   (if (<= U 8e+205)
     (* -2.0 (* J (sqrt (+ 1.0 (* 0.25 (* (/ U J) (/ U J)))))))
     (if (<= U 3e+293) (* -2.0 (* U -0.5)) (- U)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.2e-29) {
		tmp = -2.0 * (J * cos((K / 2.0)));
	} else if (U <= 8e+205) {
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * ((U / J) * (U / J))))));
	} else if (U <= 3e+293) {
		tmp = -2.0 * (U * -0.5);
	} 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 (u <= 1.2d-29) then
        tmp = (-2.0d0) * (j * cos((k / 2.0d0)))
    else if (u <= 8d+205) then
        tmp = (-2.0d0) * (j * sqrt((1.0d0 + (0.25d0 * ((u / j) * (u / j))))))
    else if (u <= 3d+293) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    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 (U <= 1.2e-29) {
		tmp = -2.0 * (J * Math.cos((K / 2.0)));
	} else if (U <= 8e+205) {
		tmp = -2.0 * (J * Math.sqrt((1.0 + (0.25 * ((U / J) * (U / J))))));
	} else if (U <= 3e+293) {
		tmp = -2.0 * (U * -0.5);
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 1.2e-29:
		tmp = -2.0 * (J * math.cos((K / 2.0)))
	elif U <= 8e+205:
		tmp = -2.0 * (J * math.sqrt((1.0 + (0.25 * ((U / J) * (U / J))))))
	elif U <= 3e+293:
		tmp = -2.0 * (U * -0.5)
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 1.2e-29)
		tmp = Float64(-2.0 * Float64(J * cos(Float64(K / 2.0))));
	elseif (U <= 8e+205)
		tmp = Float64(-2.0 * Float64(J * sqrt(Float64(1.0 + Float64(0.25 * Float64(Float64(U / J) * Float64(U / J)))))));
	elseif (U <= 3e+293)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 1.2e-29)
		tmp = -2.0 * (J * cos((K / 2.0)));
	elseif (U <= 8e+205)
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * ((U / J) * (U / J))))));
	elseif (U <= 3e+293)
		tmp = -2.0 * (U * -0.5);
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 1.2e-29], N[(-2.0 * N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 8e+205], N[(-2.0 * N[(J * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[(U / J), $MachinePrecision] * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 3e+293], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], (-U)]]]

Derivation

1. Split input into 4 regimes

2. if U < 1.19999999999999996e-29

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. associate-*l* 83.7%

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

      3. *-commutative 83.7%

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

      4. unpow2 83.7%

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

      5. sqr-neg 83.7%

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

      6. distribute-frac-neg 83.7%

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

      7. distribute-frac-neg 83.7%

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

      8. unpow2 83.7%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in J around inf 64.1%

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

   if 1.19999999999999996e-29 < U < 8.00000000000000013e205

   1. Initial program 72.1%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in K around 0 46.1%

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

      1. *-commutative 46.1%

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

      2. unpow2 46.1%

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

      3. unpow2 46.1%

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

   6. Simplified 46.1%

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

      1. times-frac 56.9%

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

   8. Applied egg-rr 56.9%

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

   if 8.00000000000000013e205 < U < 3.00000000000000013e293

   1. Initial program 36.7%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in U around -inf 63.7%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 63.7%

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

   6. Simplified 63.7%

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

   if 3.00000000000000013e293 < U

   1. Initial program 11.4%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in J around 0 83.3%

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

   5. Taylor expanded in U around 0 83.3%

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

      1. mul-1-neg 83.3%

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

   7. Simplified 83.3%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.2 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;U \leq 8 \cdot 10^{+205}:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \left(\frac{U}{J} \cdot \frac{U}{J}\right)}\right)\\ \mathbf{elif}\;U \leq 3 \cdot 10^{+293}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 6: 57.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 1.65 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J + 0.125 \cdot \frac{U}{\frac{J}{U}}\right)\right)\\ \mathbf{elif}\;U \leq 1.05 \cdot 10^{+206}:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \left(\frac{U}{J} \cdot \frac{U}{J}\right)}\right)\\ \mathbf{elif}\;U \leq 2.8 \cdot 10^{+293}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= U 1.65e-29)
   (* -2.0 (* (cos (/ K 2.0)) (+ J (* 0.125 (/ U (/ J U))))))
   (if (<= U 1.05e+206)
     (* -2.0 (* J (sqrt (+ 1.0 (* 0.25 (* (/ U J) (/ U J)))))))
     (if (<= U 2.8e+293) (* -2.0 (* U -0.5)) (- U)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 1.65e-29) {
		tmp = -2.0 * (cos((K / 2.0)) * (J + (0.125 * (U / (J / U)))));
	} else if (U <= 1.05e+206) {
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * ((U / J) * (U / J))))));
	} else if (U <= 2.8e+293) {
		tmp = -2.0 * (U * -0.5);
	} 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 (u <= 1.65d-29) then
        tmp = (-2.0d0) * (cos((k / 2.0d0)) * (j + (0.125d0 * (u / (j / u)))))
    else if (u <= 1.05d+206) then
        tmp = (-2.0d0) * (j * sqrt((1.0d0 + (0.25d0 * ((u / j) * (u / j))))))
    else if (u <= 2.8d+293) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    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 (U <= 1.65e-29) {
		tmp = -2.0 * (Math.cos((K / 2.0)) * (J + (0.125 * (U / (J / U)))));
	} else if (U <= 1.05e+206) {
		tmp = -2.0 * (J * Math.sqrt((1.0 + (0.25 * ((U / J) * (U / J))))));
	} else if (U <= 2.8e+293) {
		tmp = -2.0 * (U * -0.5);
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 1.65e-29:
		tmp = -2.0 * (math.cos((K / 2.0)) * (J + (0.125 * (U / (J / U)))))
	elif U <= 1.05e+206:
		tmp = -2.0 * (J * math.sqrt((1.0 + (0.25 * ((U / J) * (U / J))))))
	elif U <= 2.8e+293:
		tmp = -2.0 * (U * -0.5)
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 1.65e-29)
		tmp = Float64(-2.0 * Float64(cos(Float64(K / 2.0)) * Float64(J + Float64(0.125 * Float64(U / Float64(J / U))))));
	elseif (U <= 1.05e+206)
		tmp = Float64(-2.0 * Float64(J * sqrt(Float64(1.0 + Float64(0.25 * Float64(Float64(U / J) * Float64(U / J)))))));
	elseif (U <= 2.8e+293)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 1.65e-29)
		tmp = -2.0 * (cos((K / 2.0)) * (J + (0.125 * (U / (J / U)))));
	elseif (U <= 1.05e+206)
		tmp = -2.0 * (J * sqrt((1.0 + (0.25 * ((U / J) * (U / J))))));
	elseif (U <= 2.8e+293)
		tmp = -2.0 * (U * -0.5);
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 1.65e-29], N[(-2.0 * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J + N[(0.125 * N[(U / N[(J / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 1.05e+206], N[(-2.0 * N[(J * N[Sqrt[N[(1.0 + N[(0.25 * N[(N[(U / J), $MachinePrecision] * N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[U, 2.8e+293], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], (-U)]]]

Derivation

1. Split input into 4 regimes

2. if U < 1.65000000000000014e-29

   1. Initial program 83.7%

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

      1. associate-*l* 83.7%

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

      2. associate-*l* 83.7%

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

      3. *-commutative 83.7%

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

      4. unpow2 83.7%

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

      5. sqr-neg 83.7%

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

      6. distribute-frac-neg 83.7%

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

      7. distribute-frac-neg 83.7%

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

      8. unpow2 83.7%

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

   3. Simplified 91.3%

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

   4. Taylor expanded in K around 0 80.4%

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

   5. Taylor expanded in J around inf 64.0%

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

      1. fma-def 64.0%

         \[\leadsto -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{fma}\left(0.125, \frac{{U}^{2}}{J}, J\right)}\right) \]

      2. unpow2 64.0%

         \[\leadsto -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left(0.125, \frac{\color{blue}{U \cdot U}}{J}, J\right)\right) \]

   7. Simplified 64.0%

      \[\leadsto -2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \color{blue}{\mathsf{fma}\left(0.125, \frac{U \cdot U}{J}, J\right)}\right) \]
   8. Step-by-step derivation

      1. fma-udef 64.0%

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

      2. associate-/l* 64.6%

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

   9. Applied egg-rr 64.6%

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

   if 1.65000000000000014e-29 < U < 1.04999999999999993e206

   1. Initial program 72.1%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in K around 0 46.1%

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

      1. *-commutative 46.1%

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

      2. unpow2 46.1%

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

      3. unpow2 46.1%

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

   6. Simplified 46.1%

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

      1. times-frac 56.9%

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

   8. Applied egg-rr 56.9%

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

   if 1.04999999999999993e206 < U < 2.79999999999999986e293

   1. Initial program 36.7%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in U around -inf 63.7%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 63.7%

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

   6. Simplified 63.7%

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

   if 2.79999999999999986e293 < U

   1. Initial program 11.4%

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

   3. Simplified 26.1%

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

   4. Taylor expanded in J around 0 83.3%

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

   5. Taylor expanded in U around 0 83.3%

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

      1. mul-1-neg 83.3%

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

   7. Simplified 83.3%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 1.65 \cdot 10^{-29}:\\ \;\;\;\;-2 \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(J + 0.125 \cdot \frac{U}{\frac{J}{U}}\right)\right)\\ \mathbf{elif}\;U \leq 1.05 \cdot 10^{+206}:\\ \;\;\;\;-2 \cdot \left(J \cdot \sqrt{1 + 0.25 \cdot \left(\frac{U}{J} \cdot \frac{U}{J}\right)}\right)\\ \mathbf{elif}\;U \leq 2.8 \cdot 10^{+293}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 7: 66.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{if}\;J \leq -9.5 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{-25}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (cos (/ K 2.0))))))
   (if (<= J -9.5e-51)
     t_0
     (if (<= J 7.6e-297)
       (* -2.0 (- (* U -0.5) (/ (* J J) U)))
       (if (<= J 9.2e-25) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * cos((K / 2.0)));
	double tmp;
	if (J <= -9.5e-51) {
		tmp = t_0;
	} else if (J <= 7.6e-297) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 9.2e-25) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * (j * cos((k / 2.0d0)))
    if (j <= (-9.5d-51)) then
        tmp = t_0
    else if (j <= 7.6d-297) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - ((j * j) / u))
    else if (j <= 9.2d-25) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * Math.cos((K / 2.0)));
	double tmp;
	if (J <= -9.5e-51) {
		tmp = t_0;
	} else if (J <= 7.6e-297) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 9.2e-25) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J * math.cos((K / 2.0)))
	tmp = 0
	if J <= -9.5e-51:
		tmp = t_0
	elif J <= 7.6e-297:
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U))
	elif J <= 9.2e-25:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * cos(Float64(K / 2.0))))
	tmp = 0.0
	if (J <= -9.5e-51)
		tmp = t_0;
	elseif (J <= 7.6e-297)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(Float64(J * J) / U)));
	elseif (J <= 9.2e-25)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J * cos((K / 2.0)));
	tmp = 0.0;
	if (J <= -9.5e-51)
		tmp = t_0;
	elseif (J <= 7.6e-297)
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	elseif (J <= 9.2e-25)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -9.5e-51], t$95$0, If[LessEqual[J, 7.6e-297], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 9.2e-25], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -9.4999999999999998e-51 or 9.1999999999999997e-25 < J

   1. Initial program 93.7%

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

      1. associate-*l* 93.7%

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

      2. associate-*l* 93.7%

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

      3. *-commutative 93.7%

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

      4. unpow2 93.7%

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

      5. sqr-neg 93.7%

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

      6. distribute-frac-neg 93.7%

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

      7. distribute-frac-neg 93.7%

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

      8. unpow2 93.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in J around inf 74.5%

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

   if -9.4999999999999998e-51 < J < 7.6000000000000001e-297

   1. Initial program 58.0%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in K around 0 24.9%

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

      1. *-commutative 24.9%

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

      2. unpow2 24.9%

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

      3. unpow2 24.9%

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

   6. Simplified 24.9%

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

   7. Taylor expanded in U around -inf 41.3%

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

      1. +-commutative 41.3%

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

      2. mul-1-neg 41.3%

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

      3. unsub-neg 41.3%

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

      4. *-commutative 41.3%

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

      5. unpow2 41.3%

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

   9. Simplified 41.3%

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

   if 7.6000000000000001e-297 < J < 9.1999999999999997e-25

   1. Initial program 51.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in J around 0 45.9%

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

   5. Taylor expanded in U around 0 45.9%

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

      1. mul-1-neg 45.9%

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

   7. Simplified 45.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9.5 \cdot 10^{-51}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 9.2 \cdot 10^{-25}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array} \]

Alternative 8: 55.8% accurate, 24.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \left(1 + U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 4.5 \cdot 10^{-98}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (+ 1.0 (* U (/ 0.5 J)))))))
   (if (<= J -4.2e-5)
     t_0
     (if (<= J 7.6e-297) (* -2.0 (* U -0.5)) (if (<= J 4.5e-98) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (1.0 + (U * (0.5 / J))));
	double tmp;
	if (J <= -4.2e-5) {
		tmp = t_0;
	} else if (J <= 7.6e-297) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 4.5e-98) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * (j * (1.0d0 + (u * (0.5d0 / j))))
    if (j <= (-4.2d-5)) then
        tmp = t_0
    else if (j <= 7.6d-297) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    else if (j <= 4.5d-98) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (1.0 + (U * (0.5 / J))));
	double tmp;
	if (J <= -4.2e-5) {
		tmp = t_0;
	} else if (J <= 7.6e-297) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 4.5e-98) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J * (1.0 + (U * (0.5 / J))))
	tmp = 0
	if J <= -4.2e-5:
		tmp = t_0
	elif J <= 7.6e-297:
		tmp = -2.0 * (U * -0.5)
	elif J <= 4.5e-98:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * Float64(1.0 + Float64(U * Float64(0.5 / J)))))
	tmp = 0.0
	if (J <= -4.2e-5)
		tmp = t_0;
	elseif (J <= 7.6e-297)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 4.5e-98)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J * (1.0 + (U * (0.5 / J))));
	tmp = 0.0;
	if (J <= -4.2e-5)
		tmp = t_0;
	elseif (J <= 7.6e-297)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 4.5e-98)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[(1.0 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -4.2e-5], t$95$0, If[LessEqual[J, 7.6e-297], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.5e-98], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -4.19999999999999977e-5 or 4.49999999999999997e-98 < J

   1. Initial program 92.6%

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

   3. Simplified 99.0%

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

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. expm1-log1p-u 89.8%

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{0.5 \cdot \frac{U}{J}}{\cos \left(K \cdot 0.5\right)}\right)}\right)}^{3}\right)\right)}\right) \]

      2. rem-cube-cbrt 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-/l* 90.3%

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

   7. Applied egg-rr 90.3%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \frac{0.5}{\frac{\cos \left(K \cdot 0.5\right)}{\frac{U}{J}}}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. expm1-udef 90.1%

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

      2. sub-neg 90.1%

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

   9. Applied egg-rr 98.9%

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

       1. metadata-eval 98.9%

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

       2. sub-neg 98.9%

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

       3. associate--l+ 99.0%

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

       4. *-commutative 99.0%

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

       5. *-commutative 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in U around inf 51.8%

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

       1. metadata-eval 51.8%

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

       2. times-frac 51.8%

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

       3. *-commutative 51.8%

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

       4. times-frac 51.7%

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

       5. /-rgt-identity 51.7%

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

   14. Simplified 51.7%

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

   if -4.19999999999999977e-5 < J < 7.6000000000000001e-297

   1. Initial program 61.0%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in U around -inf 42.5%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 42.5%

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

   6. Simplified 42.5%

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

   if 7.6000000000000001e-297 < J < 4.49999999999999997e-98

   1. Initial program 42.0%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in J around 0 47.4%

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

   5. Taylor expanded in U around 0 47.4%

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

      1. mul-1-neg 47.4%

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

   7. Simplified 47.4%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(1 + U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 4.5 \cdot 10^{-98}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(1 + U \cdot \frac{0.5}{J}\right)\right)\\ \end{array} \]

Alternative 9: 55.8% accurate, 24.4× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} t_0 := -2 \cdot \left(J \cdot \left(1 + U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{if}\;J \leq -0.15:\\ \;\;\;\;t_0\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 2.8 \cdot 10^{-105}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (* -2.0 (* J (+ 1.0 (* U (/ 0.5 J)))))))
   (if (<= J -0.15)
     t_0
     (if (<= J 7.6e-297)
       (* -2.0 (- (* U -0.5) (/ (* J J) U)))
       (if (<= J 2.8e-105) (- U) t_0)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (1.0 + (U * (0.5 / J))));
	double tmp;
	if (J <= -0.15) {
		tmp = t_0;
	} else if (J <= 7.6e-297) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 2.8e-105) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: U should be positive before calling this function
real(8) function code(j, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-2.0d0) * (j * (1.0d0 + (u * (0.5d0 / j))))
    if (j <= (-0.15d0)) then
        tmp = t_0
    else if (j <= 7.6d-297) then
        tmp = (-2.0d0) * ((u * (-0.5d0)) - ((j * j) / u))
    else if (j <= 2.8d-105) then
        tmp = -u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

U = Math.abs(U);
public static double code(double J, double K, double U) {
	double t_0 = -2.0 * (J * (1.0 + (U * (0.5 / J))));
	double tmp;
	if (J <= -0.15) {
		tmp = t_0;
	} else if (J <= 7.6e-297) {
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	} else if (J <= 2.8e-105) {
		tmp = -U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	t_0 = -2.0 * (J * (1.0 + (U * (0.5 / J))))
	tmp = 0
	if J <= -0.15:
		tmp = t_0
	elif J <= 7.6e-297:
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U))
	elif J <= 2.8e-105:
		tmp = -U
	else:
		tmp = t_0
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	t_0 = Float64(-2.0 * Float64(J * Float64(1.0 + Float64(U * Float64(0.5 / J)))))
	tmp = 0.0
	if (J <= -0.15)
		tmp = t_0;
	elseif (J <= 7.6e-297)
		tmp = Float64(-2.0 * Float64(Float64(U * -0.5) - Float64(Float64(J * J) / U)));
	elseif (J <= 2.8e-105)
		tmp = Float64(-U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	t_0 = -2.0 * (J * (1.0 + (U * (0.5 / J))));
	tmp = 0.0;
	if (J <= -0.15)
		tmp = t_0;
	elseif (J <= 7.6e-297)
		tmp = -2.0 * ((U * -0.5) - ((J * J) / U));
	elseif (J <= 2.8e-105)
		tmp = -U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := Block[{t$95$0 = N[(-2.0 * N[(J * N[(1.0 + N[(U * N[(0.5 / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[J, -0.15], t$95$0, If[LessEqual[J, 7.6e-297], N[(-2.0 * N[(N[(U * -0.5), $MachinePrecision] - N[(N[(J * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 2.8e-105], (-U), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if J < -0.149999999999999994 or 2.8e-105 < J

   1. Initial program 92.6%

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

   3. Simplified 99.0%

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

      1. add-cube-cbrt 98.1%

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

      2. pow3 98.1%

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

   5. Applied egg-rr 98.1%

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

      1. expm1-log1p-u 89.8%

         \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\cos \left(K \cdot 0.5\right) \cdot \mathsf{hypot}\left(1, \frac{0.5 \cdot \frac{U}{J}}{\cos \left(K \cdot 0.5\right)}\right)}\right)}^{3}\right)\right)}\right) \]

      2. rem-cube-cbrt 90.3%

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

      3. *-commutative 90.3%

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

      4. associate-/l* 90.3%

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

   7. Applied egg-rr 90.3%

      \[\leadsto -2 \cdot \left(J \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \frac{0.5}{\frac{\cos \left(K \cdot 0.5\right)}{\frac{U}{J}}}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. expm1-udef 90.1%

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

      2. sub-neg 90.1%

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

   9. Applied egg-rr 98.9%

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

       1. metadata-eval 98.9%

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

       2. sub-neg 98.9%

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

       3. associate--l+ 99.0%

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

       4. *-commutative 99.0%

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

       5. *-commutative 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in U around inf 51.8%

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

       1. metadata-eval 51.8%

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

       2. times-frac 51.8%

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

       3. *-commutative 51.8%

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

       4. times-frac 51.7%

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

       5. /-rgt-identity 51.7%

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

   14. Simplified 51.7%

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

   if -0.149999999999999994 < J < 7.6000000000000001e-297

   1. Initial program 61.0%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in K around 0 29.3%

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

      1. *-commutative 29.3%

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

      2. unpow2 29.3%

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

      3. unpow2 29.3%

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

   6. Simplified 29.3%

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

   7. Taylor expanded in U around -inf 42.9%

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

      1. +-commutative 42.9%

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

      2. mul-1-neg 42.9%

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

      3. unsub-neg 42.9%

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

      4. *-commutative 42.9%

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

      5. unpow2 42.9%

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

   9. Simplified 42.9%

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

   if 7.6000000000000001e-297 < J < 2.8e-105

   1. Initial program 42.0%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in J around 0 47.4%

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

   5. Taylor expanded in U around 0 47.4%

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

      1. mul-1-neg 47.4%

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

   7. Simplified 47.4%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -0.15:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(1 + U \cdot \frac{0.5}{J}\right)\right)\\ \mathbf{elif}\;J \leq 7.6 \cdot 10^{-297}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5 - \frac{J \cdot J}{U}\right)\\ \mathbf{elif}\;J \leq 2.8 \cdot 10^{-105}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(J \cdot \left(1 + U \cdot \frac{0.5}{J}\right)\right)\\ \end{array} \]

Alternative 10: 48.5% accurate, 45.7× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;J \leq -1.9 \cdot 10^{+177}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-296}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 5.6 \cdot 10^{-19}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U)
 :precision binary64
 (if (<= J -1.9e+177)
   (* -2.0 J)
   (if (<= J 1.05e-296)
     (* -2.0 (* U -0.5))
     (if (<= J 5.6e-19) (- U) (* -2.0 J)))))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (J <= -1.9e+177) {
		tmp = -2.0 * J;
	} else if (J <= 1.05e-296) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 5.6e-19) {
		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 <= (-1.9d+177)) then
        tmp = (-2.0d0) * j
    else if (j <= 1.05d-296) then
        tmp = (-2.0d0) * (u * (-0.5d0))
    else if (j <= 5.6d-19) 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 <= -1.9e+177) {
		tmp = -2.0 * J;
	} else if (J <= 1.05e-296) {
		tmp = -2.0 * (U * -0.5);
	} else if (J <= 5.6e-19) {
		tmp = -U;
	} else {
		tmp = -2.0 * J;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if J <= -1.9e+177:
		tmp = -2.0 * J
	elif J <= 1.05e-296:
		tmp = -2.0 * (U * -0.5)
	elif J <= 5.6e-19:
		tmp = -U
	else:
		tmp = -2.0 * J
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (J <= -1.9e+177)
		tmp = Float64(-2.0 * J);
	elseif (J <= 1.05e-296)
		tmp = Float64(-2.0 * Float64(U * -0.5));
	elseif (J <= 5.6e-19)
		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 <= -1.9e+177)
		tmp = -2.0 * J;
	elseif (J <= 1.05e-296)
		tmp = -2.0 * (U * -0.5);
	elseif (J <= 5.6e-19)
		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, -1.9e+177], N[(-2.0 * J), $MachinePrecision], If[LessEqual[J, 1.05e-296], N[(-2.0 * N[(U * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 5.6e-19], (-U), N[(-2.0 * J), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -1.8999999999999999e177 or 5.60000000000000005e-19 < J

   1. Initial program 97.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in K around 0 49.3%

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

      1. *-commutative 49.3%

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

      2. unpow2 49.3%

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

      3. unpow2 49.3%

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

   6. Simplified 49.3%

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

   7. Taylor expanded in J around inf 51.4%

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

   if -1.8999999999999999e177 < J < 1.05e-296

   1. Initial program 71.0%

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

   3. Simplified 85.0%

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

   4. Taylor expanded in U around -inf 34.7%

      \[\leadsto -2 \cdot \color{blue}{\left(-0.5 \cdot U\right)} \]
   5. Step-by-step derivation

      1. *-commutative 34.7%

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

   6. Simplified 34.7%

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

   if 1.05e-296 < J < 5.60000000000000005e-19

   1. Initial program 51.1%

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

   3. Simplified 73.1%

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

   4. Taylor expanded in J around 0 45.9%

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

   5. Taylor expanded in U around 0 45.9%

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

      1. mul-1-neg 45.9%

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

   7. Simplified 45.9%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.9 \cdot 10^{+177}:\\ \;\;\;\;-2 \cdot J\\ \mathbf{elif}\;J \leq 1.05 \cdot 10^{-296}:\\ \;\;\;\;-2 \cdot \left(U \cdot -0.5\right)\\ \mathbf{elif}\;J \leq 5.6 \cdot 10^{-19}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot J\\ \end{array} \]

Alternative 11: 40.8% accurate, 83.0× speedup

Math

\[\begin{array}{l} U = |U|\\ \\ \begin{array}{l} \mathbf{if}\;U \leq 18:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \end{array} \]

FPCore

NOTE: U should be positive before calling this function
(FPCore (J K U) :precision binary64 (if (<= U 18.0) (* -2.0 J) (- U)))

C

U = abs(U);
double code(double J, double K, double U) {
	double tmp;
	if (U <= 18.0) {
		tmp = -2.0 * J;
	} 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 (u <= 18.0d0) then
        tmp = (-2.0d0) * j
    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 (U <= 18.0) {
		tmp = -2.0 * J;
	} else {
		tmp = -U;
	}
	return tmp;
}

Python

U = abs(U)
def code(J, K, U):
	tmp = 0
	if U <= 18.0:
		tmp = -2.0 * J
	else:
		tmp = -U
	return tmp

Julia

U = abs(U)
function code(J, K, U)
	tmp = 0.0
	if (U <= 18.0)
		tmp = Float64(-2.0 * J);
	else
		tmp = Float64(-U);
	end
	return tmp
end

MATLAB

U = abs(U)
function tmp_2 = code(J, K, U)
	tmp = 0.0;
	if (U <= 18.0)
		tmp = -2.0 * J;
	else
		tmp = -U;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: U should be positive before calling this function
code[J_, K_, U_] := If[LessEqual[U, 18.0], N[(-2.0 * J), $MachinePrecision], (-U)]

Derivation

1. Split input into 2 regimes

2. if U < 18

   1. Initial program 83.4%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in K around 0 39.6%

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

      1. *-commutative 39.6%

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

      2. unpow2 39.6%

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

      3. unpow2 39.6%

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

   6. Simplified 39.6%

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

   7. Taylor expanded in J around inf 34.2%

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

   if 18 < U

   1. Initial program 55.2%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in J around 0 42.2%

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

   5. Taylor expanded in U around 0 42.2%

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

      1. mul-1-neg 42.2%

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

   7. Simplified 42.2%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 18:\\ \;\;\;\;-2 \cdot J\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array} \]

Alternative 12: 27.2% accurate, 210.0× 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 Float64(-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 75.6%

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

3. Simplified 87.4%

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

4. Taylor expanded in J around 0 25.6%

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

5. Taylor expanded in U around 0 25.6%

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

   1. mul-1-neg 25.6%

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

7. Simplified 25.6%

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

8. Final simplification 25.6%

   \[\leadsto -U \]

Reproduce

herbie shell --seed 2023264 
(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)))))